Generalized fountain theorem and application to strongly indefinite semilinear problems

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by  Cyril J. Batkam

GENERALIZED FOUNTAIN THEOREM AND APPLICATION TO
STRONGLY INDEFINITE SEMILINEAR PROBLEMS
3
1 CYRILJOE¨LBATKAMANDFABRICECOLIN
0
2
Abstract. Byusingthedegreetheoryandtheτ´topologyofKryszewskiand

n Szulkin,weestablishaversionoftheFountainTheoremforstronglyindefinite
a functionals. Theabstractresultwillbeappliedforstudyingtheexistenceofin-
J finitelymanysolutionsoftwostronglyindefinitesemilinearproblemsincluding
9 thesemilinearSchro¨dingerequation.
2

]
P
A Introduction
.
h The Fountain Theorem established by T. Bartsch [1] and M. Willem [25] is a
t
a powerful tool for studying the existence of infinitely many critical points of some
m indefinite functionals, which are symmetric in the sense of admissibility posed by
[ Bartsch [1] (see also [7, 25]). This theorem has been used by several authors to
study the existence of infinitely many large energy solutions of various semilinear
1
problems, see for instance [1, 7, 8, 9, 10, 11, 25]. We also mention the paper of W.
v
Zou [12], where a variant Fountain Theorem was established without PS type
8
p q´
9 assumption.
0 . The aim of this paper is to generalize the Fountain Theorem in order to apply it
7 toawiderclassofindefinitefunctionals,speciallytostronglyindefinitefunctionals;
. 1
1 that is functionals of the form φ u 2(cid:10)Lu,u(cid:11) ψ u defined on a Hilbert space
p q “ ´ p q
0 X,whereL:X X isaselfadjointoperatorwithnegativeandpositiveeigenspace
3 Ñ
both infinite-dimensional. The study of such functionals is motivated by a number
1
of problems from mathematical physic. They arise for example in the study of
:
v periodic solutions of the one-dimensional wave equation, in the study of periodic
i
X solutions of Hamiltonian systems, or in the existence theory for systems of elliptic
equations.
r
a
. The paper is organized as follows: In the first section, the main abstract results
are outlined and future applications are discussed while in section 2, the degree
theory of Kryszewski and Szulkin is introduced and a Borsuk-Ulam type theorem
for admissible maps is stated. In section 3, we construct a deformation which will
be very helpful to establish the new fountain theorem in section 4. Sections 5
and 6 are dedicated to the application of the abstract result to a nonlinear sta-
tionary Schr¨odinger equationand to a noncooperative system of elliptic equations,
respectively.
2010 Mathematics Subject Classification. Primary35A15Secondary35J50;35J60.
Keywordsandphrases. Fountaintheorem;Kryszewski-Szulkindegree;τ´topology;Strongly
indefinitefunctional.
ThisworkwasfundedbyanNSERCgrant.
1
2 C.J.BATKAMANDF.COLIN
1. Main Results
LetX bearealHilbertspacewithinnerproduct ,norm anddecomposition
p¨q }¨}
X Y Z, where Y is closed and separable, and Z Re . Define for k 2
and“0 ‘r ρ , “‘8j“0 j ě
k k
ă ă
Y : Y k Re , Z : Re ,
k “ ‘p‘j“0 jq k “‘8j“k j
B : u Y u ρ and N : u Z u r .
k k k k k k
“ P } }ď “ P } }“
Consider on X Y Z ˇthe τ top(ology introdu ced by Kˇryszewski(and Szulkin
ˇ ˇ
in [2] (see also [2“5], C‘hapter 6),´and let ϕ : X R be a C1 functional such that
ϕ is τ upper semicontinuous, and ∇ϕ is weakÑly sequentiall´y continuous. We are
´
interested in the obtention of critical points of ϕ when the latter has the following
linking geometry
sup ϕ u inf ϕ u .
uPBBk p qăuPNk p q
In order to achieve this goal, we assume that ϕ is invariant under an admissible
action of a finite group G (see Definition 6 below), which the antipodal action of
Z2 is a particularcase. More precisely,we will introduce the following assumption:
A1 AfinitegroupGactsisometricallyandτ-isometricallyonX,andtheaction
p q
of G on every subspace of X is admissible (in the sense of Definition 6).
