High energy sign-changing solutions to Schr\"{o}dinger-Poisson type systems

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by  Cyril Joel Batkam

HIGH ENERGY SIGN-CHANGING SOLUTIONS TO
SCHRO¨DINGER-POISSON TYPE SYSTEMS
5
1
CYRILJOELBATKAM
0
2

n
a
J Abstract. Weprovetheexistenceofinfinitelymanyhighenergysign-changing

3 solutionsforsomeclassesofSchro¨dinger-Poissonsystemsinboundeddomains,
2 withnonlinearitieshavingsubcriticalorcriticalgrowth. Ourapproachisvari-
ational and relies on an application of a new sign-changing version of the

] symmetricmountainpasstheorem.
P
A
1. Introduction
.
h Thispaperisconcernedwiththeexistenceofhighenergysign-changingsolutions
t
a to some elliptic systems of Schr¨odinger-Poisson type. The Schr¨odinger-Poisson
m systemisasimplemodelusedforthestudyofquantumtransportinsemiconductor
[ devices that can be written as

1 ´ı~BBψtpt,xq“´2~m2 ∆ψpt,xq`φpt,xqψpt,xq´g ψpt,xq in Ω (1.1)
v " ´∆φpt,xq“|ψpt,xq|2 ` ˘ in Ω,
2
4 whereΩisadomaininR3,~isthePlanckconstant,mistheparticlemass,∆isthe
9 usualspatialLaplace operator,ψ :RˆΩÑC is the wavefunction, φ:RˆΩÑC
5 isthe electrostaticpotential, andg :RÑC is a nonlineartermdescribingexternal
0 perturbations and interactions between many particles. In case Ω is bounded, one
.
1 can impose the boundary conditions ψ “ φ “ 0, which physically mean that the
0 particlesareconstrainttoliveinΩ. Formoreinformationonthephysicalrelevance
5
of the Schr¨odinger-Poissonsystem, we refer to [8, 18, 19].
1
The study of standing wave solutions of (1.1), that is solutions of the form
:
v
i ψpt,xq“e´ıωtupxq,φpt,xq , upxqPR, ω ą0,
X
´ ¯
r leads to a problem of the form
a
´∆u`φu“hpx,uq in Ω
" ´∆φ“u2 in Ω, (1.2)
which has been widely study in the last decade. Many important results concern-
ing existence and non existence of solutions, multiplicity of solutions, least energy
solutions, radial and non radial solutions, semiclassicallimit and concentrations of
solution have been obtained. See for instance [2, 5, 8, 11, 10, 12, 21, 22, 23, 27]
and the references quoted there. In these papers, some of the solutions found are
nonnegative, but in many cases the sign of the solutions cannot be decided. The
existenceofasign-changingsolutionsto(1.2)wasconsideredrecentlybyAlvesand
2010 Mathematics Subject Classification. Primary35J47Secondary35B33;35J50; 35J91.
Key words and phrases. Schro¨dinger-Poisson systems, Sign-changing solutions, High energy
solutions,Infinitelymanysolutions,Criticalnonlinearity,Variationalmethods.
1
2 HIGH ENERGY SIGN-CHANGING SOLUTIONS
Souto[1],KimandSeok[13]andWangandZhou[24]. Theyobtainedanontrivial
sign-changingsolutionby using the method ofthe Neharimanifold. We recallthat
a solution pu,φq to (1.2) is said to be sign-changing if u changes its sign. The
existence of many sign-changing solutions to (1.2) was an open problem until the
recent work of Liu, Wang and Zhang [17]. In that very nice paper, they consid-
ered (1.2) in the whole space R3 and they obtained infinitely many sign-changing
solutions by using a new version of the symmetric mountain pass theorem in the
presence of invariant sets of the descending flow established in [16]. However, the
nonlineartermh wasassumedto be ofsubcriticalgrowth. Moreover,it seems that
there is no result in literature on the existence of sign-changing solutions to the
Schr¨odinger-Poissonsystem with critical growing nonlinearities.
