# Multiplicity Results forp-Laplacian with  Critical Nonlinearity of Concave-Convex Type  and Sign-Changing Weight Functions

Authors: Tsing-San Hsu
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

The multiple results of positive solutions for the following quasilinear
elliptic equation:−Δpu=λf(x)|u|q−2u+g(x)|u|p∗−2u in Ω,u=0 on ∂Ω, are established. Here, 0∈Ω is a bounded smooth domain in ℝN,Δp denotes the p-Laplacian operator, 1≤q<p<N,p∗=Np/(N−p),λ is a positive real parameter, and f,g are continuous functions on Ω¯ which are somewhere positive but which may change sign on Ω. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.


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## Body

## 1. Introduction

In this paper, we study the multiple results of positive solutions for the following quasilinear elliptic equation:(Eλf,g)-Δpu=λf(x)|u|q-2u+g(x)|u|p*-2uinΩ,u=0on∂Ω,
where λ>0, Δpu=div(|∇u|p-2∇u) is the p-Laplacian, 0∈Ω is a bounded domain in ℝN with smooth boundary ∂Ω, 1<q<p<N, p*=Np/(N-p) is the so-called critical Sobolev exponent and the weight functions f,g are satisfying the following conditions:(f1)
f∈C(Ω¯) and f+=max{f,0}≢0;(f2)
there existβ0,ρ0>0 and x0∈Ω such that B(x0,2ρ0)⊂Ω and f(x)≥β0 for all x∈B(x0,2ρ0). Without loss of generality, we assume that x0=0,(g1)
g∈C(Ω¯) and g+=max{g,0}≢0;(g2)
|g+|∞=g(0)=maxx∈Ω¯g(x);(g3)
g(x)>0 for all x∈B(0,2ρ0);(g4)
there existsβ>N/(p-1) such that(1.1)g(x)=g(0)+o(|x|β)asx→0.For the weight functionsf≡g≡1, (Eλf,g) has been studied extensively. Historically, the role played by such concave-convex nonlinearities in producing multiple solutions was investigated first in the work [1]. They studied the following semilinear elliptic equation:(1.2)-Δu=λuq-1+u2*-1inΩ,u>0inΩ,u=0on∂Ω,
for 1<q<2 and showed the existence of λ0>0 such that (1.2) admits at least two solutions for all λ∈(0,λ0) and no solution for λ>λ0. Subsequently, in the work [2, 3], the corresponding quasilinear version has been studied(1.3)-Δpu=λuq-1+up*-1inΩ,u>0inΩ,u=0on∂Ω,
where 1<p<N and 1<q<p. They obtained results similar to the results of [1] above, but only for some ranges of the exponents p and q. We summarize their results in what follows.Theorem 1.1 (see [2, 3]).
Assume that either2N/(N+2)<p<3 or p>3,p>q>p*-2/(p-1). Then there exists λ0>0 such that (1.3) admits at least two solutions for all λ∈(0,λ0) and no solution for λ>λ0.It is possible to get complete multiplicity result for problem (1.3) if Ω is taken to be a ball in ℝN. Prashanth and Sreenadh [4] have studied (1.3) in the unit ball BN(0;1) in ℝN and obtained the following results.Theorem 1.2 (see [4]).
LetΩ=BN(0;1),1<p<N,1<q<p. Then there exists λ0>0 such that (1.3) admits at least two solutions for all λ∈(0,λ0) and no solution for λ>λ0. Additionally, if 1<p<2, then (1.3) admits exactly two solutions for all small λ>0.Forp=2, Tang [5] has studied the exact multiplicity about the following semilinear elliptic equation:(1.4)-Δu=λuq-1+ur-1inBN(0;1),u>0inBN(0;1),u=0on∂BN(0;1),
where 1<q<2<r≤2N/(N-2) and N≥3. We also mention his result  below.Theorem 1.3 (see [5]).
There existsλ0>0 such that (1.4) admits exactly two solutions for λ∈(0,λ0), exactly one solution for λ=λ0, and no solution for λ>λ0.To proceed, we make some motivations of the present paper. Recently, in [6] the author has considered (1.2) with subcritical nonlinearity of concave-convex type, g≡1, and f is a continuous function which changes sign in Ω¯, and showed the existence of λ0>0 such that (1.2) admits at least two solutions for all λ∈(0,λ0) via the extraction of Palais-Smale sequences in the Nehair manifold. In a recent work [7], the author extended the results of [6] to the quasilinear case with the more general weight functions f,g but also having subcritical nonlinearity of concave-convex type. In the present paper, we continue the study of [7] by considering critical nonlinearity of concave-convex type and sign-changing weight functions f,g.In this paper, we use a variational method involving the Nehari manifold to prove the multiplicity of positive solutions. The Nehari method has been used also in [8] to prove the existence of multiple for a singular elliptic problem. The existence of at least one solution can be obtained by using the same arguments as in the subcritical case [7]. The existence of a second solution needs different arguments due to the lack of compactness of the Palais-Smale sequences. For what, we need addtional assumptions (f2) and (g2) to prove the compactness of the extraction of Palais-Smale sequences in the Nehari manifold (see Theorem 4.4). The multiplicity result is proved only for the parameter λ∈(0,(q/p)Λ1) (see Theorem 1.5) but for all 1<p<N and 1≤q<p. This is not the case in the papers referred [2, 3] where the multiplicity is global but not with the full range of p, q and with the weight functions f≡g≡1. Finally, we mention a recent contribution on p-Laplacian equation with changing sign nonlinearity by Figuereido et al. [9] which gives the global multiplicity but not with the full range of p and q. The method used in the paper by Figuereido et al. is similar to the method introduced in [1].In order to represpent our main results, we need to define the following constantΛ1. Set(1.5)Λ1=(p-q(p*-q)|g+|∞)(p-q)/(p*-p)(p*-p(p*-q)|f+|∞)|Ω|(q-p*)/p*S(N/p)-(N/p2)q+(q/p)>0,
where |Ω| is the Lebesgue measure of Ω and S is the best Sobolev constant (see (2.2)).Theorem 1.4.
