# Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions inℝN

Authors: Tsing-San Hsu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Existence and multiplicity of positive solutions for the following
semilinear elliptic equation:−Δu+u=a(x)|u|p−2u+λb(x)|u|q−2u in ℝN, u∈H1(ℝN), are established, where λ>0,1<q<2<p<2*(2*=2N/(N−2) if N≥3,2*=∞
if N=1,2), a, b satisfy suitable conditions, and b maybe changes sign in ℝN. The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.


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## Body

## 1. Introduction

In this paper, we deal with the multiplicity of positive solutions for the following semilinear elliptic equation:(Ea,λb)-Δu+u=a(x)up-1+λb(x)uq-1inRN,u>0inRN,u∈H1(RN),
where λ>0, 1<q<2<p<2* (2*=2N/(N-2) if N≥3, 2*=∞ if N=1,2) and a,b are measurable functions and satisfy the following conditions: (a1)
0<a∈L∞(ℝN), where lim|x|→∞a(x)=1, and there exist C0>0 and δ0>0 such that
(1.1)a(x)≥1-C0e-δ0|x|∀x∈RN.(b1)
b∈Lq*(ℝN)(q*=p/(p-q)), b+=max{b,0}≢0, b-=max{-b,0} is bounded and b- has a compact support K in ℝN.(b2)
There existC1>0, 0<δ1<min{δ0,q} and R0>0 such that
(1.2)b+(x)-b(x)≥C1e-δ1|x|∀|x|≥R0.Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied. For example, Ambrosetti et al. [1] considered the following equation:(Eλ)-Δu=up-1+λuq-1inΩ,u>0inΩ,u=0on⁡∂Ω,
where λ>0, 1<q<2<p<2*. They proved that there exists λ0>0 such that (Eλ) admits at least two positive solutions for all λ∈(0,λ0), has one positive solution for λ=λ0 and no positive solution for λ>λ0. Actually, Adimurthi et al. [2], Damascelli et al. [3], Korman [4], Ouyang and Shi [5], and Tang [6] proved that there exists λ0>0 such that (Eλ) in the unit ball BN(0;1) has exactly two positive solutions for λ∈(0,λ0), has exactly one positive solution for λ=λ0 and no positive solution exists for λ>λ0. For more general results of (Eλ) (involving sign-changing weights) in bounded domains; see, the work of  Ambrosetti et al. in [7], of Garcia Azorero et al. in [8], of Brown and Wu in [9], of Brown and Zhang in [10], of Cao and Zhong in [11], of de Figueiredo et al. in [12], and their references.However, little has been done for this type of problem inℝN. We are only aware of the works [13–17] which studied the existence of solutions for some related concave-convex elliptic problems (not involving sign-changing weights). Furthermore, we do not know of any results for concave-convex elliptic problems involving sign-changing weight functions except [18, 19]. Wu in [18] have studied the multiplicity of positive solutions for the following equation involving sign-changing weights:(Efλ,gμ)-Δu+u=fλ(x)uq-1+gμ(x)up-1inRN,u>0inRN,u∈H1(RN),
where 1<q<2<p<2* the parameters λ,μ≥0. He also assumed that fλ(x)=λf+(x)+f-(x) is sign chaning and gμ(x)=a(x)+μb(x), where a and b satisfy suitable conditions and proved that (Efλ,gμ) has at least four positive solutions.In a recent work [19], Hsu and Lin have studied (Ea,λb) in ℝN with a sign-changing weight function. They proved there exists λ0>0 such that (Ea,λb) has at least two positive solutions for all λ∈(0,λ0) provided that a, b satisfy suitable conditions and b maybe changes sign in ℝN.Continuing our previous work [19], we consider (Ea,λb) in ℝN involving a sign-changing weight function with suitable assumptions which are different from the assumptions in [19].In order to describe our main result, we need to define(1.3)Λ0=(2-q(p-q)‖a‖L∞)(2-q)/(p-2)(p-2(p-q)‖b+‖Lq*)Spp(2-q)/2(p-2)+q/2>0,
where ∥a∥L∞=supx∈ℝNa(x), ∥b+∥Lq*=(∫ℝN|b+(x)|q*dx)1/q* and Sp is the best Sobolev constant for the imbedding of H1(ℝN) into Lp(ℝN).Theorem 1.1.