The first goal of this paper is to prove the existence of (many) sign-changing
solutions to (1.2) by allowing the nonlinear function h to contain a critical term.
More precisely, we investigate the existence of multiple sign-changing solutions to
the system
´∆u`φu“fpx,uq`λu5 in Ω
pSPqλ $ ´∆φ“u2 in Ω
& u“φ“0 on BΩ,
where Ω is an open boun%ded subset of R3 with smooth boundary, λ ě 0, and
f :ΩÑR satisfies the following conditions:
pf1q f PC ΩˆR,R and there exists a constant cą0 such that
` ˘
|fpx,uq|ďc 1`|u|p´1 , where 4ăpă6;
` ˘
pf2q fpx,uq“˝p|u|q, uniformly in xPΩ, as uÑ0;
pf3q there exists µ ą 4 such that 0 ă µFpx,uq ď ufpx,uq for all u ‰ 0 and for
u
all xPΩ, where Fpx,uq“ fpx,sqds;
0
ş
pf4q fpx,´uq“´fpx,uq for all px,uqPΩˆR.
We shall prove the following results.
Theorem 1.1 (Subcriticalcase). Assumethat λ“0. If pf1,2,3,4q are satisfied then
pSPq0 has a sequence uk,φk kě1 of solutions such that uk is sign-changing and
` ˘
1 1
|∇u |2` φ u2 ´ Fpx,u qÑ8, as k Ñ8.
2żΩ k 4żΩ k k żΩ k
Theorem1.2(Criticalcase). Assumethatpf1,2,3,4qaresatisfied. Thenthereexists
a sequence λ ,u ,φ such that λ Ñ0`, pu ,φ q is solution to pSPq , u is
k k k kě1 k k k λk k
sign-changi`ng, and ˘
1 1 λ
|∇u |2` φ u2dx´ Fpx,u q´ k u6 Ñ8, as k Ñ8.
2żΩ k 4żΩ k k żΩ k 6 żΩ k
Remark 1.3. It was claimed by the authors of [17] that Theorem 1.1 holds see
Remark 1.1 in [17] . However, the proof we provide here, which is based o`n a
different approach, a˘ppears to be much more simpler.
HIGH ENERGY SIGN-CHANGING SOLUTIONS 3
OurapproachinprovingTheorems1.1and1.2isvariationalandreliesonanew
sign-changing critical point theorem, we established in a recent paper [7], which is
modelled on the fountain theorem of Barstch [6].
In the second part of this paper, we will use the same approach to study the
following Schr¨odinger-Poissontype system:
´∆u`ψu3 “|u|q´2u in Ω
pSP1q $ ´∆ψ “ 1u4 in Ω
2
& u“ψ “0 on BΩ,
where Ω is a bounded doma%in in RN with smooth boundary. This problem was
first introduced by Azzollini, d’Avenia and Luisi in [4]. By combining the method
ofcut-offfunctionwithvariationalarguments,theyprovedthatpSP1qpossessesat
leastonenontrivialsolutionwhenΩiscontainedinR3and1ăq ă5. In[3,14,15],
the authorsstudiedthe casewherethe nonlocaltermgrowscritically. However,in
thosepapersonlynonnegativesolutionswerefound. As farasweknow,thereis no
result concerning the existence of sign-changingsolutions of pSP1q. Therefore,the
second goal of this paper is to prove that pSP1q possesses sign-changing solutions.
Our result on this problem reads as follows:
Theorem 1.4. Let Ω be a bounded smooth domain in RN (N “ 1,2). If q ą 8
then pSP1q has a sequence u ,ψ of solutions such that u is sign-changing
k k kě1 k
and ` ˘
1 1 1
|∇u |2` ψ u4 ´ |u |q Ñ8, as k Ñ8.