Assume(f1) and (g1) hold. If λ∈(0,Λ1), then (Eλf,g) admits at least one positive solution uλ∈C1,α(Ω) for some α∈(0,1).Theorem 1.5.
Assume that(f1)-(f2) and (g1)–(g4) hold. If λ∈(0,(q/p)Λ1), then (Eλf,g) admits at least two positive solutions uλ,Uλ∈C1,α(Ω) for some α∈(0,1).This paper is organized as follows. In Section2, we give some preliminaries and some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorems 1.4 and 1.5.


## 2. Preliminaries and Nehari Manifold

Throughout this paper,(f1) and (g1) will be assumed. The dual space of a Banach space E will be denoted by E-1. W01,p(Ω) denotes the standard Sobolev space with the  following  norm:(2.1)∥u∥p=∫Ω|∇u|pdx.W01,p(Ω) with the norm ∥·∥ is simply denoted by W. We denote the norm in Lp(Ω) by |·|p and the norm in Lp(ℝN) by |·|Lp(ℝN) . |Ω| is the Lebesgue measure of Ω. B(x,r) is a ball centered at x with radius r. O(εt) denotes |O(εt)|/εt≤C, o(εt) denotes |o(εt)|/εt→0 as ε→0, and on(1) denotes on(1)→0 as n→∞. C, Ci will denote various positive constants; the exact values of which are not important. S is the best Sobolev embedding constant defined by(2.2)S=infu∈W∖{0}|∇u|pp|u|p*p.Definition 2.1.
Letc∈ℝ, E be a Banach space and I∈C1(E,ℝ).
(i)
{un} is a (PS)c-sequence in E for I if I(un)=c+on(1) and I′(un)=on(1) strongly in E-1 as n→∞.(ii)
We say thatI satisfies the (PS)c condition if any (PS)c-sequence {un} in E for I has a convergent subsequence.Associated with (Eλf,g), we consider the energy functional Jλ in W, for each u∈W,(2.3)Jλ(u)=1p∥u∥p-λq∫Ωf|u|qdx-1p*∫Ωg|u|p*dx.
It is well known that Jλ is of C1 in W and the solutions of (Eλf,g) are the critical points of the energy functional Jλ (see Rabinowitz [10]).As the energy functionalJλ is not bounded below on W, it is useful to consider the functional on the Nehari manifold(2.4)𝒩λ={u∈W∖{0}:〈Jλ′(u),u〉=0}.
Thus, u∈𝒩λ if and only if(2.5)〈Jλ′(u),u〉=∥u∥p-λ∫Ωf|u|qdx-∫Ωg|u|p*dx=0.
Note that 𝒩λ contains every nonzero solution of (Eλf,g). Moreover, we have the following results.Lemma 2.2.
The energy functionalJλ is coercive and bounded below on 𝒩λ.Proof.
Ifu∈𝒩λ, then by (f1), (2.5), and the Hölder inequality and the Sobolev embedding theorem we have
(2.6)Jλ(u)=p*-pp*p∥u∥p-λ(p*-qp*q)∫Ωf|u|qdx(2.7)≥1N∥u∥p-λ(p*-qp*q)S-q/p|Ω|(p*-q)/p*∥u∥q|f+|∞.
Thus, Jλ is coercive and bounded below on 𝒩λ.Define(2.8)ψλ(u)=〈Jλ′(u),u〉.
Then for u∈𝒩λ,(2.9)〈ψλ′(u),u〉=p∥u∥p-λq∫Ωf|u|qdx-p*∫Ωg|u|p*dx(2.10)=(p-q)∥u∥p-(p*-q)∫Ωg|u|p*dx(2.11)=λ(p*-q)∫Ωf|u|qdx-(p*-p)∥u∥p.
Similar to the method used in Tarantello [11], we split 𝒩λ into three parts:(2.12)𝒩λ+={u∈𝒩λ:〈ψλ′(u),u〉>0},𝒩λ0={u∈𝒩λ:〈ψλ′(u),u〉=0},𝒩λ-={u∈𝒩λ:〈ψλ′(u),u〉<0}.
Then, we have the following results.Lemma 2.3.
Assume thatuλ is a local minimizer for Jλ on 𝒩λ and uλ∉𝒩λ0. Then Jλ′(uλ)=0 in W-1.Proof.
Our proof is almost the same as that in Brown and Zhang [12, Theorem 2.3] (or see Binding et al. [13]).Lemma 2.4.
One has the following.
(i) Ifu∈𝒩λ+, then ∫Ωf|u|qdx>0.
(ii) Ifu∈𝒩λ0, then ∫Ωf|u|qdx>0 and ∫Ωg|u|p*dx>0.
(iii) Ifu∈𝒩λ-, then ∫Ωg|u|p*dx>0.Proof.
The proof is immediate from (2.10) and (2.11).Moreover, we have the following  result.Lemma 2.5.
Ifλ∈(0,Λ1), then 𝒩λ0=∅ where Λ1 is the same as in (1.5).Proof.
Suppose otherwise that there existsλ∈(0,Λ1) such that 𝒩λ0≠∅. Then by (2.10) and (2.11), for u∈𝒩λ0, we have
(2.13)∥u∥p=p*-qp-q∫Ωg|u|p*dx,∥u∥p=λp*-qp*-p∫Ωf|u|qdx.
Moreover, by (f1), (g1), and the Hölder inequality and the Sobolev embedding theorem, we have
(2.14)∥u∥≥(p-q(p*-q)|g+|∞Sp*/p)1/(p*-p),∥u∥≤[λp*-qp*-pS-q/p|Ω|(p*-q)/p*|f+|∞]1/(p-q).
This implies
(2.15)λ≥(p-q(p*-q)|g+|∞)(p-q)/(p*-p)(p*-p(p*-q)|f+|∞)|Ω|(q-p*)/p*S(N/p)-(N/p2)q+(q/p)=Λ1,
which is a contradiction. Thus, we can conclude that if λ∈(0,Λ1), we have 𝒩λ0=∅.By Lemma2.5, we write 𝒩λ=𝒩λ+∪𝒩λ- and define(2.16)αλ=infu∈𝒩λJλ(u),αλ+=infu∈𝒩λ+Jλ(u),αλ-=infu∈𝒩λ-Jλ(u).