Assume that(a1), (b1)-(b2) hold. If λ∈(0,(q/2)Λ0), (Ea,λb) admits at least two positive solutions in H1(ℝN).This paper is organized as follows. In Section2, we give some notations and preliminary results. In Section 3, we establish the existence of a local minimum. In Section 4, we prove the existence of a second solution of (Ea,λb).At the end of this section, we explain some notations employed. In the following discussions, we will considerH=H1(ℝN) with the norm ∥u∥=(∫ℝN(|∇u|2+u2)dx)1/2. We denote by Sp the best constant which is given by(1.4)Sp=inf⁡u∈H∖{0}‖u‖2(∫RN|u|pdx)2/p.
The dual space of H will be denoted by H*. 〈·,·〉 denote the dual pair between H* and H. We denote the norm in Ls(ℝN) by ∥·∥Ls for 1≤s≤∞. BN(x;r) is a ball in ℝN centered at x with radius r. on(1) denotes on(1)→0 as n→∞. C, Ci will denote various positive constants, the exact values of which are not important.


## 2. Preliminary Results

Associated with (1.3), the energy functional Jλ:H→ℝN defined by(2.1)Jλ(u)=12‖u‖2-1p∫RNa(x)|u|pdx-λq∫RNb(x)|u|qdx,
for all u∈H is considered. It is well-known that Jλ∈C1(H,ℝ) and the solutions of (Ea,λb) are the critical points of Jλ.SinceJλ is not bounded from below on H, we will work on the Nehari manifold. For λ>0 we define(2.2)Nλ={u∈H∖{0}:〈Jλ′(u),u〉=0}.
Note that 𝒩λ contains all nonzero solutions of (Ea,λb) and u∈𝒩λ if and only if(2.3)〈Jλ′(u),u〉=‖u‖2-∫RNa(x)|u|pdx-λ∫RNb(x)|u|qdx=0.Lemma 2.1.
Jλ is coercive and bounded from below on 𝒩λ.Proof.
Ifu∈𝒩λ, then by (b1), (2.3), and the Hölder and Sobolev inequalities, one has
(2.4)Jλ(u)=p-22p‖u‖2-λ(p-qpq)∫RNb(x)|u|qdx(2.5)≥p-22p‖u‖2-λ(p-qpq)Sp-q/2‖b+‖Lq*‖u‖q.
Since q<2<p, it follows that Jλ is coercive and bounded from below on 𝒩λ.The Nehari manifold is closely linked to the behavior of the function of the formφu:t→Jλ(tu) for t>0. Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in [20] and are also discussed by Brown and Zhang in [10]. If u∈H, we have (2.6)φu(t)=t22‖u‖2-tpp∫RNa(x)|u|pdx-tqqλ∫RNb(x)|u|qdx,φu′(t)=t‖u‖2-tp-1∫RNa(x)|u|pdx-tq-1λ∫RNb(x)|u|qdx,φu′′(t)=‖u‖2-(p-1)tp-2∫RNa(x)|u|pdx-(q-1)tq-2λ∫RNb(x)|u|qdx.
It is easy to see that (2.7)tφu′(t)=‖tu‖2-∫RNa(x)|tu|pdx-λ∫RNb(x)|tu|qdx,
and so, for u∈H∖{0} and t>0, φu′(t)=0 if and only if tu∈𝒩λ that is, the critical points of φu correspond to the points on the Nehari manifold. In particular, φu′(1)=0 if and only if u∈𝒩λ. Thus, it is natural to split 𝒩λ into three parts corresponding to local minima, local maxima, and points of inflection. Accordingly, we define (2.8)Nλ+={u∈Nλ:φu′′(1)>0},Nλ0={u∈Nλ:φu′′(1)=0},Nλ-={u∈Nλ:φu′′(1)<0},
and note that if u∈𝒩λ, that is, φu′(1)=0, then(2.9)φu′′(1)=(2-q)‖u‖2-(p-q)∫RNa(x)|u|pdx,(2.10)=(2-p)‖u‖2-(q-p)λ∫RNb(x)|u|qdx.We now derive some basic properties of𝒩λ+, 𝒩λ0, and 𝒩λ-.Lemma 2.2.
Suppose thatu0 is a local minimizer for Jλ on 𝒩λ and u0∉𝒩λ0, then Jλ′(u0)=0 in H*.Proof.
See the work of Brown and Zhang in [10, Theorem 2.3].Lemma 2.3.
Ifλ∈(0,Λ0), then 𝒩λ0=∅.Proof.