2żΩ k 8żΩ k k q żΩ k
Remark 1.5. As a consequence of the Lax-Milgram theorem, pSP1q can be trans-
formed into a single semilinear elliptic equation such that the non local term is ho-
mogeneous of degree 8 (see Section 4). Therefore, condition q ą8 in the statement
of Theorem 1.4 is needed to guaranteethat the problem possesses the mountain pass
structure, which is crucial for our argument. However, it seems that this condition
can be relaxed by means of the perturbation method used in [17].
The paper is organized as follows. In Section 2, we state the abstract critical
point theorem we will apply in the proof of our main results. In Section 3, we
provide the proof of Theorems 1.1 and 1.2. Finally, we prove Theorem 1.4 in
Section 4.
Throughoutthe paper we denoteby |¨| the normofthe Lebesgue spaceLppΩq,
q
by ”Ñ” the strong convergence and by ”á” the weak convergence.
2. Abstract preliminary
In this section, we present the critical point theorem which will be applied to
prove our main results.
Let Φ be a C1-functional defined on a Hilbert space X of the form
X :“‘8 X , with dimX ă8. (2.1)
j“0 j j
We introduce for k ě2 and mąk`2 the following notations:
Y :“‘k X , Z “‘8 X , Zm “‘m X , B :“ uPY |}u}ďρ ,
k j“0 j k j“k j k j“k j k k k
(
N :“ uPZ |}u}“r , Nm :“ uPZm|}u}“r , where 0ăr ăρ ,
k k k k k k k k
Φ :“Φ| , K :“( uPY ; Φ1 puq“0 and(E :“Y zK .
m Ym m m m m m m
(
4 HIGH ENERGY SIGN-CHANGING SOLUTIONS
Let P be a closed convex cone of Y . We set for µ ą0
m m m
0 0 0
˘D :“ uPY |dist u,˘P ăµ , D “D Yp´D qand S :“Y zD .
m m m m m m m m m m
` ˘ (
We will also denote the α-neighborhood of S ĂY by
m
V pSq:“ uPY |distpu,Sqďα , @αą0.
α m
(
The following result was established by the present author in [7]. For the sake of
completeness, the proof will be provided in this paper.
Theorem 2.1. Let Φ P C1pX,Rq be an even functional which maps bounded sets
to bounded sets. If, for k ě2 and m ąk`2, there exist 0ă r ăρ and µ ą0
k k m
such that
pA1q ak :“maxuPBBkΦpuqăbk :“infuPNkΦpuq.
pA2q Nkm ĂSm.
pA3q There exists an odd locally Lipschitz continuous vector field B : Em ÑYm
such that:
(i) B p˘D0 qXE âD0 ;
m m m
(ii) th`ere exists a con˘stant α1 ą0 such that (cid:10)Φ1mpuq,u´Bpuq(cid:11)ěα1}u´
Bpuq}2, for any uPE ;
m
(iii) for a ă b and α ą 0, there exists β ą 0 such that }u´Bpuq} ě β if
uPY is such that Φ puqPra,bs and }Φ1 puq}ěα.
m m m
Then there exists a sequence punk,mqn ĂVµ2mpSmq such that
lim Φ1 pun q“0 and lim Φ pun qP b ,maxΦpuq .
m k,m m k,m k
nÑ8 nÑ8 “ uPBk ‰
The proof of this theorem relies on the following deformation lemma.
Lemma 2.2. Let Φ P C1pX,Rq be an even functional which maps bounded sets
to bounded sets. Fix m sufficiently large and assume that the condition pA3q of
Theorem 2.1 holds. Let cPR and ε0 ą0 such that
@uPΦ´m1 rc´2ε0,c`2ε0s XVµ2mpSmq : }Φ1mpuq}ěε0. (2.2)
` ˘
Then for some εPs0,ε0r there exists η PC r0,1sˆYm,Ym such that:
(i) ηpt,uq“u for t“0 or uRΦ´1 rc`´2ε,c`2εs ; ˘
m
(ii) η 1,Φ´1ps´8,c`εsqXS Ă`Φ´1 s´8,c´˘εs ;
m m m
(iii) Φ` ηp¨,uq is not increasing˘, for any`u; ˘
m
(iv) ηpr0`,1sˆD˘mqĂDm;
(v) ηpt,¨q is odd, for any tPr0,1s.
