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This is to certify that the

dissertation entitled

Infinitely Many Periodic Solutions
of Nonlinear Wave Equations on Sn

presented by

Jin-Tae Kim

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Mathematics

4%sz 3-9

ajor professor

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Date June 20, 2000

 

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJCIRCIDateDuepsS-p 15

 

 

Infinitely many Periodic Solutions of Nonlinear Wave

Equations on S"

By

Jin- Tae Kim

AN ABSTRACT OF A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics
2001

Professor Zhengfang Zhou

ABSTRACT

Infinitely many Periodic Solutions of Nonlinear Wave Equations on 8"
By

Jin- Tae K im

The existence of time periodic solutions of nonlinear wave equations u“ — Anu +
(fig—lyu = g(u) — f (t, x) on n-dimensional spheres is considered. The corresponding
functional of the equation is studied by the convexity in suitable subspaces, minimax
arguments for almost symmetric functional, some comparison principles and Morse
theory. The existence of infinitely many time periodic solutions is obtained with suit-
able assumptions on the growth of the nonlinear term g(u) when the non-symmetric

perturbation f is not small.

To my parents.

iii

ACKNOWLEDGMENTS

I would like to express my sincere gratitude and thanks to my advisor, Professor
Zhengfang Zhou for his constant and patient help, encouragement, and excellent
advice.

I would also like to thank Professors Chichia Chiu, Dennis R. Dunninger, Michael
Huier, Darren E. Mason, and Baisheng Yan for their time and valuable sugges-
tions. Especially I would like to mention that I learned a lot about partial differential
equations from Professor Dunninger through his lectures and his lecture notes.

I also would like to thank Professor In-Bae Kim in Korea who got me started on
research in Mathematics and supplying me continual encouragement.

Finally, my warm thanks go to my wife Chin-Kyimg for her great help, daughter
Hae-In, and son Paul for having been delight and encouragement while I was preparing

my thesis.

iv

TABLE OF CONTENTS

Introduction
1 Preliminaries

2 The simple case g(u) = lulp‘zu

2.1 Variational Scheme .....

2.1.1 Introduction of a new variational formulation .........

2.1.2 Modified functional .
2.2 Minimax methods ......

OOOOOOOOOOOOOOOOOOOOOOO

2.2.1 Construction of critical points ..................
2.2.2 Comparison fimctional K (u) ...................

2.3 Critical values ,6), of K (u) .

2.3.1 Bahri-Bersstycki’s max-min value 3,, ..............
2.3.2 The relation between file and other minimax values ......

2.4 Estimate of file using Morse Index ....................

2.5 Proof of the existence of the solutions ..................

3 The existence for general nonlinearity

3.1 Variational Scheme .....

3.1.1 A new variational formulation ..................

3.1.2 Modified fimctional .

3.2 Minimax methods and existence result .................
3.3 Critical values ,8], of a comparison fimctional K (u) ..........
3.3.1 Introduction of comparison functional K (u) ..........
3.3.2 Bahri-Berestycki’s max-min value flk ..............

3.4 Morse index and file .....

3.5 Proof of the Main Theorem

BIBLIOGRAPHY

11
12
12
18
25
25
30
30
30
35
39
46

50
50
50
59
63
70
70
70
79
85

87

Introduction

During the past three decades, the initial value or Cauchy problem has played the
central role in the theory of evolutionary differential equations, which describe many
fundamental physical processes of interaction. The Cauchy problem has been studied
extensively with considerable success. In spite of a great deal of recent activity, many
physically and mathematically important difficult problems still remain, even when
global existence and unicity have been well established. Among the most interesting
problems of this type are those of the existence, regularity and stability of time-
periodic solutions.

This dissertation is focused on the nonlinear wave equation
Au = g(u) — f(t,a:), (t,a:) E S1 x S", n > 1, (1)

where Au = u“ — Ann + (Lil-Va, and f (t,:c) is 21r-periodic fimction in t. We are
concerned with the existence of multiple 27r-periodic solutions for the case where g is
superlinear, i.e., g(§)/§ —+ 00 as |§| —-> 00.

The existence of time-periodic solutions has always played an important role in the
theory of differential equations and mathematical physics. Even the existence of the
periodic solutions for nonlinear ordinary differential equations is nontrivial, requir-
ing Poincare-Bendixson theory to study the periodic orbits of general 2-dimensional
autonomous systems. In this case, periodic orbits together with the steady state

significantly influence the behavior of all other orbits. There is no question that

the existence of periodic solutions for partial differential equations is a much harder
problem.

The reason we choose the compact space S“ instead of the usual IR" is motivated
by several considerations. First, 3” is a naturally curved physical space going back
to Einstein space. There is no reason to believe that the complete flat IR" is a
better choice for consideration. Secondly, the usual Minkowski space R x R“ can be
conformally embedded into the Einstein Universe R x S", with the usual wave operator
[:10 = 0,2 — Ann is transforming into the operator A in this dissertation. Third, some
recent developments in constructive quantum field theory [13, 14, 21, 22, 26] are based
on the analysis of the Einstein Universe IR x S", which requires us to understand the
classical differential equations on it. Also, we want to point out that many simple
interactions like g(u) = 11.3 on IR x IR“ have no periodic solutions because of the
necessary decay properties of their solutions.

The main difficulty of problem (1) is the lack of compactness. When n is odd, the
null space of A is infinite dimensional, and the component of u in this eigenspace is
very dificult to control. This fact makes the problem much harder than an elliptic
equation Au = g(z, u), or than a Hamiltonian system in which every eigenspace is
finite dimensional. The associated functional of (1) is indefinite in a very strong sense.
In particular, it is not bounded from above or from below, and it does not satisfy the
Palais-Smale compactness condition in any reasonable space.

In the case of n = l, Bahri, Brezis, Coron, Nirenberg and Rabinowitz [5, 8, 9, 10,
16] have proved the existence of nontrivial periodic solutions of (1) under reasonable
assumptions on g(u) at u=0 and u at infinity, and the monotonicity of g. For n > 1,
Benci and Fortunato [7] proved by using the dual variational method that (1) possesses
infinitely many 21r-periodic solutions in LP in the case g(u) = lulp‘zu, 2 < p < 2 + %

and f = 0. The existence of a nontrivial periodic solution in the case of g(O) = O

and f = 0, and the existence of multiple, in some cases infinitely many, time periodic
solutions for several classes of nonlinear terms which satisfy symmetry and growth
conditions were established in Zhou [29, 30]. These conditions include time translation
invariance or oddness; f = 0 and g(u) ~ lulp‘zu as u —+ co, (2 < p < 321—?2).
Their proofs involved variational methods; a suitable and complicated approximation
scheme; index and pseudo-index theory; Sobolev type embedding theorem for the
operator A and the best estimate on the spherical harmonics obtained by Sogge [23].
The monotonicity of g played an essential role in their proof to compensate for the
infinite dimensional null space of A.

In this dissertation, we are going to study the effect of perturbations which are
not small, destroy the symmetry with f 75 0, and show how multiple solutions persist
despite these nonsymmetric perturbations, provided the growth of the nonlinear term
at infinity is suitably controlled. Our method is based on the following ingredients.
(1) The elimination of the null space from the underlying Hilbert space to establish the
Palais-Smale condition for a new fimctional; (2) A variational technique developed for
nonlinear nonsymmetric elliptic equations by Rabinowitz [l7]; (3) The construction of
a comparison fimctional which can be used to estimate the size of the critical values;
and (4) The estimate of Morse index at the critical points.

Our main result is the following

 

Theorem 0.1 Suppose that 2 < p < 7”“?(3151‘n’2m'9 and g(é) E C(R,R) satisfies

(91) [9(51) - 9(52)l(€1 - 52) Z alléi - 52]”;
(92) there exists 1' > 0 such that

5
o <pG<o 2p] grow 5 59(5) for I6! 2 r;
(93) there exists a2 > 0 such that

lg(€)l .<_ 02(IEIP—1 + 1) for £6 1R;

3

(94) 9(5) = 005]) at 5 = 0-
Then for any f (t,a:) E LJ"/("”1)(Sl x S“), 27r-periodic in t, the above non-linear wave

equation (1) has infinitely many periodic weak solutions in I}"(S1 x S") n H (S 1 x 5'").

Remark 0.1 By a weak solution of (1 ), we mean a function u(t, a2) satisfying

n-l

/ [nun — A.¢ +( 2 >245) + g(u)¢ — f¢l dz dt = 0
Sle"

 

for all a e (:°°(sl x 3").

Remark 0.2 For the p as in the theorem 0.1 and 1 < q < p, the following types of

functions
g(x, 2) = h(=v)l2|”"22 + g(-’L‘)|Z|”'l + k(=v)|2|"‘22,

where h(:r), g(m), k(:i:) 6 C°(S", (0, 00)), satisfy conditions (g1) -— (g4). In Chapter
2, we will deal with the simplest case g(u) = |u|P‘2u, p > 2 which shows the ideas

involved, but the estimates are much easier to obtain than the general case.

Remark 0.3 In general we cannot expect the equation (1) to have nontrivial solution

if g is not super-linear [29].

Remark 0.4 The regularity results in the case of n = 1 are obtained by Brezis and
Nirenberg [10] for asymptotically linear g and by Rabinowitz [16] for superlinear g.
For n even regularity results are obtained by Jerison, Sogge and Zhou [11] and for
n = 3 by Zhao and Zhou [28] for the spherically symmetric solutions. However for

n > 3 and n odd, the regularity of weak solutions of the equation (1) is still open.

In [29], the existence result is proved for the case g is an odd function and for 2 <

9n — 1
n 1 )1/2), where finite-dimensional approximation is used to overcome

the lack of compactness mentioned above. Using, however, Tanaka’s idea [24], we

 

l
p<-2-(1+(

get around these difficulties by maximizing the original fimctional F (u) associated

4

with the equation (1) with respect to N. That is, we consider the fimctional I (u) =
max F (u + v) for u in the orthogonal complement of N. Due to a compact embedding
theorem 1.1 for this new space, we can prove that I (u) has the desired compactness
properties. And it is easy to see that each critical point of I (u) corresponds to a
unique critical point of F(u).

We are able to improve on p without the restriction of oddness on g. In the case of
f (t, :r) E 0, the equation (1) has a natural symmetry and the fimctional F(u) is S]-
invariant. We will address the case where f (t, x) is not identically 0 as a perturbation
from symmetry by using the ideas from [17]. The situation for the wave equation
is more complicated since the operator A has infinitely many positive and infinitely
many negative eigenvalues. The idea is based on some topological linking theorems.
The key in this argument is to estimate the size of some explicitly constructed critical
values. To do this, we will introduce a symmetric comparison flmctional K (u) defined
only on the positive eigenspace. Using the symmetry we will construct critical values
of K (u), and will establish the relations between critical values of I (u) and K (u) An
argument by Morse index theory on K (u) will finally prove the needed estimates.

This dissertation will be organized as follows. We will give some preliminaries in
Chapter 1, to serve as background in understanding later presentation. In Chapter 2,
we will consider the case where g(u) = |u|”"2u, p > 2, which is relatively easier than
the more general case of Theorem 0.1 because of the explicit form of the nonlinear
term and, more importantly, that we just need to consider the Z2-action instead of
the S l-action. Finally in Chapter 3, using the ideas in Chapter 2, we carefully will
show how to modify the fimctional, check the Palais-Smale conditions of the modified
functional, establish the S1 covariant version of Borsuk-Ulam theorem, and find the

connection between the Morse index of some critical points and their critical values.

CHAPTER 1

Preliminaries

Let A the linear wave operator such that

n—l

 

Au = u“ — Anu + ( )2u,

where (t,:r) E S1 x S", n > 1. It is well known that the eigenvalues of A are

n—1 n—1

 

and the corresponding eigenfunctions in L"’(Sl x S") are
¢1,m(:c) sinjt, ¢,,m($) cos jt, m = l, 2, ..., M(l, n),

where 451,",(23), m = 1,2, ..., M (l,n), are spherical harmonics of degree 1 on S” and

= (2l+n—1)I‘(l+n—1)

I‘(l + 1) I‘(n) = Owl—ll

M(l,n)

 

Then u E L2(S1 x S") can be written as

u = Z: uz,,-,me""¢z,m(:v),

him

where “him are the Fourier coeficients with um," = u;,_j,m. Hence

(’chlul-I')L2 = Z A(I:.7.)l'u’l,.7}"'ll2'

l,j,m

And the Sobolev space we will work on is defined as
H = {u e L205“ x S") = Hunt = Z |/\(l,j)llui.j.m|2 + 23 um? < oo}.
Clearly H is a Hilbert space with the inner product
(u,v)H = Z |A(l,j)|u,,,-,mvz,,-,m + Z “l.j.m1—’l..i,m-
l,j,m A(l,j)=0

We decompose H into invariant subspaces:

N = {u 6 Hluza-m, = 0 for A(l,j) # 0},
E+ = {u E HluIJ-Jn = 0 for A(l,j) S 0},
E“ = {u E HIuIJ-Jn = 0 for /\(l,j) Z 0}.

As can be seen from the expression of the eigenvalues, if the space S“ is odd di-
mensional, i.e., n odd, the kernel N of the operator A is infinite dimensional and
HuIIH = IIUIIL2 for u E N. Consequently, a compact embedding theorem of the type
E <—+ D", (p > 2) for E = E+ QB E' the orthogonal complement of N:

 

 

Theorem 1.1 (Zhou [30]) For any 2 S p < 2: +12, E r—i LP is compact.
. . . 2n 2 . . .
Remark 1.1 The surprising fact is the exponent “:1 , which is almost optimal. Note

that ||u||H is much smaller than ||u||L2 + IquIle = ”U“Wl,2(slen). And we have the
continuous the embedding Wm H L‘p for 2 g p S 27"}13, and the compact embedding

W1'2HL" onlyfor 23p< 333%.

