# Multiple Positive Solutions for a Quasilinear Elliptic System Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Authors: Tsing-San Hsu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

LetΩ∋0 be an-open bounded domain in ℝN(N≥3) and p*=(pN/(N-p)). We consider the following quasilinear elliptic system of two equations in W01,p(Ω)×W01,p(Ω): -Δpu=λf(x)|u|q-2u+(α/(α+β))h(x)|u|α-2u|v|β,-Δpv=μg(x)|v|q-2v+(β/(α+β))h(x)|u|α|v|β-2v, where λ,μ>0, Δp denotes the p-Laplacian operator, 1≤q<p<N,α,β>1 satisfy p<α+β≤p*, and f,g,h are continuous functions on Ω¯ which are somewhere positive but which may change sign on Ω. We establish the existence and multiplicity results of positive solutions to (the above mentioned quasilinear elliptic system equations) by variational methods.


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## Body

## 1. Introduction and Main Results

LetΩ∋0 be a smooth-bounded domain in ℝN with N≥3. In this paper, we study the existence and multiplicity of positive solutions for the following quasilinear elliptic system:(Sλf,μg,h)-Δpu=λf(x)|u|q-2u+αα+βh(x)|u|α-2u|v|βinΩ,-Δpv=μg(x)|v|q-2v+βα+βh(x)|u|α|v|β-2vinΩ,u=v=0on∂Ω,
where λ,μ>0,1≤q<p<N,Δpu=div(|∇u|p-2∇u) is the p-Laplacian, α>1,β>1 satisfy p<α+β≤p*,p*=(pN)/(N-p) denotes the critical Sobolev exponent, and the weight functions f,g,h are satisfying the following assumptions:  (A1)
f,g∈C(Ω¯), f+=max{f,0}≢0,g+=max{g,0}≢0 and |f|∞=|g|∞=1;(A2)
h∈C(Ω¯),h+=max{h,0}≢0 and |h|∞=1;(A3)
there exista0,b0,r0>0 and x0∈Ω such that B(x0,2r0)⊂Ω,f(x)≥a0 and g(x)≥b0 for all x∈B(x0,2r0), without loss of generality, we assume below that x0=0;(A4)
h(x)>0 for all x∈Ω,|h|∞=h(0) and there exists δ0>N/(p-1) such that(1.1)h(x)=h(0)+o(|x|δ0)asx⟶0.
System (Sλf,μg,h) is posed in the framework of the Sobolev space W=W01,p(Ω)×W01,p(Ω) with the standard norm(1.2)‖(u,v)‖=(∫Ω|∇u|pdx+∫Ω|∇v|pdx)1/p.
Moreover, a pair of functions (u,v)∈W is said to be a weak solution of System (Sλf,μg,h) if (1.3)∫Ω|∇u|p-2∇u∇φ1dx+∫Ω|∇v|p-2∇v∇φ2dx-λ∫Ωf(x)|u|q-2uφ1dx-μ∫Ωg(x)|v|q-2vφ2dx-αα+β∫Ωh(x)|u|α-2u|v|βφ1dx-βα+β∫Ωh(x)|u|α|v|β-2vφ2dx=0
for all (φ1,φ2)∈W. Thus, the corresponding energy functional of System (Sλf,μg,h) is defined by(1.4)Iλ,μ(u,v)=1p‖(u,v)‖p-1q∫Ω(λf(x)|u|q+μg(x)|v|q)dx-1α+β∫Ωh(x)|u|α|v|βdx.Semilinear and quasilinear scalar elliptic equations with concave-convex nonlinearities are widely studied: we refer the reader to Ambrosetti et al. [1], de Figueiredo et al. [2], Azorero and Peral [3], Azorero et al. [4], EL Hamidi [5], Hirano et al. [6], Hsu [7], and Wu [8], and so forth. For the nonlinear elliptic systems, we refer to Adriouch and EL Hamidi [9], Ahammou [10], Alves et al. [11], Bozhkov and Mitidieri [12], Clément et al. [13], de Figueiredo and Felmer [14], EL Hamidi [15], Hsu and Lin [16, 17], Squassina [18], Vélin [19], and Wu [20], and so forth.Recently, in [8], the author has considered a semilinear scalar elliptic equation involving concave-convex nonlinearities and sign-changing weight functions and showed multiplicity results with respect to the parameter via the extraction of Palais-Smale sequences in the Nehair manifold, and for the definition