# Multiple Positive Solutions for Semilinear Elliptic Equations inℝN Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Authors: Tsing-San Hsu; Huei-Li Lin
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation-Δu+u=λa(x)|u|q-2u+b(x)|u|p-2u in ℝN, u∈H1(ℝN), where λ>0, 1<q<2<p<2*(2*=2N/(N-2) if N≥3, 2*=∞ if N=1,2), a(x), b(x) satisfy suitable conditions, and a(x) may change sign in ℝN.


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## Body

## 1. Introduction and Main Results

In this paper, we deal with the existence and multiplicity of positive solutions for the following semilinear elliptic equation:(Eλa,b)-Δu+u=λa(x)|u|q-2u+b(x)|u|p-2u,inℝN,u>0,inℝN,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)
a∈C(ℝN)∩Lq*(ℝN)(q*=p/(p-q)) with a+=max{a,0}≢0 in ℝN;(B1)
b∈C(ℝN)∩L∞(ℝN) and b+=max{b,0}≢0 in ℝN.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=λuq-1+up-1,inΩ,u>0,inΩ,u=0,on∂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) and has one positive solution for λ=λ0 and no positive solution for λ>λ0. Actually, Adimurthi et al. [2], Damascelli et al. [3], Ouyang and Shi [4], and Tang [5] 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) and 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 Ambrosetti et al. [6], García Azorero et al. [7], Brown and Wu [8], Brown and Zhang [9], Cao and Zhong [10], de Figueiredo et al. [11], and their references. However, little has been done for this type of problem in ℝN. We are only aware of the works [12–16] 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 [17]. Wu in [17] has studied the multiplicity of positive solutions for the following equation involving sign-changing weights:(Eaλ,bμ)-Δu+u=aλ(x)uq-1+bμ(x)up-1,inℝN,u>0,inℝN,u∈H1(ℝN),
where 1<q<2<p<2*, the parameters λ,μ≥0. He also assumed that aλ(x)=λa+(x)+a-(x) is sign-chaning and bμ(x)=c(x)+μd(x), where c and d satisfy suitable conditions, and proved that (Eaλ,bμ)  has at least four positive solutions.The main aim of this paper is to study (Eλa,b) in ℝN involving concave-convex nonlinearities and sign-changing weight functions. We will discuss the Nehari manifold and examine carefully connection between the Nehari manifold and the fibrering maps; then using arguments similar to those used in [18], we will prove the existence of two positive solutions by using Ekeland’s variational principle [19].Set(1.1)Λ0=(2-q(p-q)∥b+∥L∞)(2-q)/(p-2)(p-2(p-q)∥a+∥Lq*)Sp(p(2-q)/2(p-2))+(q/2)>0,
where ∥b+∥L∞=supx∈ℝNb+(x), ∥a+∥Lq*=(∫ℝN|a+(x)|q*dx)1/q*, and Sp is the best Sobolev constant for the imbedding of H1(ℝN) into Lp(ℝN). Now, we state the first main result about the existence of positive solution of (Eλa,b) in ℝN.Theorem 1.1.
Assume that (A1) and (B1) hold. Ifλ∈(0,Λ0), then (Eλa,b) admits at least one positive solution in H1(ℝN).Associated with (Eλa,b), we consider the energy functional Jλa,b in H1(ℝN):(1.2)Jλa,b(u)=12∥u∥H12-λq∫ℝNa(x)|u|qdx-1p∫ℝNb(x)|u|pdx,
where ∥u∥H1=(∫ℝN(|∇u|2+u2)dx)1/2. By [20, Proposition B.10], Jλa,b∈C1(H1(ℝN),ℝ). It is well known that the solutions of (Eλa,b) are the critical points of the energy functional Jλa,b in H1(ℝN).Under assumptions (A1), (B1), andλ>0, (Eλa,b) can be regarded as a perturbation problem of the following semilinear elliptic equation:(Eb)-Δu+u=b(x)up-1,inℝN,u>0,inℝN,u∈H1(ℝN),
where b(x)∈C(ℝN)∩L∞(ℝN) and b(x)>0 for all x∈ℝN. We denote by Spb the best constant which is given by(1.3)Spb=infu∈H1(ℝN)∖{0}∥u∥H12(∫ℝNb(x)|u|pdx)2/p.