Remark 1.2 Unlike the 1-dimensional case where the existence result is obtained for
all of 2 < p < oo (Tanaka [24], Zhou [29]), the above embedding theorem 1.1 presents
a crucial restriction on p for any existence results of wave equations on S", n > 1.
Note that in 1-dimension the compact embedding E H LP works for all of 2 < p < oo
([10, 27, 29]).

Remark 1.3 If n is even, than N = 0 and H = E, and hence problems are much
easier to handle [29].

Next we introduce some definitions on group actions that will be used throughout
the paper. Let G be a compact Lie-group and X a topological space. An action of
G on X is a map 43 : G x X —-> X, ¢(g, x) = gx with the following pr0perties:

(i) 1x = x for each x E X, 1 is the unit element in G,

(ii) 91(9237) = (9192):”, 91, 92 E G, 1? E X-

We denote by 0., = {gxlg E G} the orbit of x. A subspace X1 of X is called
invariant under the action of G if 02 C X1 for all x 6 X1. The closed subgroup
G3 = {g | gx = x} is called the isotropy group ofx. Isz = G, we say that x is a
fixed point under the action of G, we will denote by Fix(G) all the fixed points of
X under the action of G.

A functional F : X —> IR is said to be G-invariant if F(gx) = F(x) for each
x E X and g E G. IfX and Y are two G-spaces, we say that a function F: X —-> Y
is G-equivariant if F(gx) = gF(x) for each x E X and g E G. In this paper we will
use two groups, 22 = {id, —id} and S1 = {e‘9 [9 E [0, 2n)}. For example, any linear
topological space is Z2 space and any Hilbert space H is an S 1space if we define a

group action T9 on H as
(Tou)(x,t) = u(x + 0,t) for any 0 e [0,27r) and u E H.

Finally we set up a variational formulation for the equation (1). The ftmctional

corresponding to the equation (1) for u 6 H is given by

PM = gum)” - ] (0(a) — f - untdx,
n
where G(§) = [5 9(7) dr, fl = S1 x S", and L is the continuous self-adjoint operator

in H associated with the operator A, i.e.,

<LU,’U)H = (AU,’U) = Z: A(l’j)ulrjrmfiltjtm.

l’j’m

8

Using the Hilbert Space norm defined above, for u = u+ + u‘ E E, u+ E E+, u’ E

E“ and v E N, F(u) canbe written as
1 + 2 1 — 2
F(U+v)=§llu Ills-5Hu IIE- n(G(’u+v)-f°(U«+v))fllt0l-'E, (1-1)

which is in 02(E EB N, IR).
We close this section by introducing the notion of Palais-Smale compactness con-

dition (P. S.) which plays an essential role in applying minimax methods.

Definition 1.1 A difi'erentiable functional F(u) on a Hilbert space H is said to sat-
isfy Palais-Smale compactness condition (P.S.) if the following holds: whenever {Uj}
is a sequence in H such that F(uj) is uniformly bounded and F’ (u,-) -+ O in H " as

.l —* 00, then {uj} is precompact in H.

Remark 1.4 Note that often (P.S.) is not satisfied even by simple smooth functions,
e. g., consider F : IR —r IR with F(u) = cosu or F(u) = c, c some constant and take

u,- =j7r for eachj 6N.

CHAPTER 2

The simple case g(u) = |u|p_2u

In this chapter we consider the simpler case where g(u) = lulp‘zu and we prove the

following theorem.

 

Theorem 2.1 For the same 2 < p < 7nfl+2z3ffn4ni9 as in Theorem 0.1 and

f (t, x) E L551 27r-periodic in t, the following non-linear wave equation
Au = ]u|p_2u — f(t,x), (t,x) E S1 x S", n > 1, (2.1)
has infinitely many periodic weak solutions in L”.

The procedure for the proof of the above theorem is motivated by Tanaka [24], where
the existence of infinitely many solutions of the l-dimensional wave equation (for
2 < p < 00) is obtained using Morse theory and eigenvalue estimates. Although
most of the proofs in Tanaka works for S" with some n-dirnensional modifications,
we found that the eigenvalue estimates in his paper using interpolation theory did
not work for the n-dimensional case due to the big multiplicity of the eigenvalues of
A in n-dimensions. We take a different approach in Section 2.5 to prove the result.
We will treat this Chapter as preparation for Chapter 3.

We will assume 2 < p < 3&9}? throughout in this thesis in consideration of the
compact embedding Theorem 1.1. First we formulate the variational scheme for the

proof of the theorem 2.1.

10

2. 1 Variational Scheme

2.1.1 Introduction of a new variational formulation

As mentioned in the preliminary Chapter 1 the corresponding functional to the equa-

tion (2.1) is given by, for w =u+v E H,u E E and v E V,

l 1 _ l
F(W) “ §IIU+II2E - 5”“ His — 1-0 IIU+vII£+ (f,u+v)- (2-2)

We instead study the functional I (u) on E,

I(u) =gga13¢F(u+v) = gurus - gnu-Hi —Q(u), (2.3)
where
can = 1151i; Ilu + vii; — (f, u + 22>], (2.4)

which is easier to handle due to the compact embedding Theorem 1.1 on E. In
Section 2.5, we will show that critical points of I (u) are also those of F (u) First, in
the following lemmas we study the functional Q(u) in detail to prepare for the proof

of a compactness result for the functional I (u)
Lemma 2.1 (i) For all u E L1"+ 1, there exists a unique v(u) E N such that
l
Q(U) = 1.9 IIU+v(U)||£ - (f,u+ 0(a)) (25)

(ii) The map v : LP —+ N is continuous.

(iii) Q: E—+IR is in C'1 andfor allu,h E E,
(Q'(u)ih) = (|u+v(u)|P'2(u+ v(11)) — f, h) (2-6)

Moreover, Q’ : E ——> E“ is compact and there are constants 01,02 > 0 depending on

Ilfllp/(p_1) such that for all u E E,

 

 

“62’ E. 5 Queen?" +1), (2.7)
I<Q'(u).u> -pc2(u>| 3 Cinema + 1). (2.8)

11

Proof: (i) Because the map v i—r 119”” + v”; — (f,u + v) is strictly convex
and coercive on N, there is a minimum at, say, v(u) by the generalized Weierstrass
Theorem.

(ii) Suppose that Uj -—* u in L”. We will show that v(uj) —> v(u) strongly in

N 0 LP. Since v(uj) is the minimizer for Uj, we have

1 P— u- vu -l-u- vu- 9— u. vu-
Elluj+v(U)Hp (f, 2+ ()).>_ pll 1+ (2)||p (f, 1+ (1))-

Then {v(u,-)} is bounded in L9 and hence there is a subsequence {up} of {u,-} such
that v(u,-:) -* a in N. We will denote {up} by {Uj} for simplicity. Letting j —> oo
in the above inequality, we get

1 —.—— 1

5H” + v(‘U)|l‘D - (f, u + v(u» Z 1.1530(1-Dlluj + v(u))”; - (f, Us + v(uj»)

1
> E|Iu+v||§— (f,u+v).

This implies BEE ||u + v(uj)||p = Hu+ v(u)||p and v = v(u) by the uniqueness of v(u).
Hence v(uj) —r v(u) strongly in N 0 LP.
(iii) By the convexity of the function v H illu + vllz — (f, u + v), we have for all

u,hEEand T>O,

Q(u + Th) - Q01) = (ll'u + Th + v(u + Th)||£ - “U + v(U)||,‘§3)

l
P
—(f, Th + v(u + Th) — v(u))
Z (lu + v(u)]p‘2(u + v(u)) — f, Th + v(u + Th) — v(u)).

Noting that v(u + Th) — v(u) E N, we get

Q(u + Th) — Q(u) Z T(|u + v(u)]p'2(u + v(u)) — f, h).
By interchanging the role of Q(u + Th) and Q(u), we have

Q(u + Th) — Q(u) S T(|u + Th + v(u + Th) [1"2
(u + Th + v(u) + Th)) — f, h).

12

Taking limit T -—i 0 in the above two inequalities, we obtain the derivative formula
(2.6). Therefore Q E C1 (E, R). Moreover from the compact embedding Theorem 1.1
and the continuity of v(u) : LP“ ——> N, we conclude that Q’ (u) : E —> E" is compact.

On the other hand by (2.6) and Theorem 1.1,

IIQ'(U)||2.~ = “grilflu + v(u)|”'2(u + v(u» - f, h)

S Cpl] l'u + v(it)|”"2(u + v(’11)) - fllP/(p—1)-
Applying Héilder’s inequality and (2.5), we get
“cum. 3 or; llu + v(u)ll£‘1 + 1) s cums-WP + 1).
Inequality (2.8) can be easily obtained from (2.6), (2.7) and Holder’s inequality. In

all we have obtained the desired results. [I]

For later use we introduce Q0 6 C1 (E, R) defined by
, 1 p 1 p
Qo(U) = £95119; II“ + ”Hp = 5 H“ + vo(")||p, (29)

where v0 (u) can be given uniquely as in Lemma 2.1. In the following we list some
properties of Q0 that will be needed in constructing a modified functional in Section

2.1.2. First by setting f = 0 in Lemma 2.1, we obtain, for u, h e E,

(Qua). h) = (I'u + vo(U)|"‘2(u + vo(U)), h), (2.10)
llQb(u)llE‘ S C(Qo(U)‘P'1)/” +1), (2-11)
<Qz,(u).u> s on(u) + C(Qo(u)‘/P + 1)- (212)

Similarly, as in the proof of the previous lemma, we can easily show the following

relations between Q(U) and Q00“)-

13

Lemma 2.2 There is a constant C > 0 depending on I] f llp/(p_1 such that for u e E,

IQ(U)I S C(Qo(U)+1), (2-13)

IQ(u) - Qo(u)l .<_ C(Qo(u)‘/P + 1). (2.14)

Now we verify the Palais-Smale compactness condition (P. S.) for I (u) which plays

a crucial role in applying minimax methods to I (u).
Proposition 2.1 I (u) 6 Cl(E, IR) satisfies (P.S.).

Proof: Let M > 0. Suppose I(uj) g M for all j and I’(u,-) —+ 0 in E". We have

fOIUj=U;+U;eE+®E-=E,

(I’M), h> = (u; — u

3')

h) — (Q’(u,-), h) for h E E.
First we will show that {uj} is bounded in E. Then the compactness of Q’ will
immediately prompt the existence of a convergent subsequence of {Uj}. Setting h = u,-
+

orh=u- —u7

J J,we get

”Infill - “uj—llli‘ - (Q’(uj),uj)| S mllujlle, (2-15)

|||uj||213- (01%),“? - u;)| S mllujller (2-16)

 

where m = sup ||I’(u,-)| 3.. From I(Uj) g M, we have
1 + 2 1 —- 2
gnu,- HE — 5w»,- IIE — as.) s M,
which combined with (2.15) leads to
1
§(Q’(u,),u,-) - QM“) S M + mllujlle-
Thus by (2.8), we get
(3 —1)Q(u.-)— 02(|Q(uj)|"” +1) 3 M + muujug.

l4

which implies
QM) S C(Ilujlls +1), for all 3', (217)
where C > 0 is independent of j. Now it follows from (2.7) and (2.17) that

I(Q’(uj),'u;-F - ufll S IIQ’(uj)l|E-||ujlla
C(|Q(uj)l(’"1”" + DHUJ'HE

IA

-1
s C(IlujHE ”Handle.

This, substituted into (2.16), yields

Hug-Ilia; S mllujIIE + (00%),“? - Hi)

—1
< mnujuwcmujng ”’+1)lluj||E.

Thus {uj} is bounded in E.
Finally note that I ’ (u,) = u; — u; — Q’ (uj) where Q’ : E —-> E“ is compact and

I’(u,-) —> 0 as j —+ 00. We can easily see that {Uj} is precompact in E. [:1

2.1.2 {Modified functional

Next we replace I (u) by a modified fimctional J (u) for which it is easier to construct
the critical values. Let x E C°°(lR, R) be such that X(T) = 1 for T S 1, X(T) = 0
for T22 and—2Sx’(r)$0, OSx(T)_<_l, forTEIR. Foru=u++u‘€
E+ GB E‘ = E and a = max{1, 12%}, let
<NU) = “(1002 + 1)1’2, MU) = X(‘1’(U)'1Qo(U))-
Define
1 + 2 1 — 2
J(U) = '2-llu ”E - 5”“ “E - Q00!) - ¢(U)(Q(U) - 62001)),

15

where Qo(u) is as defined in Section 2.1.1.
The reason for introducing J (u) is that the first assertion of the following propo-
sition which says J (u) is almost an even ftmctional, holds for J (u) but not for I (u).

Using the following proposition, we will then show that large critical values of J (u)
are also critical values of I (u)

Proposition 2.2 (Proposition 1.2 of Tanaka [24]) The functional J (u) E CI(E, R)
satisfies the following:
(i) there is a = a(||f||p/(p_1)) > 0 such that for u 6 E,

W») — J<—u>| s a(|J(u)|i + 1). and

(ii) there is Mo > 0 such that J(u) 2 Mo and [[J’(u)|

 

E. _<_ 1 implies J (u) = I (u)

Proof: (i) Ptom the definition of J (u), we have

|J(-U) - J(U)| S i/J(U)|Q(U) - Qo(u)| + ¢(-U)IQ(—U) - Qo(-U)|- (2-18)
Suppose that —u E supp w, i.e., Qo(u) g 2€l5(—u) = 2a(I(—u)2 + 1)1/2. Ptom the
definition of J (u),

I(-U) = J(U) + (0001) - Q(-U)) - WUXQW) - Qo(U))-
By Lemma 2.2, we get
II (—u)l s IJ(u)I + C(Qo(u)1/p + 1)
< [J(u)] + Cd5(-u)1/".

