# Existence of Nontrivial Solutions for Perturbedp-Laplacian Equation in ℝN with Critical Nonlinearity

Authors: Huixing Zhang; Xiaoqian Liu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider a perturbedp-Laplacian equation with critical nonlinearity in ℝN. By using variational method, we show that it has at least one positive solution under the proper conditions.


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## Body

## 1. Introduction and Main Results

In this paper, we are concerned with the existence of nontrivial solutions for the following nonlinear perturbedp-Laplacian equation with critical nonlinearity:
(1)-ɛpΔpu+V(x)|u|p-2u=K(x)|u|p*-2u+f(x,u),x∈ℝN,u(x)>0,u(x)⟶0as|x|⟶∞,
where Δpu=div⁡(|∇u|p-2∇u) is the p-Laplacian operator with 1<p<N, N≥3, p*=Np/(N-p) denotes the Sobolev critical exponent, V(x) is a nonnegative potential, K(x) is a bounded positive function, and f(x,u) is a superlinear but subcritical function.Forp=2, (1) turns into the following Schrödinger equation of the form
(2)-ɛ2Δu+V(x)u=K(x)|u|2*-2u+f(x,u),x∈ℝN.
The equation (2) has been studied extensively under various hypotheses on the potential and nonlinearity by many authors including Ambrosetti and Rabinowitz [1], Bartsch and Wang [2], Brézis and Lieb [3], Brėzis and Nirenberg [4], and Del Pino and Felmer [5] in bounded domains. Meanwhile, we recall some works in unbounded domains which contain Cingolani and Lazzo [6], Clapp and Ding [7], Ding and Lin [8], Floer and Weinstein [9], Grossi [10], Jeanjean and Tanaka [11], Kang and Wei [12], Oh [13], Pistoia [14], Rabinowitz [15], and Tang [16].For generalp>1, most of the work (see [17–19] and the reference therein) dealt with (1) with ɛ=1, K(x)≡0 and a certain sign potential V(x). Liu and Zheng [20] considered the above mentioned problem with sign-changing potential and subcritical p-superlinear nonlinearity. Cao et al. [21] also studied the similar problem. However, to our best knowledge, it seems that there is almost no work on the existence of semi-classical solutions to the equation in ℝN with critical nonlinearities. This paper will study the critical nonlinearity case in whole space.Throughout the paper, we make the following assumption:(H1)
V∈C(ℝN), V(0)=inf⁡x∈ℝNV(x)=0 and there exists b>0 such that the set νb:={x∈ℝN:V(x)<b} has finite Lebesgue measure;(H2)
K(x)∈C(ℝN,ℝ+),  0<inf⁡K≤sup⁡K<∞;(H3)
f∈C(ℝN×ℝ,ℝ) and f(x,t)=o(|t|p-2t) uniformly in x as t→0;(H4)
there area0>0 and p<q<p* such that |f(x,t)|≤a0(1+|t|q-1) for all (x,t);(H5)
there existb0>0, α>p and μ∈(p,p*) such that F(x,t)≥b0|t|α and μF(x,t)≤f(x,t)t for all (x,t), where F(x,t)=∫0t‍f(x,s)ds.Our main result reads as follows.Theorem 1.
Assume that (H1)–(H5) hold. Then for any σ>0, there exists εσ>0 such that if ɛ≤εσ, (1) has at least one positive solution uɛ of least energy which satisfied the following estimate:
(3)μ-ppμ∫ℝN(ɛp|∇uɛ|p+V(x)|uɛ|p)≤σɛN.
The main tool used in the proof of Theorem 1 is variational method which was mainly developed in [8]. The main difficulty in the case is to overcome the loss of the compactness of the energy functional related to (1) because of unbounded domain ℝN and critical nonlinearity. Although the energy functional does not satisfy the (PS) condition, we can prove that it possesses (PS)c condition at some energy level c.This outline of the paper is organized as follows. In Section2, we give the variational settings and preliminary results. In Section 3, we show that the corresponding energy functional satisfies (PS)c condition at the levels less than α0λ1-N/p with some α0>0 independent of λ. Furthermore, it possesses the mountain geometry structure. Section 4 is devoted to the proof of the main result.