A typical approach for solving problem of this kind is to use the Minimax method:(1.4)αΓb=infγ∈Γmaxt∈[0,1]J0b(γ(t)),
where(1.5)Γ={γ∈C([0,1],H1(ℝN)):γ(0)=0,γ(1)=e},J0b(e)=0, and e≠0. By the Mountain Pass Lemma due to Ambrosetti and Rabinowitz [21], we called the nonzero critical point u∈H1(ℝN) of J0b a ground state solution of (Eb) in ℝN if J0b(u)=αΓb. We remark that the ground state solutions of (Eb) in ℝN can also be obtained by the Nehari minimization problem(1.6)α0b=infv∈ℳ0bJ0b(v),
where ℳ0b={u∈H1(ℝN)∖{0}:∥u∥H12=∫ℝNb(x)|u|pdx}. Note that ℳ0b contains every nonzero solution of (Eb) in ℝN (see Willem [22])(1.7)αΓb=α0b=p-22p(Spb)p/(p-2)>0.Whenb(x)≡b∞ is a constant function in ℝN, the existence of ground state solutions of (Eb) in ℝN has been established by Berestycki and Lions [23]. Actually, Kwong [24] proved that the positive solution of (Eb) in ℝN is unique.Whenb(x)≢b∞ and b(x)∈C(ℝN)∩L∞(ℝN), it is well known that the existence of ground state solutions of (Eb) has been established by the condition b(x)≥b∞=lim|x|→∞b(x) and the existence of ground state solutions of limit equation(Eb∞)-Δu+u=b∞up-1,inℝN,u>0,inℝN,u∈H1(ℝN).In order to get the second positive solution of (Eλa,b) in ℝN, we need some additional assumptions for a(x) and b(x). We assume the following conditions on a(x) and b(x):(B2)
b(x)>0 for all x∈ℝN, and b(x) satisfies suitable conditions such that (Eb) in ℝN has a positive ground state solution w0, that is, J0b(w0)=α0b;(A2)
∫ℝNa(x)|w0|qdx>0, where w0 is a positive ground state solution of (Eb) in ℝN.Theorem 1.2.
Assume that (A1)-(A2) and (B1)-(B2) hold. Ifλ∈(0,(q/2)Λ0), (Eλa,b) admits at least two positive solutions in H1(ℝN).Remark 1.3.
(i) In [17, Theorem 1.1], the author has proved that if
(1.8)bμ(x)=c(x)+μd(x),aλ(x)=λa+(x)+a-(x),lim|x|→∞c(x)=1,lim|x|→∞d(x)=0,1≥c(x)≥1-c0exp(-ra|x|),forc0<1,∀x∈ℝN,d(x)≥d0exp(-rb|x|),ford0>0,∀x∈ℝN,a-(x)≥-ĉexp(-rf-|x|),∀x∈ℝN,
where rb<min{ra,rf-,q}, then for sufficiently small λ and μ, (Eaλ,bμ),
(1.9)-Δu+u=aλ(x)uq-1+bμ(x)up-1,inℝN,u>0,inℝN,u∈H1(ℝN),
admits at least two positive solutions in ℝN. In particular, bμ satisfies the following condition:
(1.10)bμ(x)=c(x)+μd(x)≥1=lim|x|→∞bμ(x),forlarge|x|.
(ii) According to Lions' paper, if b(x)≥b∞=lim|x|→∞b(x) for any x∈ℝN, then there is a positive ground state solution w0 of (Eb) in ℝN. Supposing ∫ℝNa(x)w0pdx=∫ℝN[a+(x)+a-(x)]w0pdx>0, we can prove that for sufficiently small λ, (Eλa,b),
(1.11)-Δu+u=λa(x)uq-1+b(x)up-1,inℝN,u>0,inℝN,u∈H1(ℝN)
admits at least two positive solutions in ℝN. We give an example of a(x) as follows. Let η1:ℝN→[0,1] be a Cc∞-function on ℝN such that 0≤η1≤1 and
(1.12)η1(x)={1,for|x|≤1,0,for|x|≥2.
Since w0∈H1(ℝN), there is a positive number R>2 such that
(1.13)∫{|x|≥R}w0pdx<∫{|x|≤1}w0pdx.
Let η2:ℝN→[0,1] be a C∞-function on ℝN such that 0≤η2≤1 and
(1.14)η2(x)={0,for|x|≤R,1,for|x|≥2R.
Define
(1.15)a(x)=η1(x)-η2(x)|x|-r,wherer>0,N-rq*<0,
then by (1.13), we have that
(1.16)∫ℝNa(x)w0pdx≥∫{|x|≤1}w0pdx-∫{|x|≥R}|x|-rw0pdx≥∫{|x|≤1}w0pdx-R-r∫{|x|≥R}w0pdx>0.
In this case, a-(x)=-η2(x)|x|-r and b(x) do not satisfy the assumptions of exponential decay in [17].Throughout this paper, (A1) and (B1) will be assumed.H1(ℝN) denotes the standard Sobolev space, whose norm ∥·∥H1 is induced by the standard inner product. The dual space of H1(ℝN) will be denoted by H-1(ℝN). 〈·,·〉 denotes the usual scalar product in H1(ℝN). We denote the norm in Ls(ℝN) by ∥·∥Ls for 1≤s≤∞. on(1) denotes on(1)→0 as n→∞. C, Ci will denote various positive constants, the exact values of which are not important. This paper is organized as follows. In Section 2, we give some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorems 1.1 and 1.2.


## 2. Nehari Manifold

In this section, we will give some properties of Nehari manifold. As the energy functionalJλa,b is not bounded below on H1(ℝN), it is useful to consider the functional on the Nehari manifold(2.1)ℳλa,b={u∈H1(ℝN)∖{0}:〈(Jλa,b)′(u),u〉=0}.
Thus, u∈ℳλa,b if and only if(2.2)〈(Jλa,b)′(u),u〉=∥u∥H12-λ∫ℝNa(x)|u|qdx-∫ℝNb(x)|u|pdx=0.