Roughlyspeaking,theabove-mentionedadmissibilityauthorizesanextensionofthe
Borsuk-Ulamtheoremforacertainclassoffunctionsconvenientlycalledσ admissible
´
later in the text. By taking advantage of the notion of admissible group, we show
that B and N link in the following sense: If γ : B X is τ continuous and
k k k
Ñ ´
equivariant, γ id, and every u B has a τ neighborhood N in Y such
|BBk “ P k ´ u k
that id γ N int B iscontainedinafinite-dimensionalsubspaceofX,then
u k
p ´ q X p q
γ Bk Nk ` . Since in˘our purpose Y canbe infinite-dimensional, the Brouwer
p qX ‰H
degree which is usually used in the finite-dimensional case will be replaced with
the Kry`szewski-Szulkindegree (see Definition 3 below) in th˘e establishment of the
abovelinking. Byconstructinganequivariantdeformationwiththeflowofacertain
gradient vector field, we finally show that if in addition
supϕ u ,
p qă8
Bk
then there is a sequence un X such that
p kqĂ
ϕ un 0, ϕ un inf sup ϕ un as n ,
1p kqÑ p kqÑγPΓkuPBk p kq Ñ8
whereΓ isaclassofmapsγ :B X definedinTheorem11. Theabstractresult
k k
Ñ
will be used to study the existence of infinitely many large energy solutions of the
two following strongly indefinite semilinear problems.
1.1. SemilinearSchro¨dingerequation. Forthefirstapplicationweconsiderthe
following nonlinear stationary Schr¨odinger equation
∆u V x u f x,u ,
P
p q " ´u H`1 RpNq ,“ p q
P p q
under the following assumptions:
V0 The function V :RN R is continuousand1 periodicin x1,...,xN and0
p q Ñ ´
lies in a gap of the spectrum of ∆ V.
´ `
GENERALIZED FOUNTAIN THEOREM AND APPLICATION 3
f1 The function f : RN R R is continuous and 1 periodic with respect
p q ˆ Ñ ´
to each variable x ,j 1,...,N.
j
“
f2 There is a constant c 0 such that
p q ą
f x,u c 1 up 1
´
| p q|ď p `| | q
for all x RN and u R, where p 2 for N 1,2 and 2 p 2N N 2
P P ą “ ă ă { ´
if N 3.
ě
f3 f x,u u uniformly with respect to x as u 0.
pf4q Tphereqe“xis˝tps|γ|q 2 such that for all x RN and|u|ÑR 0
p q ą P P zt u
0 γF x,u uf x,u ,
ă p qď p q
u
where F x,u : f x,s ds.
0
p q “ p q
f5 For all x RN anşd u R, f x, u f x,u .
p q P P p ´ q“´ p q
The natural energy functional associated to P is given by
p q
ϕ u : ∇u2 V x u2 F x,u dx, u H1 RN .
p q “żRN`| | ` p q ´ p q˘ P p q
By V0 the Schr¨odinger operator ∆ V in L2 RN has purely continuous
specptruqm, and the space H1 RN ca´n be`deco`mposepd intqo˘H1 RN Y Z such
p q p q “ ‘
that the quadratic form
u H1 RN ∇u2 V x u2 dx,
P p qÞÑżRN`| | ` p q ˘
is negative and positive definite on Y and Z respectively. Both Y and Z are
infinite-dimensional, so the functional ϕ is strongly indefinite. This case has been
of large interest in the last two decades, existence and multiplicity results have
been obtained by different methods, see for instance [2, 13, 14, 15, 16, 17, 18]. We
prove in this paper that, under the above assumptions, P has infinitely many
p q
large energy solutions.