## 2. Preliminaries

Letλ=ɛ-p in (1). The equation (1) reads as
(4)-Δpu+λV(x)|u|p-2u=λK(x)|u|p*-2u+λf(x,u),x∈ℝN,u(x)>0,u(x)⟶0,as|x|⟶∞.
In order to prove Theorem 1, we are going to prove the following result.Theorem 2.
Assume that (H1)–(H5) is satisfied. Then for any σ>0, there exists λσ>0 such that if λ>λσ, (4) has at least one positive solution uλ satisfying the following estimate:
(5)μ-ppμ∫ℝN(|∇uλ|p+λV(x)|uλ|p)≤σλ1-N/p.Next, we introduce the space(6)Eλ(ℝN,V)={∫ℝNλV(x)|u|p<∞,λ>0u∈W1,p(ℝN):∫ℝNλV(x)|u|p<∞,λ>0}
equipped with the norm
(7)∥u∥Eλ=(∫ℝN(|∇u|p+λV(x)|u|p))1/p.
Note that the norm ∥·∥E1 is equivalent to the one ∥·∥Eλ for any λ>0. It follows from (H1) that Eλ(ℝN,V) continuously is embedded in W1,p(ℝN). To prove Theorem 2, one considers the C1 functional I:W1,p(ℝN)→ℝ defined by
(8)Iλ(u)=1p∫ℝN(|∇u|p+λV(x)|u|p)-λp*∫ℝNK(x)|u|p*-λ∫ℝNF(x,u)=1p∥u∥Eλp-λ∫ℝNG(x,u),
where G(x,u)=(1/p*)K(x)|u|p*+F(x,u).Under the assumptions of Theorem2, standard arguments [22] show that Iλ∈C1(Eλ,ℝ) and its critical points are weak solutions of (4).