Using Young’s inequality, we deduce that
|I(-U)| S 2|J(U)| + 0-
Hence we get for —u E supp w,
Qo(u) S 245(—u) = 2a(I(—u)2 + 1)”2
< G|J(u)l + C. (2.19)

16

Similarly we have for u E supp w,
Qo(u) S G|J(u)| + C. (2.20)
It follows (2.13), (2.18), (2.19) and (2.20) that for u E E

|J(-u)-J(u)l s C(w(u)+z/z(—u))(oo(u)1/P+1)

< a(|J(u)|1/” +1).

This proves the first assertion of the proposition.

(ii) To prove the second assertion of the proposition, we need the following two

lemmas.

Lemma 2.3 There is a constant M1 = M1(||f||p/(p_1)) > 0 such that J(u) _>_ M1 and

u e supp v imply I(u) 2 §J(u).
Proof: From the definition of J(u),
1(a) = I (U) - (1 - v(u))(Qa) - Qo(U))
S I(u) + C(Qo(u)1/P +1).
By definition of v, we get for u e supp w,

J(u) S I(u) + C(II(u)l1/’”+ 1)

< I(u) + %|I(u)| + 0,.

Choosing M1 = 201, we get the desired result. El

Lemma 2.4 For allu=u++u‘ EE=E+EBE“ anthE,

(1'01), h) = (1 + T1(U))(u+ - U“. h) - (1+ T2(U))(Q6(U), h)
-(¢(U) + T1(U))(Q’(U) - QM“), h),

17

where T1 (u), T2 (u) E C(E, R) are functionals satisfying
sup{|T2(u)|;u E E, J(u) 2 M2, i = 1,2} -—> 0 as M2 —> oo. (2.21)
Proof: For all u = u+ + u" E E and h E E, we have
(NU), h) = (11+ - U'ah) - (Q6(U),h) - (i/J'(U),h)(Q(u) — Qo(U))
- 1P(U)(Q'(U)- 0601),”, where
(WWW) = X'(4’(U)'1Qo(U))4’(u)’3
X {-02101) (1'01). h)QoW) + 2(U)2(Q6(U)i h)l.
(1'01), h) = (‘u+ - u‘. h) - (9260‘), h) - (Q'(U) - QM“), h)-

By regrouping terms, we get the desired expression for (J ’ (u), h) for

T101) = 02X'(-)¢(U)’3I(U)Qo(U)(Q(U)-Qo(U)),
T2(u) = Ti(u)+X'(°)4’(U)'l(Q(u) "Qo(u))-

Now suppose that u E E satisfies J (u) 2 M2. Ftom (2.14), we get
|T1(U)| S C IX'(°)| 4500—2920(11) (62001)”? +1)-

If u 91 supp w, then T1 = 0. Otherwise, by the definition of ’l/J(’u), we have Qo(u) S
245(u). On the other hand we get from Lemma 2.3, @(u) 2 I (u) Z %J (u) 2 §M2.

Hence we obtain
[2130.)] s C§D(u)’(P“1)/P g CMflP'lW —» o as M2 —» 00.
Similarly we have T2(u) —+ 0 as M2 ——> 00. Cl
Let us now prove the second assertion of the proposition. Recall that by definition
of J (u) it suffices to show that w(u) = 1. Thus we need to show
Qo(u) S 9(a). (2.22)

18

for u E E such that J (u) 2 Mo and ||J’(u)||E. _<_ 1. For sufficiently large Mo > 0, we
can assume by Lemma 2.4 that J (u) 2 Mo implies |T1(u)| 5 %, |T2(u)| S l and

 

 

 

P(1 + T2(“)) P — 2 _
2(1 +T1(u)) — 1 > T = I"
From (2.21), we obtain
= —Q ul + T201.)

)+ 2(1 +T1(u))<Q0(u)1u>

(u
7g(u) +T1(U ) I I

_ p2(l+T2(u)) _ u _ u _ u

11’0“) + T1 (u) , I
+ 2(1 + T1(u)) (Q ("l — QM), 11)

S (I) + (II) + (III).

 

 

 

But by (2.14) we easily see that
K”)! S C(Qo(U)1”’+ 1)-

On the other hand it follows from (2.8), (2.13), (2.14), and (2.12) that

I<Q’(U) — Q6(u),u)l S IpQ(u) — (Q’(u).u)| + p|Q(U) — Qo(u)l
+G(Qo(u)1/” + 1)
S C(Qo(u)"" + 1).
which implies
I(III)| s C(Qo(U)l"’ + 1).

Thus we have

1 I
2(1+ T1(u)) (J (“Ml

(3812):]; — noun) — C(Qo(U)1"’ + 1)

2 bQo(u) — C. (2.23)

 

I(U) -

IV

 

19

Now letting h = 11*” - u“ in Lemma 2.4, we get

(1'01),u+ - u') = (1+ T1(u))IIUII21:- (1+ T2(U))(Q6(u),u+ - U‘)

-(1/)(U) + T1(u))(Q'(u) - Q1101), 11+ - 1F)-

We estimate the second and third term on the right hand side of the previous equation.

By (2.11), we have
I(C2€1(U),u+ - u’)l S llQb(u)E‘llullE S C(Qo(U)(’"”’” + 1)||UIIE-
Similarly by (2.7) and (2.13),
I(Q’(U),u+ - u’)| S C(Qo(U)(’”1)”’ +1)llullE-

Recalling that |T1(u)| S % and the assumption that ||J’(u)||E. g 1, we then have

|/\

IIJ'(U)I ulli: + C(Qo(U)(”‘”"’ + 1)|Iu|le~

macaw-1W + unuue,

 

 

 

1
51111113. e-

l/\

which leads us to
llullE S C(Qo(u)(fl)/P +1). (2.24)

It follows from (2.23) and (2.24) that

1 I
2(1 + 11(5)“ (“)1">+ b62001) - c

-C||J'(U)l Ulla + bQo(U) - C

 

I (u)

IV

IV

 

 

 

b62001) —- C(Qo(u)"“’/” + 1)
bQo(u)/2 — C'0-

IV

IV

Finally we remark that

inf {Qo(“); llJ'(U)|

 

3:31 and J(u)ZM}—>oo as M-+oo.

20

This follows from (2.24) since J (u) —> 00 implies b-Q—‘éfl — Co 2 0; hence I (u) 2

bQo(u) / 3. Combining these estimates yields
Q0(u) g aI(u) S @(u).

Thus the proof of the lemma is completed. CI

Immediate consequences of the above proposition are the following two corollaries
which ensures that large critical values of J (u) are also critical values of I (u), and
that the (RS) condition holds for large values of J (u).

Corollary 2.1 If J’(u) = 0 and J(u) 2 M0 for u E E, then I(u) = J(u) and
I’(u) = 0.

Corollary 2.2 J (u) satisfies (RS) on the set {u | J (u) 2 M0}.

2.2 Minimax methods

2.2.1 Construction of critical values

We rearrange the positive eigenvalues of the wave operator A as 0 < #1 S [12 S M S
- - - , and let e1, e2, e3, - - - be the corresponding orthonormal eigenfunctions. Then the

positive eigenspace E+ can be written as
E+ =span {ej :j E N}.

Define

E: =span{e,- : l Sj _<_ k}.

21

Note that ||u||E g 11],” ||u||L2 for u E E;. For u = u+ +u‘ E E; 69 E", by Lemma

2.1 and Lemma 2.2, we have

1 1 _
J(u) = 51121212 — 5111 Hi: — one) - waxes) - 620(1))

1 l _

s gums—gnu Iii-20(u)+0(oo<u>l/P+1)
1 1 1 -

s §l|u+lli3-§Qo(U)--2-llu ”2+0
1 l 1 _

= §|lu+lli~-%IIU+vo(u)II£-§llu ”2+0
1 + 2 + — P l - 2

S 5“11 Ilia-ellu u +vo(u)|l2-§|lu “3+0
1 l _

s yant-wmss—pm12+c
1 _ 2 l _

s gurus-cu." ||u+ll%-§llu “2+0.

.Hence there is an R]. > 0 such that J(u) _<_ 0 for all u E E; 69 E" with ||u||E 2 R1,.
We may assume that R]. < R1.“ for each k E N.

Now we construct minimax values following Rabinowitz’s procedure [17]. Let B R
denote the closed unit ball of radius R in E, 0,, = BR“ n (E: 619 E”) and

I‘k = {'7 E C(Dk, E);7 satisfies (71) — (73)}, where

(71) '7 is odd in D1,,

(72) v(u) = u for all u E 6D,“

(73) 7(u) = a+(u)u+ + a’(u)u" + k(u), where 01+ E C(Dk, [0,1]) and a" E
0(0),, [1, 62]) are even fimctionals (62 _>_ 1 depends on 7 ) and Is: is a compact operator
such that on 0D)“ a(u) = a+(u) + a‘(u) = 1 and rc(u) = 0.

Define
bl. = inf sup J('y(u)), k E N.

761‘ 06D):

If f E 0 and J is even, it can be shown as in [1] that the numbers bk are critical
values of J. If f is not identically 0, that need not be the case. However we will use
these numbers as the basis for a comparison argument. To construct a sequence of

critical values of J, we must define another set of minimax values. Let

22

Uk = Dk+l O {U E E; (“131:“) Z 0};

A,c = {A E C(Uk, E); A satisfies (A1) — (A3)}, where

(A1) AID). 6 Pk:

(A2) A(u) = u on 6U). \ Dk,

(A3) A(u) = c'iJ'(u)u+ + 6F(u)u‘ + ii(u), where 61+ E C(Uk,[0,1]) and c"!— E
C(Uk, [1, 61]) are even fimctionals (61 2 1 depends on A ) and Fe is a compact operator
such that &(u) = l and iri(u) = 0 on 0U,c \ Die.

Now define
ch = inf sup J(A(u)) k E N.

By definition of bk and ck we easily see that ck Z bk. The key to this construction is
that we have the following existence result.
First recall that J satisfies the (RS) condition (Corollary 2.2) on {u E E; J (u) 2

Mo} and J’ (u) is an operator of the form:
J’(u) = (1 + T1(u))(u+ — u’) + compact,

where |T1(U)l S 1/2 on {u E E; J (u) _>_ Mo} (see proof of Pr0position 2.2). Thus we

have the following deformation lemma.

Lemma 2.5 (of. [18, 19]) Suppose that c > M0 is a regular value of J (u), that is,
J’(u) 74 0 when J(u) = c. Then for any 5 > 0, there exist an e E (0,5] and 17 E
C([0, 1] x E, E) such that

(i) r)(t, .) is odd for all t E [0,1] if f(t, x) E 0;

(ii) r)(t, ) is a homeomorphism of E onto E for all t;

(iii) 11(0, u) = ufor all u E E;

(iv) n(t,u) = u if J(u) E [c — €,c+ E];

(v) J(fl(1,u)) S c- 6 if J(U) S 6+6;

(vi) n(l,u) satisfies (A3).

23

Proposition 2.3 Suppose ck > bk 2 Mo. Let 6 E (0, ck — bk) and
Add) = {A E Ak; J(A) S bk + 6 on Dk}.
Then

06(6) = “1:13;” 3215‘ J(A(u)) ( 2 cl.)

is a critical value of I (u)

Proof: By Corollary 2.1, it is enough to show that ck(6) is a critical value of
J (u) First note that by definition of bk and A1,, Ak(6) 74 0. Choose E = %(ck—bk—6) >
0. Now suppose that ck(6) is not a critical value of J. Then by Lemma 2.5 there exist

5 E (0, E] and n as in the lemma. Choose H E Ak(6) such that

“(If J(1701)) S 61(5) + 8-

Let H = 71(1, H). We need to show H E Ah. Clearly H E C(Uk, E). (A1) and (A2)
easily follow from the choice of H and (iv) of Lemma 2.5. Since H satisfies (A3), so
does H by Lemma 2.5. Moreover on Dk, J(H(u)) S ck(6) — E and hence J(H(u)) =
J (H (u)) S be + 6 on Dk, again by (iv) of Lemma 2.5. Therefore H (u) E Ak(6) and
by (v) of Lemma 2.5,

111,311: J (H (11)) S 61(5) - e,

which contradicts to the definition of ck(6). Hence ck(6) is a critical value of J D
Therefore, to establish the existence of critical values, it suffices to show that there
exists a subsequence {k,} such that

ckj>bijMo forjEN and bkj—>oo asj—->oo. (2.25)
Arguing indirectly we have the following proposition.

24

Proposition 2.4 If ck = bk for all k 2 k0, then there exists a constant C > 0 such

that
bk S ka/(P-I) for all k E N. (2.26)
Proof: We refer [17] for the proof using the property of almost symmetry of

J (u) ((i) of Proposition 2.2). [:1

Our goal in the rest of Chapter 2 is showing the existence of subsequence {kj} with the
property (2.25). In fact, by Proposition 2.4, we will prove that there exists {kj}, e > 0
and 0,3 satisfying

b., > Cekf/(P'l‘el for all j e N. (2.27)

2.2.2 Comparison functional K (u)

To show (2.27), we introduce a comparison functional. By the definition of Q0(u) and

(2.14) for u= u+ +u‘ E E: E+ EBE‘,

1(1) = gurus — gnu-11. — 20(111— v(uxoe) - one»
2 gnawi-éllu-IIt-zoo(u)—a1

1 1 _ 2 _
= glltf’llia-gll'uIlls-13W“?u +vo(U)||i3—al

IV

1 + 2 1 - 2 2 + -
— u —— u —— u +u P—al
2|| llE 2l| llp pll llp

IV

1 1 _ a0 00 _

§llu+llzp - gllu Ilia - gllui’ll; - —-|l'u ll; - 111.

where a0 > 0, a1 > 0 are constants independent of u. For u E E+, set
1 ac

KW) = §llu+|li~ - FIIU‘LIIS E C2(E+. IR).

Then we can easily see the following.

25

Lemma 2.6 i) J(u) Z K(u) — al for all u E E+.
ii) K (n) satisfies the (P.S'.) on E+.

In the next section, we will construct critical values fik of K (n) such that fik g

bk + a1 and we will deal with file instead of bk to prove (2.27).