We argue by contradiction. Suppose that there existsλ∈(0,Λ0) such that 𝒩λ0≠∅. Then for u∈𝒩λ0 by (2.9) and the Sobolev inequality, we have
(2.11)2-qp-q‖u‖2=∫RNa(x)|u|pdx≤‖a‖L∞Sp-p/2‖u‖p,
and so
(2.12)‖u‖≥(2-q(p-q)‖a‖L∞)1/(p-2)Spp/2(p-2).
Similarly, using (2.10), Hölder and Sobolev inequalities, we have
(2.13)‖u‖2=λp-qp-2∫RNb(x)|u|qdx≤λp-qp-2‖b+‖Lq*Sp-q/2‖u‖q
which implies
(2.14)‖u‖≤(λp-qp-2‖b+‖Lq*)1/(2-q)Sp-q/2(2-q).
Hence, we must have
(2.15)λ≥(2-q(p-q)‖a‖L∞)(2-q)/(p-2)(p-2(p-q)‖b+‖Lq*)Spp(2-q)/2(p-2)+q/2=Λ0
which is a contradiction.In order to get a better understanding of the Nehari manifold and fibering maps, we consider the functionψu:ℝ+→ℝ defined by (2.16)ψu(t)=t2-q‖u‖2-tp-q∫RNa(x)|u|pdxfort>0.
Clearly, tu∈𝒩λ if and only if ψu(t)=λ∫ℝNb(x)|u|qdx. Moreover,(2.17)ψu′(t)=(2-q)t1-q‖u‖2-(p-q)tp-q-1∫RNa(x)|u|pdxfort>0,
and so it is easy to see that if tu∈𝒩λ, then tq-1ψu′(t)=φu′′(t). Hence, tu∈𝒩λ+ (or tu∈𝒩λ-) if and only if ψu′(t)>0 (or ψu′(t)<0).Letu∈H∖{0}. Then, by (2.17), ψu has a unique critical point at t=tmax(u), where(2.18)tmax⁡(u)=((2-q)‖u‖2(p-q)∫RNa(x)|u|pdx)1/(p-2)>0,
and clearly ψu is strictly increasing on (0,tmax(u)) and strictly decreasing on (tmax(u),∞) with limt→∞ψu(t)=-∞. Moreover, if λ∈(0,Λ0), then (2.19)ψu(tmax⁡(u))=[(2-qp-q)(2-q)/(p-2)-(2-qp-q)(p-q)/(p-2)]‖u‖2(p-q)/(p-2)(∫RNa(x)|u|pdx)(2-q)/(p-2)=‖u‖q(p-2p-q)(2-qp-q)2-q/p-2(‖u‖p∫RNa(x)|u|pdx)(2-q)/(p-2)≥‖u‖q(p-2p-q)(2-qp-q)(2-q)/(p-2)Spp(2-q)/2(p-2)>λ‖b+‖Lq*Sp-q/2‖u‖q≥λ∫RNb+(x)|u|qdx≥λ∫RNb(x)|u|qdx.Therefore, we have the following lemma.Lemma 2.4.
Letλ∈(0,Λ0) and u∈H∖{0}. (i)
Ifλ∫ℝNb(x)|u|qdx≤0, then there exists a unique t-=t-(u)>tmax(u) such that t-u∈𝒩λ-, φu is inceasing on (0,t-) and decreasing on (t-,∞). Moreover,
(2.20)Jλ(t-u)=sup⁡t≥0Jλ(tu).(ii)
Ifλ∫ℝNb(x)|u|qdx>0, then there exist unique 0<t+=t+(u)<tmax(u)<t-=t-(u) such that t+u∈𝒩λ+, t-u∈𝒩λ-, φu is decreasing on (0,t+), inceasing on (t+,t-) and decreasing on (t-,∞)(2.21)Jλ(t+u)=inf⁡0≤t≤tmax⁡(u)Jλ(tu),Jλ(t-u)=sup⁡t≥t+Jλ(tu).(iii)
𝒩λ-={u∈H∖{0}:t-(u)=(1/∥u∥)t-(u/∥u∥)=1}.(iv)
There exists a continuous bijection betweenU={u∈H∖{0}:∥u∥=1} and 𝒩λ-. In particular, t- is a continuous function for u∈H∖{0}.Proof.