## 3. Necessary Lemmas

This section will show some lemmas which are important for the proof of the main result.Lemma 3.
Assume that (H1)–(H5) is satisfied. For the (PS)c sequence {un}⊂Eλ for Iλ, we get that c≥0 and {un} is bounded in the space Eλ.Proof.
By direct computation and the assumptions(H2) and (H5), one has
(9)Iλ(un)-1μIλ′(un)un=(1p-1μ)∥un∥Eλp+(1μ-1p*)λ∫ℝNK(x)|un|p*+λ∫ℝN(1μf(x,u)u-F(x,u)).
Together with Iλ(un)→c and Iλ′(un)→0 as n→∞, we easily get that the (PS)c sequence is bounded in Eλ and the energy level c≥0.By Lemma3, there is u∈Eλ such that un⇀u in Eλ. Furthermore, passing to a subsequence, we have un→u in Lloc⁡b(ℝN) for any b∈[p,p*) and un→ua.e. in ℝN.Lemma 4.
For anys∈[p,p*), there is a subsequence {uni} such that, for any ɛ>0, there exists rɛ>0 with
(10)lim⁡i→∞⁡sup⁡∫Bi∖Br|uni|s≤ɛforanyr≥rɛ,
where Br:={x∈ℝN:|x|≤r}.Proof.
Fromun→u in Lloc⁡s(ℝN), we have
(11)∫Bi|un|s⟶∫Bi|u|sasn⟶∞.
Thus, there exists n~i∈ℕ such that
(12)∫Bi(|un|s-|u|s)<1i,∀n=n~i+j,j=1,2,….
In particular, for ni=n~i+i, we have
(13)∫Bi(|uni|s-|u|s)<1i.
Note that there exists rɛ>0 satisfying
(14)∫ℝN∖Br‍(|u|s)<ɛ∀r≥rɛ.
Then
(15)∫Bi∖Br|uni|s=∫Bi∖Br(|uni|s-|u|s)+∫Bi∖Br|u|s=∫Bi(|uni|s-|u|s)+∫Br(|u|s-|uni|s)+∫Bi∖Br|u|s≤1i+∫ℝN∖Br|u|s+∫Br(|u|s-|uni|s)≤ɛ,asi⟶∞.
This completes the proof of Lemma 4.Letη∈C∞(ℝ+) be a smooth function satisfying 0≤η(t)≤1, η(t)=1 if t≤1 and η(t)=0 if t≥2. Define u~i(x)=η(2|x|/i)u(x). It is clear that
(16)∥u-u~i∥Eλ⟶0asi⟶∞.Lemma 5.
One has(17)lim⁡i→∞sup⁡|∫ℝN(f(x,uni)-f(x,uni-u~i)-f(x,u~i))φ|=0
uniformly in φ∈Eλ with ∥φ∥Eλ≤1.Proof.
From (16) and the local compactness of Sobolev embedding, for any r≥0, we have
(18)lim⁡i→∞sup⁡|∫Br(f(x,uni)-f(x,uni-u~i)-f(x,u~i))φ|=0
uniformly in ∥φ∥Eλ≤1. For any ɛ>0, it follows from (14) that
(19)lim⁡i→∞sup⁡∫Bi∖Br|u~i|s≤∫ℝN∖Br|u|s≤ɛ
for all r≥rɛ. By Lemma 4 and (H3)-(H4), we obtain
(20)lim⁡i→∞sup⁡|∫ℝN(f(x,uni)-f(x,uni-u~i)-f(x,u~i))φ|=lim⁡i→∞sup⁡|∫Bi∖Br(f(x,uni)-f(x,uni-u~i)∫Bi∖BrF-f(x,u~i))φ|≤a1lim⁡i→∞sup⁡∫Bi∖Br(|uni|p-1+|u~i|p-1)|φ|+a2lim⁡i→∞sup⁡∫Bi∖Br(|uni|q-1+|u~i|q-1)|φ|≤a1lim⁡i→∞sup⁡(∥uni∥Lp(Bi∖Br)p-1∥uni∥Lpp-1+∥u~i∥Lp(Bi∖Br)p-1)∥φ∥Lp(Bi∖Br)+a2lim⁡i→∞sup⁡(∥uni∥Lq(Bi∖Br)q-1+∥u~i∥Lq(Bi∖Br)q-1)×∥φ∥Lq(Bi∖Br)≤a3ɛ(p-1)/p+a4ɛ(q-1)/q.
This shows that the desired conclusion holds.Lemma 6.
One has along a subsequence(21)Iλ(un-u~n)⟶c-Iλ(u),Iλ′(un-u~n)⟶0inEλ-1(thedualspaceofEλ).Proof.
By Lemma 2.1 of [23] and the arguments of [24], we have
(22)Iλ(un-u~n)=1p∫ℝN(|∇un-∇u~n|p+λV(x)|un-u~n|p)-λp*∫ℝNK(x)|un-u~n|p*-λ∫ℝNF(x,un-u~n)=Iλ(un)-Iλ(u~n)+λp*∫ℝNK(x)(|un|p*-|un-u~n|p*-|u~n|p*)+λ∫ℝN(F(x,un)-F(x,un-u~n)-F(x,u~n))+o(1).
By (16) and the similar idea of proving the Brézis-Lieb Lemma [3], we easily get
(23)lim⁡n→∞∫ℝNK(x)(|un|p*-|un-u~n|p*-|u~n|p*)=0,lim⁡n→∞∫ℝN(F(x,un)-F(x,un-u~n)-F(x,u~n))=0.
Together with the fact Iλ(un)→c and Iλ(u~n)→Iλ(u), one has
(24)Iλ(un-u~n)⟶c-Iλ(u).
Next, we will check the fact Iλ′(un-u~n)→0 in Eλ-1. For any φ∈Eλ, we have
(25)Iλ′(un-u~n)φ=Iλ′(un)φ-Iλ′(u~n)φ+λ∫ℝNK(x)(|un|p*-2un-|un-u~n|p*-2(un-u~n)-|u~n|p*-2u~n)φ+λ∫ℝN(f(x,un)-f(x,un-u~n)-f(x,u~n))φ+o(1).
By the standard argument, it follows that
(26)lim⁡n→∞∫ℝNK(x)(|un|p*-2un-|un-u~n|p*-2(un-u~n)-|u~n|p*-2u~n)φ=0
uniformly in ∥φ∥Eλ≤1. Together with Lemma 5, we get the desired conclusion.Setun1=un-u~n; then, un-u=un1+(u~n-u). From (16), it shows that un→u in Eλ if and only if un1→0 in Eλ.Furthermore, we have(27)Iλ(un1)-1pIλ′(un1)un1=(1p-1p*)λ∫ℝNK(x)|un1|p*+λ∫ℝN(1pun1f(x,un1)-F(x,un1))≥λN∫ℝNK(x)|un1|p*≥λNKmin⁡∥un1∥p*p*,
where Kmin⁡=inf⁡x∈ℝNK(x)>0.By the facts thatIλ(un1)→c-Iλ(u) and Iλ′(un1)→0inEλ-1, one has