Then we get the following result.Theorem 2.6.
(i) Ifλ∈(0,Λ1) and u∈𝒩λ+, then one has Jλ(u)<0 and αλ≤αλ+<0.
(ii) Ifλ∈(0,(q/p)Λ1), then αλ->d0 for some positive constant d0 depending on λ,p,q,N,S,|f+|∞,|g+|∞, and |Ω|.Proof.
(i) Letu∈𝒩λ+. By (2.10), we have
(2.17)p-qp*-q∥u∥p>∫Ωg|u|p*dx,
and so
(2.18)Jλ(u)=(1p-1q)∥u∥p+(1q-1p*)∫Ωg|u|p*dx<[(1p-1q)+(1q-1p*)p-qp*-q]∥u∥p=-p-qqN∥u∥p<0.
Therefore, from the definition of αλ, αλ+, we can deduce that αλ≤αλ+<0.
(ii) Letu∈𝒩λ-. By (2.10), we have
(2.19)p-qp*-q∥u∥p<∫Ωg|u|p*dx.
Moreover, by (g1) and the Sobolev embedding theorem, we have
(2.20)∫Ωg|u|p*dx≤S-p*/p∥u∥p*|g+|∞.
This implies
(2.21)∥u∥>(p-q(p*-q)|g+|∞)1/(p*-p)SN/p2,∀u∈𝒩λ-.
By(2.7) in the proof of Lemma 2.2, we have
(2.22)Jλ(u)≥∥u∥q[p*-pp*p∥u∥p-q-λS-q/pp*-qp*q|Ω|(p*-q)/p*|f+|∞]>(p-q(p*-q)|g+|∞)q/(p*-p)SqN/p2×[p*-pp*pS(p-q)N/p2(p-q(p*-q)|g+|∞)(p-q)/(p*-p)-λS-q/pp*-qp*q|Ω|(p*-q)/p*|f+|∞].
Thus, if λ∈(0,(q/p)Λ1), then
(2.23)Jλ(u)>d0,∀u∈𝒩λ-,
for some positive constant d0=d0(λ,p,q,N,S,|f+|∞,|g+|∞,|Ω|). This completes the proof.For eachu∈W with ∫Ωg|u|p*dx>0, we write(2.24)tmax=((p-q)∥u∥p(p*-q)∫Ωg|u|p*dx)1/(p*-p)>0.
Then the following lemma holds.Lemma 2.7.
Letλ∈(0,Λ1). For each u∈W with ∫Ωg|u|p*dx>0, one has the following:
(i) if∫Ωf|u|qdx≤0, then there exists a unique t->tmax such that t-u∈𝒩λ- and
(2.25)Jλ(t-u)=supt≥0Jλ(tu),
(ii) if∫Ωf|u|qdx>0, then there exists unique 0<t+<tmax<t- such that t+u∈𝒩λ+, t-u∈𝒩λ-, and
(2.26)Jλ(t+u)=inf0≤t≤tmaxJλ(tu);Jλ(t-u)=supt≥0Jλ(tu).Proof.
Fixu∈W with ∫Ωg|u|p*dx>0. Let
(2.27)k(t)=tp-q∥u∥p-tp*-q∫Ωg|u|p*dxfort≥0.
It is clear that k(0)=0, k(t)→-∞ as t→∞. From
(2.28)k′(t)=(p-q)tp-q-1∥u∥p-(p*-q)tp*-q-1∫Ωg|u|p*dx,
we can deduce that k′(t)=0 at t=tmax, k′(t)>0 for t∈(0,tmax) and k′(t)<0 for t∈(tmax,∞). Then k(t) that achieves its maximum at tmax is increasing for t∈[0,tmax) and decreasing for t∈(tmax,∞). Moreover,
(2.29)k(tmax)=((p-q)∥u∥p(p*-q)∫Ωg|u|p*dx)(p-q)/(p*-p)∥u∥p-((p-q)∥u∥p(p*-q)∫Ωg|u|p*dx)(p*-q)/(p*-p)∫Ωg|u|p*dx=∥u∥q[(p-qp*-q)(p-q)/(p*-p)-(p-qp*-q)(p*-q)/(p*-p)](∥u∥p*∫Ωg|u|p*dx)(p-q)/(p*-p)≥∥u∥q(p*-pp*-q)(p-q(p*-q)|g+|∞Sp*/p)(p-q)/(p*-p).
We have ∫Ωf|u|qdx≤0. There exists a unique t->tmax such that k(t-)=λ∫Ωf|u|qdx and k′(t-)<0. Now,
(2.30)(p-q)(t-)p∥u∥p-(p*-q)(t-)p∫Ωg|u|p*dx=(t-)1+q[(p-q)(t-)p-q-1∥u∥p-(p*-q)(t-)p*-q-1∫Ωg|u|p*dx]=(t-)1+qk′(t-)<0,〈Jλ′(t-u),t-u〉=(t-)p∥u∥p-(t-)p*∫Ωg|u|p*dx-(t-)qλ∫Ωf|u|qdx=(t-)q[k(t-)-λ∫Ωf|u|qdx]=0.
Then we have that t-u∈𝒩λ-. For t>tmax, we have
(2.31)(p-q)∥tu∥p-(p*-q)∫Ωg|tu|p*<0,d2dt2Jλ(tu)<0,ddtJλ(tu)=tp-1∥u∥p-tp*-1∫Ωg|u|p*dx-tq-1λ∫Ωf|u|qdx=0fort=t-.
Thus, Jλ(t-u)=supt≥0Jλ(tu).
(ii) We have∫Ωf|u|qdx>0. By (2.29) and
(2.32)k(0)=0<λ∫Ωf|u|qdx≤λS-q/p|Ω|(p*-q)/p*∥u∥q|f+|∞<∥u∥q(p*-pp*-q)(p-q(p*-q)|g+|∞Sp*/p)(p-q)/(p*-p)≤k(tmax)forλ∈(0,Λ1),
there are unique t+ and t- such that 0<t+<tmax<t-,
(2.33)k(t+)=λ∫Ωf|u|qdx=k(t-),k′(t+)>0>k′(t-).