of Nehair manifold we refer the reader to see Nehari [21] or Willem [22].More recently, in [20] the author extends the method of [8] to system (Sλf,μg,h) in the semilinear case p=2 with the subcritical case 2<α+β<2* and the sign-changing weight functions f,g. In [16] the author also extends the method of [8] to system (Sλf,μg,h) in the quasilinear case 1<p<N with critical case α+β=p* and the constant weight functions f≡g≡h≡1. In the present paper, motivated by [16, 20] we extend and improve the papers by Hsu [16] and Wu [20]. First, we deal with more general weight functions f,g,h which may be changing sign, and second, we also deal with quasilinear elliptic systems involving subcritical or critical Sobolev exponents.LetS be the best Sobolev constant for the embedding of W01,p(Ω) in Lα+β(Ω) defined by(1.5)S=inf⁡u∈W01,p(Ω)∖{0}∫Ω|∇u|pdx(∫Ω|u|α+βdx)p/(α+β),
and set(1.6)Λ1=(p-qα+β-q)p/(α+β-p)(α+β-qα+β-p|Ω|(α+β-q)/(α+β))-p/(p-q)Sp(α+β-q)/(p-q)(α+β-p)>0,
where |Ω| is the Lebesgue measure of Ω. Our main results are as follows.Theorem 1.1 (existence of one positive solution).
Assume that(A1)-(A2) hold. If 1≤q<p<N,p<α+β≤p*, and λ,μ>0 satisfy 0<λp/(p-q)+μp/(p-q)<Λ1, then system (Sλf,μg,h) has at least one positive solution in W01,p(Ω)×W01,p(Ω).Theorem 1.2 (second positive solution in the subcritical case).
Assume that(A1)-(A2) hold. If 1≤q<p<N,p<α+β<p*, and λ,μ>0 satisfy 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then System (Sλf,μg,h) has at least two positive solutions in W01,p(Ω)×W01,p(Ω).Theorem 1.3 (second positive solution in the critical case).
Assume that(A1)–(A4) hold. If 1≤q<p<N,α+β=p*, and λ,μ>0 satisfy 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then System (Sλf,μg,h) has at least two positive solutions in W01,p(Ω)×W01,p(Ω).This paper is organized as follows. In Section2, we give some notations and preliminaries. The proofs of Theorems 1.1 and 1.2 are in Section 3. In Section 4, we manage to give the proof of Theorem 1.3. Throughout this paper, (A1) and (A2) will be assumed.


## 2. Notations and Preliminaries

In this section, we give some notations and necessary preliminary results.Notations.  We make use of the following notation. Ls(Ω),1≤s<∞, denote Lebesgue spaces; the norm Ls is denoted by |·|s for 1≤s≤∞;W=[W01,p(Ω)]2, endowed with norm ∥z∥p=∥(u,v)∥p=|∇u|pp+|∇v|pp;The dual space of a Banach spaceW will be denoted by W-1;tz=t(u,v)=(tu,tv) for all z=(u,v)∈W and t∈ℝ;|z|=(|u|,|v|) for all z=(u,v)∈W;z=(u,v) is said to be nonnegative in Ω if u≥0 and v≥0 in Ω;z=(u,v) is said to be positive in Ω if u>0 and v>0 in Ω;|Ω| is the Lebesgue measure of Ω;B(x0,r)={x∈ℝN∣|x-x0|<r} is the ball in ℝN;O(ɛt) denotes |O(ɛt)|/ɛt≤C as ɛ→0 for t≥0;on(1) denotes on(1)→0 as n→∞;C, Ci will denote various positive constants, the exact values of which are not important;p*=pN/(N-p)(1<p<N) is the critical Sobolev exponent;LetJ,Kλ,μ:W→ℝ be the functionals defined by(2.1)J(z)=∫Ωh(x)|u|α|v|βdx,Kλ,μ(z)=∫Ω(λf(x)|u|q+μg(x)|v|q)dxfor allz=(u,v)∈W.As the energy functionalIλ,μ is not bounded below on W, it is useful to consider the functional on the Nehari manifold(2.2)Nλ,μ={z∈W∖{0}∣〈Iλ,μ′(z),z〉=0}.