1.2. Noncooperative elliptic system. As the second application we study the
following system:
∆u H x,u,v in Ω,
u
P1 $ ∆v“ Hvpx,u,vq in Ω,
p q & ´u “v 0pon Ωq,
“ “ B
where Ω is an open bounded%subset of RN. It is well known that P has a
1
p q
variational structure, and the associated Euler-Lagrangefunctional is given by
1 1
Φ u,v : ∇v 2 ∇u2 H x,u,v dx u,v H1 Ω .
0
p q “żRN ´2| | ´ 2| | ´ p q¯ P p q
BytakingadvantageofthenewFountainTheorem,weobtaininfinitelymanylarge
energy solutions of P , provided the following conditions are satisfied:
1
h1 H C1 Ωp Rq R and H x,0,0 0, x Ω.
p q P p ˆ ˆ q p q“ @ P
h2 c 0 such that
p q D ą
H x,u,v H x,u,v c 1 up 1 v p 1 ,
u v ´ ´
| p q|`| p q|ď `| | `| |
for all x RN and u,v R, where p 2 for`N 1,2 and 2 p˘ 2N N 2 if
P P ą “ ă ă { ´
N 3.
ě
h3 0 pH x,u,v uHu x,u,v vHv x,u,v , for u,v 0,0 , x Ω.
p q ă p qď p q` p q p q‰p q @ P
4 C.J.BATKAMANDF.COLIN
h4 H x, u, v H x,u,v , u,v R, x Ω.
p q p ´ ´ q“ p q @ P @ P
. The solutions of P represent the steady state solutions of reaction-diffusion
1
p q
systems which are derived from severalapplications, such as mathematical biology
or chemical reactions (see [19] and reference therein). Recently the existence and
multiplicity of solutions for noncooperative elliptic systems of the form P have
1
p q
beenprovedbyseveralauthors,seeforinstance[3,4,5,6,19]andreferencestherein.
2. Kryszewski-Szulkin degree theory
Let H, , beaseparableHilbertspace. Let bj j 0 beatotalorthonormal
sequencpe inp¨qH}a¨n}qd let p q ě
}u}1 :“ 8 2j11 pu, bjq, uPH.
jÿ0 ` ˇ ˇ
“ ˇ ˇ
We will denote by σ the topology generated by the norm 1. Clearly y 1 y
for every y H, and moreover if y is a bounded sequen}ce¨}in H then } } ď} }
n
P p q
σ
y y y y.
n n
á ðñ Ñ
Let U be an open bounded subset of H such that U is σ closed. The following
´
definitions are due to Kryszewski and Szulkin (see [25]).
Definition 1. A map f : U H is said to be σ-admissible if it meets the two
Ñ
following conditions.
(1) f is σ continuous,
´
(2) each point u U has a σ neighborhood N such that id f N U is
u u
contained in aPfinite-dime´nsional subspace of H. p ´ qp X q
Definition 2. An application h: 0,1 U H is a σ-admissible homotopy if:
r sˆ Ñ
(a) h 1 0 0,1 U ,
´
p qX r sˆB “H
(b) h is σ-con`tinuous, ˘
(c) Every t,u 0,1 U hasaσ-neighborhoodN suchthat s h τ,s τ,s
t,u
N p 0q,P1r Usˆ is contained in a finite-dpimqensional su bs´pacpe ofqH{p. qP
t,u
p qXpr sˆ q
If f is a σ-admissible ma(p such that 0 f U , then f 1 0 is σ closed and
´
consequently,σ compact. Letus considertRhe cpBoveqringoff 1p0qbyse´tsN which
´ u
are σ open nei´ghborhoods of u f 1 0 such that id f Np q U is contained
´ u
inafi´nite-dimensionalsubspaceoPfH.Spoq,wecanfindpm´poiqnpts u1X, ...,qum f´1 0
such that f 1 0 V : m N U and id f N U is contPainedpinq
a finite-dime´nspioqnaĂl subs“paŤcen“F1 ofunHX, k 1,p...´,m.qpTuhkeXconqtraction and the
@ “
excision properties of the Brouwer’s degree imply that deg f ,V F does
B V F
not depend on the choice of V and F, where deg is the Bropu|weXr’s degXree.q This
B
leads to the following definition due to Kryszewski and Szulkin:
Definition 3 (Kyszewski-Szulkin degree). Let f be a σ-admissible map such that
0 f U . The degree of f is defined by
R pB q
deg f,U : deg f ,V F ,
B V F
p q “ p | X X q
where V and F are defined above.