2.3 Critical values fik of K (a)

2.3.1 Bahri-Berestycki’s max-min value 31,; [3, 4]
For m>k, k, mEN, set

:3 = {0‘ e cam-hm); 0(—a:) = —o(a:) for all a: e sm-k}
and

3,2": sup min K(o(x)).

06A? IESm-k

We list some properties of 3;" in the following proposition.

Proposition 2.5 (i) 0 S 3},” S fig, < 00 for all m, k E N;
(it) for all k E N, there exists u(k) and 17(k) such that

OSV(k)_<_fl,'c”gl7(k)<oo forall m2k+l;
(iii) moreover, u(k) —+ 00 as j —’ 00.

Proof: (i) For any a E AL", it is clear that there is a 6 E A?“ with
6(Sm'k'1) C a(S"‘""). Hence we have 5}," S 37:11- To prove (ii) and (iii) of the

proposition, we need the following lemmas.
Lemma 2.7 For all a E AL",
0(Sm—k) 0 E2" 75 (0.

26

Proof: Applying Lemma 2.9 in Section 2.3.2 to h = o : Sm”c —> E; and g =

id : E; -§ Eg, we easily get the result. [:1

Lemma 2.8 For all 0 E (0,1/p), there is a Ca > 0 independent of k E N such that
”’qu S CofliglluHE for U E (Emi,
where (Eg)i = {v E E+; (u,e,-) = Oforz' = 1,2, ...,k}.
Proof: We have by the definition of H - II E and pk
”qu s 741/2“qu for u e (Em
On the other hand, by Theorem 1.2 (Compact Embedding)
Hunt, 5 quluHE for all u E E and q E [2, (2n+ 2)/(n —1)).
Using Holder’s inequality, we get for q E (p, (2n + 2) / (n — 1))
IIUIlp S HUHEIIUIIl’T for u 6 E+,
where 1' = §g—:§% E (0, fi). Thus
llullp s Uni/flung for u e (Ema

which is the dasired rasult. E]

(ii) We now prove the existence of 17(k). By the linking Lemma 2.7 we have for
all a E A7,",

min K(o(:1:)) g sup K(u). (2.28)

—l¢

Recalling that ||u||E S ui/2||u||2 on E3, we have on E;
K <1 2—C p<_1_ 2—C’ -p/2 p
(u) _ zllulls Hullz - 2Hulls Mt Hulls-

27

Thus the right-hand side of (2.28) is finite and independent of a and m. Set
17(k) = sup K(u) < oo,
uEE:
which implies
fife" = sup min K(0’(.’E)) S 50:).

06A? 365m-“

Then we prove the existence of u(k). First we define a map a : Sm‘k —> Eff,\{0}

by
u(x) = a31/‘P‘2’Hw<x>n;P/<P-2>w(x),
m
where w(a:) = 2 me, and Sm’k is understood as
i=k

sm-k = {:12 = (33k, ...,:cm) 6 Rm-'°+1; 22:? = 1}.
i=1:

Then obviously a E AL". Since ||w(a;)||E = 1 on Sm‘k, we have

K(a(m)) = (é — gnaw—2’1wants/(H).

Since w(a:) E (E;_1)i, ||w(:c)||E = 1 for all a: E Sm‘k, it follows from Lemma 2.8
that

||w($)||p S 00 #1231 for 27 6 SW”,
where 6’ E (O, l/ p) and C9 is a constant independent of k and 13. Thus
K(a(:c)) 2 C; Mia/(”2) for all a: E Sm'k.
The right-hand side is independent of m. Set u(k) = C; [1293/ (p4). Then we have

5}," 2 min K(o(a:)) _>_ u(k) for m > k,

xESm‘*
which completes the proof of (ii) of the proposition.
(iii) From the definition of u(k), it is easy to see that u(k) —> 00 as n —> 00 since

u(k)—»ooask—+oo. Cl

28

As in Proposition 2.1 we can verify the following compactness conditions

(P.S.)., (P.S.),,, for Km).

(P.S.).: If {u,,,},i',‘,’=1 C E+ satisfies um E 13;, K(um) S C and

||(KIE;)'(%)IIE;' —> 0 as m —> 00, then {um} is relatively compact in E+;
(P.S.)m: If {uj};-’;1 C E; satisfies K(u,-) S C and (KlEfl'Wj) —> 0 asj ——> 00,

then {Uj} is relatively compact in E;

Since K is an even flmctional satisfying above (RS). and (P.S.), we have the follow-

ing result via standard argument.

Proposition 2.6 Suppose u(k) > 0. Then 3;," is a critical value of K l 3,1; , and the

limit of any convergent subsequence of fig“ as m -—+ 00 is a critical value of K.

By (ii) of Proposition 2.5 choose a sequence {m,} such that m,- —> 00 as j —+ 00 and
[3,, = 11in; fig" exists for all k E N.

Then we have the following facts about the ,Bk’s due to Proposition 2.5 and 2.6:

Corollary 2.3 i) file ’3 are critical values ofK E C2(E+, IR) for each h E N;
ii) .61: S 5H1 for all k 6 N;

iii)flk—>ooask—voo.

2.3.2 The relation between H], and other minimax values
To estimate bk we establish the following relation between bk and m.

Proposition 2.7 For all k E N,
bk 2 file — 01, (2-29)

where a1 is the number in Lemma 2. 6.

29

To prove this proposition, we need several topological linking lemmas. We first state

a version of the Borsuk-Ulam theorem.

Lemma 2.9 Let a,b E N. Suppose h E C(S“, Ra”) and g E C(Rb, Ra”) are odd

functions and there exists r0 > 0 such that g(y) = y for lyl 2 r0. Then h(S“) fl
g(R") at 0-

Proof: We choose R Z '70 such that R > maxxesa |h(:z:)|. Write
D"+1 ={t:1: E Ra+1;t E [0, 1],:1: E 5“}, Db = {y E Rb; |y| S R}.
Define F E C(6(D‘l'+1 x Db),lR"+b) by

F (tx, y) = Wm) - g(u)-

This is well defined and odd on 6(Da+1 x Db). Note that 6(D"+1 x Db) c: 8"”
(odd homeomorphic). Thus by the Borsuk-Ulam theorem, there is a (tonic, yo) E
0(0‘”l x D”) such that

F(toxo,yo) = 0, 336-, toh($o) = g(yol-

Since 6(D"+l x Db) = S“ x Db U D“+1 x 00", the following two cases should be
considered:

i) to = 1,30 e S‘1 and :10 e 0";

ii) to e [0,1),a:o e S“ and yo e 60”.

Case 1. We have h(xo) = g(yo). So we have h(S“) fl g(lR”) 79 (0. This is the desired
result.

Case 2. Since g(y) = y on 6D”, we have |g(yo)| = lyol = R. On the other
hand, by the choice of R, we get |t0h($o)| < R. These are incompatible with

toh(zo) = g(yo). So this case cannot take place. [3

From the above lemma, we can deduce the following.

30

Lemma 2.10 For all '7 E I", and for all a E A1,",
((pm'r)(Dk) U {u e E; e e- : Hulls 2 Rk}) n g(sm-k) 7e 0,

where Pm : E —> E; EB E” is the usual orthogonal projection.

Proof: Let 7 E l";c = {7 E C(Dk,E);7 satisfies (’y1)—('73)},D,c = BRkfl(Ef€B
E"). We extend '7 to ’7 E C(E; EBE‘,E) by 7(u) = 7(u) if ||u||E S Rk, and '7(u) =
u if ||u||E Z Rk. Obviously, flu) is well defined and odd in E; EB E' and since
m > k,

gnaw; e E“) = Pm 7(1),.) u {u e E; ea E‘; IIuHE 2 12k}.

Therefore, it suffices to prove Pm “E;L 69 E“) n 0(Sm‘k) 74 (0. We rearrange
{¢¢,m(a:)cosjt,¢I,m(:c)sinjt : A(l,j) < O,m = 1,... ,M(l,n)} as follows, denoted
by f1,f2,f3,-~ . We set for l E N,

E1. = span{f,-;l S j S l}

and let Pm) : E = E" EB E+ -i E; 8—) E," be the orthogonal projection. Consider the

operators
0 : S’""‘ —» E; c E;®Ef, Pm,.:y : E; 99E; a EgeaEf.

Applying Lemma 2.9 for h = a and g = Pmyfir, there exists 2:; E Sm‘k and u; E
E; 63 B; such that

0(2),) = Pm,1'7(ul). (2.30)
Since Sm'k is compact, there is a subsequence 2:11. such that
2:11. —i a: in SW4“, 0(a).) —> 0(23) in E;
On the other hand, by ('73),

Pm) '7(u,) = Pm) [chr (u)u,+ + a—(u)u,‘ + n(u;)] = a+ (u)u,+ + a' (u)u,' + Pm) n(u1),

31

 

where a‘(u) 2 1 on E- 6 E: and K.(E" 6 ED = K(Dk) is compact. Hence we have

_-—1— —0’$¢-K.uz
ul —a_.(ul)PE,[( ) ( )l

and {uf} has a convergent subsequence {ug }. From the boundedness of u; and
Dim(E,*,;) < 00, u, has a convergent subsequence. Passing to the limit in (2.30), we

obtain

Pm flu) = u(x), i.e., Pm “E: 6 E’) 00(Sm'k) aé 0.

This completes the proof. [:1

Let us define
bi“ = inf sup J(meu»

7€Fk 156D);

and recall that bk = igf sup J (7(u)). Then we have
'7

'3 uEDg

Lemma 2.11 For k E N, bk = lim bi".

m—boo

Proof: Since Pml‘,c = {Pam 'y E Pk} C h, it is clear that bk S by," for all
m > k. Let’s prove the other direction i.e., bk 2 limsup b}? for k E N. From the

m—ioo

definition of bk, for any 5 > 0 there is a 'y E I‘k such that

sup J(7(u)) S bk + e.
1160'.

By (73), 7(u) = a+ (u)u+ + a“(u)u‘ + n(u), where ozi satisfies the condition in (73)

and u(Dk) is compact. Since
Pmn(u) —-) n(u) as m —> oo uniformly in Dk,
we have

Pm7(u) = a+(u)u++a’(u)u_+Pmrt(u) —> a+(u)u++a‘(u)u_+n(u) = v(u) uniformly in Dk.

32

Hence

sup J(Pm'y(u)) —i sup J(ry(u)) as m —i 00.
uEDk uEDk

Thus we obtain

limsup bf," S limsup sup J(Pm'y(u)) = sup J(7(u)) S bk + e.

m-+oo m—ioo uE Dy. uEDk

Since the above inequality holds for any 5 > 0, we get the desired result. C]

Using above lemmas, we now prove Proposition 2.7.
Proof: Since J(u) S 0 on {u E E,‘,L,6E' : ||u||E 2 Rk}, Lemma 2.10 concludes

that

min 107(3))- < sup J( Pm'7(U)),

zESm-k uED h

forall 76H, andall aEAL". Thus

min K(o(:r)) — a1 S sup J(Pm'y(u)),

zESm“* uED k
which implies

sup gan(o(:c)) — a1 S inf sup J(Pm'y(u)).
aEA’" "“k

Thus ,6}? — a1 S b}? and by letting m = m, —> 00, we get
fik — a1 S limsupr‘ = bk.

This establishes the proof. [I

2.4 Estimate of B], using Morse Index

In this section some index properties of 3,, are discussed. The lower bound for the
index of K" obtained here and the upper bound estimate in the next section give the

growth estimate (2.27 ) that we are looking for.

33

Definition 2.1 For u E E+, we define an index of K”(u) by

index K ”(u) = the number of nonpositive eigenvalues of K ” (u)

= max {dim S; S S E+ such that (K”(u)h, h) S 0 for allh E S}.
Here ”A S B” in the bracket means A is a subspace of B.

Proposition 2.8 Suppose ,3], < Sh“. Then there exists uk E E+ such that

K(uk) S 16k)
K’(uk) = 0,

indexK”(uk) 2 It.

By definition of 3,, is a critical value of K (u), the result without the last assertion

is obvious. To prove the last assertion, we first consider finite dimensional case.

Proposition 2.9 Suppose Sf," < @211, m > k+ 1. Then there exists a u}? E E); such

that

K04?) S [312",
(K|E$)I(ulcn) = 0:

index (K I E; )” (uZ‘) Z It.

To prove the above proposition, we will use a theorem from Morse theory, i.e., a
result concerning the relationship between certain homotopy groups of level sets of
a flmctional and its critical points. First we need a theorem to treat the case where

critical points may be degenerate.

Proposition 2.10 (Marino-Prodi [15]) Let U be a 02 open subset in some Hilbert
space H and p E C2(U, IR). Assume 45" is a Fredholm operator ( of null index) on the
critical set Z (9’)) = {x E U; d)’ (x) = 0}, (1) satisfies (PS) and Z (43) is compact. Then,

34

for any 8 > 0, there exists a 6 WW, IR) satisfying (P.S.) and with the following
properties:

(i) W) = 45(93) if distance {13, Z (45)} 2 5;

(ii) I1/J(=v) - ¢(m)l, lid/(x) - ¢’(-’B)ll, III/Hm) - ¢"($)ll S e for all x 6 U;

(iii) the critical points of w are finite in number and nondegenerate.

It is easy to see that K | at. satisfies all the assumptions of the above Proposition.
That is,

1) K's; E C2(E,‘,*,, 1R) satisfies (RS) and Fredholrn.

2) All critical value of K | E; are non-negative because

K<u> = M) - generate = (g —1—1,>aonun;2 o.

3) Z (K | E; ) is compact. In fact, note that there exists R,,, > 0 such that K (u) < 0

for u E E; with ||u||E Z R.,,.; hence Z(K|E$) is bounded.
Applying Proposition 2.10 to Kl E; , for all e > 0 there exists (be E C2(E,*,‘,, 1R) satis-
fying (RS) and for all E E;
|¢e(U) - K(U)| < 6,
||¢2(U) - (KIE;)’(U)|| < 8,

||¢’e’(U) - (Kle;)”(U)ll < 8; (2-31)

the critical points of the are finite and non-degenerate. (2.32)

Form>k and e>0,let

fl?(€)= sup min «be-(0(3))-

06A}? 368m_h

Then by (2.31),

sr—esx(e)sx+e.