Note that ℳλa,b contains every nonzero solution of (Eλa,b). Moreover, we have the following results.Lemma 2.1.
The energy functionalJλa,b is coercive and bounded below on ℳλa,b.Proof.
Ifu∈ℳλa,b, then by (A1), (2.2), and Hölder and Sobolev inequalities
(2.3)Jλa,b(u)=p-22p∥u∥H12-λ(p-qpq)∫ℝNa(x)|u|qdx(2.4)≥p-22p∥u∥H12-λ(p-qpq)Sp-(q/2)∥a+∥Lq*∥u∥H1q.
Thus, Jλa,b is coercive and bounded below on ℳλa,b.The Nehari manifold is closely linked to the behavior of the function of the formφu:t→Jλa,b(tu) for t>0. Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in [25] and are also discussed in [9]. If u∈H1(ℝN), we have (2.5)φu(t)=t22∥u∥H12-tqqλ∫ℝNa(x)|u|qdx-tpp∫ℝNb(x)|u|pdx;φu′(t)=t∥u∥H12-tq-1λ∫ℝNa(x)|u|qdx-tp-1∫ℝNb(x)|u|pdx;φu′′(t)=∥u∥H12-(q-1)tq-2λ∫ℝNa(x)|u|qdx-(p-1)tp-2∫ℝNb(x)|u|pdx.
It is easy to see that (2.6)tφu′(t)=∥tu∥H12-λ∫ℝNa(x)|tu|qdx-∫ℝNb(x)|tu|pdx,
and so, for u∈H1(ℝN)∖{0} and t>0, φu′(t)=0 if and only if tu∈ℳλa,b, 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∈ℳλa,b. Thus, it is natural to split ℳλa,b into three parts corresponding to local minima, local maxima and points of inflection. Accordingly, we define (2.7)ℳλa,b+={u∈ℳλa,b:φu′′(1)>0},ℳλa,b0={u∈ℳλa,b:φu′′(1)=0},ℳλa,b-={u∈ℳλa,b:φu′′(1)<0}
and note that if u∈ℳλa,b, that is, φu′(1)=0, then
(2.8)φu′′(1)=(2-p)∥u∥H12-(q-p)λ∫ℝNa(x)|u|qdx(2.9)=(2-q)∥u∥H12-(p-q)∫ℝNb(x)|u|pdx.We now derive some basic properties ofℳλa,b+,ℳλa,b0, and ℳλa,b-.Lemma 2.2.
Assume thatuλ is a local minimizer for Jλa,b on ℳλa,b and uλ∉ℳλa,b0. Then (Jλa,b)′(uλ)=0 in H-1(ℝN).Proof.
Our proof is almost the same as that in Brown and Zhang [9, Theorem 2.3] (or see Binding et al. [26]).Lemma 2.3.
One has the following.(i)
Ifu∈ℳλa,b+∪ℳλa,b0, then ∫ℝNa(x)|u|qdx>0;(ii)
Ifu∈ℳλa,b-, then ∫ℝNb(x)|u|pdx>0.Proof.
The proof is immediate from (2.8) and (2.9).Moreover, we have the following result.Lemma 2.4.
Ifλ∈(0,Λ0), then ℳλa,b0=∅, where Λ0 is the same as in (1.1).Proof.
Suppose the contrary. Then there existsλ∈(0,Λ0) such that ℳλa,b0≠∅. Then for u∈ℳλa,b0 by (2.8) and Sobolev inequality, we have
(2.10)2-qp-q∥u∥H12=∫ℝNb(x)|u|pdx≤∥b+∥L∞Sp-(p/2)∥u∥H1p
and so
(2.11)∥u∥H1≥(2-q(p-q)∥b+∥L∞)1/(p-2)Spp/2(p-2).
Similarly, using (2.9) and Hölder and Sobolev inequalities, we have
(2.12)∥u∥H12=λp-qp-2∫ℝNa(x)|u|qdx≤λp-qp-2∥a+∥Lq*Sp-(q/2)∥u∥H1q
which implies that
(2.13)∥u∥H1≤(λp-qp-2∥a+∥Lq*)1/(2-q)Sp-(q/2(2-q)).
Hence, we must have
(2.14)λ≥(2-q(p-q)∥b+∥L∞)(2-q)/(p-2)(p-2(p-q)∥a+∥Lq*)Sp(p(2-q)/2(p-2))+(q/2)=Λ0,
which is a contradiction. This completes the proof.In order to get a better understanding of the Nehari manifold and fibering maps, we consider the functionψu:ℝ+→ℝ defined by (2.15)ψu(t)=t2-q∥u∥H12-tp-q∫ℝNb(x)|u|pdx,fort>0.