We have t+u∈𝒩λ+, t-u∈𝒩λ-, and Jλ(t-u)≥Jλ(tu)≥Jλ(t+u) for each t∈[t+,t-] and Jλ(t+u)≤Jλ(tu) for each t∈[0,t+]. Thus,
(2.34)Jλ(t+u)=inf0≤t≤tmaxJλ(tu),Jλ(t-u)=supt≥0Jλ(tu).
This completes the proof.


## 3. Proof of Theorem1.4

First, we will use the idea of Tarantello [11] to get the following results.Lemma 3.1.
Ifλ∈(0,Λ1), then for each u∈𝒩λ, there exist ϵ>0 and a differentiable function ξ:B(0;ϵ)⊂W→ℝ+ such that ξ(0)=1, the function ξ(v)(u-v)∈𝒩λ, and
(3.1)〈ξ′(0),v〉=p∫Ω|∇u|p-2∇u∇vdx-λq∫Ωf|u|q-2uvdx-p*∫Ωg|u|p*-2uvdx(p-q)∥u∥p-(p*-q)∫Ωg|u|p*dx
for all v∈W.Proof.
Foru∈𝒩λ, define a function F:ℝ×W→ℝ by
(3.2)Fu(ξ,w)=〈Jλ′(ξ(u-w)),ξ(u-w)〉=ξp∫Ω|∇(u-w)|pdx-ξqλ∫Ωf|u-w|qdx-ξp*∫Ωg|u-w|p*dx.
Then Fu(1,0)=〈Jλ′(u),u〉=0 and
(3.3)ddξFu(1,0)=p∥u∥p-λq∫Ωf|u|qdx-p*∫Ωg|u|p*dx=(p-q)∥u∥p-(p*-q)∫Ωg|u|p*dx≠0.
According to the implicit function theorem, there exist ϵ>0 and a differentiable function ξ:B(0;ϵ)⊂W→ℝ such that ξ(0)=1,
(3.4)〈ξ′(0),v〉=p∫Ω|∇u|p-2∇u∇vdx-λq∫Ωf|u|q-2uvdx-p*∫Ωg|u|p*-2uvdx(p-q)∥u∥p-(p*-q)∫Ωg|u|p*dx,Fu(ξ(v),v)=0,∀v∈B(0;ϵ),
which is equivalent to
(3.5)〈Jλ′(ξ(v)(u-v)),ξ(v)(u-v)〉=0,∀v∈B(0;ϵ),
that is, ξ(v)(u-v)∈𝒩λ.Lemma 3.2.
Letλ∈(0,Λ1), then for each u∈𝒩λ-, there exist ϵ>0 and a differentiable function ξ-:B(0;ϵ)⊂W→ℝ+ such that ξ-(0)=1, the function ξ-(v)(u-v)∈𝒩λ-, and
(3.6)〈(ξ-)′(0),v〉=p∫Ω|∇u|p-2∇u∇vdx-λq∫Ωf|u|q-2uvdx-p*∫Ωg|u|p*-2uvdx(p-q)∥u∥p-(p*-q)∫Ωg|u|p*dx
for all v∈W.Proof.
Similar to the argument in Lemma3.1, there exist ϵ>0 and a differentiable function ξ-:B(0;ϵ)⊂W→ℝ such that ξ-(0)=1 and ξ-(v)(u-v)∈𝒩λ for all v∈B(0;ϵ). Since
(3.7)〈ψλ′(u),u〉=(p-q)∥u∥p-(p*-q)∫Ωg|u|p*dx<0.
Thus, by the continuity of the function ξ-, we have
(3.8)〈ψλ′(ξ-(v)(u-v)),ξ-(v)(u-v)〉=(p-q)∥ξ-(v)(u-v)∥p-(p*-q)∫Ωg|ξ-(v)(u-v)|pdx<0,
if ϵ sufficiently small, this implies that ξ-(v)(u-v)∈𝒩λ-.Proposition 3.3.
(i) Ifλ∈(0,Λ1), then there exists a (PS)αλ-sequence {un}⊂𝒩λ in W for Jλ.
(ii) Ifλ∈(0,(q/p)Λ1), then there exists a (PS)αλ--sequence {un}⊂𝒩λ- in W for Jλ.Proof.
(i) By Lemma2.2 and the Ekeland variational principle [14], there exists a minimizing sequence {un}⊂𝒩λ such that
(3.9)Jλ(un)<αλ+1n,Jλ(un)<Jλ(w)+1n∥w-un∥foreachw∈𝒩λ.
By αλ<0 and taking n large, we have
(3.10)Jλ(un)=(1p-1p*)∥un∥p-(1q-1p*)λ∫Ωf|un|qdx<αλ+1n<αλp.
From (2.7), (3.10), αλ<0, and the Hölder inequality, we deduce that
(3.11)|f+|∞λS-q/p|Ω|(p*-q)/p*∥un∥q≥λ∫Ωf|un|qdx>-p*qp(p*-q)αλ>0.
Consequently, un≠0 and putting together (3.10), (3.11), and the Hölder inequality, we obtain
(3.12)∥un∥>[-p*qpλ(p*-q)|f+|∞αλSq/p|Ω|(q-p*)/p*]1/q,∥un∥<[p(p*-q)q(p*-p)λS-q/p|Ω|(p*-q)/p*|f+|∞]1/(p-q).
Now, we show that
(3.13)∥Jλ′(un)∥W-1→0,asn→∞.
Apply Lemma 3.1 with un to obtain the functions ξn:B(0;ϵn)→ℝ+ for some ϵn>0, such that ξn(w)(un-w)∈𝒩λ. Choose 0<ρ<ϵn. Let u∈W with u≢0 and let wρ=ρu/∥u∥. We set ηρ=ξn(wρ)(un-wρ). Since ηρ∈𝒩λ, we deduce from (3.9) that
(3.14)Jλ(ηρ)-Jλ(un)≥-1n∥ηρ-un∥,
and by the mean value theorem, we have
(3.15)〈Jλ′(un),ηρ-un〉+o(∥ηρ-un∥)≥-1n∥ηρ-un∥.