Thus, z=(u,v)∈𝒩λ,μ if and only if(2.3)〈Iλ,μ′(z),z〉=‖z‖p-Kλ,μ(z)-J(z)=0.Note that𝒩λ,μ contains every nonzero solution of System (Sλf,μg,h). Moreover, we have the following resultsLemma 2.1.
The energy functionalIλ,μ is coercive and bounded on 𝒩λ,μ.Proof.
Ifz=(u,v)∈𝒩λ,μ, then by (2.3), the Hölder inequality, and the Sobolev embedding theorem,
(2.4)Iλ,μ(z)=α+β-pp(α+β)‖z‖p-α+β-qq(α+β)Kλ,μ(z)(2.5)≥α+β-pp(α+β)‖z‖p-α+β-qq(α+β)S-q/p|Ω|(α+β-q)/(α+β)(λp/(p-q)+μp/(p-q))(p-q)/p‖z‖q.
Thus, Iλ,μ is coercive and bounded on 𝒩λ,μ.Define(2.6)Φλ,μ(z)=〈Iλ,μ′(z),z〉.
Then for u∈𝒩λ,μ,(2.7)〈Φλ,μ′(z),z〉=p‖z‖p-qKλ,μ(z)-(α+β)J(z)=(p-q)‖z‖p-(α+β-q)J(z)(2.8)=(α+β-q)Kλ,μ(z)-(α+β-p)‖z‖p.
Similar to the method used in Tarantello [23], we split 𝒩λ,μ into three parts: (2.9)Nλ,μ+={z∈Nλ,μ:〈Φλ,μ′(z),z〉>0},Nλ,μ0={z∈Nλ,μ:〈Φλ,μ′(z),z〉=0},Nλ,μ-={z∈Nλ,μ:〈Φλ,μ′(z),z〉<0}.
Then, we have the following results.Lemma 2.2.
Assume thatz0 is a local minimizer for Iλ,μ on 𝒩λ,μ and z0∉𝒩λ,μ0. Then Iλ,μ′(z0)=0 in W-1.Proof.
Our proof is almost the same as that in [24, Theorem 2.3].Lemma 2.3.
One has the following:(i)
ifz∈𝒩λ,μ+, then Kλ,μ(z)>0;(ii)
ifz∈𝒩λ,μ0, then Kλ,μ(z)>0 and J(z)>0;(iii)
ifz∈𝒩λ,μ-, then J(z)>0.Proof.
The proof is immediate from (2.7) and (2.8).Moreover, we have the following result.Lemma 2.4.
If0<λp/(p-q)+μp/(p-q)<Λ1, then 𝒩λ,μ0=∅ where Λ1 is the same as in (1.6).Proof.
We argue by contradiction. Assume that there existλ,μ>0 with
(2.10)0<λp/(p-q)+μp/(p-q)<Λ1
such that 𝒩λ,μ0≠∅. Then by (2.7) and (2.8), for u∈𝒩λ,μ0, on has the following:
(2.11)‖z‖p=α+β-qp-qJ(z),‖z‖p=α+β-qα+β-pKλ,μ(z).
By |f|∞=|g|∞=|h|∞=1, the Hölder inequality, and the Sobolev embedding theorem, we have
(2.12)‖z‖≥(p-qα+β-qS(α+β)/p)1/(α+β-p),‖z‖≤(α+β-qα+β-pS-q/p|Ω|(α+β-q)/(α+β))1/(p-q)(λp/(p-q)+μp(p-q))1/p.