Clearly tu∈ℳλa,b if and only if ψu(t)=λ∫ℝNa(x)|u|qdx. Moreover,(2.16)ψu′(t)=(2-q)t1-q∥u∥H12-(p-q)tp-q-1∫ℝNb(x)|u|pdx,fort>0,
and so it is easy to see that, if tu∈ℳλa,b, then tq-1ψu′(t)=φu′′(t). Hence, tu∈ℳλa,b+ (or tu∈ℳλa,b-) if and only if ψu′(t)>0 (or ψu′(t)<0).Let𝒵={u∈H1(ℝN):∫ℝNb(x)|u|pdx=0}. Suppose that u∈H1(ℝN)∖𝒵. Then, by (2.16), ψu has a unique critical point at t=tmax(u), where(2.17)tmax(u)=((2-q)∥u∥H12(p-q)∫ℝNb(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 (2.18)ψu(tmax(u))=[(2-qp-q)(2-q)/(p-2)-(2-qp-q)(p-q)/(p-2)]∥u∥H12(p-q)/(p-2)(∫ℝNb(x)|u|pdx)(2-q)/(p-2)=∥u∥H1q(p-2p-q)(2-qp-q)(2-q)/(p-2)(∥u∥H1p∫ℝNb(x)|u|pdx)(2-q)/(p-2)≥∥u∥H1q(p-2p-q)(2-qp-q)(2-q)/(p-2)Spp(2-q)/2(p-2)>λ∥a+∥Lq*Sp-(q/2)∥u∥H1q∥b+∥L∞≥λ∫ℝNa+(x)|u|qdx≥λ∫ℝNa(x)|u|qdx.
Therefore, we have the following lemma.Lemma 2.5.
Letλ∈(0,Λ0). For each u∈H1(ℝN)∖𝒵, one has the following. (i)
Ifλ∫ℝNa(x)|u|qdx≤0, then there exists a unique t-=t-(u)>tmax(u) such that t-u∈ℳλa,b-, φu is increasing on (0,t-) and decreasing on (t-,∞). Moreover,
(2.19)Jλa,b(t-u)=supt≥0Jλa,b(tu).(ii)
Ifλ∫ℝNa(x)|u|qdx>0, then there exists unique 0<t+=t+(u)<tmax(u)<t-=t-(u) such that t+u∈ℳλa,b+, t-u∈ℳλa,b-, φu is decreasing on (0,t+), increasing on (t+,t-), and decreasing on (t-,∞)(2.20)Jλa,b(t+u)=inf0≤t≤tmax(u)Jλa,b(tu);Jλa,b(t-u)=supt≥t+Jλa,b(tu).(iii)
ℳλa,b-={u∈H1(ℝN)∖𝒵:(1/∥u∥H1)t-(u/∥u∥H1)=1}.(iv)
There exists a continuous bijection betweenU={u∈H1(ℝN)∖𝒵:∥u∥H1=1} and ℳλa,b-. In particular, t- is a continuous function for u∈H1(ℝN)∖𝒵.Proof.
Fixu∈H1(ℝN)∖𝒵. (i)
Supposeλ∫ℝNa(x)|u|qdx≤0. Then ψu(t)=λ∫ℝNa(x)|u|qdx has a unique solution t->tmax(u) such that ψu′(t-)<0 and φu′(t-)=0. Thus, by tq-1ψu′(t)=φu′′(t), φu has a unique critical point at t=t- and φu′′(t-)<0. Therefore, t-u∈ℳλa,b- and (2.19) holds.(ii)
Supposeλ∫ℝNa(x)|u|qdx>0. Since ψu(tmax(u))>λ∫ℝNa(x)|u|qdx, the equation ψu(t)=λ∫ℝNa(x)|u|qdx has exactly two solutions t+<tmax(u)<t- such that ψu′(t+)>0 and ψu′(t-)<0. Thus, there exist exactly two multiples of u lying in H1(ℝN), that is, t+u∈ℳλa,b+ and t-u∈ℳλa,b-. Therefore, by tq-1ψu′(t)=φu′′(t), φu has critical points at t=t+ and t=t- with φu′′(t+)>0 and φu′′(t-)<0. Therefore, φu is decreasing on (0,t+), increasing on (t+,t-) and decreasing on (t-,∞). This implies that (2.20) holds.(iii)
Foru∈ℳλa,b-. By Lemma 2.3(ii) and, considering w=u/∥u∥H1, we have u∈H1(ℝN)∖𝒵. By (i) and (ii), there exists a unique t-(w)>0 such that t-(w)w∈ℳλa,b-, that is, t-(u/∥u∥H1)(1/∥u∥H1)u∈ℳλa,b-. Since u∈ℳλa,b-, we have (1/∥u∥H1)t-(u/∥u∥H1)=t-(u)=1. Therefore
(2.21)ℳλa,b-⊂{u∈H1(ℝN)∖𝒵:1∥u∥H1t-(u∥u∥H1)=1}.