(28)∥un1∥p*p*≤N(c-Iλ(u))λKmin⁡+o(1).
Let Vb(x):=max⁡{V(x),b}, where b is the positive constant in the assumption (H1). Since the set νb has finite measure and un1→0 in Lloc⁡p(ℝN), we get
(29)∫ℝNV(x)|un1|p=∫ℝNVb(x)|un1|p+o(1).
From (H2)–(H5) and Young inequality, there exists Cb>0 such that
(30)∫ℝN‍(K(x)|un1|p*+un1f(x,un1))≤b∥un1∥pp+Cb∥un1∥p*p*.
Next, we consider the energy level of the functional Iλ below which the (PS)c condition held.Lemma 7.
Assume that the assumptions of Theorem2 are satisfied. There exists α0>0 (independent of λ) such that, for any (PS)c sequence {un}⊂Eλ for Iλ with un⇀u, either un→u in Eλ or c-Iλ(u)≥α0λ1-(N/p).Proof.
Assume thatun↛u; then,
(31)lim⁡inf⁡n→∞∥un1∥Eλ>0,c-Iλ(u)>0.
By the Sobolev inequality, (29), and (30), we get
(32)S∥un1∥p*p≤∫ℝN|∇un1|p=∫ℝN(|∇un1|p+λV(x)|un1|p)-λ∫ℝNV(x)|un1|p=λ∫ℝNK(x)|un1|p*+un1f(x,un1)-λ×∫ℝNVb(x)|un1|p+o(1)≤λb∥un1∥pp+λCb∥un1∥p*p*-λb∥un1∥pp+o(1)=λCb∥un1∥p*p*+o(1),
where S is the best Sobolev constant of the immersion
(33)S∥u∥p*p≤∫ℝN|∇u|p∀u∈W1,p(ℝN).
This gives
(34)S≤λCb∥un1∥p*p*-p+o(1)≤λCb(N(c-Iλ(u))λKmin⁡)p/N+o(1)=λ1-p/NCb(NKmin⁡)p/N(c-Iλ(u))p/N+o(1).
Setα0=SN/pCb-N/pN-1Kmin⁡; then,
(35)α0λ1-N/p≤c-Iλ(u)+o(1).
This proof is completed.From Lemma7, we will show that Iλ satisfies the following local (PS)c condition.Lemma 8.
Assume that (H1)–(H5) is satisfied. There exists a constant α0>0 (independent of λ) such that, if a (PS)c sequence {un}⊂Eλ for Iλ satisfies c≤α0λ1-N/p, the sequence {un} has a strongly convergent subsequence in Eλ.Proof.
By Lemma7, we easily obtain the required conclusion.Now, we considerλ≥1. The following standard arguments show that the energy functional Iλ possesses the mountain-pass structure.Lemma 9.
Under the assumptions of Theorem2, there exist αλ,ρλ>0 such that
(36)Iλ(u)>0if0<∥u∥Eλ<ρλ,Iλ(u)≥αλif∥u∥Eλ=ρλ.Proof.
By (30) and (H5), for any δ>0, there is Cδ>0 such that
(37)∫ℝNG(x,u)≤δ∥u∥pp+Cδ∥u∥p*p*.
Thus
(38)Iλ(u)=1p∥u∥Eλp-λ∫ℝNG(x,u)≥1p∥u∥Eλp-λδ∥u∥pp-λCδ∥u∥p*p*.
Observe that ∥u∥pp≤a5∥u∥Eλp. Choosing δ≤(2pλa5)-1,
(39)Iλ(u)≥12p∥u∥Eλp-λCδ∥u∥p*p*.
The fact p*>p implies the desired conclusion.Lemma 10.
For any finite dimensional subspaceF⊂Eλ, we have
(40)Iλ(u)⟶-∞,u∈Eλas∥u∥Eλ⟶∞.Proof.
By the assumption (H5), one has
(41)Iλ(u)≤1p∥u∥αp-λb0∥u∥αα∀u∈Eλ.
Since all norms in a finite-dimensional space are equivalent and α>p, this implies the desired conclusion.Lemma8 shows that Iλ satisfies (PS)cλ condition for λ large enough and cλ small sufficiently. In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small minimax levels.Define the functional(42)Φλ(u)=1p∫ℝN(|∇u|p+λV(x)|u|p)-λb0∫ℝN|u|α.
It is apparent that Φλ∈C1(Eλ) and Iλ(u)≤Φλ(u) for all u∈Eλ.Note that(43)inf⁡⁡{∫ℝN‍|∇ϕ|p:ϕ∈C0∞(ℝN,ℝ),∥ϕ∥Lα(ℝN)=1}=0.
For any δ>0, there is ϕδ∈C0∞(ℝN,ℝ) with ∥ϕδ∥Lα(ℝN)=1 and supp⁡ϕδ⊂Brδ(0) such that ∥∇ϕδ∥pp<δ. Let eλ(x)=ϕδ(λpx), then supp⁡eλ⊂Bλ-1/prδ(0). For any t≥0, we have
(44)Φλ(teλ)=tpp∥eλ∥Eλp-b0λtα∫ℝN|ϕδ(λpx)|α=λ1-N/pJλ(tϕδ),
where
(45)Jλ(u)=1p∫ℝN(|∇u|p+V(λ-(1/p)x)|u|p)-b0∫ℝN|u|α.
By direct computation, we easily get
(46)max⁡t≥0⁡Jλ(tϕδ)≤α-ppα(αb0)p/(α-p)(∫ℝN(|∇ϕδ|p+V(λ-(1/p)x)|ϕδ|p)+V(λ-1/px)|ϕδ|p)∫)α/(α-p).
In connection with V(0)=0 and ∥∇ϕδ∥pp<δ, it shows that there exists Λδ>0 such that for all λ≥Λδ, we have
(47)max⁡t≥0Iλ(tϕδ)≤(α-ppα(αb0)p/(α-p)(2δ)α/(α-p))λ1-N/p.
It follows from (47) thatLemma 11.
Assume that (H1)–(H5) is satisfied. For any σ>0, there is Λσ>0 such that λ≥Λσ; there exists e¯λ∈Eλ with ∥e¯λ∥Eλ>ρλ; we have Iλ(e¯λ)≤0 and
(48)max⁡t≥0Iλ(te¯λ)≤σλ1-N/p,
where ρλ is defined in Lemma 9.Proof.
This proof is similar to the one of Lemma 4.3 in [8], so we omit it.