The proof of this lemma is given in the appendix.
Proof of Theorem 2.1. pA1q and pA2q imply that for k big enough we have ak ă
bk ďinfuPNmΦmpuq. Let
k
Γm :“ γ PCpB ,Y q; γ is odd, γ| “id and γpD qĂD .
k k m BBk m m
(
Γm is clearly non empty and for any γ PΓm the set U :“ uPB ; }γpuq}ăr is
k k k k
an open bounded and symmetric i.e. ´U “U neighbor hood of the origin in(Y .
k
By the Borsuk-Ulam theorem, the`continuous o˘dd map Πk ˝γ : BU Ă Yk Ñ Yk´1
has a zero,where Πk :X ÑYk´1 is the orthogonalprojection. It then follows that
HIGH ENERGY SIGN-CHANGING SOLUTIONS 5
γpB qXNm ‰ H and, since Nm Ă S , that γpB qXS ‰ H. This intersection
k k k m k m
property implies that
c :“ inf max Φ puqě inf Φpuqěb .
k,m m k
γPΓmk uPγpBkqXSm uPNkm
We would like to show that for any ε0 Ps0,ck,m2´akr, there exists u P Φ´m1 rck,m´
2ε0,ck,m`2ε0s XVµ2mpSmq such that }Φ1mpuq}ăε0. `
Arguing by con˘tradiction, we assume that we can find ε0 Ps0,ck,m2´akr such that
}Φ1mpuq}ěε0, @uPΦ´m1 rck,m´2ε0,ck,m`2ε0s XVµ2mpSmq.
` ˘
Apply Lemma 2.2 with c “c and define, using the deformation η obtained the
k,m
map
θ :B ÑY , θpuq:“ηp1,γpuqq,
k m
where γ PΓm satisfies
k
max Φ puqďc `ε, (2.3)
m k,m
uPγpBkqXSm
with ε also given by Lemma 2.2.
Using the properties of η (see Lemma 2.2), one can easily verify that θ PΓm.
k
On the other hand, we have
η 1,γpB q XS Ăη 1,Φ´1ps´8,c `εsqXS . (2.4)
k m m k,m m
` ˘ ` ˘
Infact, if uPη 1,γpB q XS thenu“η 1,γpvq PS forsome v PB . Observe
k m m k
that γpvqPSm`. Indeed,˘if this is not true`then we˘have γpvqPDm, and by (iv) of
Lemma 2.2 we have u “ ηp1,γpvqq P D which contradicts the fact that u P S .
m m
Since by (2.3) γpvqPΦ´1ps´8,c `εsq,we deduce using (ii) ofLemma 2.2that
m k,m
u“ηp1,γpvqqPη 1,s´8,c `εsXS . Hence (2.4) holds.
k,m m
Using (2.4) and (i`i) of Lemma 2.2, we obta˘in
max Φ puq“ max Φ puq
m m
uPθpBkqXS uPη 1,γpBkq XSm
ď ` ˘max Φ puq
m
uPη 1,Φ´m1ps´8,ck,m`εsqXSm
ďc ` ´ε, ˘
k,m
contradicting the definition of c .
k,m
The above contradiction assures that for any ε0 Ps0,ck,m2´akr, there exists u P
Φ´m1 rck,m´2ε0,ck,m`2ε0s XVµ2mpSmq such that }Φ1mpuq}ăε0.
Wet`hendeducebylettingε0˘goesto0thatthereisasequencepunk,mqn ĂVµ2mpSmq
such that
Φ1 pun qÑ0 and Φ pun qÑc , as nÑ8.
m k,m m k,m k,m
(cid:3)
3. Proof of Theorems 1.1 and 1.2
In this section, we treat the system
´∆u`φu“fpx,uq`λu5 in Ω
pSPqλ $ ´∆φ“u2 in Ω
& u“φ“0 on BΩ,
We will assume throughou%t this section that pf1,2,3,4q are satisfied.