Conversely, if u∈H1(ℝN)∖𝒵 is such that (1/∥u∥H1)t-(u/∥u∥H1)=t-(u)=1, then by the uniqueness of t-(u), we get that u∈ℳλa,b-. Thus, we have
(2.22)ℳλa,b-={u∈H1(ℝN):1∥u∥H1t-(u∥u∥H1)=1}.(iv)
Fixu∈U arbitrary. Define Gu:(0,∞)×U→ℝ by
(2.23)Gu(t,w)=〈(Jλa,b)′(tw),tw〉=ϕλa,b(tw),
where ϕλa,b:H1(ℝN)→ℝ is defined by ϕλa,b(u)=〈(Jλa,b)′(u),u〉. Since Gu(t-(u),u)=〈(Jλa,b)′(t-(u)u),t-(u)u〉=0, and
(2.24)∂Gu∂t(t-(u),u)=[t-(u)]-1〈(ϕλa,b)′(t-(u)u),t-(u)u〉<0,
then by the implicit function theorem, there is a neighborhood Wu of u in U and a unique continuous function Hu:Wu→(0,∞) such that Gu(Hu(w),w)=0 for all w∈Wu, in particular Hu(u)=t-(u). Since u∈U is arbitrary, we obtain that the function H:U→(0,∞), given by H(u)=t-(u), is continuous and one to one. By H-:U→ℳλa,b-, where H-(u)=t-(u)u, we have that H- is continuous and one to one. Now if u∈ℳλa,b-, then by (iii) we have that H-(w)=u, where w=u/∥u∥H1. Since t- is continuous on U, it follows that t- is continuous on H1(ℝN)∖𝒵. This completes the proof.


## 3. Proof of Theorem1.1

First, we remark that it follows from Lemma2.4 that(3.1)ℳλa,b=ℳλa,b+∪ℳλa,b-
for all λ∈(0,Λ0). Furthermore, by Lemma 2.5 it follows that ℳλa,b+ and ℳλa,b- are nonempty, and by Lemma 2.1 we may define(3.2)αλa,b=infu∈ℳλa,bJλa,b(u);αλa,b+=infu∈ℳλa,b+Jλa,b(u);αλa,b-=infu∈ℳλa,b-Jλa,b(u).
Then we get the following result.Theorem 3.1.
One has the following.(i)
Ifλ∈(0,Λ0), then one has αλa,b+<0.(ii)
Ifλ∈(0,(q/2)Λ0), then αλa,b->d0 for some d0>0. In particular, for each λ∈(0,(q/2)Λ0), one has αλa,b+=αλa,b.Proof.
(i) Letu∈ℳλa,b+. By (2.8)
(3.3)2-qp-q∥u∥H12>∫ℝNb(x)|u|pdx
and so
(3.4)Jλa,b(u)=(12-1q)∥u∥H12+(1q-1p)∫ℝNb(x)|u|pdx<[(12-1q)+(1q-1p)(2-qp-q)]∥u∥H12=-(p-2)(2-q)2pq∥u∥H12<0.
Therefore, αλa,b+<0.
(ii) Letu∈ℳλa,b-. By (2.8)
(3.5)2-qp-q∥u∥H12<∫ℝNb(x)|u|pdx.
Moreover, by (B1) and Sobolev inequality
(3.6)∫ℝNb(x)|u|pdx≤∥b+∥L∞Sp-(p/2)∥u∥H1p.
This implies that
(3.7)∥u∥H1>(2-q(p-q)∥b+∥L∞)1/(p-2)Spp/2(p-2),∀u∈ℳλa,b-.
By (2.4) and (3.7), we have
(3.8)Jλa,b(u)≥∥u∥H1q[p-22p∥u∥H12-q-λ(p-qpq)Sp-(q/2)∥a+∥Lq*]>(2-q(p-q)∥b+∥L∞)q/(p-2)Sppq/2(p-2)×[p-22pSpp(2-q)/2(p-2)(2-q(p-q)∥b+∥L∞)(2-q)/(p-2)-λ(p-qpq)Sp-(q/2)∥a+∥Lq*].
Thus, if λ∈(0,(q/2)Λ0), then
(3.9)Jλa,b(u)>d0,∀u∈ℳλa,b-,
for some positive constant d0. This completes the proof.Remark 3.2.
(i) Ifλ∈(0,Λ0), then by (2.8) and Hölder and Sobolev inequalities, for each u∈ℳλa,b+ we have
(3.10)∥u∥H12<p-qp-2λ∫ℝNa(x)|u|qdx≤p-qp-2λ∥a∥Lq*Sp-(q/2)∥u∥H1q<p-qp-2Λ0∥a∥Lq*Sp-(q/2)∥u∥H1q,
and so
(3.11)∥u∥H1≤(p-qp-2Λ0∥a∥Lq*Sp-(q/2))1/(2-q),∀u∈ℳλa,b+.
(ii) If λ∈(0,(q/2)Λ0), then by Lemma 2.5(i), (ii) and Theorem 3.1(ii), for each u∈ℳλa,b- we have
(3.12)Jλa,b(u)=supt≥0Jλa,b(tu).We define the Palais-Smale (simply by(PS)) sequences, (PS) values, and (PS) conditions in H1(ℝN) for Jλa,b as follows.Definition 3.3.
(i) For c∈ℝ, a sequence {un} is a (PS)c sequence in H1(ℝN) for Jλa,b if Jλa,b(un)=c+on(1) and (Jλa,b)′(un)=on(1) strongly in H-1(ℝN) as n→∞.
(ii)  c∈ℝ is a (PS) value in H1(ℝN) for Jλa,b if there exists a (PS)c sequence in H1(ℝN) for Jλa,b.
(iii)
Jλa,b satisfies the (PS)c-condition in H1(ℝN) if any (PS)c sequence {un} in H1(ℝN) for Jλa,b contains a convergent subsequence.Now, we use the Ekeland variational principle [19] to get the following results.Proposition 3.4.