See the work of Hsu and Lin in [19, Lemma 2.5].We remark that it follows Lemma2.4, 𝒩λ=𝒩λ+∪𝒩λ- for all λ∈(0,Λ0). Furthermore, by Lemma 2.4 it follows that 𝒩λ+ and 𝒩λ- are non-empty and by Lemma 2.1 we may define(2.22)αλ=inf⁡u∈NλJλ(u),αλ+=inf⁡u∈Nλ+Jλ(u),αλ-=inf⁡u∈Nλ-Jλ(u).Theorem 2.5.
(i) Ifλ∈(0,Λ0), then we have αλ≤αλ+<0.
(ii) Ifλ∈(0,(q/2)Λ0), then αλ->d0 for some d0>0.
In particular, for eachλ∈(0,(q/2)Λ0), we have αλ+=αλ<0<αλ-.Proof.
See the work of Hsu and Lin in [19, Theorem 3.1].Remark 2.6.
(i) Ifλ∈(0,Λ0), then by (2.9), Hölder and Sobolev inequalities, for each u∈𝒩λ+ we have
(2.23)‖u‖2<p-qp-2λ∫RNb(x)|u|qdx≤p-qp-2λ‖b‖Lq*Sp-q/2‖u‖q≤p-qp-2Λ0‖b‖Lq*Sp-q/2‖u‖q,
and so
(2.24)‖u‖≤(p-qp-2Λ0‖b‖Lq*Sp-q/2)1/(2-q)∀u∈Nλ+.
(ii) Ifλ∈(0,(q/2)Λ0), then by Lemma 2.4(i), (ii) and Theorem 2.5(ii), for each u∈𝒩λ- we have(2.25)Jλ(u)=sup⁡t≥0Jλ(tu)≥αλ->0.


## 3. Existence of a Positive Solution

First, we define the Palais-Smale (simply by(PS)) sequences, (PS)-values, and (PS)-conditions in H for Jλ as follows.Definition 3.1.
(i) Forc∈ℝ, a sequence {un} is a (PS)c-sequence in H for Jλ if Jλ(un)=c+on(1) and Jλ′(un)=on(1) strongly in H* as n→∞.
(ii)c∈ℝ is a (PS)-value in H for Jλ if there exists a (PS)c-sequence in H for Jλ.
(iii)Jλ satisfies the (PS)c-condition in H if any (PS)c-sequence {un} in H for Jλ contains a convergent subsequence.Now we will ensure that there are(PS)αλ+-sequence and (PS)αλ--sequencein on 𝒩λ and 𝒩λ-, respectively, for the functional Jλ.Proposition 3.2.
Ifλ∈(0,(q/2)Λ0), then (i)
there exists a(PS)αλ-sequence {un}⊂𝒩λ in H for Jλ.(ii)
there exists a(PS)αλ--sequence {un}⊂𝒩λ- in H for Jλ.Proof.
See Wu [21, Proposition 9].Now, we establish the existence of a local minimum forJλ on 𝒩λ+.Theorem 3.3.
Assume(a1) and (b1) hold. If λ∈(0,(q/2)Λ0), then there exists uλ∈𝒩λ+ such that (i)
Jλ(uλ)=αλ=αλ+<0,(ii)
uλ is a positive solution of (Ea,λb),(iii)
∥uλ∥→0 as λ→0+.Proof.
From Proposition3.2(i) it follows that there exists {un}⊂𝒩λ satisfying
(3.1)Jλ(un)=αλ+on(1)=αλ++on(1),Jλ′(un)=on(1)inH*.
By Lemma 2.1 we infer that {un} is bounded on H. Passing to a subsequence (Still denoted by {un}), there exists uλ∈H such that as n→∞(3.2)un⇀uλweaklyinH,un⟶uλalmosteverywhereinRN,un⟶uλstronglyinLloc⁡s(RN)∀1≤s<2*.
By (b1), Egorov theorem and Hölder inequality, we have
(3.3)λ∫RNb(x)|un|qdx=λ∫RNb(x)|uλ|qdx+on(1)asn⟶∞.
By (3.1) and (3.2), it is easy to see that uλ is a solution of (Ea,λb). From un∈𝒩λ and (2.4), we deduce that
(3.4)λ∫RNb(x)|un|qdx=q(p-2)2(p-q)‖un‖2-pqp-qJλ(un).
Let n→∞ in (3.4). By (3.1), (3.3) and αλ<0, we get
(3.5)λ∫RNb(x)|uλ|qdx≥-pqp-qαλ>0.