6 HIGH ENERGY SIGN-CHANGING SOLUTIONS
Let X :“H1pΩq be the usual Sobolev space endowed with the inner product
0
(cid:10)u,v(cid:11)“ ∇u∇v
żΩ
and norm }u}2 “(cid:10)u,u(cid:11), for u,v PH01pΩq.
For any fixed uPH01pΩq, the Lax-Milgram theorem implies that the problem
2 1
´∆φ“u , φPH pΩq
0
has a unique solution φ . Moreover, φ has the following properties see e.g [20]
u u
for a proof : `
Propositio˘n 3.1. For uPH1pΩq we have
0
(i) φ ě0 and there exists C ą0 such that }φ }ďC}u}2;
u u
(ii) φ “t2φ , for all tě0;
tu u
(iii) If u áu in H1pΩq, then φ áφ in H1pΩq and
n 0 un u 0
2 2
φ u Ñ φ u . (3.1)
żΩ un n żΩ u
As a consequence of this proposition, pu,φq P H1pΩqˆH1pΩq is a solution to
0 0
pSPq if, and only if φ“φ and u is solution to the non local problem
λ u
5 1
´∆u`φ u“fpx,uq`λu , uPH pΩq. (3.2)
u 0
Problem (3.2) is variational and its solutions are critical points of the functional
defined in H01pΩq by
1 1 λ
2 2 6
I puq:“ }u} ` φ u ´ Fpx,uq´ u . (3.3)
λ u
2 4żΩ żΩ 6 żΩ
By using standard argument one can verify that I PC1pX,Rq and
λ
(cid:10)Iλ1puq,v(cid:11)“ ∇u∇v` φuuv´ vfpx,uq´λ vu5. (3.4)
żΩ żΩ żΩ żΩ
Let 0 ă σ1 ă σ2 ă σ3 ă ¨¨¨ be the distinct eigenvalues of the Laplacian. Then
each σj has finite multiplicity. It is well known that the principal eigenvalue σ1 is
simple witha positive eigenfunctione1,andthe eigenfunctions ej correspondingto
σ (j ě2) are sign-changing. Let X be the eigenspace of σ .
j j j
We set for k ě2
Yk :“‘kj“1Xj and Zk “‘8j“kXj. (3.5)
Lemma 3.2.
(1) For any uPY , we have I puqÑ´8, uniformly in λ as }u}Ñ8.
k λ
(2) There exist Λ1 ą0, r ą0 with r Ñ8 as k Ñ8, such that
k k k
1
2
I puqě r ´c (3.6)
λ 8 k
for all λPr0,Λ1r and for all uPZ such that }u}“r , where cą0 is constant.
k k k
Proof. (1) Werecallthatpf3qimpliesthatFpx,uqěc1|u|µ´c2 forsomeconstants
c1,c2 ą 0. Since all norms are equivalent in the finite-dimensional space Yk, it is
easy to conclude.
(2) Using pf1q we obtain
1 λ
Iλpuqě 2}u}2´c3|u|pp´ 6|u|66´c4, @uPX,
HIGH ENERGY SIGN-CHANGING SOLUTIONS 7
where c3,c4 ą0 are constant. If we set
|∇u|2
2
S :“ inf and β :“ sup |u| , (3.7)
uPuH‰010pΩq |u|26 k }uuP}Z“k1 p
then we obtain
1 λ
Iλpuqě 2}u}2´c3βkp}u}p´ 6S3}u}6´c4, @uPZk.