Thus,
(3.16)〈Jλ′(un),-wρ〉+(ξn(wρ)-1)〈Jλ′(un),(un-wρ)〉≥-1n∥ηρ-un∥+o(∥ηρ-un∥).
Since ξn(wρ)(un-wρ)∈𝒩λ and (3.16) it follows that
(3.17)-ρ〈Jλ′(un),u∥u∥〉+(ξn(wρ)-1)〈Jλ′(un)-Jλ′(ηρ),(un-wρ)〉≥-1n∥ηρ-un∥+o(∥ηρ-un∥).
Thus,
(3.18)〈Jλ′(un),u∥u∥〉≤∥ηρ-un∥nρ+o(∥ηρ-un∥)ρ+(ξn(wρ)-1)ρ〈Jλ′(un)-Jλ′(ηρ),(un-wρ)〉.
Since ∥ηρ-un∥≤ρξn(wρ)+|ξn(wρ)-1|∥un∥ and
(3.19)limρ→0|ξn(wρ)-1|ρ≤∥ξn′(0)∥,
if we let ρ→0 in (3.18) for a fixed n, then by (3.12) we can find a constant C>0, independent of ρ, such that
(3.20)〈Jλ′(un),u∥u∥〉≤Cn(1+∥ξn′(0)∥).
The proof will be complete once we show that ∥ξn′(0)∥ is uniformly bounded in n. By (3.1), (3.12), (f1), (g1), and the Hölder inequality and the Sobolev embedding theorem, we have
(3.21)〈ξn′(0),v〉≤b∥v∥|(p-q)∥un∥p-(p*-q)∫Ωg|un|p*dx|forsomeb>0.
We only need to show that
(3.22)|(p-q)∥un∥p-(p*-q)∫Ωg|un|p*dx|>C
for some C>0 and n large enough. We argue by contradiction. Assume that there exists a subsequence {un} such that
(3.23)(p-q)∥un∥p-(p*-q)∫Ωg|un|p*dx=on(1).
By (3.23) and the fact that un∈𝒩λ, we get
(3.24)∥un∥p=p*-qp-q∫Ωg|un|p*dx+on(1),∥un∥p=λp*-qp*-p∫Ωf|un|qdx+on(1).
Moreover, by (f1), (g1), and the Hölder inequality and the Sobolev embedding theorem, we have
(3.25)∥un∥≥[p-q(p*-q)|g+|∞Sp*/p]1/(p*-p)+on(1),∥un∥≤[λ(p*-q)|f+|∞p*-pS-q/p|Ω|(p*-q)/p*]1/(p-q)+on(1).
This implies λ≥Λ1 which is a contradiction. We obtain
(3.26)〈Jλ′(un),u∥u∥〉≤Cn.
This completes the proof of (i).
(ii) Similarly, by using Lemma3.2, we can prove (ii). We will omit detailed proof here.Now, we establish the existence of a local minimum forJλ on 𝒩λ+.Theorem 3.4.
Ifλ∈(0,Λ1), then Jλ has a minimizer uλ in 𝒩λ+ and it satisfies that
(i)Jλ(uλ)=αλ=αλ+;
(ii)uλ is a positive solution of (Eλf,g) in C1,α(Ω) for some α∈(0,1).Proof.
By Proposition3.3(i), there exists a minimizing sequence {un} for Jλ on 𝒩λ such that
(3.27)Jλ(un)=αλ+on(1),Jλ′(un)=on(1)inW-1.
Since Jλ is coercive on 𝒩λ (see Lemma 2.2), we get that {un} is bounded in W. Going if necessary to a subsequence, we can assume that there exists uλ∈W such that
(3.28)un⇀uλweaklyinW,un→uλalmosteverywhereinΩ,un→uλstronglyinLs(Ω)∀1≤s<p*.
First, we claim that uλ is a nontrivial solution of (Eλf,g). By (3.27) and (3.28), it is easy to see that uλ is a solution of (Eλf,g). From un∈𝒩λ and (2.6), we deduce that
(3.29)λ∫Ωf|un|qdx=q(p*-p)p(p*-q)∥un∥p-p*qp*-qJλ(un).
Let n→∞ in (3.29), by (3.27), (3.28), and αλ<0, we get
(3.30)∫Ωf|uλ|qdx≥-p*qp*-qαλ>0.
Thus, uλ∈𝒩λ is a nontrivial solution of (Eλf,g). Now we prove that un→uλ strongly in W and Jλ(uλ)=αλ. By (3.29), if u∈𝒩λ, then
(3.31)Jλ(u)=p*-pp*p∥u∥p-p*-qp*qλ∫Ωf|u|qdx.
In order to prove that Jλ(uλ)=αλ, it suffices to recall that uλ∈𝒩λ, by (3.31), and applying Fatou's lemma to get
(3.32)αλ≤Jλ(uλ)=p*-pp*p∥uλ∥p-p*-qp*qλ∫Ωf|uλ|qdx≤liminfn→∞(p*-pp*p∥un∥p-p*-qp*qλ∫Ωf|un|qdx)≤liminfn→∞Jλ(un)=αλ.
This implies that Jλ(uλ)=αλ and limn→∞∥un∥p=∥uλ∥p. Let vn=un-uλ, then Brézis and Lieb lemma [15] implies that
(3.33)∥vn∥p=∥un∥p-∥uλ∥p+on(1).