Thus, uλ∈𝒩λ is a nonzero solution of (Ea,λb).
Next, we prove thatun→uλ strongly in H and Jλ(uλ)=αλ. From the fact un,uλ∈𝒩λ and applying Fatou's lemma, we get(3.6)αλ≤Jλ(uλ)=p-22p‖uλ‖2-p-qpqλ∫RNb(x)|uλ|qdx≤lim⁡inf⁡n→∞(p-22p‖un‖2-p-qpqλ∫RNb(x)|un|qdx)≤lim⁡inf⁡n→∞Jλ(un)=αλ.
This implies that Jλ(uλ)=αλ and limn→∞∥un∥2=∥uλ∥2. Standard argument shows that un→uλ strongly in H. By Theorem 2.5, for all λ∈(0,(q/2)Λ0) we have that uλ∈𝒩λ and Jλ(uλ)=αλ+<αλ- which implies uλ∈𝒩λ+. Since Jλ(uλ)=Jλ(|uλ|) and |uλ|∈𝒩λ+, by Lemma 2.2 we may assume that uλ is a nonzero nonnegative solution of (Ea,λb). By Harnack inequality [22] we deduce that uλ>0 in ℝN. Finally, by (2.10), Hölder and Sobolev inequlities,
(3.7)‖uλ‖2-q<λp-qp-2‖b+‖Lq*Sp-q/2,
and thus we conclude the proof.


## 4. Second Positive Solution

In this section, we will establish the existence of the second positive solution of (Ea,λb) by proving that Jλ satisfies the (PS)αλ--condition.Lemma 4.1.
Assume that(a1) and (b1) hold. If {un}⊂H is a (PS)c-sequence for Jλ, then {un} is bounded in H.Proof.
See the work of Hsu and Lin in [19, Lemma 4.1].Let us introduce the problem at infinity associated with (Ea,λb):(E∞)-Δu+u=up-1inRN,u∈H,u>0inRN.
We state some known results for problem (E∞). First of all, we recall that by Lions [23] has studied the following minimization problem closely related to problem (E∞):(4.1)S∞=inf⁡{J∞(u):u∈H,u≢0,(J∞)′(u)=0}>0,
where J∞(u)=(1/2)∥u∥2-(1/p)∫ℝN|u|pdx. Note that a minimum exists and is attained by a ground state w0>0 in ℝN such that (4.2)S∞=J∞(w0)=sup⁡t≥0J∞(tw0)=(12-1p)Spp/(p-2),
where Sp=infu∈H∖{0}∥u∥2/(∫ℝN|u|pdx)2/p. Gidas et al. [24] showed that for every ɛ>0, there exist positive constants Cɛ, C2 such that for all x∈ℝN,(4.3)Cɛexp⁡(-(1+ɛ)|x|)≤w0(x)≤C2exp⁡(-|x|).
We define(4.4)wn(x)=w0(x-ne),wheree=(0,0,…,0,1)isaunitvectorinRN.
Clearly, wn(x)∈H.Lemma 4.2.
LetΩ be a domain in ℝN. If f:Ω→ℝ satisfies
(4.5)∫Ω|f(x)eσ|x||dx<∞forsomeσ>0,
then
(4.6)(∫Ωf(x)e-σ|x-x̃|dx)eσ|x̃|=∫Ωf(x)eσ〈x,x̃〉/|x̃|dx+o(1)as|x̃|⟶∞.Proof.
We knowσ|x̃|≤σ|x|+σ|x-x̃|. Then,
(4.7)|f(x)e-σ|x-x̃|eσ|x̃||≤|f(x)eσ|x||.
Since -σ|x-x̃|+σ|x̃|=σ〈x,x̃〉/|x̃|+o(1) as |x̃|→∞, then the lemma follows from the Lebesgue dominated convergence theorem.Lemma 4.3.
Under the assumptions(a1), (b1)-(b2) and λ∈(0,Λ0). Then there exists a number n0∈ℕ such that for n≥n0(4.8)sup⁡t≥0Jλ(twn)<S∞.
In particular, αλ-<S∞ for all λ∈(0,Λ0).Proof.
(i) First, since∥wn∥=∥w0∥ for all n∈ℕ and Jλ is continuous in H and Jλ(0)=0, we infer that there exists t1>0 such that
(4.9)Jλ(twn)<S∞∀n∈N,t∈[0,t1].