We define
r :“ 1 p´12β´p´p2 and Λ1 :“ 3S3. (3.8)
k `8c3˘ k k 2rk4
One can easily verify that the inequality (3.6) is satisfied. The fact that r Ñ 8,
k
as kÑ8, is a consequence of Theorem 3.8 in [26]. (cid:3)
Remark 3.3. Lemma 3.2-(2) implies that
b :“ inf I puqÑ8, uniformly in λ, as k Ñ8. (3.9)
k λ
uPZk
}u}“rk
Now we fix k large enough and we set for mąk`2
I :“I | , K :“ uPY ; I1 puq“0 , E :“Y zK ,
λ,m λ Ym m m λ,m m m m
(
P :“ uPY ; upxqě0 , Zm :“‘m X and Nm :“ uPZm|}u}“r .
m m k j“k j k k k
( (
RemarkthatforalluPPmz 0 wehave Ωue1 ą0,whileforalluPZk, Ωue1 “0,
wheree1istheprincipaleige nfu(nctionofştheLaplacian. ThisimpliesthatşPmXZk “
0 . Since Nm is compact, it follows that
k
(
δ :“dist Nm,´P YP ą0. (3.10)
m k m m
` ˘
Remark 3.4. If 0ăµ ăδ then Nm ĂS .
m m k m
For uPY fixed, we consider the functional
m
1 1
2 2
κ pvq“ }v} ` φ v ´ vfpx,uq, v PY .
u u m
2 2żΩ żΩ
Onecaneasilyverifythatκ iscontinuous,coercive,boundedbelow,weaklysequen-
u
tially continuous, and strictly convex. Therefore, κ possesses a unique minimizer,
u
namely Au, which is the unique solution to the problem
5
´∆v`φ v “fpx,uq`λu , v PY .
u m
Clearly, the set of fixed points of A coincide with K . Moreover, the operator
m
A:Y ÑY has the following properties.
m m
Lemma 3.5.
(1) A is continuous and it maps bounded sets to bounded sets.
(2) For any uPY we have
m
(cid:10)Iλ1,mpuq,u´Au(cid:11)ěc1}u´Au}2, (3.11)
}Iλ1,mpuq}ďc2 1`}u}2 }u´Au}. (3.12)
(3) There exists µ Ps0,δ r such`that Ap˘˘D0 q Ă ˘D0 , where δ is defined
m m m m m
by (3.10).
8 HIGH ENERGY SIGN-CHANGING SOLUTIONS
The three assertions of this lemma can be proved in the same way as Lemma
3.1,Lemma 3.2 andLemma 3.4 in [17], respectively. We shall provide more details
when we shall prove their analogues in Section 4 below.
It should be noted that the vector field A itself does not satisfy the assumption
pA3qofTheorem2.1asitisnotlocallyLiptschitzcontinuous. However,itisthefirst
step in the construction of a vector field satisfying the above mentioned condition.
Lemma 3.6 ([17], Lemma 3.5). There exists an odd locally Lipschitz continuous
operator B :E ÑY such that
m m
(1) (cid:10)Iλ1,mpuq,u´Bpuq(cid:11)ě 21}u´Apuq}2, for any uPEm.
(2) 1}u´Bpuq}ď}u´Apuq}ď2}u´Bpuq}, for any uPE .
2 m
(3) B p˘D0 qXE âD0 .
m m m
` ˘
Lemma 3.7 ([17], Lemma 3.3). Let c ă d and α ą 0. For all u P Y such that
m
I puqPrc,ds and }I1 puq}ěα, there exists β ą0 such that }u´Bpuq}ěβ.
λ,m λ,m
Lemma 3.8. For any λ P r0,Λ1kr, there exists a sequence uk,m P Vµ2mpSmq such
that
Iλ1,mpuk,mq“0 and Iλpuk,mqPrbk,maxI0puqs. (3.13)
uPBk
Proof. By Lemmas 3.2, 3.6 and 3.7 and Remark 3.4, the assumptions of Theorem
2.1 are satisfied. Therefore,applying Theorem2.1 we obtain a sequence pun q Ă
k,m n
VµmpSmq such that
2
lim Iλ1,mpunk,mq“0 and lim Φmpunk,mqP bk,maxI0puq .
nÑ8 nÑ8 “ uPBk ‰
Using pf1q and pf3q and the fact that φu ě0, we obtain
1 1 1
Iλpuq´ µ(cid:10)Iλ1,mpuq,u(cid:11)ě 2 ´ µ }u}2, @uPYm. (3.14)
` ˘
Wethendeducethatthesequencepun q aboveisboundedinY . SincedimY ă
k,m n m m
8clo,sietdfaonlldowIs thaistsumnk,omotÑh, tuhke,mc,onucplutsoiona fsoulblosweqsu.ence, in Ym. Since Vµ2mpSmq (cid:3)is
λ,m
We are now ready to give the proof of Theorem 1.1.