Therefore, un→uλ strongly in W. Moreover, we have uλ∈𝒩λ+. On the contrary, if uλ∈𝒩λ-, then by Lemma 2.7, there are unique t0+ and t0- such that t0+uλ∈𝒩λ+ and t0-uλ∈𝒩λ-. In particular, we have t0+<t0-=1. Since
(3.34)ddtJλ(t0+uλ)=0,d2dt2Jλ(t0+uλ)>0,
there exists t0+<t̅≤t0- such that Jλ(t0+uλ)<Jλ(t̅uλ). By Lemma 2.7,
(3.35)Jλ(t0+uλ)<Jλ(t̅uλ)≤Jλ(t0-uλ)=Jλ(uλ),
which is a contradiction. Since Jλ(uλ)=Jλ(|uλ|) and |uλ|∈𝒩λ+, by Lemma 2.3 we may assume that uλ is a nontrivial nonnegative solution of (Eλf,g). Moreover, from f,g∈L∞(Ω), then using the standard bootstrap argument (see, e.g., [16]) we obtain uλ∈L∞(Ω); hence by applying regularity results [17, 18] we derive that uλ∈C1,α(Ω) for some α∈(0,1) and finally, by the Harnack inequality [19] we deduce that uλ>0. This completes the proof.Now, we begin the proof of Theorem1.4. By Theorem 3.4, we obtain (Eλf,g) that has a positive solution uλ in C1,α(Ω) for some α∈(0,1).


## 4. Proof of Theorem1.5

Next, we will establish the existence of the second positive solution of (Eλf,g) by proving that Jλ satisfies the (PS)αλ- condition.Lemma 4.1.
Assume that(f1) and (g1) hold. If {un}⊂W is a (PS)c-sequence for Jλ, then {un} is bounded in W.Proof.
We argue by contradiction. Assume that∥un∥→∞. Let ûn=un/∥un∥. We may assume that ûn⇀û in W. This implies that ûn→û strongly in Ls(Ω) for all 1≤s<p* and
(4.1)λq∫Ωf|ûn|qdx=λq∫Ωf|û|qdx+on(1).
Since {un} is a (PS)c-sequence for Jλ and ∥un∥→∞, there hold
(4.2)1p∫Ω|∇ûn|pdx-λ∥un∥q-pq∫Ωf|ûn|qdx-∥un∥p*-pp*∫Ωg|ûn|p*dx=on(1),∫Ω|∇ûn|pdx-λ∥un∥q-p∫Ωf|ûn|qdx-∥un∥p*-p∫Ωg|ûn|p*dx=on(1).
From (4.1)-(4.2), we can deduce that
(4.3)∫Ω|∇ûn|pdx=p(p*-q)q(p*-p)∥un∥q-pλ∫Ωf|û|qdx+on(1).
Since 1≤q<2 and ∥un∥→∞, (4.3) implies
(4.4)∫Ω|∇ûn|pdx→0,asn→∞,
which is contrary to the fact ∥ûn∥=1 for all n.Lemma 4.2.
Assume that(f1) and (g1) hold. If {un}⊂W is a (PS)c-sequence for Jλ with c∈(0,(1/N)|g+|∞-(N-p)/pSN/p), then there exists a subsequence of {un} converging weakly to a nontrivial solution of (Eλf,g).Proof.
Let{un}⊂W be a (PS)c-sequence for Jλ with c∈(0,(1/N)|g+|∞-(N-p)/pSN/p). We know from Lemma 4.1 that {un} is bounded in W, and then there exists a subsequence of {un} (still denoted by {un} and u0∈W such that
(4.5)un⇀u0weaklyinW,un→u0almosteverywhereinΩ,un→u0stronglyinLs(Ω)∀1≤s<p*.
It is easy to see thatJλ′(u0)=0 and
(4.6)λ∫Ωf(x)|un|qdx=λ∫Ωf(x)|u0|qdx+on(1).
Next we verify thatu0≢0. Arguing by contradiction, we assume u0≡0. Setting
(4.7)l=limn→∞∫Ωg|un|p*dx.
Since Jλ′(un)=on(1) and {un} is bounded, then by (4.6), we can deduce that
(4.8)0=〈limn→∞Jλ′(un),un〉=limn→∞(∥un∥p-∫Ωg|un|p*)=limn→∞∥un∥p-l,
that is,
(4.9)limn→∞∥un∥p=l.
Ifl=0, then we get c=limn→∞Jλ(un)=0, which contradicts with c>0. Thus we conclude that l>0. Furthermore, the Sobolev inequality implies that
(4.10)∥un∥p≥S(∫Ω|un|p*)p/p*≥S(∫Ωg|g+|∞|un|p*)p/p*=S|g+|∞-(N-p)/N(∫Ωg|un|p*)p/p*.
Then as n→∞ we have
(4.11)l=limn→∞∥un∥p≥S|g+|-(N-p)/Nlimn→∞(∫Ωg|un|p*)p/p*=S|g+|∞-(N-p)/Nlp/p*,
which implies that
(4.12)l≥|g+|∞-(N-p)/pSN/p.
Hence, from (4.6) to (4.12) we get
(4.13)c=limn→∞Jλ(un)=1plimn→∞∥un∥p-λqlimn→∞∫Ωf|un|qdx-1p*limn→∞∫Ωg|un|p*dx=(1p-1p*)l≥1N|g+|∞-(N-p)/pSN/p.
This is a contradiction to c<(1/N)|g+|∞-(N-p)/pSN/p. Therefore u0 is a nontrivial solution of (Eλf,g).Lemma 4.3.
Assume that(f1)-(f2) and (g1)–(g4) hold. Then for any λ>0, there exists vλ∈W such that
(4.14)supt≥0Jλ(tvλ)<1N|g+|∞-(N-p)/pSN/p.
In particular,αλ-<(1/N)|g+|∞-(N-p)/pSN/p for all λ∈(0,Λ1) where Λ1 is as in (1.5).Proof.
For convenience, we introduce the following notations:(4.15)I(u)=∫Ω{1p|∇u|p-1p*g|u|p*}dx,χB(0,2ρ0)={1ifx∈B(0,2ρ0),0ifx∉B(0,2ρ0),Q(u)=|∇u|pp|(gχB(0,2ρ0))1/p*u|p*p.
From (g3) to (g4), we know that there exists δ0∈(0,ρ0) such that for all x∈B(0,2δ0),
(4.16)g(x)=g(0)+o(|x|β)forsomeβ>Np-1.