(i) If λ∈(0,Λ0), then there exists a (PS)αλa,b sequence {un}⊂ℳλa,b in H1(ℝN) for Jλa,b.
(ii) If λ∈(0,(q/2)Λ0), then there exists a (PS)αλa,b- sequence {un}⊂ℳλa,b- in H1(ℝN) for Jλa,b.Proof.
The proof is almost the same as that in [27, Proposition 9].Now, we establish the existence of a local minimum forJλa,b on ℳλa,b+.Theorem 3.5.
Assume that (A1) and (B1) hold. Ifλ∈(0,Λ0), then Jλa,b has a minimizer uλ in ℳλa,b+, and it satisfies the following:(i)
Jλa,b(uλ)=αλa,b=αλa,b+;(ii)
uλ is a positive solution of (Eλa,b) in ℝN;(iii)
∥uλ∥H1→0 as λ→0+.Proof.
By Proposition3.4(i), there is a minimizing sequence {un} for Jλa,b on ℳλa,b such that
(3.13)Jλa,b(un)=αλa,b+on(1),(Jλa,b)′(un)=on(1),inH-1(ℝN).
Since Jλa,b is coercive on ℳλa,b (see Lemma 2.1), we get that {un} is bounded in H1(ℝN). Going if necessary to a subsequence, we can assume that there exists uλ∈H1(ℝN) such that
(3.14)un⇀uλ,weaklyinH1(ℝN),un→uλ,almosteverywhereinℝN,un→uλ,stronglyinLlocs(ℝN),∀1≤s<2*.
By (A1), Egorov theorem, and Hölder inequality, we have
(3.15)λ∫ℝNa(x)|un|qdx=λ∫ℝNa(x)|uλ|qdx+on(1),asn→∞.
First, we claim that uλ is a nonzero solution of (Eλa,b). By (3.13) and (3.14), it is easy to see that uλ is a solution of (Eλa,b). From un∈ℳλa,b and (2.3), we deduce that
(3.16)λ∫ℝNa(x)|un|qdx=q(p-2)2(p-q)∥un∥H12-pqp-qJλa,b(un).
Let n→∞ in (3.16); by (3.13), (3.15), and αλa,b<0, we get
(3.17)λ∫ℝNa(x)|uλ|qdx≥-pqp-qαλa,b>0.
Thus, uλ∈ℳλa,b is a nonzero solution of (Eλa,b). Now we prove that un→uλ strongly in H1(ℝN) and Jλa,b(uλ)=αλa,b. By (3.16), if u∈ℳλa,b, then
(3.18)Jλa,b(u)=p-22p∥u∥H12-p-qpqλ∫ℝNa(x)|u|qdx.
In order to prove that Jλa,b(uλ)=αλa,b, it suffices to recall that un,uλ∈ℳλa,b, by (3.18) and applying Fatou’s lemma to get
(3.19)αλa,b≤Jλa,b(uλ)=p-22p∥uλ∥H12-p-qpqλ∫ℝNa(x)|uλ|qdx≤liminfn→∞(p-22p∥un∥H12-p-qpqλ∫ℝNa(x)|un|qdx)≤liminfn→∞Jλa,b(un)=αλa,b.
This implies that Jλa,b(uλ)=αλa,b and limn→∞∥un∥H12=∥uλ∥H12. Let vn=un-uλ, then Brezis-Lieb lemma [28] implies that
(3.20)∥vn∥H12=∥un∥H12-∥uλ∥H12+on(1).
Therefore, un→uλ strongly in H1(ℝN). Moreover, we have uλ∈ℳλa,b+. On the contrary, if uλ∈ℳλa,b-, then by Lemma 2.5, there are unique t0+ and t0- such that t0+uλ∈ℳλa,b+ and t0-uλ∈ℳλa,b-. In particular, we have t0+<t0-=1. Since
(3.21)ddtJλa,b(t0+uλ)=0,d2dt2Jλa,b(t0+uλ)>0,
there exists t0+<t¯≤t0- such that Jλa,b(t0+uλ)<Jλa,b(t¯uλ). By Lemma 2.5,
(3.22)Jλa,b(t0+uλ)<Jλa,b(t¯uλ)≤Jλa,b(t0-uλ)=Jλa,b(uλ),
which is a contradiction. Since Jλa,b(uλ)=Jλa,b(|uλ|) and |uλ|∈ℳλa,b+, by Lemma 2.2 we may assume that uλ is a nonzero nonnegative solution of (Eλa,b). By Harnack inequality [29] we deduce that uλ>0 in ℝN. Finally, by (2.3) and Hölder and Sobolev inequalities,
(3.23)∥uλ∥H12-q<λp-qp-2∥a+∥Lq*Sp-(q/2)
and so ∥uλ∥H1→0 as λ→0+.Now, we begin the proof of Theorem1.1: By Theorem 3.5, we obtain that (Eλa,b) has a positive solution uλ in H1(ℝN).


## 4. Proof of Theorem1.2

In this section, we will establish the existence of the second positive solution of (Eλa,b) by proving that Jλa,b satisfies the (PS)αλa,b--condition.Lemma 4.1.