(ii) Sincelim|x|→∞a(x)=1, there exists n1∈ℕ such that if n≥n1, we get a(x)≥1/2 for x∈BN(ne;1). Then, for n≥n1(4.10)Jλ(twn)=t22‖wn‖2-tpp∫RNa(x)|wn|pdx-tqq∫RNλb(x)|wn|qdx≤t22‖w0‖2-tpp∫BN(0;1)a(x+ne)|w0|pdx+tqqλ‖b-‖L∞∫RN|wn|qdx≤t22‖w0‖2-tp2p∫BN(0;1)|w0|pdx+tqqλ‖b-‖L∞∫RN|w0|qdx⟶-∞ast⟶∞.
Thus, there exists t2>0 such that for any t>t2 and n>n1 we get
(4.11)Jλ(twn)<0.
(iii) By (i) and (ii), we need to show that there existsn0 such that for n≥n0(4.12)sup⁡t1≤t≤t2Jλ(twn)<S∞.
We know that supt≥0J∞(tw0)=S∞. Then, t1≤t≤t2, we have
(4.13)Jλ(twn)=12‖twn‖2-1p∫RNa(x)(twn)pdx-1q∫RNλb(x)(twn)qdx≤t22‖w0‖2-tpp∫RNw0pdx+tpp∫RN(1-a(x))wnpdx-tqq∫RNλb(x)wnqdx≤S∞+t2pp∫RN(1-a)+(x)wnpdx-t1qq∫RNλb+(x)wnqdx+t2qq∫RNλb-(x)wnqdx.
Suppose a satisfies (a1), we get (1-a)+(x)≤C0e-δ0|x| for all x∈ℝN and some positive constant δ0. By (4.3) and Lemma 4.3, there exists n2>n1 such that for any n≥n2(4.14)∫RN(1-a)+(x)wnpdx≤C3e-min⁡{δ0,p}n.
By (b1) and (4.3), we get
(4.15)∫RNλb-(x)wnqdx≤λ‖b-‖L∞C2∫Ke-q|x-ne|dx≤λC3e-qn.
By (b2), (4.3) and Lemma 4.3, we have
(4.16)∫RNλb+(x)wnqdx≥λC1Cɛ∫|x|≥R0e-δ1|x|e-q(1+ɛ)|x-ne|dx≥λCɛ¯e-δ1n.
Since 0<δ1<min{δ0,q}≤min{δ0,p} and λ∈(0,Λ0) and using (4.13)–(4.16), we have there exists n0>n2 such that for all n≥n0, then
(4.17)sup⁡t1≤t≤t2Jλ(twn)<S∞,λ∫RNb(x)|wn|qdx>0.
This implies that if λ∈(0,Λ0), then for all n≥n0 we get
(4.18)sup⁡t≥0Jλ(twn)<S∞.
From a(x)>0 for all x∈ℝN and (4.17), we have
(4.19)∫RNa(x)|wn0|pdx>0,∫RNb(x)|wn0|qdx>0.
Combining this with Lemma 2.4(ii), from the definition of αλ- and supt≥0Jλ(twn0)<S∞, for all λ∈(0,Λ0), we obtain that there exists t0>0 such that t0wn0∈𝒩λ- and
(4.20)αλ-≤Jλ(t0wn0)≤sup⁡t≥0Jλ(twn0)<S∞.Lemma 4.4.
Assume that(a1) and (b1) hold. If {un}⊂H is a (PS)c-sequence for Jλ with c∈(0,S∞), then there exists a subsequence of {un} converging weakly to a nonzero solution of (Ea,λb) in ℝN.Proof.
Let{un}⊂H be a (PS)c-sequence for Jλ with c∈(0,S∞). We know from Lemma 4.1 that {un} is bounded in H, and then there exist a subsequence of {un} (still denoted by {un}) and u0∈H such that
(4.21)un⇀u0weaklyinH,un→u0almosteverywhereinRN,un→u0stronglyinLloc⁡s(RN)∀1≤s<2*.
It is easy to see that Jλ′(u0)=0 and by (b1), Egorov theorem and Hölder inequality, we have
(4.22)λ∫RNb(x)|un|qdx=λ∫RNb(x)|u0|qdx+on(1).
