Volume 2010, Article ID 579387,13pages doi:10.1155/2010/579387
Research Article
Existence and Multiplicity of Positive Solutions to
a Class of Quasilinear Elliptic Equations in
R
N
Tsing-San Hsu
Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan
Correspondence should be addressed to Tsing-San Hsu,[email protected]
Received 9 October 2009; Accepted 12 February 2010
Academic Editor: Kanishka Perera
Copyrightq2010 Tsing-San Hsu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the following class of quasilinear elliptic equations−hpΔpuVεx|u|p−2u|u|q−2u,
ux>0 for allx∈RN, whereh >0,Δpudiv|∇u|p−2∇u, 2≤p < N,p < q < p∗Np/N−p.
We allow the potentialVεto be unbounded below and prove the existence and multiplicity for
positive solutions.
1. Introduction
In this paper we are concerned with the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations:
−hpΔ
puVεx|u|p−2u|u|q−2u in RN,
u∈W1,pRN with 2≤p < N,
ux>0, ∀x∈RN,
Ph,ε
whereh >0,p < q < p∗ Np/N−p,andΔpu div|∇u|p−2∇u.Moreover, we consider
the perturbed potentialVεsatisfying
whereε : 0,∞ → 0,∞, W : RN → 0,∞is a measurable function such that, for someα1>0 andα2≥0,the inequality
RNWx|u|
p≤α1∇up
pα2upp 1.2
holds for anyu ∈ W1,pRN and the “unperturbed” potentialV is a continuous function
satisfying
0< V0inf
RN V <lim inf|x| → ∞ Vx. 1.3
The last hypothesis was introduced by Rabinowitz in1.
For the casep2, equations of the kind
−h2ΔuVxu|u|q−2u inRN P
∗
in different models, for example, are related with the existence of standing waves of the
nonlinear Schr ¨odinger equation
ih∂ψ
∂t −h2Δψ Vx−λψ−ψ
q−2ψ, ∀x∈RN, NLS
whereλ ∈Rand 2 < q < 2N/N−2.A standing wave ofNLSis a solution of the form
ψx, t exp−iλh−1tux. In this case,uis a solution ofP ∗.
Existence and concentration of positive solutions for P∗ have been extensively
studied in the recent years; see, for example, Ambrosetti et al.2,3, Cingolani and Lazzo
4,5, Floer and Weinstein6, Oh7–9, Rabinowitz1, Serrin and Tang10, Wang11, and their references. In12, Lazzo considers the potential inP∗perturbed by adding a negative
potential. Under the assumptions 1.1–1.3 she obtained the existence and multiplicity
results for positive solutions of the equation
−h2ΔuV
εxu|u|q−2u inRN, 1.4
whereh >0, 2< q <2N/N−2.
In this paper, we will adapt some variational arguments explored by Lazzo12and
extend the results of 12 to the quasilinear case. In order to state our results we need
the following standard notation: if Y is a closed subset of a topological space Z,catZY
is the Ljusternik-Schnirelman category of Y in Z, namely, the least number of closed and
contractible sets inZwhich coverY. IfY Z, we set catZZ catY. Let
ε0lim sup
h→0 εh
hp ,
Mx∈RN :Vx V0.
1.5
Now we can describe our main results.
Theorem 1.1. Suppose that the assumptions1.1–1.3hold. There existsε∗>0such that ifε0< ε∗, thenPh,εhas a positive solution forhsufficiently small.
Theorem 1.2. Suppose that the assumptions1.1–1.3hold. For anyδ >0there existsε∗δ>0
such that ifε0< ε∗δ, thenPh,εhas at least catMδMpositive solutions forhsufficiently small.
2. Existence of Solutions
In this section, we will give an existence result forPh,ε. We need some notations, definitions, and auxiliary results. Let us recall the definition ofW1,pRN,
W1,pRNu∈LpRN:∂ iu∈Lp
RN, i1,2, . . . , N,
u1,pup∇up,
2.1
where · pdenotes the norm inLpRN.The spaceW1,pRNis the completion of the space
DRNofC∞-functions with compact support with respect to the norm ·
1,pand
X
u∈W1,pRN:Vx|u|p<∞ , 2.2
X∗is the dual space ofXand the integration setRNwill be understood.
InXwe define the functionals
Jh,εu
hp|∇u|pV
εx|u|p,
Jh,0u
hp|∇u|pVx|u|p.