Motivated by some ideas of selecting cut-off functions in [20, Lemma 4.1], we take such cut-off function η(x) that satisfies η(x)∈C0∞(B(0,2δ0)), η(x)=1 for |x|<δ0,η(x)=0 for |x|>2δ0,0≤η≤1, and |∇η|≤C. Define, for ε>0,
(4.17)uε(x)=ε(N-p)/p2η(x)(ε+|x|p/(p-1))(N-p)/p.
Step 1.
Show thatsupt≥0I(tuε)≤(1/N)|g+|∞-(N-p)/pSN/p+O(ε(N-p)/p).
On that purpose, we need to establish the following estimates (asε→0):
(4.18)|(gχB(0,2ρ0))1/p*uε|p*p=|g+|∞-(N-p)/N|U|Lp*(ℝN)p+O(εN/p),(4.19)|∇uε|pp=|∇U|Lp(ℝN)p+O(ε(N-p)/p),
where U(x)=(1+(x)p/(p-1))-(N-p)/p∈W1,p(ℝN) is a minimizer of {|∇u|pp/|u|p*p}u∈W1,p(ℝN)∖{0}, that is,
(4.20)|∇U|Lp(ℝN)p|U|Lp*(ℝN)p=S=infu∈W1,p(ℝN)∖{0}|∇u|Lp(ℝN)p|u|Lp*(ℝN)p,
and ωN=2πN/2/NΓ(N/2) which is the volume of the unit ball B(0,1) in ℝN. We only show that equality (4.18) is valid; proofs of (4.19) are very similar to [20]. In view of (4.17), we get that
(4.21)|(gχB(0,2ρ0))1/p*uε|p*p*=∫B(0,2δ0)g(x)|uε|p*dx=∫ℝNεN/pηp*(x)g(x)(ε+|x|p/(p-1))Ndx.
On the other hand, let x=ε(p-1)/py, we can deduce that
(4.22)∫ℝN1(ε+|x|p/(p-1))Ndx=ε-N/p∫ℝN1(1+|y|p/(p-1))Ndy=ε-N/p|U|Lp*(ℝN)p*.
Combining with g(0)=g+∣∞ and the equalities above, we have
(4.23)ε-N/p|g+|∞|U|Lp*(ℝN)p*-ε-N/p|(gχB(0,2ρ0))1/p*uε|p*p*=∫ℝN∖B(0,δ0)g(0)-ηp*(x)g(x)(ε+|x|p/(p-1))Ndx+∫B(0,δ0)g(0)-g(x)(ε+|x|p/(p-1))Ndx,
hence
(4.24)0≤ε-N/p|g+|∞|U|Lp*(ℝN)p*-ε-N/p|(gχB(0,2ρ0))1/p*uε|p*p*≤∫ℝN∖B(0,δ0)g(0)(ε+|x|p/(p-1))Ndx+∫B(0,δ0)o(|x|β)(ε+|x|p/(p-1))Ndx≤∫ℝN∖B(0,δ0)g(0)|x|Np/(p-1)dx+∫B(0,δ0)o(|x|β)|x|Np/(p-1)dx=NωN∫δ0∞rN-1g(0)rpN/(p-1)dr+∫0δ0o(rβ)rN-1rpN/(p-1)dr=(p-1)ωNδ0-N/(p-1)g(0)+o(1)δ0β-(N/(p-1))β-(N/(p-1))≤C1=Const.,
which leads to
(4.25)0≤1-|g+|∞-1|(gχB(0,2ρ0))1/p*uε|p*p*|U|Lp*(ℝN)-p*≤C1|g+|∞-1|U|Lp*(ℝN)-p*εN/p,
that is,
(4.26)1-C1|g+|∞-1|U|Lp*(ℝN)-p*εN/p≤|g+|∞-1|(gχB(0,2ρ0))1/p*uε|p*p*|U|Lp*(ℝN)-p*≤1.
Now, let ε be small enough such that C1|g+|∞-1|U|p*-p*εN/p<1, then from (4.26) we can deduce that
(4.27)1-C1|g+|∞-1|U|Lp*(ℝN)-p*εN/p≤(1-C1|g+|∞-1|U|Lp*(ℝN)-p*εN/p)p/p*≤|g+|∞-(N-p)/N|(gχB(0,2ρ0))1/p*uε|p*p|U|Lp*(ℝN)-p≤1,
which yields that
(4.28)|g+|∞(N-p)/N|U|Lp*(ℝN)p-C1|g+|∞-p/N|U|Lp*(ℝN)p-p*εN/p≤|(gχB(0,2ρ0))1/p*uε|p*p≤|g+|∞(N-p)/N|U|Lp*(ℝN)p,
equivalently, equality (4.18) is valid.
Combining (4.18) and (4.19), we obtain that
(4.29)Q(uε)=|∇U|Lp(ℝN)p+O(ε(N-p)/p)|g+|∞(N-p)/N|U|Lp*(ℝN)p+O(εN/p)=|g+|∞-(N-p)/N|∇U|Lp(ℝN)p+O(ε(N-p)/p)|U|Lp*(ℝN)p+O(εN/p).
Hence
(4.30)Q(uε)-|g+|∞-(N-p)/NS=|g+|∞-(N-p)/N[|∇U|Lp(ℝN)p+O(ε(N-p)/p)|U|Lp*(ℝN)p+O(εN/p)-|∇U|Lp(ℝN)p|U|Lp*(ℝN)p]=|g+|∞-(N-p)/N[|U|Lp*(ℝN)pO(ε(N-p)/p)-|∇U|Lp(ℝN)pO(εN/p)(|U|Lp*(ℝN)p+O(εN/p))|U|Lp*(ℝN)p]=O(ε(N-p)/p).
Using the fact that
(4.31)maxt≥0(tppa-tp*p*b)=1N(abp/p*)N/pforanya,b>0,
we can deduce that
(4.32)supt≥0I(tuε)=1N(Q(uε))N/p.
From (4.30), we conclude that supt≥0I(tuε)≤(1/N)|g+|∞-(N-p)/pSN/p+O(ε(N-p)/p).Step 2.
We claim that for anyλ>0 there exists a constant ελ>0 such that supt≥0Jλ(tuελ)<(1/N)|g+|∞-(N-p)/pSN/p.