Proposition 4.
(i) If y U then deg id y,U 1.
P p ´ q“
GENERALIZED FOUNTAIN THEOREM AND APPLICATION 5
(ii) Let f be a σ-admissible map such that 0 f U . If deg f,U 0 then there
R pB q p q‰
exists u U such that f u 0.
P p q“
(iii) If h is a σ-admissible homotopy, then deg h t, ,U does not depend on the
p p ¨q q
choice of t.
For the proof of Proposition 4, we refer the reader to [25].
The following theorem was inspired by [2] Theorem 2.4 iv .
p q
Theorem5(Borsuk-Ulamtheoremforadmissiblemaps). LetU beanopenbounded
symmetric neighborhood of 0 in H such that U is σ closed. Let f : U H be a
σ-admissible odd map. If f U is contained in a pro´per subspace of H, Ñthen there
p q
exists u0 U such that f u0 0.
PB p q“
Proof. Assume by contradiction that f 1 0 U . Since f is odd, we may
´
assume that V is a symmetric (i.e V pVq)X. BLet“FHbe a proper subspace of H
´ “
such that f U F, and let z H F. Define h : 0,1 U H by h t,u :
p q Ă P z r sˆ Ñ p q “
f u tz. Onecaneasilyverifythathis aσ-admissiblehomotopy. ByProposition
p q´
4 iii , deg h 0, ,U deg h 1, ,U . The classical Borsuk theorem implies that
p q p p ¨q q “ p p ¨q q
deg h 0, ,U deg f,U 0, so deg h 1, ,U deg f z,U 0. It then
p p ¨q q “ p q ‰ p p ¨q q “ p ´ q ‰
follows from Proposition 4 ii that there exists u0 U such that z f u0 , which
is a contradiction since z pHqf U . P “ p q (cid:3)
P z p q
Now we need to precise what kind of symmetries we will consider in the sequel.
WerecallthattheactionofatopologicalgroupGonanormedvectorspace E,
p }¨}q
is a continuous map G E E, g,u gu, such that:
ˆ Ñ p qÞÑ
(1) eu u, u E.
“ @ P
(2) h gu hg u, u E, h,g G.
p q“p q @ P @ P
(3) The map u gu is linear for every g G.
ÞÑ P
The action of G is isometric if gu u , u E,g G. A subset F of E is
invariant if gF F for every g} }G“. }A}fu@nctPional ϕP: E R is invariant if
“ P Ñ
ϕ gu ϕ u , for every u E and g G. A map f : X X is equivariant if
p q “ p q P P Ñ
f gu gf u , for every u E and g G.
p q“ p q P P
Definition 6. Let G be a finite group acting on H. The action of G is said to be
admissibleif u H gu u, g G 0 andeveryσ-admissible andequivariant
P “ @ P “t u
map f :U H0, whˇere U is an ope(n bounded invariant neighborhood of the origin
in H such tÑhat U is ˇσ closed and H0 is a proper subspace of H, has a zero on U.
´ B
Example 7. Theorem 5 implies that the antipodal action of Z2 on every separable
Hilbert space is admissible.
3. A deformation lemma
Let Y be a closed separable subspace of a Hilbert space X endowed with the
inner product and the associated norm . Let P : X Y and Q : X Y
K
p¨q }¨} Ñ Ñ
be the orthogonal projections. Let θ be an orthonormal basis of Y. On X we
j
p q
consider a new norm
8 1
u : max Pu, θ , Qu , (1)
~ ~ “ ´jÿ02j`1|p jq| } }¯
“
6 C.J.BATKAMANDF.COLIN
andwedenotebyτ thetopologyandalltopologicalnotionsrelatedtothetopology
generated by see [2] or [25] . It is clear that Qu u u . Moreover,
~¨~ } } ď ~ ~ ď } }
if un is a bounded`sequence in X˘ then
p q
τ
u u Pu Pu and Qu Qu.
n n n
Ñ ðñ á Ñ
We recall some standard notations:
Let S X and ϕ C1 X,R . dist u,S : u S , dist u,S : u S ,
τ
Ă P p q p q “ } ´ } p q “ ~ ´ ~
S : u X dist u,S α , ϕa : u X ϕ u a .