Proof of Theorem 1.1. Here we assume that λ“0. We consider the elements u
k,m
obtained in Lemma 3.8. In view of Lemma 3.14, the sequence pu q is bounded
k,m m
in X. Hence, up to a subsequence, u á u in X and u Ñ u in LqpΩq
k,m k k,m k
(1ďq ă6), as mÑ8.
LetusdenotebyΠ :X ÑY theorthogonalprojection. ItisclearthatΠ u Ñ
m m m k
u in X, as mÑ8. We have
k
(cid:10)I01,mpuk,mq,uk,m´Πmuk(cid:11)“(cid:10)uk,m,uk,m´Πmuk(cid:11)
` φ u u ´Π u ´ u ´Π u fpx,u q. (3.15)
żΩ uk,m k,m k,m m k żΩ k,m m k k,m
` ˘ ` ˘
Since the sequence uk,m m is bounded, we deduce from pf1q that the sequence
fpx,u q is`also˘bounded. It follows, using the Ho¨lder inequality that
k,m p m
`ˇ ˇp´1˘
ˇ ˇ
u ´Π u fpx,u q ď|u ´Π u | fpx,u q Ñ0.
k,m m k k,m k,m m k p k,m p
ˇżΩ` ˘ ˇ ˇ ˇp´1
ˇ ˇ ˇ ˇ
HIGH ENERGY SIGN-CHANGING SOLUTIONS 9
On the other hand, we also obtain using the Ho¨lder inequality
ˇżΩφuk,muk,m`uk,m´Πmuk˘ˇď|φuk,m|3|uk,m|3|uk,m´Πmuk|3 Ñ0.
Since Iˇ1 pu q “ 0, we deduce froˇm (3.15) that }u } Ñ }u }, and hence that
0,m k,m k,m k
u Ñu inX,asmÑ8. Atthis point,itis straightforwardtoverifythatu is
k,m k k
a critical point of I0 such that I0pukqěbk. Since bk Ñ8, as k Ñ8 (see Remark
3.3), the proof will be completed if we show that u is sign-changing.
k
As usual, we denote u˘ :“maxt0,˘uu, for any uPX. Observe that
(cid:10)I01,mpuk,mq,u˘k,m(cid:11)“0 ñ }u˘k,m}2 ďżΩu˘mjfpx,u˘k,mq.
pf1q and pf2q imply
@εą0, Dc ą0 ; |fpx,tq|ďε|t|`c |t|p´1, @px,tqPΩˆR. (3.16)
ε ε
We then obtain by using the Sobolev embedding theorem
}u˘ }2 ď u˘ fpx,u˘ qďc ε}u˘ }2`c }u˘ }p ,
k,m żΩ k,m k,m k,m ε k,m
` ˘
for some constant c ą 0. Since u is sign-changing, u˘ are not equal to 0.
k,m k,m
Choosing ε small enough it follows that p}u˘ }q are bounded below by strictly
positive constants which do not depend on m.mHj ence u is sign-changing. (cid:3)
k
We now prove our result in the critical case.