This implies
(2.13)λp/(p-q)+μp/(p-q)λp/(p-q)+μp/(p-q)≥(p-qα+β-q)p/(α+β-p)(α+β-qα+β-p|Ω|(α+β-q)/(α+β))-p/(p-q)×S(α+β)/(α+β-p)+q/(p-q)=Λ1,
which contradicts 0<λp/(p-q)+μp/(p-q)<Λ1.By Lemma2.4, we write 𝒩λ,μ=𝒩λ,μ+∪𝒩λ,μ- and define(2.14)θλ,μ=inf⁡z∈Nλ,μIλ,μ(z);θλ,μ+=inf⁡z∈Nλ,μ+Iλ,μ(z);θλ,μ-=inf⁡z∈Nλ,μ-Iλ,μ(z).
Then we get the following result.Theorem 2.5.
(i) If0<λp/(p-q)+μp/(p-q)<Λ1, then one has θλ,μ≤θλ,μ+<0;
(ii) if0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then θλ,μ->d0 for some positive constant d0 depending on λ,μ,p,q,N,S,|Ω|,|f|∞,|g|∞ and |h|∞.Proof.
(i) Letz=(u,v)∈𝒩λ,μ+. By (2.7)(2.15)p-qα+β-q‖z‖p>J(z)
and so
(2.16)Iλ,μ(z)=(1p-1q)‖z‖p+(1q-1α+β)J(z)<[(1p-1q)+(1q-1α+β)p-qα+β-q]‖z‖p=-(p-q)(α+β-p)pq(α+β)‖z‖p<0.
Therefore, from the definitions of θλ,μ, θλ,μ+, we can deduce that θλ,μ≤θλ,μ+<0.
(ii) Letz∈𝒩λ,μ-. By (2.7)(2.17)p-qα+β-q‖z‖p<J(z).
Moreover, by |h|∞=1, the Hölder inequality, and the Sobolev embedding theorem,
(2.18)J(z)≤S-(α+β)/p‖z‖α+β.
This implies
(2.19)‖z‖>(p-qα+β-q)1/(α+β-p)S(α+β)/p(α+β-p)∀z∈Nλ,μ-.
By (2.5), one has the following:
(2.20)Iλ,μ(z)≥‖z‖q[α+β-pp(α+β)‖z‖p-q-α+β-qq(α+β)S-q/p|Ω|(α+β-q)/(α+β)(λp/(p-q)+μp/(p-q))(p-q)/p]>(p-qα+β-q)q/(α+β-p)Sq(α+β)/p(α+β-p)×[α+β-pp(α+β)S(p-q)(α+β)/p(α+β-p)(p-qα+β-q)p-q/(α+β-p)-α+β-qq(α+β)S-q/p|Ω|(α+β-q)/(α+β)(λp/(p-q)+μp/(p-q))(p-q)/p].
Thus, if 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then
(2.21)Iλ,μ(z)>d0∀z∈Nλ,μ-,
for some positive constant d0=d0(λ,μ,p,q,N,S,|Ω|,|f|∞,|g|∞,|h|∞). This completes the proof.For eachz∈W with J(z)>0, we write(2.22)tmax⁡=((p-q)‖z‖p(α+β-q)J(z))1/(α+β-p)>0.
Then the following lemma holds.Lemma 2.6.
Letλp/(p-q)+μp/(p-q)∈(0,Λ1). For each z∈W with J(z)>0, one has the following:  (i)
ifKλ,μ(z)≤0, then there exists a unique t->tmax such that t-z∈𝒩λ,μ- and(2.23)Iλ,μ(t-z)=sup⁡t≥0Iλ,μ(tz);(ii)
ifKλ,μ(z)>0, then there exist unique 0<t+<tmax<t- such that t+z∈𝒩λ,μ+, t-z∈𝒩λ,μ- and(2.24)Iλ,μ(t+z)=inf⁡0≤t≤tmax⁡Iλ,μ(tz);Iλ,μ(t-z)=sup⁡t≥0Iλ,μ(tz).Proof.