Assume that (A1) and (B1) hold. If{un}⊂H1(ℝN) is a (PS)c sequence for Jλa,b, then {un} is bounded in H1(ℝN).Proof.
We argue by contradiction. Assume that∥un∥H1→∞. Let ûn=un/∥un∥H1. We may assume that ûn⇀û weakly in H1(ℝN). This implies that ûn→û strongly in Llocs(ℝN) for all 1≤s<2*. By (A1), Egorov theorem, and Hölder inequality, we have
(4.1)λq∫ℝNa(x)|ûn|qdx=λq∫ℝNa(x)|û|qdx+on(1).
Since {un} is a (PS)c sequence for Jλa,b and ∥un∥H1→∞, there hold
(4.2)12∥ûn∥H12-λ∥un∥H1q-2q∫ℝNa(x)|ûn|qdx-∥un∥H1p-2p∫ℝNb(x)|ûn|pdx=c+on(1),∥ûn∥H12-λ∥un∥H1q-2∫ℝNa(x)|ûn|qdx-∥un∥H1p-2∫ℝNb(x)|ûn|pdx=on(1).
From (4.1)-(4.2), we can deduce that
(4.3)∥ûn∥H12=2(p-q)q(p-2)∥un∥q-2λ∫ℝNa(x)|û|qdx+on(1).
Since 1<q<2 and ∥un∥H1→∞, (4.3) implies that
(4.4)∥ûn∥H12→0,asn→∞,
which contradicts with the fact ∥ûn∥H1=1 for all n.We assume that condition (B2) holds and recall(4.5)Spb=infu∈H1(ℝN)∖{0}∥u∥H12(∫ℝNb(x)|u|pdx)2/p.Lemma 4.2.
Assume that (A1) and (B1)-(B2) hold. If{un}⊂H1(ℝN) is a (PS)c sequence for Jλa,b with c∈(0,α0b), then there exists a subsequence of {un} converging weakly to a nonzero solution of (Eλa,b) in ℝN.Proof.
Let{un}⊂H1(ℝN) be a (PS)c sequence for Jλa,b with c∈(0,α0b). We know from Lemma 4.1 that {un} is bounded in H1(ℝN), and then there exist a subsequence of {un} (still denoted by {un}) and u0∈H1(ℝN) such that
(4.6)un⇀u0,weaklyinH1(ℝN),un→u0,almosteverywhereinℝN,un→u0,stronglyinLlocs(ℝN),∀1≤s<2*.
It is easy to see that (Jλa,b)′(u0)=0, and by (A1), Egorov theorem, and Hölder inequality, we have
(4.7)λ∫ℝNa(x)|un|qdx=λ∫ℝNa(x)|u0|qdx+on(1).
Next we verify thatu0≢0. Arguing by contradiction, we assume that u0≡0. We set
(4.8)l=limn→∞∫ℝNb(x)|un|pdx.
Since (Jλa,b)′(un)=on(1) and {un} is bounded, then by (4.7), we can deduce that
(4.9)0=limn→∞〈(Jλa,b)′(un),un〉=limn→∞(∥un∥H12-∫ℝNa(x)|un|qdx-∫ℝNb(x)|un|pdx)=limn→∞∥un∥H12-l,
that is,
(4.10)limn→∞∥un∥H12=l.
Ifl=0, then we get c=limn→∞Jλa,b(un)=0, which contradicts with c>0. Thus we conclude that l>0. Furthermore, by the definition of Spb we obtain
(4.11)∥un∥H12≥Spb(∫ℝNb(x)|un|pdx)2/p.
Then as n→∞ we have
(4.12)l=limn→∞∥un∥H12≥Spbl2/p,
which implies that
(4.13)l≥(Spb)p/(p-2).
Hence, from (1.7) and (4.7)–(4.13) we get
(4.14)c=limn→∞Jλa,b(un)=12limn→∞∥un∥H12-λqlimn→∞∫ℝNa(x)|un|qdx-1plimn→∞∫ℝNb(x)|un|pdx=(12-1p)l≥p-22p(Spb)p/(p-2)=α0b.
This is a contradiction to c<α0b. Therefore u0 is a nonzero solution of (Eλa,b).Lemma 4.3.
Assume that (A1)-(A2) and (B1)-(B2) hold. Letw0 be a positive ground state solution of (Eb), then(i)
supt≥0Jλa,b(tw0)<α0b for all λ>0;(ii)
αλa,b-<α0b for all λ∈(0,Λ0).Proof.
(i) First, we consider the functionalQ:H1(ℝN)→ℝ defined by
(4.15)Q(u)=12∥u∥H12-1p∫ℝNb(x)|u|pdx,∀u∈H1(ℝN).
Then, from (1.3) and (1.7), we conclude that
(4.16)supt≥0Q(tw0)=p-22p(∥w0∥H12(∫ℝNb(x)|w0|pdx)2/p)p/(p-2)=p-22p(Spb)p/(p-2)=α0b,
where the following fact has been used:
(4.17)supt≥0(t22A-tppB)=p-22p(AB2/p)p/(p-2),whereA,B>0.
Using the definitions of Jλa,b, w0, and b(x)>0 for all x∈ℝN, for any λ>0 we have
(4.18)Jλa,b(tw0)→-∞,ast→∞.