35

Moreover we have

Lemma 2.12 Suppose that a,5 E IR satisfies 6;"(8) < as — 25 < a5 < 633,1(5). Then

7rm—lc—1([¢e Z aeImap) 74 0 for some P 6 [$6 2 aEImr

where [4% Z 05]", = {u E Egg; ¢E(u) 2 as} and 1rm is the m - th homotopy group.
Proof: We argue by contradiction. Suppose that

7rm—k—1([¢e Z Calm?) = 0 for all p E I¢E 2 aelm-

By the definition of 6,23,1(5), there is a o E A)?“ such that 0(Sm‘k‘1) C [ng > a5]m.

Since rrm_k_1([¢e Z a5]m, p) = 0, there is a homotopy
H: [0,1] x 3”“ _. [¢. 2 a.]...
such that
H(O,x) = u(x), H(1,x) = p for all x E Sm'k‘l.

Write
5"” = {(t,x);x 6 RW", t 6 1R, |:z:|2 + t2 = 1}.

Define 6 : S'""‘ —> E; by

p if t=1, x=0,
H(t,x/|x|) if 0<tS l,
—H(—t, —x/|x|) if — 1 < t S 0,

_p ift=—1,$=0.

Obviously 5(SL'H‘) C [die > a5]m, where we denote 83"“ = {(t,x) E Sm'k;t >

0(< 0)}. On the other hand, we obtain from (2.31) and evenness of K (it) that

|¢5(—u) — ¢E(u)| S 25 for u E E;

36

So we have (HST—k) C [9256 2 a5 — Ze]m. Consequently, we have 6(Sm'k) C [th 2

a5 — 25],". 131-0111 the definition of 6;,”(5),

which contradicts with the assumption. Thus the proof is completed. C]

Using property (2.32), we can apply a classical theorem from Morse theory to (be

and we obtain

Lemma 2.13 For a regular value a E IR of 435 , set
L(€; 0) = maX{ indewflx); ¢e($) S a. 452-03) = 0}-
Then
«([455 2 a]m,p) = 0 for all p E [atE Z a]m, l S m — L(e; a) — 2.

Proof: Let b E IR, b < a be such that (be has no critical values in (-oo, b]. By
the “noncritical neck principle” (cf. Theorem 4.67 of Schwartz [20]), [436 2 b]m is a
deformation retract of 13;. Hence

«1([435 2 b]m,p) = 0 for all I E N and for all p.
Using theorem 7.3 in Schwartz [20] ,
rrl([gbe Z b]m, [d6 > a]m) = 0 for l S m — L(e; a) — 1.

Considering the homotopy exact sequence:

’7 7r!+1([¢5 Z bIma [455 Z aIm) _’ 7Tl([¢e Z almap) _’ 7rl([¢6 Z bImip)

_’ 7rl([¢£ 2 bImr I¢e Z aIm) _‘i - ' '3

we get the result. CI

37

Now we can prove Proposition 2.9.
Proof: Since 6;," < 67;, and the critical points of ¢s are finite and nondegen-

erate, by Sard’s theorem there exists a sequence as E IR (0 < e S 50) such that as is

a regular value of abs and
63(8) < ae—2e < a.: <fi7.’.‘.1(e). a. 45;," as e—»o.

By Lemma 2.12 and 2.13, we have L(e; as) 2 k for 0 < e < 50 and hence there exists

us E E; such that
¢e(ue) S. are, ¢;(ue) = 0: index ¢:(ue) Z ’9'
It follows from (2.31) that (us) satisfies

K(us) is boundedas e—>0,
(KIE;)’(us)—*0 as 5—40.

Since K | at, satisfies (P. S.) on E;, we can choose a convergent subsequence us J. —-+ u}?

for some u}? E E; Then we have
Ker) s a". Kisser) = o and index<KIEs)"(ur) 2 k.

This completes the proof. Cl

Finally we prove Proposition 2.8, the main result in this section:
Since 6)., < 6k“, we have 2" < 6231 for sufficiently large j. By Proposition 2.9, there
exists ur’ E E; satisfying, K (u?) S SI," and (K | E; )’ (u?) = 0. Since K E CQ(E+, IR)
satisfies (P.S.)., (it?) —> its for some subsequence m,: of mj, we have K (as) S 6,,
and K’ (uk) = 0

Let us prove the last assertion: index K " (us) 2 It.

First of all, we have
index K ” (u?) 2 index (K | E; )” (u?) for all m E N.

38

On the other hand, we observe that K ” (us) is an operator of type: K ” (uk) = id + r:

where It is a compact operator. Note that
(K”(uk)h,h) S 0 if and only if (n(uk)h,h) S —(h, h)

and X,- -—; 0 where A: are eigenvalues of K. Hence there exists an e > 0 such that
for h E E+,
index K”(uk) = index (K"(uk) - 5).

Since K E C2(E+, IR), we have for some 3'6,

IlK”(uL"3') — was“ < .- for 2" 21:.
Thus for j’ 2 3'6 and h E E+,

<K"(u.)h,h> — euhnt s <K"(u’;‘9‘>h,h>.

i.e.,

index (K ” (us) - e) Z index K ” (a?)
N ow by Proposition 2.9, we have

index K”(uk) 2 k,

which completes the proof of Preposition 2.8

2.5 Proof of the existence of the solutions

By Proposition 2.3 and Proposition 2.4, we know that (2.27), the growth estimate
on fik’s, ensures the existence of an unbounded sequence of critical values. We now

prove (2.27). First note by Proposition 2.8 that there exists {ukj} such that

1 a0 1 1
fit,- 2 K(uk,-) = §llukjllfe - —p—|Iu:.,||£ = (-2- - EIMIIUJ'IIS- (2.33)

39

Thus, by Proposition 2.8 again, we need to get an upper bound of index K ” (ukj) in
terms of [[1149]]; in proving (2.27).

For u, h,w E E+, K”(u) is given by
(K”(UIw,h> = (wnh) - (P - 1)ao(IUI‘”—2h,h)-
Thus by the definition of index,
indexK”(u) = max{dimS; S S E+, (p— 1)ao(|u[”_2h, h) 2 ||h||23,h E S}.
Define an Operator D : L2 —+ E+ such that for v(x, t) = Z v,,,-,m¢;,me‘j‘,

(Dv)(x.t)=Z Z M(l,j)l"‘/2n,.-,mgtme‘j‘.

m A(l,j)>0
Remark 2.1 D is an isometry fmm L1 = WL2{¢1,me‘j‘;A(l,j) > 0} to E+ and
D = 0 on WL2{¢l,meijti A(I.J‘) S 0}-

Remark 2.2 Setting h = Dv in the above expression of index, we get

indexK"(u) = max{ dimS; S S L2 at (p — 1)ao(|u[p‘2Dv,Dv) 2 ||v||§,v E S}

#{n- zn. 2 1, ngnwueeaz (wag -1)aoIUI”‘2)D}-

Proposition 2.11 There exist C > 0 such that for u E E+,
indexK”(u,-) S C||u||:,
2 2 —2
wherer=filfilfi ands=5%lm.

Proof: We try to find a big enough l such that

(p - 1)ao(IUI"'2Dv, 00) S llvllg. 0n E+\Ez‘:1,

40

which implies index KI ' (u) S I. First we have the following estimate on E+\E,‘:1

AlePIulp-2 S C(fQIDvI2q)%(/Qlu](P—2)q—ZT)1;—l

= Cllelliqllull‘;_2,q_3,.

|/\

2 1—
CIIDvII§°IIDvII.‘ ”Ilu ””3322,

I/\

2(1— 8)
Cfillvlh 2||v||2 II“”(p-2)fi—’

1 2 p-2
C,—,IIvII.IIuII(,.,,,s,,

2 2 1 1 —
where (j = -:—:—, — = 3 + _cj_3 and to get the second last inequality, we used

the facts [leHZE S [All-lllvllis on E+\E,*_'_l and ”Dung. = Hung, and the compact
embedding theorem 1.1. Thus to have f |Dv|2|u|”‘2 S ||v||§, we need
Ilull‘g'; 27:.- < IAII‘ ~ C|l|’/". s = (n +1 — (n — 1)g)/2g, i.e.,
(P‘ _2)n(n )—In- )
a _ —CIIuII(p_:I)_II_+l 1 q ~1-
Let Z = [a + 1]. Then

/ lelzlulp“ s Ilvllia for an v e E+\E;:.-

2n
and therefore index K (u) S l— — [oz + 1] < Ca— — C||u||:_ gal—("”5" CI

We now prove bk, > C k,- P-H (2.27) : From Proposition 2.11 and Proposition 2.8

we have
jS indexK”(ukj) g one,c ”(£3W, 2 < < Egg—1a.
Note that
Hutu; C_>.||utll"_2).g, ifg 2.;
so that

IInI. ”:22“? ’r—rt—r if g2§

41

In order to have (2.27 ) it needs

 

Wm
(p—2)(n+1)—(n—1)q (00-1)“

Since m is an increasing function of q, choose q = ‘23. Then we finally obtain

 

7n+1+\/25n2-2n+9

2<<
p 2Gn—D ’

 

for which (2.27) is satisfied.

Remark 2.3 This upper bound of p may not be optimal and we are still trying to

improve it.

Now there exists a sequence uk C E of critical points of I (u) such that as k —+ 00

1,2 1 _, 1 p
[(1%) = illus IIE — 5”“): ”E - 5 HUI: + ”(uklllp — (f,uk + ”(1%))“ 00-
Since I ’ (uk) = 0, we have
(I,(uk)ruk) = “Willis - “villi; — (Ink + “(UMP—2(1“: + “(um + f, u]. + v(uk)) = 0-
Above two equations combined gives
1 l 1
(‘2- —1-9)|]Uk + v(uk)|[; + -2—(f,uk + v(uk)) —* 00 as Tl —) 00. (2.34)

By direct calculation we can easily see that the {uIc + v(uk)} are critical points of
F(u), so it follows from (2.34) that

||u,s + v(uk)||p —» 00 as k —2 00.

This ensures the existence of a unbounded sequence of critical points for F (u), which

is a unbounded sequence of the weak solutions of the nonlinear wave equation (1) on

S".

We have proved the result for the simple case where g(u) = |u|P‘2u. Now we turn

to the more general case where g(u) satisfies the conditions (91) — (94) of Theorem

0.1.

42

 

CHAPTER 3

The existence for general

nonlinearity

Here we apply similar ideas as in Chapter 2 to prove Theorem 0.1, but we have to use
S1 index theory to replace Zg—action and hence estimates are much more complicated.

We first state Theorem 0.1 again.

 

Theorem 3.1 Suppose that 2 < p < 7"+1i2("32n5:‘1)"2"+9 and 9(5) E C(IR, IR) satisfies

(91) [9(51) — 9(52)I(€1 — £2) 2 “1'51 — 52V;
(92) there exists r > 0 such that

f
o <pG(€) age/o gmde s 69(5) for IEI 2 r;
(93) there exists a2 > 0 such that
|9(€)| S ae(|€|“"1 + 1) for 66 R;

(94) 9(5) = 0(|€|) at 5 = 0-
Then for any f (t,x) E If” (”-1)(S1 x S"), 2rr-periodic in t, the above non-linear wave

equation (1) has infinitely many periodic weak solutions in L”(S1 x S") FIH(Sl x S").

3. 1 Variational Scheme

43

3.1.1 A new variational formulation

As we did in Chapter 2.1, we introduce a new functional on E.

I(u) = wan + g) = gums — gnu-Ht — g(u), (3.1)
where
Q(u)=mihrri/(G(u+v)—f-(u+v))dtdx, 9:31 x3". (3.2)
v6 0

It turns out that the functional I (u) is in Cl(E, IR) and much easier to handle in
proving the Palais—Smale (P. S.) condition due to the compact embedding Theorem
1.1. Moreover, it is shown in Section 3.5 that the critical points of I (u) are also the
critical points of F (u).

By Properties (92) and (93), we have the following facts:
Remark 3.1 (g2’) cIItIP s 0(5) + e2 5 gage) + cg).
Remark 3-2 (93’) Ig(€)|”“‘"” S C4(£g(€) + 1)-
We will use (g2’) and (g3’) to verify the (RS) condition for I (u)
Lemma 3.1 (i) For all u E L”, there exists a unique v(u) E Np such that
Q(u) = fnmu + v(u)) — f . (u + v(u))) dt dx, (3.3)

(ii) Suppose u,- -—> u in E. Then v(u)) _. v(u) in I)" and g(uj + v(uj)) __.
g(u + v(u)) in L25,
(iii) Q(u) is of class C1 on E and for all u, h E E,

(Q’(u), h) = (“(902 + v(u)) — f) . h dt an. (3.4)

In particular, Q’ (u) : E —> E‘ is a compact operator.

44

From now on we denote by C various constants which depend on ||f||,,/(p_1) and are
independent of u E E.

Proof: (i) Fix u E E. Then the fimctional v I—+ / [C(u + v) — f - (u + v)] dt dx
is strictly convex and coercive and hence there exists un‘inue v(u) that minimizes this
functional.

(ii) Suppose u,- —2 u in E. Then Uj —* u in L” by the compact embedding Theorem

1.1. Since v(uj) is the minimizer for {Uj}, we have

[n [C(uj + v(u)) '— f ° (“j + v(u))] dt dx

2 / Ian.- + v(u.» — f - (He + n<n.->>I «Item. (3.5)
0

Since (g3) and (g4) imply

0(6) S (6 9(6) +0) S C(IEI”+ |€|) +01.