α
“ P p qď “ P p qď
The followinˇg lemma is som(ewhat a combinaˇtion of Lem(ma 3.1 and Lemma 6.8
ˇ ˇ
in [25].
Lemma 8 (Deformation lemma). Assume that a finite group G acts isometrically
and τ-isometrically on the Hilbert space X. Assume also that ϕ C1 X,R is
invariant and τ-upper semicontinuous and ∇ϕ is weakly sequentiallPy conptinuoqus.
Let S X be invariant and c R, ǫ,δ 0 such that
Ă P ą
8ǫ
1
u ϕ´ c 2ǫ,c 2ǫ S2δ, ϕ1 u . (2)
@ P pr ´ ` sqX } p q}ě δ
Then there exists η C 0,1 ϕc 2ǫ,X such that:
`
P pr sˆ q
(i) η t,u u if t 0 or if u ϕ´1 c 2ǫ,c 2ǫ S2δ,
p q“ “ R pr ´ ` sqX
(ii) η 1,ϕc ǫ S ϕc ǫ,
` ´
(iii) ηp t,u Xu qĂδ u ϕc 2ǫ, t 0,1 ,
2 `
(iv) }ϕpη ,uq´is}nďon i@ncrPeasing, @u Pϕrc 2sǫ,
`
(v) Epacph¨poqiqnt t,u 0,1 ϕc@2ǫPhas a τ-neighborhood N such that v
` t,u
η s,v s,vp NqPr sˆ0,1 ϕc 2ǫ is contained in apfinqite-dimensi on´al
t,u `
supbspaqcˇepof qXP, p qXpr sˆ q(
ˇ
(vi) η is τ-continuous,
(vii) η t, is equivariant t 0,1 .
p ¨q @ Pr s
Proof. Let us define
w v : 2 ∇ϕ v ´2∇ϕ v , v ϕ´1 c 2ǫ,c 2ǫ .
p q “ } p q} p q @ P r ´ ` s
Since ∇ϕ is weakly sequentially continuous, for ev`ery v ϕ 1 c˘ 2ǫ,c 2ǫ
´
there exists a τ-open invariant neighborhood Nv of v suchPthat ∇`rϕ´u ,w v` s1˘
p p q p qq ą
u N . Since ϕ is τ-upper semicontinuous, N : ϕ 1 ,c 2ǫ is τ-open.
v ´
@ P “ s´8 ` r
It follows then that the family ` ˘
r
N N c 2ǫ ϕ v c 2ǫ N
v
“ ´ ď p qď ` Y
is a τ-open covering of ϕc` 2ǫ. Sˇˇince ϕc`2ǫ,τ is metr(ic, anrd hence paracompact,
thereexistsaτ-locallyfinite τ-opencpoveringMq : M :i I ofϕc 2ǫ finerthan
i `
N. Define “ P (
V : M .
i
“
ďi I
P
For every i I we have only the possibilities M N for some v or M N. In
i v i
P Ă Ă
the first case we define v : w v and in the second case v : 0. Let λ i I
i i i
be a τ-Lipschitz continuous“parptitqion of unity subordinated to“M and detfineˇ roPn Vu
ˇ
h u : λ u v .
i i
p q “ p q
ÿi I
P
The map h satisfies the following properties (see Lemma 6.7 of [25]):
GENERALIZED FOUNTAIN THEOREM AND APPLICATION 7
(a) h is τ-locally Lipschitz continuous and locally Lipschitz continuous,
(b) each point u V has a τ-neighborhood V such that h V is contained in
u u
P p q
a finite dimensional subspace of X,
(c) ∇ϕ u ,h u 0 u V,
(d) pu pϕq1 pcqqě2ǫ,c@ 2Pǫ , ∇ϕ u ,h u 1.