Proof Theorem 1.2. We suppose here that 0ăλăΛ1. We consider again the ele-
k
ments u obtained in Lemma 3.8. In view of Lemma 3.14, the sequence pu q
k,m k,m m
is bounded in X. Hence, up to a subsequence, we have as mÑ8:
u áu in X;
k,m k
u Ñu a.e. in Ω; (3.17)
k,m k
u Ñu in LrpΩq p1ďr ă6q; (3.18)
k,m k
6
u áu in L pΩq; (3.19)
k,m k
φ áφ in X;
uk,m uk
φ Ñφ in LrpΩq p1ďr ă6q;
uk,m uk
2 2
φ u Ñ φ u . (3.20)
żΩ uk,m k,m żΩ uk k
We claim that
2 3
uk,m Ñuk if λăΛk :“S { 3maxI0 . (3.21)
` Bk ˘
We fix λ “ λ such that 0 ă λ ă mintΛ1,Λ2u. Using the same argument as in
k k k k
the proof of Theorem 1.1 above, we show that u is a sign-changing critical point
k
ofI suchthatI pu qěb . Sinceb Ñ8,ask Ñ8,theconclusionofTheorem
λk λk k k k
1.2 follows.
We complete the proof of Theorem 1.2 by proving our above claim. Let us then
assume that λăΛ2.
k
(3.19), (3.17) and Theorem 10.36 in [25] imply that
6 6 6
u áu in L5pΩq.
k,m k
10 HIGH ENERGY SIGN-CHANGING SOLUTIONS
For any v PX we obtain, using the Ho¨lder inequality
φ u ´φ u v “ φ pu ´u qv ` pφ ´φ qu v
ˇżΩ` uk,m k,m uk k˘ ˇ ˇżΩ uk,m k,m k ˇ ˇżΩ uk,m uk k ˇ
ˇ ď|φuk,m|3|ˇuk,mˇ ´uk|3|v|3`|φuk,m ´ˇφuˇk|3|uk|3|v|3 Ñ0, mÑˇ 8.
This implies that φ u áφ u in D1pΩq, the space of distributions.
uk,m k,m uk k
It is clear that ´∆u á ´∆u and fpx,u q á fpx,u q in D1pΩq. Therefore,
k,m k k,m k
we obtain
´∆u `φ u “fpx,u q`λu5 in D1pΩq.
k uk k k k
Multiplying the two members of this equation by u and integrating, we obtain
k
2 2 6
}u } ` φ u “ u fpx,u q`λ u . (3.22)
k żΩ uk k żΩ k k żΩ k
On the other hand, (cid:10)Iλ1,mpuk,mq,uk,m(cid:11)“0 is equivalent to
2 2 6
}uk,m} `żΩφuk,muk,m´żΩuk,mfpx,uk,mq´λ|uk,m|6 “0. (3.23)
It is clear that
2 2 2
}u } “}u } `}u ´u } `˝p1q (3.24)
k,m k k,m k
By Brezis-Lieb lemma [9] we have
6 6 6
|u | “|u | `|u ´u | `˝p1q. (3.25)
k,m 6 k 6 k,m k 6
Onecanverifyeasilythat(3.16)implies that,foralmosteveryxPΩ,the functions
s ÞÑ sfpx,sq and s ÞÑ Fpx,sq satisfy the conditions of Theorem 2 in [9]. It then
follows that
u fpx,u q“ u fpx,u q` u ´u f x,u ´u `˝p1q (3.26)
k,m k,m k k k,m k k,m k
żΩ żΩ żΩ
` ˘ ` ˘
Fpx,u q“ Fpx,u q` F x,u ´u `˝p1q. (3.27)
k,m k k,m k
żΩ żΩ żΩ
` ˘
Using (3.16) and (3.18), it is readily seen that
u ´u f x,u ´u Ñ0 (3.28)
k,m k k,m k
żΩ
` ˘ ` ˘
F x,u ´u Ñ0. (3.29)
k,m k
żΩ
` ˘
We then deduce from (3.23), using (3.18), (3.20), (3.24), (3.25), (3.26), and (3.28)
that
2 6
}uk,m´uk} ´λ|um,k´uk|6 “˝p1q. (3.30)
This implies that
}u ´u }2 ďλS´3}u ´u }6`˝p1q, (3.31)
k,m k m,k k
where S is defined in (3.7). If }um,k´uk}Ñs0 ą0, then by (3.31) we would have
s0 ě S3{λ 14. (3.32)
` ˘