The proof is almost the same as that in [25,Lemma 2.6] and is omitted here.For eachz∈W with Kλ,μ(z)>0, we write(2.25)t¯max⁡=((α+β-q)Kλ,μ(z)(α+β-p)‖z‖p)1/(p-q)>0.
Then we have the following lemma.Lemma 2.7.
Letλp/(p-q)+μp/(p-q)∈(0,Λ1). For each z∈W with Kλ,μ(z)>0, one has the following:  (i)
ifJ(z)≤0, then there exists a unique 0<t+<t¯max such that t+z∈𝒩λ,μ+ and(2.26)Iλ,μ(t+z)=inf⁡t≥0Iλ,μ(tz);(ii)
ifJ(z)>0, then there exist unique 0<t+<t¯max<t- such that t+z∈𝒩λ,μ+, t-z∈𝒩λ,μ- and(2.27)Iλ,μ(t+z)=inf⁡0≤t≤tmax⁡Iλ,μ(tz);Iλ,μ(t-z)=sup⁡t≥0Iλ,μ(tz).Proof.
The proof is almost the same as that in [25, Lemma 2.7] and is omitted here.


## 3. Proofs of Theorems1.1 and 1.2

First, we give the following definitions about(PS)c-sequence.Definition 3.1.
Letc∈ℝ, W be a Banach space and I∈C1(W,ℝ).      (i)
{zn} is a (PS)c-sequence in W for I if I(zn)=c+on(1) and I′(zn)=on(1) strongly in W-1 as n→∞.(ii)
We say thatI satisfies the (PS)c condition if any (PS)c-sequence {zn} in W for I has a convergent subsequence.Now, we use the idea of Tarantello [23] to get the following results.Proposition 3.2.
Let1≤q<p<N and p<α+β≤p*, we have  (i)
if0<λp/(p-q)+μp/(p-q)<Λ1, then there exists a (PS)θλ,μ-sequence {zn}⊂𝒩λ,μ in W for Iλ,μ;(ii)
if0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then there exists a (PS)θλ,μ--sequence {zn}⊂𝒩λ,μ in W for Iλ,μ,where Λ1 is the positive constant given in (1.6).Proof.
The proof is almost the same as that in [8, Proposition 9].Now, we establish the existence of a local minimum forIλ,μ on 𝒩λ,μ+.Theorem 3.3.
LetΛ1 be the same positive constant as in (1.6). If 1≤q<p<N,p<α+β≤p*, and 0<λp/(p-q)+μp/(p-q)<Λ1, then Iλ,μ has a minimizer zλ,μ1 in 𝒩λ,μ+ and it satisfies  the following:(i)
Iλ,μ(zλ,μ1)=θλ,μ=θλ,μ+<0;(ii)
zλ,μ1 is a positive solution of System (Sλf,μg,h);(iii)
Iλ,μ(zλ,μ1)→0  as  λ→0+, μ→0+.Proof.
By Proposition3.2 (i), there exists a minimizing sequence {zn} for Iλ,μ on 𝒩λ,μ such that
(3.1)Iλ,μ(zn)=θλ,μ+on(1),Iλ,μ′(zn)=on(1)inW-1.
Since Iλ,μ is coercive on 𝒩λ,μ (see Lemma 2.1), we get that {zn} is bounded in W. Then there exist a subsequence {zn=(un,vn)} and zλ,μ1=(uλ,μ1,vλ,μ1)∈W such that
(3.2)un⇀uλ,μ1,vn⇀vλ,μ1weaklyinW01,p(Ω),un⟶uλ,μ1,vn⟶vλ,μ1almosteverywhereinΩ,un⟶uλ,μ1,vn⟶vλ,μ1stronglyinLs(Ω)∀1≤s<p*.
This implies
(3.3)Kλ,μ(zn)=Kλ,μ(zλ,μ1)+on(1)asn⟶∞.
First, we claim that zλ,μ1 is a nontrivial solution of System (Sλf,μg,h). By (3.1) and (3.2), it is easy to verify that zλ,μ1 is a weak solution of System (Sλf,μg,h). From zn∈𝒩λ,μ and (2.4), we deduce that
(3.4)Kλ,μ(zn)=q(α+β-p)p(α+β-q)‖zn‖p-q(α+β)α+β-qIλ,μ(zn).