´
@ P r ´ ` s p q p q ą
` ˘ ` ˘
Let us define the equivariant vector field h on V by:
1 r 1
h u : g´ h gu .
p q “ G p q
| |gÿG
r P
We claim that h satisfies properties a , b , c and d above. In fact, since ϕ is
p q p q p q p q
invariant we have
r
∇ϕ gu ,v ∇ϕ u ,g 1v g G u,v X.
´
p q “ p q @ P @ P
` ˘ ` ˘
Thus
1
∇ϕ gu ,h u ∇ϕ u ,g 1h gu
´
p q p q “ G p q p q
` ˘ | |gÿG` ˘
r P
1
∇ϕ gu ,h gu ,
“ G p q p q
| |gÿG` ˘
P
sothathsatisfies c and d . Letu V,foreveryg Gthereexistsaτ neighborhood
p q p q P P ´
V of gu suchthat h V is containedin a finite dimensional subspace of X,con-
gu gu
r p q
sequently W : V is a τ neighborhood of u such that h W is contained
u gu u
“ ´ p q
gŤPG r
in a finite dimensional subspace of X and then h satisfies b . Finally h obviously
p q
satisfies a .
p q r r
Let us define
1
A: ϕ´ c 2ǫ,c 2ǫ S2δ,
“ r ´ ` s X
` ˘
1
B :ϕ c ǫ,c ǫ S ,
´ δ
“ r ´ ` s X
` ˘
1
ψ u : dist u,V A dist u,V A dist u,B ´ on V,
τ τ τ
p q “ ` z ˘” ` z ˘` ` ˘ı
and
f u : ψ u h u , u V.
p q “ p q p q P
It is clear that f is τ-locally Lipschitz, τr-continuous, continuous, locally Lipschitz
andequivariant. Byassumption 2 , w u δ , whichimplies f u δ onV.
For each u ϕc 2ǫ, the Cauchy pproqbl}emp q}ď 4ǫ } p q}ď 4ǫ
`
P
dµ t,u f µ t,u
dt p q“´ p p qq
" µ 0,u u
p q“
hasauniquesolutionµ ,u definedonR . MoreoverµiscontinuousonR ϕc 2ǫ.
` ` `
p¨ q ˆ
Since f u δ , we have for t 0
} p q}ď 4ǫ ě
δt
µ t,u u .
} p q´ }ď 4ǫ
8 C.J.BATKAMANDF.COLIN
We also have
d d
ϕ µ t,u ∇ϕ µ t,u , µ t,u
dt p p qq“´ p p qq dt p q¯
∇ϕ µ t,u ,f µ t,u
“´´ p p qq p p q¯
ψ µ t,u ∇ϕ µ t,u ,h µ t,u 0
“´ p p qq´ p p qq p p q¯ď
It is easy to verify that if we define η on 0,1 ϕc 2ǫrby η t,u : µ 2ǫt,u , then
`
r sˆ p q “ p q
i , ii , iii and iv are satisfied. The proof of v and vi is the same as that
p q p q p q p q p q p q
of b and c in Lemma 6.8 of [25] (with T 2ǫ). Since f is equivariant, vii is a
q q “ p q
directconsequence ofthe existence andthe uniqueness ofthe solutionof the above
Cauchy problem. (cid:3)
4. An infinite-dimensional version of the fountain theorem
In the sequel we assume that Z : YK 8 Rej.