Using the definitions ofJλ,uε and by (f2), (g3), we get
(4.33)Jλ(tuε)≤tpp|∇uε|pp,∀t≥0,∀λ>0.
Combining this with (4.19), let ε∈(0,1), then there exists t0∈(0,1) independent of ε such that
(4.34)sup0≤t≤t0Jλ(tuε)<1N|g+|∞-(N-p)/pSN/p,∀λ>0,∀ε∈(0,1).
Using the definitions of Jλ,uε, and by the results in  Step 1 and (f2), we have
(4.35)supt≥t0Jλ(tuε)=supt≥t0(I(tuε)-tqqλ∫f(x)|uε|qdx)≤1N|g+|∞-(N-p)/pSN/p+O(ε(N-p)/p)-t0qqβ0λ∫B(0,δ0)|uε|qdx.
Let 0<ε≤δ0p/(p-1), we have
(4.36)∫B(0,δ0)|uε|qdx=∫B(0,δ0)εq(N-p)/p2(ε+|x|p/(p-1))((N-p)/p)qdx≥∫B(0,δ0)εq(N-p)/p2(2δ0p/(p-1))((N-p)/p)qdx=C2(N,p,q,δ0)ε(q(N-p))/p2.
Combining (4.35) and (4.36), for all ε∈(0,δ0p/(p-1)), we get
(4.37)supt≥t0Jλ(tuε)≤1N|g+|∞-(N-p)/pSN/p+O(ε(N-p)/p)-t0qqβ0C2λεq(N-p)/p2.
Hence, for any λ>0, we can choose small positive constant ελ<min{1,δ0p/(p-1)} such that
(4.38)O(ελ(N-p)/p)-t0qqβ0C2λελq(N-p)/p2<0.
From (4.34), (4.37), (4.38), we can deduce that for any λ>0, there exists ελ>0 such that
(4.39)supt≥0Jλ(tuελ)<1N|g+|∞-(N-p)/pSN/p.Step 3.
Prove thatαλ-<(1/N)SN/p for all λ∈(0,Λ1).
By(f2),(g2), and the definition of uε, we have
(4.40)∫Ωf(x)|uε|qdx>0,∫Ωg(x)|uε|p*dx>0.
Combining this with Lemma 2.7(ii), from the definition of αλ- and the results in Step 2, for any λ∈(0,Λ1), we obtain that there exists tελ>0 such that tελuελ∈𝒩λ- and
(4.41)αλ-≤Jλ(tελuελ)≤supt≥0Jλ(tuελ)<1N|g+|∞-(N-p)/pSN/p.
This completes the proof.Now, we establish the existence of a local minimum ofJλ on 𝒩λ-.Theorem 4.4.
Ifλ∈(0,(q/p)Λ1), then Jλ satifies the (PS)αλ- condition. Moreover, Jλ has a minimizer Uλ in 𝒩λ- and satisfies that
(i)Jλ(Uλ)=αλ-;
(ii)Uλ is a positive solution of (Eλf,g) in C1,α(Ω) for some α∈(0,1),
whereΛ1 is as in (1.5).Proof.
Ifλ∈(0,(q/p)Λ1), then by Theorem 2.6(ii), Proposition 3.3(ii), and Lemma 4.3, there exists a (PS)αλ--sequence {un}⊂𝒩λ- in W for Jλ with αλ-∈(0,(1/N)|g+|∞-(N-p)/pSN/p). From Lemma 4.2, there exists a subsequence still denoted by {un} and nontrivial solution Uλ∈W of (Eλf,g) such that un⇀Uλ weakly in W. Now we prove that un→Uλ strongly in W and Jλ(Uλ)=αλ-. By (3.29), if u∈𝒩λ, then
(4.42)Jλ(u)=p*-pp*p∥u∥p-p*-qp*qλ∫Ωf|u|qdx.
First, we prove that Uλ∈𝒩λ-. On the contrary, if Uλ∈𝒩λ+, then by 𝒩λ- closed in W, we have ∥Uλ∥<lim infn→∞∥un∥. By Lemma 2.7, there exists a unique tλ- such that tλ-Uλ∈𝒩λ-. Since un∈𝒩λ-, Jλ(un)≥Jλ(tun) for all t≥0 and by (4.42), we have
(4.43)αλ-≤Jλ(tλ-Uλ)<limn→∞Jλ(tλ-un)≤limn→∞Jλ(un)=αλ-,
and this is contradiction.
In order to prove thatJλ(Uλ)=αλ-, it suffices to recall that un,Uλ∈𝒩λ- for all n, by (4.42), and applying Fatou's lemma to get
(4.44)αλ-≤Jλ(Uλ)=p*-pp*p∥Uλ∥p-p*-qp*qλ∫Ωf|Uλ|qdx≤liminfn→∞(p*-pp*p∥un∥p-p*-qp*qλ∫Ωf|un|qdx)≤liminfn→∞Jλ(un)=αλ-.
This implies that Jλ(Uλ)=αλ- and limn→∞∥un∥p=∥Uλ∥p. Let vn=un-Uλ, then Brézis and Lieb lemma [15] implies that
(4.45)∥vn∥p=∥un∥p-∥Uλ∥p+on(1).
Therefore, un→Uλ strongly in W.
SinceJλ(Uλ)=Jλ(|Uλ|) and |Uλ|∈𝒩λ-, by Lemma 2.3 we may assume that Uλ is a nontrivial nonnegative solution of (Eλf,g). Finally, by using the same arguments as in the proof of Theorem 3.4, for all λ∈(0,(q/p)Λ1), we have that Uλ is a positive solution of (Eλf,g) in C1,α(Ω) for some α∈(0,1).Now, we complete the proof of  Theorem1.5. By Theorems 3.4 and 4.4, if λ∈(0,(q/p)Λ1), then we obtain (Eλf,g) that has two positive solutions uλ and Uλ such that uλ∈𝒩λ+, Uλ∈𝒩λ-, and uλ,Uλ∈C1,α(Ω) for some α∈(0,1). Since 𝒩λ+∩𝒩λ-=∅, this implies that uλ and Uλ are distinct.



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*Source: 652109-2009-10-19.xml*