Let n→∞ in (3.4), by (3.1), (3.3), and θλ,μ<0, we get
(3.5)Kλ,μ(zλ,μ1)≥-q(α+β)α+β-qθλ,μ>0.
Thus, zλ,μ1∈𝒩λ,μ is a nontrivial solution of System (Sλf,μg,h). Now we prove that zn→zλ,μ1 strongly in W and Iλ,μ(zλ,μ1)=θλ,μ. By (3.4), if z∈𝒩λ,μ, then
(3.6)Iλ,μ(z)=α+β-pp(α+β)‖z‖p-α+β-qq(α+β)Kλ,μ(z).
In order to prove that Iλ,μ(zλ,μ1)=θλ,μ, it suffices to recall that zλ,μ1∈𝒩λ,μ, by (3.6), and apply Fatou’s lemma to get
(3.7)θλ,μ≤Iλ,μ(zλ,μ1)=α+β-pp(α+β)‖zλ,μ1‖p-α+β-qq(α+β)Kλ,μ(zλ,μ1)≤lim⁡inf⁡n→∞(α+β-pp(α+β)‖zn‖p-α+β-qq(α+β)Kλ,μ(zn))≤lim⁡inf⁡n→∞Iλ,μ(zn)=θλ,μ.
This implies that Iλ,μ(zλ,μ1)=θλ,μ and limn→∞∥zn∥p=∥zλ,μ1∥p. Let z̃n=zn-zλ,μ1, then by Brézis and Lieb lemma [26] implies
(3.8)‖z̃n‖p=‖zn‖p-‖zλ,μ1‖p.
Therefore, zn→zλ,μ1 strongly in W. Moreover, we have zλ,μ1∈𝒩λ,μ+. Thus θλ,μ=θλ,μ+. On the contrary, if zλ,μ1∈𝒩λ,μ-, then by (2.17), (3.5), we have that J(zλ,μ1)>0 and Kλ,μ(zλ,μ1)>0. Thus, from Lemma 2.6 (ii), there exist unique t1+ and t1- such that t1+zλ,μ1∈𝒩λ,μ+ and t1-zλ,μ1∈𝒩λ,μ-. In particular, we have t1+<t1-=1. Since
(3.9)ddtIλ,μ(t1+zλ,μ1)=0,d2dt2Iλ,μ(t1+zλ,μ1)>0,
there exists t1+<t¯≤t1- such that Iλ,μ(t1+zλ,μ1)<Iλ,μ(t¯zλ,μ1). By Lemma 2.6,
(3.10)Iλ,μ(t1+zλ,μ1)<Iλ,μ(t¯zλ,μ1)≤Iλ,μ(t1-zλ,μ1)=Iλ,μ(zλ,μ1)=θλ,μ,
which is a contradiction. Since Iλ,μ(zλ,μ1)=Iλ,μ(|zλ,μ1|) and |zλ,μ1|∈𝒩λ,μ+, by Lemma 2.2 we may assume that zλ,μ1 is a nontrivial nonnegative solution of System (Sλf,μg,h).
In particularuλ,μ1≢0,vλ,μ1≢0. Indeed, without loss of generality, we may assume that vλ,μ1≡0. Then as uλ,μ1 is a nontrivial nonnegative solution of(3.11)-Δpu=λf(x)|u|q-2uinΩ,u=0on∂Ω.
By the standard regularity theory, we have uλ,μ1>0 in Ω and
(3.12)‖(uλ,μ1,0)‖p=Kλ,μ(uλ,μ1,0)>0.
Moreover, by conditions (A1),(A2) and uλ,μ1>0 in Ω, we may choose w∈W01,p(Ω)∖{0} such that
(3.13)‖(0,w)‖p=Kλ,μ(0,w)>0,J(uλ,μ1,w)≥0.