.1.
u
the left hand side of 3.5 is bounded. Fhrther (g1) concludes that
M 2 [[006 + ”('12))— f ' (u) + ”(“2)” 6“ div.
0
Z [[CIUJ' + ’U(’Uj)]p+l — f ' (Uj + v(uj)) + Cy] dt dx.
0

Thus v(uj) is bounded in L" and hence v(uj) —-* a in U’. Also, the left hand side of

(3.5) converges to fn[G(u + v(u)) —— f . (u + v(u))] dt dx as j —> 00 and so we have
fn [C(u + v(u)) — f - (n + v(u))] dtde
2 liminf / [G(ug + v(u.» — f - (n.- + n<u.-))I dtde
J-’°° a

2/n[C(u+v)—f-(u+‘v)]dtdx,

where last inequality is due to the weak lower semi-continuity of w I—> In [G (w) - f -

w] dt dx. Thus v(u) = E by the uniqueness of v(u), which implies v(uj) —* v(u) in L1".

45

Now we will show g(uj +v(u,-)) —e g(u+v(u)) in LID/0"”. Note that g(uj +v(u,-))
is bounded in LP/(P‘ll by (g3). Thus g(uj + v(uj)) _. n in LP/(P‘I). We have to show
g(u + v(u)) = n, which will be done by Minty’s trick:

For any w E L”, we have from the monotonicity of g({)
(9(“1 + v(uj)) — g(uj + w), 11(6) — w) 2 0- (3-6)

Since g(uj + ”(“2” — f E [Vi/(10-1), (g(uj + v(ujl) - f, v(uj» = 0 and hence
T, _ f E N13L/(p_1). Thlls

(g(ug- + v(Ug)).v(Ug)) = (f.v(ug-)) -> (f.v(U)) = (n,v(UI)-
Taking limit in (3.6), we have
(n —g(u+w),v(u) — w) 2 0 for all w e D”.
Set w = v(u) — new > 0,222 6 LP), divide by g and let T —. o,
(n — g(u+ v(u)),w) 2 o for all if) 6 LP.

Therefore 17 = g(u + v(u)).
(iii) By the convexity of C(é), we have for all u, h E E and z E IR

Q(u + Th) — Q(u) = fn[G(u + Th + v(u + Th))
—G’(u + v(u)) — f - (Th + v(u + Th) — v(u)] dt dx

ll
q
>
:2
g
+
2
E,
I
"H
v
3‘
&
9..
96-?

since g(u + v(u)) — f E NFL/(1H) and v(u + Th) — v(u) E Np.
Similarly we have by interchanging the role of Q(u + Th) and Q(u)

Q(u + Th) — Q(u) S T/n(g(u + Th + v(u + Th)) — f)hdt dx.

46

Letting T —2 0, we get

(Q’(u),h) = A(g(u + v(u) — f)hdtdx for all u, h e E.

Hence by (ii) and the compact embeddings of E ¢—+ LP and its dual

(0’)" = LP/(P'I) =—-+ E‘, Q’ : E —+ E‘ is continuous and compact.

Cl

Proposition 3.1 Under the conditions (g1 )-(g3) and f E HAP—1), I (u) E Cl(E, IR)

satisfies the Palais-Smale compactness condition (P.S.).
Proof: From the assumptions of (RS), we have

1
I(u)) = 5 (IllG-“ll2 - Hug—Hz) - Q(uj) S M.

I(I'(uj).h)| = I(Uf - will) - (Q'(uj),h)| S mllhllsi

 

 

where m = sup ||I’(u,-) 3..

Setting h = u,- in (3.8), we have
IIIUQ-Ill2 - lluill2 - (Q’(’ui),’uj>| S mllujH-
This then combined (3.7) with gives
1
|§(Q'(ui),ui) - Q(U)| S M + mllug'll-

Since fn(g(u,~ + v(uj)) — f)v(u,-) dt dx = 0, it follows from (3.9) that

[$06 + v(uj))9(uj + ”(Th)) - C(uj + v(uj)) + g f - (u,- + v(u,))] dt dx
:2

g M+m||u,~||. (3.10)

We want to get an estimation of the left-hand side of (3.10).

47

(3.9)

From (g2) the first term of left-hand side is estimated as
/n (u, + v(u.))g<n.- + v(u.» dtde
l
s 0 /n[§(ug- + v(u.>)g(n.- + v(u») — an».- + v(n.>>I me + c.
and from (95) the second term as
I /‘2 Mn.- +v<n.-)>I s Ilfllp/e—1>|lug+v(uj)llg
s C(fnnn + ..(..,))g(.., + v(u.» new):
Two terms combined yield
C/flh‘j + v(u,))g(u,- + ”(“1” dtde — C(fflWj + v(uj))g(uj + "(“1” dt div);
Thus
(g(u. + n<u.-)>g(n.- + v(u.» s CIIuglIe + 0.
Using (93’), we have
~57
”g(ug' + v(ug))II;s, S C/QWJ' + v(uj))g(uj + 0(6)) dt d3? + C S Cllujlls + 0.
Consequently,
“g(u.- + v(ug))llg/Ip—1) s Gangs—1”" + 1). (3.11)
Let h = u: — u].- in (3.8) to get
Hujllig - (Q’(u,-),uf - u;) = Hujlli: — /n(g(uj + 0(6)) — f)( f— 21;) dtdx
S mllujlls

Thus
llujllig — ”g(uj + v(uj)) - fle/(p—lflluf - “Illp S mllug'llE-

48

By (3.11) and the embedding theorem 1.1,
Hug-H22 — dangle—1”" + 1)||ug-||E s mllug-lle-
Thus {u,-} is bounded in E. Now from
u) — “j— = 1'06) + Q’(u,-),

where I ’ u- -—> 0 and Q’ is compact, we can conclude that u. has a convergent
J J

subsequence. C!

For later use we define

can) = 12,15: [n on. + g) dtdx.

Note that Q0 (u) is an S 1-invariant and satisfies the following properties as Q(u) does:

(i) for all u E E, there exists a unique vo(u) E Np“ such that
Qo(u) = (a G<n+go(n>)dtde,
(ii) if un —> u in E, then vo(u,,) —> vo(u) in Np and
g(ufl + vo(u,,) —e g(u + vo(u)) in Lp/(P‘I),
(iii) Qo(u) is of class C1 on E, Q, : E —-> E“ is compact and for all u, h E E,
<Qt(n),h> = [a g(u + n<u>>hdtde
We show the following relations between Q(u) and Q0(u):

Lemma 3.2 There are constants C1,C2, - - - , C7 > 0 such that for all u E E,

||u + v(u)“; S CIQ(u) + C2, (3.12)

||u + v0(u)||£ S C1Q0(u) + C2, (3.13)

49

(Q’Mm) Z pQ(u) - 03(|Q(UIll/” +1), (3-14)

<Qt(u>.n> 2goo(n) -03, (3.15)

|Q(u)l _<.. 04(Qo(u)+1). (3.16)

Wit) 3 c.<IQ(n)I + 1). (3.17)

IQ(u) — Qo<n)I s 05(Qo(U)1/" + 1), (3.18)

Iona — Qo(n>I s (2502an + 1). (3.19)

IIg<n + v(u))! 28:1; 5 Ce<Q’(u). u) + 07(Qo(u)“’ + 1). (3.20)
”g(u + vo(n))II;;g:3 5 cents). u> + 0.. (321)

Proof: We will prove (3.12), (3.14), (3.16), (3.18) and (3.20). The rest imme-

diately follows from these with f E 0. To prove (3.12) it is enough to get
lél‘” S C(0(6) - f-E) + CHIP/(”‘1’ + C for E 6 R.
which follows from (g5) and Young’s inequality on f - 5. We again use (9’2) to get

mm = gntGw + v(u)) — f . (n + v(undtde

[new v(u)) -g(u + v(u)) —pf- (u + v(u))dtdx + c
S (001),“) + Cllpr/(p-DHU + v(U)||p + C

S (Q'Miu) + CIQ(U)|1/p + C by (3-12),

l/\

which implie (3.14). Next we prove (3.16). First by the definition of Q(u), we get
Q(’U) — Q00!)
S [C(u + vo(u)) — f - (u + v0(u)) dt dx - / C(u + v0(u)) dt dx
0 n
= —/f ' (u+ ’Uo(’U.))dtd$

S IIfIIp/(p—1)IIu + ”0(a)”?

s C(Qo(u)”‘°+1) by (3.13).

50

Thus we have (3.16) by Young’s inequality. And (3.18) can be similarly proved . It
follows from (93’), (3.12) and (3.16) that

“g(u + v(u))nzlgji = [a |g(u + v(umfi dtde
C/(u+ v(u) -g(u + v(u)) dtdx + c
Q
C<Q'(n),n> + C ||f||p/<p-1> - He + v(u)”. + C
C(Q’(u),u> + 0 I001)!” + C

|/\ |/\ |/\

|/\

C(Q’(u),u) + CQo(u)1/” + C,

which yields (3.20). a

3.1.2 Modified functional

As in Rabinowitz [17], we replace I (u) by a modified functional J (u) E C1(E, IR).

Foru=u++u' E E, weset
1 l _
A(n) = 5 IIn+IIig — 5 In H22

and a1 E 4/(p+2) E (0,1). Let 6 > 0 be a constant such that 60 E a1(1 +6)3 E (0,1)
and set a0 E a1(1 + (5)2. Let x E C°°(IR, IR) be a function such that X(T) = 1 for
TSl, X(T)=0 for T21+6 and 0Sx(T) S1 forall TEIR. Furtherweset

__ Q(u) + b
1111(11.) - X aI(A(’u)2 +1)1/2)i

Q00!) + bo )
ao(A(u)2 + I)“2 ’

 

 

I(0(a) = 1 — X(
where b, b0 > 0 are constants such that
Qo(U) +b0 Z 1. Q(u) +b 2 1,

Qo(u) + be S (1 + 6)(Q(u) + b) for 211 u e E.

51

Note that the existence of b, be > 0 is ensured by (3.18). By the choice of b, b0 and
the definitions of w1(u) and 'l/Jo(‘u), we observe that I

supp 1/20(u) fl supp 1/11 (u) = (I) (3.22)
and for u E suppw1(u) U supp(l — wo(u)),

|Qo(U)|, |Q(U)| S 5olA(U)| + C- (323)

We now define for u E E,

J(u) = gums-gun-Hi.—§<1+go<n>—g.(n>)oo<n>

_%(1—go(n)+vi(u))Q(u) 6 OWE, 1R)-

First we state an inequality that will be often used. For all u E E, it follows from
(3.18) that

We) - (guru; - gun-Hi — 620nm s C(Qo(u)”” +1). (324)

where C > 0 is a constant independent of u E E. The reason for introducing J (u) is
that the first assertion of the following proposition, which says J is almost invariant,
holds for J (u) but not for I (it) Using the following proposition, we will show that
large critical values of J (u) are also critical values of I (u)

Proposition 3.2 The functional J (u) E C1(E, IR) satisfies:

(i) there is a constant a > 0 such that for u E E and 0 E [O,2rr),
[J(Tou) - J(u)| S a(|J(u)|1/p + 1),

where (Tou)(t,x) = u(t + 0, x) for 0 E [0,2rr) 2 S1.
(ii) there is a constant Mo > 0 such that J(u) 2 Mo and ||J’(u)||;.3 S

min [J (u)"”/ (1"1), 1] imply that J (u) = I (u)

52

Proof: Since Q0(T9u) = Qo(u) and i/J0(Tgu) = wo(u) for all 0 and u, we have
from the definition of J (u),

l

J(ToU) — J(u) = — ,(1 -- go<n))(Q(Ttn) — 2(a)) + ggduxcxu) — one»

— ggdnuxomn) — Qo(U))-

By (3.18), we have
|Q(Tou) — Q(u)l, |Q(u) — Qo(u)l. lama) — Qo(u)I s C(QgW/P + 1).
Hence we get
|J(Tou) — J(u)I s C((1 — now» + g(u) + gi(Tgu)(Qo(u)1/P + 1).

We may suppose that u E supp (1 — 1/J0(°)) U supp 1p,“ U supp ¢1(T9-). Otherwise
we have J (Tau) = J (u) It follows from the definition of J (u) and (3.24) (note that
60 E (0,1)) that

|J(U)l

IV

IA(u)| - Qo(U) — C(Qo(u) + town
2 IA<u>I — 60|A(u)l — C(|A(u)l +1>W

Z CilA(u)l - 02 Z CiQoM - 05.
which leads us to the conclusion:
|J(ToU) - J(U)| S C"(Qo(U)‘/” +1) S C"(|J(u)ll/“’ + 1)-

Proof of (ii) of the proposition using the following lemma can be similarly done

as in [25] (see also the proof of Proposition 2.2).
Lemma 3.3 (Tanaka/25]) For u = u+ + u" E E and h E E,
(J'(’U), h) = (1+ T2(u))(’u+ — “_ih)
—§(1 + gen) — g(u) + To(u))(Qb(u), I»)
—§(1 — gen) + Wu) + T1 (u))<Q'<n), n).

53

where To(u), T1(u), T2(u) E C(E, IR) and
suppT, C suppi/J, fori=0,1,

sup {I’T}(u)|; i=0,1,2,J(u) Z M} —e 0 as M—i 00.

As corollaries to (ii) of Proposition 3.2 and Proposition 3.1, we have

Corollary 3.1 Whenever u E E satisfies J’ (u) = 0 and J (u) 2 M0, then I (u) =
J(u) and I’(u) = 0.

Corollary 3.2 J (u) satisfies the (RS) condition on {u E E; J (u) 2 Mo}.

Corollary 3.1 ensures that large critical values of J (u) are critical values of I (u)

Hence in what follows we will seek for critical points of J (u) with large critical values.

3.2 Minimax methods and existence result

First we define an S 1-action on E which plays an important role in constructing

critical values of the functional I (u)

Definition 3.1 We define a group action To an E by (Tau)(t,x) = u(t + 0,x) for
9 E [0,2rr) 2 S1 and u E E. Suppose E is a subspace of E, then E is said to be

invariant under the action T9 if T9(E) = E.