From this we know that there exists t0>0 such that
(4.19)supt≥0Jλa,b(tw0)=sup0≤t≤t0Jλa,b(tw0).
By the continuity of Jλa,b(tw0) as a function of t≥0 and Jλa,b(0)=0, we can find some t1∈(0,t0) such that
(4.20)sup0≤t≤t1Jλa,b(tw0)<α0b.
Thus, we only need to show that
(4.21)supt1≤t≤t0Jλa,b(tw0)<α0b.
To this end, by (A2) and (4.16), for all λ>0 we have
(4.22)supt1≤t≤t0Jλa,b(tw0)≤supt≥0Q(tw0)-t1qqλ∫ℝNa(x)|w0|qdx<α0b.
Hence (i) holds.
(ii) By (A1), (A2), and the definition ofw0, we have
(4.23)∫ℝNb(x)|w0|pdx>0,∫ℝNa(x)|w0|qdx>0.
Combining this with Lemma 2.5(ii), from the definition of αλa,b- and part (i), for all λ∈(0,Λ0), we obtain that there exists t0>0 such that t0w0∈ℳλa,b- and
(4.24)αλa,b-≤Jλa,b(t0w0)≤supt≥0Jλa,b(tw0)<α0b.
Therefore (ii) holds.Now, we establish the existence of a local minimum ofJλa,b on ℳλa,b-.Theorem 4.4.
Assume that (A1)-(A2), (B1), and(ℝbN) hold. If λ∈(0,(q/2)Λ0), then Jλa,b has a minimizer Uλ in ℳλa,b-, and it satisfies the following:(i)
Jλa,b(Uλ)=αλa,b-;(ii)
Uλ is a positive solution of (Eλa,b) in ℝN.Proof.
Ifλ∈(0,(q/2)Λ0), then by Theorem 3.1(ii), Proposition 3.4(ii), and Lemma 4.3(ii), there exists a (PS)αλa,b- sequence {un}⊂ℳλa,b- in H1(ℝN) for Jλa,b with αλa,b-∈(0,α0b). From Lemma 4.2, there exist a subsequence still denoted by {un} and a nonzero solution Uλ∈H1(ℝN) of (Eλa,b) such that un⇀Uλ weakly in H1(ℝN). Now we prove that un→Uλ strongly in H1(ℝN) and Jλa,b(Uλ)=αλa,b-. By (3.18), if u∈ℳλa,b, then
(4.25)Jλa,b(u)=p-22p∥u∥H12-p-qpqλ∫ℝNa(x)|u|qdx.
First, we prove that Uλ∈ℳλa,b-. On the contrary, if Uλ∈ℳλa,b+, then by the definition of
(4.26)ℳλa,b-={u∈ℳλa,b:φu′′(1)<0}
and Lemma 2.4, we have ∥Uλ∥H12<liminfn→∞∥un∥H12. From Lemma 2.3(i) and b(x)>0 for all x∈ℝN, we get
(4.27)∫ℝNa(x)|Uλ|qdx>0,∫ℝNb(x)|Uλ|pdx>0.
By Lemma 2.5(ii), there exists a unique tλ- such that tλ-Uλ∈ℳλa,b-. Since un∈ℳλa,b-, Jλa,b(un)≥Jλa,b(tun) for all t≥0, and by (4.25), we have
(4.28)αλa,b-≤Jλa,b(tλ-Uλ)<limn→∞Jλa,b(tλ-un)≤limn→∞Jλa,b(un)=αλa,b-,
and this is a contradiction.
In order to prove thatJλa,b(Uλ)=αλa,b-, it suffices to recall that un,Uλ∈ℳλa,b- for all n, by (4.25) and applying Fatou’s lemma to get
(4.29)αλa,b-≤Jλa,b(Uλ)=p-22p∥Uλ∥H12-p-qpqλ∫ℝNa(x)|Uλ|qdx≤liminfn→∞(p-22p∥un∥H12-p-qpqλ∫ℝNa(x)|un|qdx)≤liminfn→∞Jλa,b(un)=αλa,b-.
This implies that Jλa,b(Uλ)=αλa,b- and limn→∞∥un∥H12=∥Uλ∥H12. Let vn=un-Uλ, then Brézis and Lieb lemma [28] implies that
(4.30)∥vn∥H12=∥un∥H12-∥Uλ∥H12+on(1).
Therefore, un→Uλ strongly in H1(ℝN).
SinceJλa,b(Uλ)=Jλa,b(|Uλ|) and |Uλ|∈ℳλa,b-, by Lemma 2.2 we may assume that Uλ is a nonzero nonnegative solution of (Eλa,b). Finally, By the Harnack inequality [29] we deduce that Uλ>0 in ℝN.
Now, we complete the proof of Theorem1.2. By Theorems 3.5 and 4.4, we obtain that (Eλa,b) has two positive solutions uλ and Uλ such that uλ∈ℳλa,b+, Uλ∈ℳλa,b-. Since ℳλa,b+∩ℳλa,b-=∅, this implies that uλ and Uλ are distinct. It completes the proof of Theorem 1.2.



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*Source: 658397-2010-07-19.xml*