2.3
From1.1–1.3and if 0< hp ≤V0α1α2−1 no restrictions onhifα2 0, then for anyu∈X, we have
1−α1εh
hp
Jh,0u≤Jh,εu≤Jh,0u. 2.4
Indeed,
Wx|u|p≤α1
|∇u|pα2V0
Vx|u|p≤ α1hpJh,0u. 2.5
As a consequence,
Jh,0u Jh,εu εh
whence2.4follows. From2.4, if lim suph→0 εhh−p< α−11there existα0, h∗0>0 such that
Jh,εu≥min{hp, V0}α0up1,p 2.7
for anyu∈X,for any 0< h < h∗0.As a result the setX,endowed with the normuphJh,εu,
is a Banach space and it is continuously embedded inW1,pRN.
Weak solution toPh,ε can be found by looking for critical points ofJh,εu on the
manifoldΣ {u ∈X :u|q 1}.Indeed,Jh,ε is well defined and smooth onΣ; moreover,
for any critical pointuofJh,εonΣ,Jh,εu1/q−puis a weak solution forPh,ε. Therefore, in
order to prove existence of solutions toPh,εit suffices to solve the following minimization problem:
chuinf∈ΣJh,εu. P
ProblemPis affected by a lack of compactness, due to the noncompact Sobolev embedding
W1,pRN → LqRN. One way is to guarantee that c
h is attained and to prove that Jh,ε
satisfies the Palais-Smale condition below ch α, for some positive α. This is indeed the
case: as we prove below, the Palais-Smale condition holds below some level, related to lim inf|x| → ∞Vx.In order to state this result more precisely, we need some notations. First, let us recall some facts about ground state solution of the equation
−hpΔ
puλ|u|p−2u|u|q−2u inRN, Q
whereh, λ >0. By13, Propositions 2.1 and 2.2, there is a positive radially symmetric ground
state solution wh, λ of Q. By adopting arguments similar to those in Li and Yan 14,
Theorem 3.1, we obtain thatwh, λ ∈ L∞RN∩C1,αRNfor some 0 < α < 1 and that
wh, λdecays exponentially at infinityalso see Alves and Carriao15, Lemma 2.1. The
infimum
mh;λ inf
hp∇up
pλupp
upq
:u∈W1,pRN, u /≡0
2.8
is achieved bywh;λ wh, λ/wh, λq.It is easy to see that
mh;λ hθm1;λ withθ N
q−p
q . 2.9
By1.3, we can chooseV∞∈Rsuch that
V0< V∞≤lim inf
|x| → ∞ Vx. 2.10
Let us denote
being the mapλ → m1;λstrictly increasing,2.10implies
m0< m∞. 2.12
We are ready to state our compactness result.
Proposition 2.1. Suppose that assumptions1.1–1.3hold and
ε0< α11
1− m0
m∞
. 2.13
Then there existsk1∗∈0, m∞−m0andh∗1 >0such thatJh,ε satisfies the Palais-Smale condition in the sublevel{u∈Σ:Jh,εu<m0k1∗hθ},for any0< h < h∗1.
Proof. Letβ∈m0,1−α1ε0m∞and fixη0>0 such that
βα1η0m∞<1−α1ε0m∞, 2.14
obviously, forhsmall we have
εh
hp ≤ε0η0. 2.15
Next, letγ < βand let{un} ⊂Σbe a Palais-Smale sequence forJh,εonΣat the levelγh≡γhθ,
namely,
Jh,εun γho1, 2.16
−hpΔ
punVεx|un|p−2un−λn|un|q−2uno1 inX∗, 2.17
asn → ∞,it is easily seen thatλn γho1.By standard calculations, we can see that{un}
is bounded inX.Therefore there existsu∈Xsuch that, up to a subsequence,un uweakly
inX.Moreover, adapting arguments found in16–18, it follows thatuis a weak solution of the following equation:
−hpΔ
puVεx|u|p−2uγh|u|q−2u inRN. E