We rearrange positive eigenvalues of A in the following order counting multiplicity,
denoted by

0<fl1Sfl2SM3S°”-

54

Recall that ¢¢,m(x) sin jt and ¢,,m(x) cos jt corresponds to the same eigenvalues of A.
We may arrange eigenflmctions corresponding to the positive eigenvalues in such a

way e9) , e?) , e9), e512) , - - - that

(1)
k

i) e = ¢1,m(x) sinjt; e?) = ¢l,m(x) cosjt for same l,m,j, and k = 1,2,3, . -- ,

and

ii) corresponding eigenvalues are such that
0<uil’ =u§2’ <29) =29” < ~---
We define subspaces E; (k E N) by
e: =W{ei". es, en es... en es}.

Then E; is Sl-invariant, UEIE; = E+, and [IuIIE S mellull2 for u E E;. For

u = u+ + u" E E}: ® E‘, we have, by (3.24) and (3.13),

1 1 _
J(u) s intent—sun IIi—Qo(n)+0(oo<u)l/P+1> by (3.24)

l 1 _ 1

S. EII“+II%‘§II“ II23-5Qo(u)+c
1 l _

s summit—sun IIi-CIIIn+go(n)IIi+cg by (3.13)
1 l __

S §|Iu+llb—§Ilu [lb—CIIIU'I'UOWNIQ‘I'Cz
1 _ l _

s ,IIu+IIi-c;n.P/2IIn+IIi-§IIn niece.

Hence there is a constant B), > 0 such that
J(u) < 0 for all u E E}: 6E” with ||u||E 2 Rs.

We may assume Rk < Rk+1 for all It. To construct a family of minimax sets, we

introduce another(simpler) Sl-action To on E by

(Tau) (t, x) = Z e,,,,mei98*9"<i>¢,,m(e)e"fi.
1

:jim

55

Recall that ¢;,m(x) sin jt and dim, (x) cos jt corresponds to the same eigenvalues of A.

We may arrange eigenfunctions corresponding to the positive eigenvalues in such a

way e9) , e9, egl), e53), - - - that
i) e9) = ¢1,m(x) sinjt; e)? = ¢¢,m(x)cosjt for same l,m,j, and k = 1,2,3, - -- ,
and

ii) corresponding eigenvalues are such that
1 2 1 2
0<M)=A)SA’=%’S~~

We define subspaces E; (k E N) by

E; = span{e[l), e32), egl), e52),--- ,eg), 69)}.

Then E; is Sl-invariant, U214 E}: = E+, and ||u||E S uk||u||2 for u E E3. For

u = u+ + u" E E: $ E", we have, by (3.24) and (3.13),

J(u) s gIIw‘IIi-,3,-IIn-IIi-Qo(n)+c<oo(u)1/P+1) by (3.24)
s gurus—guru;—§Qo<u>+c
s éllui‘ll'i’g—$-IIu‘IIi;—Cillu+vo(u)llt+02 by (3.13)
s gurus—gurus— illu+vo(u)ll’2'+02
s gunni— ins/”unali—glIu-‘Iliwe.

Hence there is a constant R), > 0 such that
J(u) < 0 for all u E E: 6E‘ with Hull]; 2 Rs.

We may assume Rs < Rk+1 for all It. To construct a family of minimax sets, we

introduce another(simpler) S 1-action To on E by

(Tau)(t, x) = Z u,,j,me’98i9"(j)(b),m(x)e’j’.

1.13m

55

We denote by X = (E, To) the space E with Sl-action To and E = (E, T9) the space
E with Sl-action T9. We also denote by X1", X3“, X ‘ the spaces E)“, E3”, E‘ with

Sl-action To. Let
FixS1 = {u E X : Tau = u for all 9 E [0,27r)}.
Definition 3.2 A mapping h : X —> E is said to be Sl- equivariant if and only if
(h o Tg)(u) = (To 0 h)(u) for u E X and 0 E [0,2rr] x S’.

The usual identity map is not S 1 (X, E)-equivariant. Let us define a new map r) :
X -> E which is S 1 (X, E)-equivariant and will play the role of the identity map. For

it = 21,33... ’Yz,j,m€’9"""”¢I,m($)€’jtt let

(M) )(,t 1‘)=Z71,],mei91.j.mljl¢l,m($)eijt

l ,j, m
where
710,", = 7,,_,-m non-negative for all l, j,m

01733", = -01J,m for all l,j,m and 0%", E [0,21r] for j > 0.

Note that the mapping 1) : X —2 E is linear and isometry and it is easy to see the

following properties of 17:

Lemma 3.4 (i) g(u) E C(X, E);
(ii) 17(u) is Sl-equivariant;
(in) g(XI: e X-) = E: e E- and IIn<n>IIe = Hunt for an n e x;
(iv) If K is precompact in E, the 17"1(K) is also precompact in X.

Now, similarly as in Chapter 2 we can define a family of minimax sets. Let B R is
the closed unit ball of radius R in E about 0, Dk = (BR, 0 (X: 6 X')}, and

Pk ___ {27 E C(Dk,E) : 7 satisfies (’71)—(73)}i

56

where

(71) 7 is Sl- equivariant,

(72) 7(u) = g(u) for all u E (D,c n 633,) U (D,c fl FixSl),

(73) 7(u) = a‘“(u)r)(u)+ + a“ (u)n(u)‘ + 5(u) for all u E 0),, where (1+ E
C(Dk, [0, 1]) and a’ E C(Dk,[1,c'i)) is an Sl-invariant functional (62 > 1 depends on
7) and H E C(Dk, E) is a compact and Sl-equivariant mappings such that a(u) = 1
and H(u) = 0 on (D;c fl 6BRk) U (D,c fl FixSl).

Moreover, set
U), = {u E Dk+1;u=x+pe),1_:1,x E XIEBX', p 2 0},
A), = {A E C(Uk, E) : A satisfies (A1) — (A3) in the following },

(A1) Alp, 6 Pk,

(A2) /\(u) = g(u) for all u E (UkflBBRH,)U(U,sfl(X,':6X-)\Dk)U(UkflFixS1),

(A3) Mu) = a+(u)n(u)+ + a‘(u)r)(u)‘ + 3(u) for all u E U”, where or" E
C(Uk, [0,1]) and of E C(Uk,[l,C—¥)) (a > 0 depends on 7) and ,3 E C(Uk,E) is a
compact mapping such that a(u) = 1 and fl(u) = 0 on (U,c n 68,“) U ((U;c n (X; 6
x-) \ Bk) 0 (U,c n FixSl).

Note that 1",, 75 0 and A), 75 0 Since 1le,c E I", and my, E Ak. Define

bk = inf sup J(7(u)), ch = Ainf sup J(A(u)).

761‘). uEDh 61": uEUps

Then we easily see that ck Z bk, moreover if ck > bk, we have the following existence
result.
First we have the same Deformation Lemma as in Chapter 2 since J satisfies

(P.S.) condition (Corollary 3.2) and J’ (u) is an operator of the form
J’(u) = (1+ T1(u))(u+ — u”) + compact,

Where [T1(u)| S 1/2 on {u E E; J (u) 2 Mo} (see proof of Lemma 3.3). We state it

again here.

57

Lemma 3.5 (cf. [18], [19]) (Deformation Lemma) Suppose C > M0 is a regular value
of J (u) Then for any E > 0 there exist an e E (0,2?) and an one parameter family of
homeomorphisms 45(t, ) of E, 0 S t S 1 with the properties:

(i) @(t,u) = u, if t = 0, or [J(u) — cl 2 E;

(ii) 45(1,As+s) S As_s where Ac = {u E E: J(u) S c};

(iii) 45(1,u) = a+(u)u+ + a'(u)u‘ + rt(u), where 01+ E C(E, [0,1)), 0‘ E

C(E, [1,&)) (61 > 1 constant) and K. is a compact operator.

By standard contradiction argument using this Lemma, we get the critical values

{ck(6)} of J (u) as in the following lemma.

Lemma 3.6 Suppose ck > bk 2 M2. Let 6 E (O,m, — bk) and
A146) = {A 6 Ah 2 J(A) S bled-60’". D’s}.
Let

6:45) = iefigfib’f J(A(UD (2 Ct)-

Then ck(6)is a critical value of I (it)

Therefore the existence of a subsequence of {cflfiil which satisfy ck, > bk, 2 Mo
ensures the existence of critical values of I (u). In what follows, we will show that

there is a subsequence {kj } :32, such that

ij>bkj fOT‘jEN,
bkj—ioo asj—+oo.

Arguing alternatively, we have

Proposition 3.3 Assume ch = bk for all k 2 he, then there is a constant C > 0
such that
blc g ore/W) for all k e N.

58

Proof: Note that Dk+1 = U T9(Uk) and for any u E DH; \ (X: 6 X’)
OE[0,21r)
there is a unique (x, 0) E (Uk\Dk) x [0, 2r) such that Tgx = u. For any given A E As,
we define 7 : Dk+1 —+ E by
A(u) = T9(A(x)) for u = Tax E Dk+1 where (x,0) E Us x [0,2rr).

We can see A is well-defined (by (A1)), continuous and belongs to Pk“. Moreover

by (i) of Proposition 3.2, we have

bk“ S sup J(A(u))= sup J(T9A(x))

“6 Dls+1 xEU;s ,OE [0,21r)

S 8:5[J(/\($)) + a (|J(/\($))|’/" +1)I-
: I:
Since A E A), is arbitrary, we deduce
I),c+1 g ck + a(e,‘,/P + 1) for all k.
Ifck=bkfork2ko ,weobtain
em 3 bk + 2(1),?” + 1) for k 2 ’60.
An induction argument yields the desired result. CI
Our goal in the next two sections is proving that there exists a subsequence
{lg-highs > 0 and Cs > 0 satisfying

bk, > Csky/ (JD—1‘5) for all j e N. (3.25)

59

3.3 Critical values 6;, of a comparison functional

K00

3.3.1 Introduction of comparison functional K (it)

To estimate bk, we introduce a new comparison fimctional K (u) here. By (3.24), the

definition of Q0(u) and (93), we have for u+ E E+

l
J(U+) Z 5 ||u+||23 - CQo(u+) — C
2 %||u+||§.3—C/G(u+)dtdx—C
n
l 51 _
Z §Hu+||2E—“p—Hu+||£“a2t

where C, (‘11, 52 > 0 are constants independent of u+ E E+. We define a comparison

functional K on E+ by
Knr=hnfls—§Wewi
2 p I"

Then K (u) E C2(E+, IR) and it is easy to show that K (it) satisfies the (RS) condi-

tion. So we have the following lemma:

Lemma 3.7 (i) J(u) 2 K(u) — 62 for all u E E+.
(ii) K satisfies the Palais-Smale condition (P.S.).

3.3.2 Bahri-Berestycki’s max-min value 6;,

First let us define a family of max-min sets for K (u) Recall that
52m-2k-I-1 = {2' E Cm—k+1;lzl = 1},

and the group S1 = {e‘o} acts naturally on it by

'9 '9 '9 i0 _ 2m—2k+1
e' z = (e’ zl,e' 22, ...,e zm_k+1) for z — (21,22,...,Zm_k+1) E S .

60

Form>k, k, mEN, set
L" = {o E C(S2m‘2"+1,E,‘;) : 0(e’9x) = T90'(CC) for all x E Szm-2k+1},

= su min K x .

flirt 06:41:" zeSQvn-Qk-l-l (0( ))
We will prove that 6,, = lim.,-_.s,o 6,?” is a sequence of critical values of K (u) and
bk 2 6;; + C. To get some estimates on films, we need several lemmas. First we state

a version of a Borsuk-Ulam lemma.

Lemma 3.8 Let a, b,N E N. Suppose that g E C(IRN x C“, IRN x C“+”) and h E
C(S2"+ 1, IRN x C“+“) satisfy the following conditions:
(1)9 = (gl--':gNrgN-l-l: "'igN-i-a-I-b) and h =(h1:"-rhN+a+b) are SI -equivariant in

the following sense: for all 1 S j S N and l S l S a + b,

9i(=c. e‘ay) = 9206.21). g~+z(z. e‘oy) = e“""9~+z(1‘,y),

hj(ewz) = hj(Z)a hN+1(¢'3”9Z) = eik'ohN+l(Z)

for all (x,y) E IRN x C“ and z E S2“+1, where k, at 0 are integers;
(ii) g(x,0) = (x, O) for all x E IR”;

(iii) there is a 70 > 0 such that
lg(a:.y)|2 = IIL‘I2 + It)!2 for Incl2 + lyl2 Z 73-
Then

h(SZb+1) flg(IRN x C“) at 0.

Proof: Consider the following S 1-equivariant continuous mapping.
F : IRN x C“ x C"+1 —+ IRN x C“ x Cb; F(x,y,tz) = g(x,y) — th(z), where x E
IR", y E C“ and tz E C"+1 = {tz;t 2 0,2 E S2b+1}. Set R = max{7o,max{|h(z)|; z E

Sm1}}+ l and

9 = {(z.g,tz); le2 + IzI2 < R'flt e [0.1m e 52"“).

61

Applying S 1-version of Borsuk-Ulam theorem to F : 652 —> IRN x C“ x C”, there

exists (x0, yo, tozo) E 00 = {(x,y,tz) E S2; [x]2 + Iyl2 = R2 or t = 1} such that
F(xo, yo, tozo) = 0, i.e., g(xo, yo) = t0h(zo).
From the choice of R, F(x, y, tz) 76 0 on 652 (I {(x, y, tz); [x]2 + Iyl2 = R2}. Therefore
we have to = 1 and g(xo,y0) = h(zo). D
We also need the following technical lemma(same as Lemma 2.8).
Lemma 3.9 For all 0 E (0,1/p), there is a Ca > 0 independent of k E N such that
IIUIIp S CoflZOIIUIIE f0r u E (133i,
where (E:)J- = {v E E+; (v,e,-) = 0 for i = 1,2, ...,k}.
N ow we can prove the following estimates on ,BL"’S(see Proposition 2.5).