Next we verify thatu0≢0. Arguing by contradiction, we assume u0≡0. By (a1), for any ɛ>0, there exists R0>0 such that |a(x)-1|<ɛ for all x∈[BN(0;R0)]C. Since un→0 strongly in Llocs(ℝN) for 1≤s<2*, {un} is a bounded sequence in H, therefore ∫ℝN(a(x)-1)|un|p≤C∫BN(0;R0)|un|p+ɛC. Setting n→∞, then ɛ→0, we have(4.23)lim⁡n→∞∫RNa(x)|un|pdx=lim⁡n→∞∫RN|un|pdx.
We set
(4.24)l=lim⁡n→∞∫RNa(x)|un|pdx=lim⁡n→∞∫RN|un|pdx.
Since Jλ′(un)=on(1) and {un} is bounded, then by (4.22), we can deduce that
(4.25)0=lim⁡n→∞〈Jλ′(un),un〉=lim⁡n→∞(‖un‖2-∫RNa(x)|un|pdx)=lim⁡n→∞‖un‖2-l,
that is,
(4.26)lim⁡n→∞‖un‖2=l.
Ifl=0, then we get c=limn→∞Jλ(un)=0, which contradicts to c>0. Thus we conclude that l>0. Furthermore, by the definition of Sp we obtain(4.27)‖un‖2≥Sp(∫RN|un|pdx)2/p.
Then, as n→∞, we have
(4.28)l=lim⁡n→∞‖un‖2≥Spl2/p,
which implies that
(4.29)l≥Spp/(p-2).
Hence, from (4.2) and (4.22)–(4.29), we get
(4.30)c=lim⁡n→∞Jλ(un)=12lim⁡n→∞‖un‖2-1plim⁡n→∞∫RNa(x)|un|pdx-λqlim⁡n→∞∫RNb(x)|un|qdx=(12-1p)l≥p-22pSpp/(p-2)=S∞.
This is a contradiction to c<S∞. Therefore, u0 is a nonzero solution of (Ea,λb).Now, we establish the existence of a local minimum ofJλ on 𝒩λ-.Theorem 4.5.
Assume that(a1) and (b1)-(b2) hold. If λ∈(0,(q/2)Λ0), then there exists Uλ∈𝒩λ- such that (i)
Jλ(Uλ)=αλ-,(ii)
Uλ is a positive solution of (Ea,λb).Proof.
Ifλ∈(0,(q/2)Λ0), then by Theorem 2.5(ii), Proposition 3.2(ii) and Lemma 4.3(ii), there exists a (PS)αλ--sequence {un}⊂𝒩λ- in H for Jλ with αλ-∈(0,S∞). From Lemma 4.4, there exist a subsequence still denoted by {un} and a nonzero solution Uλ∈H of (Ea,λb) such that un⇀Uλ weakly in H.
First, we prove thatUλ∈𝒩λ-. On the contrary, if Uλ∈𝒩λ+, then by 𝒩λ- is closed in H, we have ∥Uλ∥2<liminfn→∞∥un∥2. From (2.9) and a(x)>0 for all x∈ℝN, we get(4.31)∫RNb(x)|Uλ|qdx>0,∫RNa(x)|Uλ|pdx>0.
By Lemma 2.4(ii), there exists a unique tλ- such that tλ-Uλ∈𝒩λ-. If u∈𝒩λ, then it is easy to see that
(4.32)Jλ(u)=p-22p‖u‖2-p-qpqλ∫RNb(x)|u|qdx.
From (3.1), un∈𝒩λ- and (4.32), we can deduce that
(4.33)αλ-≤Jλ(tλ-Uλ)<lim⁡n→∞Jλ(tλ-un)≤lim⁡n→∞Jλ(un)=αλ-
which is a contradiction. Thus, Uλ∈𝒩λ-.
Next, by the same argument as that in Theorem3.3, we get that un→Uλ strongly in H and Jλ(Uλ)=αλ->0 for all λ∈(0,(q/2)Λ0). Since Jλ(Uλ)=Jλ(|Uλ|) and |Uλ|∈𝒩λ-, by Lemma 2.2 we may assume that Uλ is a nonzero nonnegative solution of (Ea,λb). Finally, by the Harnack inequality [22] we deduce that Uλ>0 in ℝN.Now, we complete the proof of Theorem1.1. By Theorems 3.3, 4.5, we obtain (Ea,λb) has two positive solutions uλ and Uλ such that uλ∈𝒩λ+, Uλ∈𝒩λ-. Since 𝒩λ+∩𝒩λ-=∅, this implies that uλ and Uλ are distinct. It completes the proof of Theorem 1.1.



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*Source: 864296-2011-06-28.xml*