In order to prove that{un}converges toustrongly inXwe apply Lions
Concentration-Compactness Lemmasee19,20to the sequence of measuresρnhp|∇un|pVεx|un|p. By
20, Lemma I.1, and the fact thatun∈Σ,we can exclude that vanishing occurs. If dichotomy
occurs, there existsδ1, δ2 >0, withδ1δ2 γhsuch that for anyξ >0 there areyn∈RN, R >
0, Rn → ∞such that
|x−yn|<R
ρn≥δ1−ξ,
|x−yn|>2Rn
As a consequence,
R<|x−yn|<2Rn
ρn≤2ξ. 2.19
Letζ :0,∞ → 0,1be a smooth, nonincreasing function, such thatζt 1 if 0 ≤t≤1,
ζt 0 ift≥2.If we define
u1
nx unxζ
x−y
n
R
, u2
nx unx−unxζ
x−y
n
Rn
, 2.20
then2.18yields
hp|∇ui
n|pVεx|uin|p≥δi−ξ, i1,2. 2.21
From the definition ofuin, i1,2,and2.19we get
|∇un|p−2∇un· ∇uin ∇uin p
Oξ,
Vεx|un|p−2unuin
Vεxuin p
Oξ,
|un|q−2unuin uin q
Oξ,
2.22
whence, by taking2.17into account,
Jh,ε ui n hp|∇ui
n|pVεx|uin|pγh
|ui
n|qo1 Oξ. 2.23
Now, if the sequence{yn}is unbounded inRN,for largenwe haveVx≥ V∞−ξ for any
x∈BRyn. Thus from2.4,2.15, the definition ofmh;V∞,and2.23we have
Jh,ε
u1
n
≥1−α1εh
hp hp|∇u1n|pVx|u1n|p
≥Oξ 1−α1ε0η0 hp|∇u1n|pV∞|u1n|p
≥Oξ 1−α1ε0η0mh;V∞u1
n p q
Oξ o1 1−α1ε0η0mh;V∞
Jh,εu1n
γh
p/q
,
whence
Jh,ε
u1
n
≥Oξ o1 1−α1ε0η0q/q−pmh;V∞q/q−pγhp/p−q. 2.25
From2.16and2.25we can deduce
γho1≥Jh,ε
u1
n
Oξ
≥Oξ o1 1−α1ε0η0q/q−pmh;V∞q/q−pγhp/p−q,
2.26
lettingξ → 0,n → ∞and dividing byhθyields
γ≥1−α1ε0η0m∞ 2.27
and, from2.14,γ > β,a contradiction. If the sequence{yn}is bounded inRN,for largenwe
haveVx ≥ V∞−ξfor anyxsuch that|x−yn| > Rn,and we get again a contradiction by takingu2
ninto account. Dicotomy is therefore ruled out in any case. As a result, the sequence
{ρn}is tight; there exists{yn} ⊂RNsuch that for anyξ >0
|x−yn|<R
hp|∇u
n|pVεx|un|p≥γh−ξ 2.28
for a suitableR >0.If the sequence{yn}is unbounded inRN,we could defineu1nas in2.20
and, noticing that
hp|∇u1
n|pVεx|un1|p≥γh−ξ, 2.29
we could get a contradiction exactly as before. So{yn}is bounded inRN,and for someRwe have
|x|>Rh p|∇u
n|pVεx|un|p< ξo1. 2.30
By the compactness of the embeddingW1,p →Lq on bounded domains implies that{u
n} →
ustrongly inLqanduis a weak solution ofE, we get
hp|∇u
n|pVεx|un|pγh
|un|qo1 γh
|u|qo1
hp|∇u|pV
εx|u|pOξ o1.
2.31
In other words,unph → uph.Finally, by using the Brezis-Lieb’s lemma21and arguing as
Remark 2.2. By Proposition 2.1and the choice ofV∞it follows that ifV is coercive, namely,
Vx → ∞ as|x| → ∞, then Jh,ε satisfies the Palais-Smale condition onΣ at any level. Without loss of generality, we will henceforth assumeV∞lim inf|x| → ∞Vx<∞.
We are interested in positive solutions forPh,ε. Now, we state our result on the sign of solutions forPh,ε.
Proposition 2.3. Suppose that assumptions1.1–1.3hold and
ε0< α11 1−2p−q/q. 2.32
Then there existsk2∗, h∗2 >0such that, for any0< h < h∗2,every critical pointuofJh,εonΣsatisfying
Jh,εu≤m0k2∗
hθ 2.33
does not change sign, whereθis the same as in2.9.