Proposition 3.4 (i) 0 S ,8}? S 6,211 < 00 for all m, k E N;
(ii) For all k E N, there exists u(k) and 17(lc) such that

0SV(k)S6,',"Sz7(k)<oo forallm2k+1;
(iii) V(k)—+oo ask—+00.

PI‘OOf: (i) For any 0 6 AL", OISZm—Wc—l E A?“ and Ulszm—zrt—1(S2m—2k_1) C
0(Szm‘2k“). Hence we have 6;,” S 67,11.
(ii) First we prove the existence of 17(k). Applying Lemma 3.8 to h = o :

S2""2"+1 —> E; and g = id : E; —> E; we can see that

0(S2m"2k+1) I) E; 51$ (I) for all a E AL".

62

Thus we have for all a E AL",

zegglnlgk-lel K(o(x)) S 8111: K(u). (3.26)
uEEk
For u E E:, we have
_ 1 2 0’0 p 1 2 P
K(U) — 5 IIUIIE - FIIUIIP S '2-II’UIIE - C|l1t||2

|/\

l _
5 IIuII'i — Cfltp/leullig-
Thus the right-hand side of (3.26) is finite and independent of o and m. Set
17(lc) = sup K(u) < oo,
uEE:

then we obtain

fl]: = 08341;: xesglnj-EHI K(U($)) S 17(k).

Now we show the existence of u(lc). We construct a special a E AL“ as follows:
write
2m+1

S2m—2k+l = {x = (32," ...,$2m+1) E R2m-2k+2; Z x? = 1}
i=2lc

and set a : S2""'2"+1 ——> E;\0 by
g(e) = gal/“‘2’”wens/(ring),
2m+1
where w(x) is defined by w(x) = Z x,-e,-. Obviously we have a E A7,". Since

i=2lc
||w(x)||E = l on Szm‘ZkH, we have

1 1 _ _ _ _
K(e(e)) = (5 — yet”? 2’Ilw(w)llp”’/“’ 2’.
On the other hand w(x) E (E;_1)i, ||w(x)||E = 1 for all x E Szm‘2k“, and hence it

follows from Lemma 3.9 that

||w($)||p S Cir/1:1 for x E S2m’2k“.

63

where 0 E (0, 1 / p) and Co is a constant independent of k and x. Therefore
K (o(x)) 2 C; pig/(r2) for all x E Szm'zk“.

The right-hand side the above inequality is independent of m. Set u(k) =

C9, Ilia/(”’2’ . Then we have

as" Z “$121,13ng K(a(x)) 2 u(k) for m > n.

(iii) Since u(k) —-> 00 as n —> 00, we obtain u(k) —> 00 as k —> 00. D

As in Proposition 3.1, we can prove the following compactness conditions
(P.S.)m, (P.S.). for K(u):

(P.S.)m: If {Uj} C E; satisfies K(Uj) S C and (K|E$)’(u,-) —> 0 as j —+ 00, then
{u,} is relatively compact in E;

(P.S.).: If {um} C E+ satisfies um E E;, K(um) S C and [I(K|E$)’(um)||(E$). —»
0 as m —2 00, then {um} is relatively compact in E+.

Since K is an even fimctional, we have the following results via standard argument.

(Bahri and Berestycki [4])

Proposition 3.5 Suppose u(k) > 0. Then 6;" is a critical value of K | 33;. And the

limit of any convergent subsequence of 6;," as rn —> 00 is a critical value of K.

By (ii) of Proposition 3.4, choose a sequence {m,-} such that m, —> 00 as j —+ co
and

fl), = lim (3:7 exists for all k E N.
J"’°°
Then by the above Proposition 3.5, we have the following properties for 6),.
Proposition 3.6 i) m is a critical value of K E C2(E+, IR) for each k E N;

1.2') 3k S ,Bk-I-l for all k E N;

iii),Bk—+ooask—>oo.

64

Here we establish the comparison result between critical values of J (u) and K (it).
Proposition 3.7 For all k E N,
bk 2 fire - 512,
where 62 is the number appeared in Lemma 3. 7.

First we state a linking lemma which can be proved using Borsuk-Ulam Lemma
3.8.

Lemma 3.10 For all 7 E I", and for all 0‘ E A2“,
((Pm’Y)(DIc) U {u E E}: €19 E“ = IIUHE 2 Bid) 0 “Sm—2"“) 75 0.
where Pm : E —> E; 6 E‘ is an orthogonal projection.

Proof: Let 7 E I"c and extend 7 to 7 E C(X: 6 X+,X) by 7(u) =
7(u) if Hulls S Rt. and 7(u) = We) if HUIIE .>_ Rk- Obviously, ’70!) is W611
defined and Sl-equivariant. Since m > 'k, by definition of 17(u) we have

Pew: e X’) =- met) u {n e E: e E2 llulle 2 Rt}.
Therefore it suflices to prove Pm 7(X: 6 X ‘) fl o(S’""2"+1) aé 0. We rearrange
negtive eigenfunctions and denote by f1, f2, f3, . We set for l E N,
E“ = Spanlfj;1SjSl}

and let Pm; : E = E+ 6 E‘ -—> E; 6 E,’ be the orthogonal projection. Consider the

operators

e : sir-2"+1 —» E; C E; e 13;. Pm,17 : x; e X,‘ .2 s; e 13,-.
Applying Lemma 3.8 for h = o and g = PM “Xian—a we get some x, E 32m-2k—1
and u; e E; a; By,

0(a) = ng 7(uI)- (3.27)

Since Sm"2lc+l is compact, there is a subsequence {931,-} such that

65

2k+l

x), -—» x in Szm‘ , u(xlj) —> 0(x) in E3,.

Now, using (73) similarly as in the proof of Lemma 2.10 we can show that there

exists it E X3 6 X ‘ such that Pm 7(u) = o(x). This completes the proof. CI

Now we prove Proposition 3.7, the main result of this section.
Proof: First we recall that J(7(u)) S 0 for u E E3 6 E‘ with Hull}; 2 R), by

the choice of Rs. Using Lemma 3.10 and Lemma 3.7, we can see
b'": — nf J m > J
7163361115: (P v(u)) _ 08613413, $651,319,”, (0(3))

> sup min K(o(x))—c’tg,

aEA'" zES‘Zm—Zh+l

that is,

bEZW—c‘tg forall m>k.

Hence we have

liminf b7," 2 ,8), — 62. (3.28)
On the other hand, we have
limsup b]? S bk. (3.29)

In fact, it follows from (73) that for 7 E B,

Pm7(U) = a+n(U)+a’n(U)‘ + Pmfl(U) —> a+n(U)+a‘n(U)’ + 6(a) = 7(a).
uniformly in D), as m ——> 00. Hence we have
811p J(Pm7(U)) -* sup J(7(u )) as m -> 00-

1560). “GD I:

Choosing 7 E 1“,, such that sup J (7(u)) < bk + e, we obtain

uEDh

IimStIp b7." S IimSUP sup J(Pm7(u )) = sup J(7(u )) S be + 8-

m—voo m—ooo uED;s uED 5

Thus (3.29) holds since the above inequality holds for any 5 > 0. Combining (3.28)

and (3.29), we get the estimate of the proposition. CI

66

3.4 Morse index and B],

We want to get 6k, 2 Cskg/ 00—1—6) for all j E N. Estimates of Morse index at Ski’s

will give the result. We proceed similarly as in Chapter 2.
Definition 3.3 For u E E+, we define a index of K”(u) by

index K ”(u) = the number of nonpositive eigenvalues of K ” (u)

= max {dimS;S S E+ such that (K"(u)h, h) S 0, h E S}.
Here “A S B” in the bracket means A is a subspace of B.

Proposition 3.8 Suppose S), < (3H1. Then there exists us E E+ such that

K(uk) S Bk)
K’(Uk) = 0,
indexK”(u,s) 2 2k — 1.

By definnitin of Bk, the result without the last assertion is obvious. To get the

last assertion, we first consider finite dimensional case.

Proposition 3.9 Suppose 63' < 6311, m > n + 1. Then there exists it}? E E3, such
that

(Kls;)'(uie") = 0.
index(KlE3,)”(uI-.n) 2 2k — 1.

To prove the above proposition, we will use a theorem from Morse theory, i.e., a
result concerning the relationship between certain homotopy groups of level sets of
a fimctional and its critical points. We proceed as in Chapter 2. First we need a

theorem to treat the case where critical points may be degenerate.

67

PrOposition 3.10 (cf. Marina-Prodi [15]) Let U be a C2 open subset in some Hilbert
space H and 4') E CQ(U, IR). Assume a)" is a Fredholm operator (of null index) on the
critical set Z ((1)) = {x E U; 45’ (x) = O}, 4) satisfies (RS) and Z (43) is compact. Then,
for any 8 > 0, there exists if) E C2(U, IR) satisfying (RS) and with the following
properties :

(1') Wm) = 43(50) z'f distance {3, Z (45)} 2 5;

(ii) Wit) - ¢($)|, [WM - ¢'($)lli III/((50) - ¢"($)|| S 8 f0?" all-1‘ E U;

(iii) the critical points of 11) are finite in number and nondegenerate.

We can easily prove that K | E; satisfies all the assumptions of the above proposi-
tion, that is,

1) K | E; E C’(E3,, IR) satisfies (PS) and Fiedholm.

2) All critical value of K | E; are non-negative because

K(u) = K(u) - §<<Klegr<n>n> = (3 — 1%,) no IIuII; 2 o.

3) Z (K | E; ) is compact. In fact, note that there exists Rm > 0 such that K (u) < 0
for u e E3, with Hens 2 IL; hence Z(K|E;) is bounded.

Thus by Proposition 3.10, for all e > 0 there exists ¢s E C2(E3,, IR) satisfying
(PS) and
|¢e(U) - K(u)l < e,
||¢'e(“) — (KIE;)'(")|| < 6.

||¢’e’(U) - (Kle;)”(U)ll < 6; (3-30)

the critical points of obs are finite and non-degenerate. (3.31)

Form>kande>0,let

AF(€)= sup min ¢e(0($))-

m-2k+l
06A? 2682

68

By (2.31),
BS-IESBLWE) Sfir+e
Moreover, we have
Lemma 3.11 Suppose that as E IR satisfies 63(5) < as — 25 < as < Balk) Then

rr2m_2k_1([q§s Z as]m,w) 31$ 0 for some w E [cps Z as]m,
where [¢s _>_ as]m = {u E E3; ¢s(u) 2 as}.
Proof: We argue by contradiction. Suppose that
71'2m_2k_1([¢5 Z as]m,w) = 0 for all w E We 2 as]m.
Then there is a homotopy
H : [0,1] x [92"“2'”1 -2 l¢e 2 at]...
such that H(0,x) = u(x), H(l, x) = we for all x E S2m'2k'1. Write

32m‘2k+1={£ = (9/26”); C 6 Cm‘k. p G 1R. ICI2 +122 =1}-

By the definition of 63;,(5), there is a o E AI,”+1 such that 0(S2m’2k‘1) C [43s 2 as]m.

Define 5‘ : Sim-2""1 —+ E3; by

0(C) if p= 1. ICI =1.
6(4. p6”) = T9H(e“9;l’;&-l) if p 74 o, c at 0,
Tgwo if p= —1, C = 0.

Then we can easily check that 6 E AL". Since K is invariant under the action T9, by
(3.30), we have
|¢s(u) — ¢s(Tgu)| S 26 for u E E3,.

69

Thus ¢s(&(C,pe’9)) Z as—2e, i.e., 6(S2m‘2k“) C [tbs Z as—2e]m. From the definition
Of 3.1" (5).

33(6) 2 min ¢s(&(x)) 2 as — 2e.
:1:E.Sr""""'2"+1

But this contradicts with the assumption. Thus the proof is completed. [I

Now the proofs of Proposition 3.9 and Proposition 3.8 can be similarly done as
those of Propositon 2.9 and Pr0position 2.9 using Lemma 3.11 and the following

Lemma 3.12.

Lemma 3.12 For a regular value a E IR of its , set
Me: a) = max{ we... gun); w(x) s a, nun) = 0}.
Then

rn([¢s 2 a]m,w) = 0 for all p E [45s 2 as]m, l S 2m — L(e; a) — 2.

3.5 Proof of the Main Theorem

By Lemma 3.6 and Proposition 3.3, we know that (3.25), the growth estimate on
Bk’s, ensures the existence of an unbounded sequence of critical values. We now

prove (3.25). First note by Proposition 3.8 that there exits us, such that
1 2 a0 1 1
file,- 2 K0119): §||ukg||s — glluellt = (5 — 1;)00 Huts-ll?-

Due to Proposition 3.8, we can get an upper b01md of index K ” (uj) same as in

Proposition 2.11.
Proposition 3.11 There exist C > 0 such that for u E E+,
indexK”(u,-) S C||u||:,

—2 —2
2 "q and s - SID—)2,

n+1—(n—1 q _

where r = q_,

70

Then by the same proof as in the case of g(u) = |u|P“2u, we get (3.25) for the

same p’s satisfying 2 < p < 7"+1+2V(32:fl)‘2"+9.

 

This establishes the existence of a sequence {uk} C E of critical points of I (u)
such that as k —-» 00,

Km) —-> co and I’(uk) = 0.

Let 21,. = uk + v(uk). Then it can be shown that 21,. is a critical point of F(u) by

direct calculation. On the other hand since I ’ (uk) = O, we have
1 _ _ _ l _
I(uk) = / 59(11):) uk — G(uk) + ifuk (111) (it —) 00.
0

Finally it follows from (93) that {me} is a unbounded sequence in D". We have
proved that there exists a unbounded sequence of critical points for F (u), which is a

unbounded sequence of the weak solutions of the nonlinear wave equation (1) on S".

71

BIBLIOGRAPHY

72

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75