“ “
jà0
“
We use the following notation:
k
Y : Y Re , Z : 8 Re ,
k j k j
“ ‘p q “
jà0 jàk
“ “
B : u Y u ρ , N : u Z u r where 0 r ρ , k 2.
k k k k k k k k
“t P || ||ď u “t P || ||“ u ă ă ě
We denote P ˇ:X Y and Q :X Z ˇthe orthogonal projections. We define
kˇ Ñ k k Ñ kˇ
e for j 0,1,...,k
e1j :“" θjj k 1 fo“r j k 1,
´ ´ ě `
where θ is the orthonormal basis of Y we chose at the beginning of section 2 .
j
p q
W` e can define a new norm on X by setting ˘
8 1
~u~k “max´jÿ02j`1|pPku,e1jq|,||Qk`1u||¯.
“
Remark 9. Since a direct calculation shows that
3
u u and u 2k 1 u , u Y ,
k ` k k
~ ~ ď 2~ ~ ~ ~ď ~ ~ @ P
then for every k, the norms and are equivalent on Y . In addition, it is
k k
~¨~ ~¨~
easy to show that the projections P are τ-continuous, for every k.
k
Lemma 10 (Intersection lemma). Under assumption A1 , let γ : Bk X such
p q Ñ
that
(a) γ is equivariant and γ id,
|BBk“
(b) γ is τ-continuous,
(c) every u int B has a τ-neighborhood N in Y such that id γ N
k u k u
P p q p ´ q X
int B is contained in a finite-dimensional subspace of X. `
k
p q
Then γ B N˘ .
k k
p qX ‰H
Proof. Let U u B γ u r . Since ρ r and γ 0 0, U is an open
k k k k
“ t P || p q||ă u ą p q“
bounded and invariant nˇeighborhood of 0 in Yk. It is clear that Bk is τ-closed, so
ˇ
we deduce from (b) that U is also τ-closed. Consider the equivariant map
Pk 1γ :U Yk 1.
´ Ñ ´
GENERALIZED FOUNTAIN THEOREM AND APPLICATION 9
τ τ
(i) Pk 1γ is τ-continuous. In fact, if un u, then from b γ un γ u and
´ Ñ p q p q Ñ p q
by Remark 9 we have Pk 1 γ un γ u 0 as n .
(ii) Letu U. From c u~has´aτp-npeigqhb´orhpooqqd~NÑsuchthÑat8id γ N U
u u
P p q p ´ qp X qĂ
W, where W is a finite-dimensional subspace of X. Let v N U Y
u k
Yk 1 Rek, then id Pk 1γ v Pk 1 v γ v λekP W XRekĂwhic“h
´ ‘ p ´ ´ qp q “ ´ p ´ p qq` P `
is finite-dimensional.
Thus Pk 1γ : U Yk 1 is σ admissible in the sense of Definition 1, where the
pair H,´b in sÑectio´n 2 is re´placedby th`e pair Y , e . Since the action of G
p jq k p 1jq
on Yk` is adm˘issible, there exists u0 U such tha`t Pk 1γ˘˘u0 0. This ends the
proof of the lemma since X Yk 1 P BZk. ´ p q “ (cid:3)
“ ´ ‘
Theorem 11. Under assumption A1 , let ϕ C1 X,R be invariant and τ-upper
semicontinuous such that ∇ϕ is wepaklyqsequenPtiallyp contqinuous.
For k 2, define
ě
a : sup ϕ u ,
k
“ p q
}uu}P“Yρkk
b : inf ϕ u ,
k
“}uu}P“Zrkk p q
c : inf sup ϕ γ u ,
k
“γPΓkuPBk ` p q˘
where
Γ : γ :B X γ is equivariant, τ continuous and γ id.
k “! k Ñ ˇ ´ |BBk“
Every u int B ˇhas a τ neighborhood N in Y such that id γ N int B
k u k u k
P p q ´ p ´ q X p q
` ˘
is contained in a finite-dimensional subspace of X. Futhermore ϕ γ u ϕ u u B . ,
k
p p qqď p q @ P )
and
d : sup ϕ u .
k
“ p q
}uu}PďYρkk
If
d and b a ,
k k k
ă8 ą
then c b and, for every ǫ 0, c a 2 , δ 0 and γ Γ such that
k k k k k
ě Ps p ´ q{ r ą P
supϕ γ c ǫ, (3)
k
˝ ď `
Bk
there exists u X such that
P
(1) c 2ǫ ϕ u c 2ǫ,
k k
´ ď p qď `
(2) dist u, γ B 2δ,
k
p p qqď
(3) ϕ u 8ǫ δ.