Proof. Fix η0 > 0 such that 0 < α1ε0η0 < 1−2p−q/q and let h∗2 ∈ 0, h∗0 be such that
εh<ε0η0hpfor any 0< h < h∗
2,whereh∗0is the same as in2.7. Finally, choose
0< k∗2<2q−p/q1−α1ε0η0−1m0. 2.34
Now, let 0< h < h∗2and letuu−u−be a critical point ofJh,εonΣsuch thatu, u−/≡0, where
umax{u,0}andu−max{−u,0}.We recallc
hinfu∈ΣJh,εu. If we multiply
−hpΔ
puVεx|u|p−2uJh,εu|u|q−2u 2.35
byuand integrate onRN,we get
Jh,εuuqqJh,εu≥chupq, 2.36
thus
uq q≥
c
h
Jh,εu
q/q−p
. 2.37
Similarly, the same inequality holds foru−,thus
1uqqu−qq ≥2
c
h
Jh,εu
q/q−p
, 2.38
whence
Then2.4,2.9,2.33, and the definition ofm0give
m0k∗
2
hθ≥J
h,εu≥2q−p/q1−α1ε0η0m0hθ, 2.40
if we divide byhθ,the last inequality contradicts2.34. This completes the proof.
Proof ofTheorem 1.1. Letδ >0 be fixed and letη:0,∞ → 0,1be a smooth, nonincreasing
function, such thatηt 1 if 0 ≤t ≤δ/2 andηt 0 ift≥ δ. Letw w1;V0, fix anyx0
such thatVx0 V0and set
ψh,x0x μhw
x−x0 h
η|x−x0|, 2.41
the constantμhis chosen in such a way thatψh,x0q 1.Then,ψh,x0 ∈Σand it is easy to see
that
Jh,εψh,x0
≤Jh,0
ψh,x0
hp|∇ψ h,x0|
pVx|ψ h,x0|
p
hN ∇
wxηh|x|pVhxx0wxηh|x|p
hN wxηh|x|qp/q
|∇wx|pVx0|wx|o1
|wx|qo1p/q h
θ m0o1hθ.
2.42
As a consequence, forhsmall we havech<m0k∗1hθ; ifε0 < ε∗1/α1min{1−2p−q/q,1−
m0/m∞},Propositions2.1and2.3apply and implyJh,εu chfor someu∈Σandudoes
not change sign. We can therefore assume thatuis positive and, up to a Lagrange multiplier,
Jh,εu1/q−puis a positive solution ofPh,ε.
3. Multiplicity of Solutions
We begin our discussion by giving some definitions and some known results. For any
constanta,we define
Ja
h,ε {u∈Σ:Jh,εu≤a}. 3.1
We recall that M denotes the set of global minima points of V and, for any positive δ,
let Mδ {x ∈ RN: distx, M ≤ δ}. In order to prove our multiplicity result, we need
the following proposition. For the proof, based on the very definition of category and homotopical equivalence, we refer, for instance, to23.
Proposition 3.1. Leta >0and letJ∗be a closed subset ofJh,εa . LetΦh :M → J∗, β:Jh,εa → Mδ
be continuous maps such thatβ◦Φh is homotopically equivalent to the embeddingj : M → Mδ.
Then catJa
In our setting, the construction of the mapΦhis very simple. Indeed, for anyx0 ∈M
and for anyhwe defineΦhx0 ψh,x0cf.2.41, whereψh,x0was introduced.
For anyδ > 0, letρ ρδ >0 be such thatMδ ⊂ Bρ0. Letχ :RN → RN be defined
asχx xfor|x|< ρandχx ρx/|x|for|x| ≥ρ.Finally, we define the barycenter map
β:Σ → RNby settingβu χx|ux|q.SinceMδ⊂Bρ0,we can use the definition ofχ
and the Lebesgue theorem to conclude that
lim
h→0βΦhx0 x0 uniformly forx0∈M. 3.2
The content of the following proposition is that barycenters of low energy functions are close toM.
Proposition 3.2. Suppose that assumptions1.1–1.3hold. For anyδ > 0there existsε∗1δ >0
such that if
ε0< ε∗
1δ, 3.3
then there existk3∗, h∗3 > 0 such thatβu ∈ Mδ for anyu ∈ ΣsatisfyingJh,ε ≤ m0k∗3hθfor
0< h < h∗3, whereθis the same as in2.9.
Proof. By contradiction, let us assume that for someδ >0 we can findεm≥0 such thatεm → 0 asm → ∞, lim suph→0εhh−p≤εm,and the claim inProposition 3.2does not hold.
Forhsmall we haveεhh−p< εm1/mand by2.4
1−α1
εmm1
Jh,0u≤Jh,εu. 3.4
Lethn, kn → 0asn → ∞andun ∈ Σbe such thatJh,εun ≤m0knhθn andβun/∈Mδ.