## 4. Proof of Theorem2

In the following, we will give the proof of Theorem2.Proof.
By Lemma11, for any σ>0 with 0<σ<α0, there is Λσ>0 such that for λ≥Λσ, we obtain
(49)cλ=inf⁡γ∈Γλmax⁡t∈[0,1]Iλ(γ(t))≤σλ1-N/p,
where Γλ={γ∈C([0,1],Eλ):γ(0)=0,γ(1)=e¯λ}.
It follows from Lemma8 that Iλ satisfies (PS)cλ condition. Hence, by the mountain-pass theorem, there exists uλ∈Eλ which satisfies Iλ(uλ)=cλ and Iλ′(uλ)=0. Actually, uλ is a weak solution of (4). Similar to the argument in [8], we also get that uλ is a positive least energy solution.
In the end, we show that the solutionuλ satisfies the estimate (5). We easily get
(50)Iλ(uλ)=Iλ(uλ)-1μIλ′(uλ)(uλ)=(1p-1μ)∥uλ∥Eλp+(1μ-1p*)λ∫ℝN‍K(x)|uλ|p*+λ∫ℝN(1μuλf(x,uλ)-F(x,uλ))≥(1p-1μ)∥uλ∥Eλp.
Note that Iλ(uλ)=cλ and cλ≤σλ1-N/p and it implies the required conclusion. The proof is complete.



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