1
} p q}ď {
Proof. It follows from Lemma 10 that c b . Assume by contradiction that the
k k
ě
thesis is false. We apply Lemma 8 with S : γ B . We may assume that
k
“ p q
c 2ǫ a . (4)
k k
´ ą
We define on B the map β u : η 1,γ u . We claim that β Γ .
k k
p q “ p p qq P
(i) It follows from 4 and i of the deformation lemma that β id.
(ii) Itisclearthatβpisqequivparqiantandτ-continuous,andϕ β u |BBkϕ“u u B .
k
p p qqď p q@ P
10 C.J.BATKAMANDF.COLIN
(iii) Let u int B . Since γ Γ , u has a τ-neighborhood N in Y such that
k k u k
P p q P
id γ Nu int Bk W1,whereW1 isafinite-dimensionalsubspaceofX.
p ´ q X p q Ă
From v` of the defor˘mation lemma the point 1,γ u has a τ-neighborhood
p q p p qq
M 1,γ u M1 Mγu such that z η s,z s,z M 1,γ u 0,1
vϕcpk`N2pǫqq(qγis“c1oMntaiˆnedinBa,fiwneithea-dviemei dns´ioβnapvlsubqsˇˇpipdaceγqWP2vofpXγ.vpTqhqηuXs1fp,orγrevvserˆy
u ´ γ u k
WP1 WX2 whpich ipsqfiqnXite-dimensionpal.´ qp q“p ´ qp q` p q´ p p qqP
`
Thus β Γ .
k
P
Now by using 3 and ii of Lemma 8 we obtain
p q p q
c sup ϕ β u sup ϕ η 1,γ u c ǫ,
k k
ď p p qq“ p p p qqqď ´
}uu}P“Yρkk }uu}P“Ykρk
which contradicts the definition of c . (cid:3)
k
Theorem 12. Under assumption A1 , let ϕ C1 X,R be invariant and τ-upper
semicontinuous such that ∇ϕ is wpeakqly sequePntiaplly coqntinuous. If there exists
ρ r 0 such that
k k
ą ą
A2 ak : sup ϕ u 0 and dk : sup ϕ u .
p q “ p qď “ p qă8
}uu}P“Yρkk }uu}PďYρkk
A3 bk : inf ϕ u , k .
p q “ }uu}P“Zrkk p qÑ8 Ñ8
Then there exists a sequence un X such that
p kqn Ă
ϕ un 0, ϕ un c : inf sup ϕ γ u as n .
1p kqÑ p kqÑ k “γPΓkuPBk ` p q˘ Ñ8
Proof. Choose k sufficiently large and apply the preceding theorem. (cid:3)
Recallthatthefunctionalϕsatisfiesthe PS -condition Palais-Smalecondition
c
p q
at level c , if every sequence u X such that `
n
p qĂ
˘
ϕ un c and ϕ1 un 0 as n ,
p qÑ p qÑ Ñ8
has a convergentsubsequence.
Corollary 13. Under the assumptions of the preceding theorem, if ϕ satisfies in
addition the PS condition for every c 0, then ϕ has an unbounded sequence of
c
p q ą
critical values.
In the sequel is the usual norm in Lp.
p
|¨|
5. Semilinear Schro¨dinger equation
In this sectionwe apply our abstracttheoremto the resolutionof the semilinear
Schr¨odinger equation
∆u V x u f x,u ,
" u´ H`1 RNp q. “ p q (5)
P p q
We define the functional
1
ϕ u : ∇u2 V x u2 dx F x,u dx, u H1 RN . (6)
p q “ 2żRN`| | ` p q ˘ ´żRN p q P p q
It is well knownthat if ϕ is of classC1 thenits criticalpoints are weaksolutions
of (5). Observe also that, due to the periodicity of f and v, if u is a solution of