Letvnx hN/qn unhnxand from3.4we have
|∇vn|pVhnx|vn|p≤ 1−α1m0εkn
m1/m. 3.5
We apply Lions’ lemma to the sequence of probability measuresσn|vn|q.Vanishing is easily ruled out. If dichotomy occurs, there existδ1, δ2>0, withδ1δ2 1 such that for anyξ >0 there areyn∈RN,R >0,Rn → ∞such that
|x−yn|<R
σn≥δ1−ξ,
|x−yn|>2Rn
σn≥δ2−ξ. 3.6
Let us considerζas in the proof ofProposition 2.1and definev1
n, vn2accordingly as in2.20.
Inequalities3.6give
|vi
From3.5and3.7we get
m0kn
1−α1εm1/m ≥
|∇v1
n|pV0|v1n|p
|∇v2
n|pV0|v2n|pOξ
≥m0v1
n p q
v2
n p q
Oξ
≥m0δ1−ξp/q δ2−ξp/q.
3.8
Asm, n → ∞andξ → 0 we deduce 1≥δ1p/qδp/q2 ,a contradiction. Thus{σn}is tight; there
exists{yn} ⊂RNsuch that for anyξ >0
|x−yn|<R
|vnx|q≥1−ξ 3.9
for a suitableR >0.The sequencevn vn·ynis bounded inW1,pRN,hence it weakly
converges to some vin W1,pRNand, due to the compactness property 3.9, strongly in
LqRN.If the sequencex
n≡hnyn → ∞asn → ∞,then3.5gives
m0≥
|∇v|plim inf
n→ ∞
Vhnxxn|vn|p≥
|∇v|pV∞|v|p≥m∞, 3.10
which contradicts 2.12. Thus we can assume that xn converges to some x up to a
subsequence, and arguing as before we obtain
m0≥
|∇v|pVx|v|p≥m1;Vx≥m0. 3.11
From this we haveVx V0and|∇v|pV0x|v|p m0,hencem0 m1;V0is achieved
byv∈Σ. Furthermore, since|∇vn|pV0|vn|p≥ m0,from3.5we get|∇vn|pV0|vn|p →
m0|∇v|pV0|v|pasn → ∞. By using the Brezis-Lieb’s lemma21and as in22, Lemma
2.4, we get thatvn converges tovstrongly inW1,pRN. Finally, letδ > 0 be fixed and let
η:0,∞ → 0,1be a smooth, nonincreasing function, such thatηt 1 if 0≤t≤δ/2 and
ηt 0 ift≥δ. Set
ψnx μnv
x−x
n
hn
η|x−xn|, 3.12
where the constantμn is chosen in such a way thatψnq 1.Then,ψn ∈Σand it is easy to
see that
βun−β
By xn → x ∈ M and the factMδ ⊂ Bρ0and Lebesgue theorem, it follows that|βψn−
xn| o1. Therefore,|βun−xn| o1,which contradictsβun/∈Mδ.This completes the
proof.
Proof ofTheorem 1.2. Letδ >0 be fixed and letε∗1δbe as inProposition 3.2. Let
ε∗δ min 1
α1
1−2p−q/q, 1
α1
1− m0
m∞
, ε∗
1δ , 3.14
and assumeε0 < ε∗δ. Let 0< h∗ ≤min{h∗i :i 1,2,3}andk∗ min{ki∗ :i1,2,3}, with the constantsh∗i, ki∗being defined in Propositions2.1,2.3, and3.2. Let 0 < h < h∗; we can assume thatah ≡m0k∗hθ is not a critical value forJh,ε onΣ. For convenience, we set
Σh{u∈Σ:Jh,εu≤ah},Σh {u∈Σh:u≥0},andΣ−h{u∈Σh:u≤0}.
Ifhis small enough,2.42givesJh,εΦhx0 ≤m0k∗hθfor anyx0 ∈ M.In other
words, Φhx0 ∈ Σh for anyx0 ∈ M. Furthermore,Proposition 3.2impliesβu ∈ Mδ for
anyu∈ Σh.Finally, as a consequence of3.2it is easy to see thatβ ◦ Φhis homotopically
equivalent to the embeddingj :M → Mδ.ThusProposition 3.1gives catΣhΣh≥catMδM. If we use the map−Φhwe also get catΣhΣ−h≥catMδM,whence catΣh≥2catMδM,for
hsmall.
Proposition 2.1 guarantees that the Palais-Smale condition holds in a sublevel
containing Σh. Thus Ljusternik-Schnirelman theory applies and we deduce thatJh,ε has at
least 2catMδMcritical points onΣ, satisfyingJh,εu ≤ ah < m0k1∗hθ. Therefore, by Proposition 2.3they do not change sign and we can assume that at least catMδMcritical points are positive.
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