Now
(3.14)Kλ,μ(uλ,μ1,w)=Kλ,μ(uλ,μ1,0)+Kλ,μ(0,w)>0,
and so by Lemma 2.7 there is unique 0<t+<t¯max such that (t+uλ,μ1,t+w)∈𝒩λ,μ+. Moreover,
(3.15)t¯max⁡=((α+β-q)Kλ,μ(uλ,μ1,w)(α+β-p)‖(uλ,μ1,w)‖p)=(α+β-qα+β-p)>1,Iλ,μ(t+uλ,μ1,t+w)=inf⁡0≤t≤t¯max⁡Iλ,μ(tuλ,μ1,tw).
This implies
(3.16)θλ,μ+≤Iλ,μ(t+uλ,μ1,t+w)≤Iλ,μ(uλ,μ1,w)<Iλ,μ(uλ,μ1,0)=θλ,μ+
which is a contradiction.
By a standard bootstrap argument, it is proved that a weak solution for System (Sλf,μg,h) is in the space C2(Ω¯)×C2(Ω¯), and it is really a classical solution. Finally, by the Harnack inequality [27] we deduce that zλ,μ1 is a positive solution of System (Sλf,μg,h). Moreover, by Theorem 2.5 (i) and (2.5) we have(3.17)0>θλ,μ+=Iλ,μ(zλ,μ1)>-α+β-qq(α+β)S-q/p|Ω|(α+β-q)/(α+β)(λp/(p-q)+μp/(p-q))(p-q)/p‖zλ,μ1‖q.
This implies that Iλ,μ(zλ,μ1)→0 as λ→0+, μ→0+.Next, we establish the existence of a local minimum forIλ,μ on 𝒩λ,μ- in the subcritical case p<α+β<p*. This implies that there exists the second positive solution in the subcritical case p<α+β<p*.Theorem 3.4.
LetΛ1 be the same positive constant as in (1.6). If 1≤q<p<N,p<α+β<p*, and 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then Iλ,μ has a minimizer zλ,μ2 in 𝒩λ,μ- and it satisfies  the following:(i)
Iλ,μ(zλ,μ2)=θλ,μ-;(ii)
zλ,μ2 is a positive solution of System (Sλf,μg,h);Proof.
Let{zn} be a minimizing sequence for Iλ,μ on 𝒩λ,μ-. Then by Iλ,μ coercive on 𝒩λ,μ and the compact imbedding theorem, there exist a subsequence {zn=(un,vn)} and zλ,μ2=(uλ,μ2,vλ,μ2)∈W such that
(3.18)un⇀uλ,μ2,vn⇀vλ,μ2weaklyinW01,p(Ω),un⟶uλ,μ2,vn⟶vλ,μ2stronglyinLq(Ω),Lα+β(Ω).
This implies
(3.19)Kλ,μ(zn)=Kλ,μ(zλ,μ2)+on(1),J(zn)=J(zλ,μ2)+on(1).
By (2.17) and (2.19) there exists a positive number C¯ such that
(3.20)J(zn)>C¯.
This implies
(3.21)J(zλ,μ2)≥C¯.
Now, we prove that zn→zλ,μ2 strongly in W. Suppose otherwise, then ∥zλ,μ2∥<liminfn→∞∥zn∥. By Lemma 2.6, there exists a unique t2- such that t2-zλ,μ2∈𝒩λ,μ-. Since zn∈𝒩λ,μ-,Iλ,μ(zn)≥Iλ,μ(tzn) for all t≥0, we have
(3.22)θλ,μ-≤Iλ,μ(t-zλ,μ2)<lim⁡n→∞Iλ,μ(t-zn)≤lim⁡n→∞Iλ,μ(zn)=θλ,μ-,
and this is a contradiction. Hence
(3.23)zn⟶zλ,μ2stronglyinW.
This implies
(3.24)Iλ,μ(zλ,μ2)=lim⁡n→∞Iλ,μ(zn)=θλ,μ-.
Since Iλ,μ(zλ,μ2)=Iλ,μ(|zλ,μ2|) and |zλ,μ2|∈𝒩λ,μ-, by Lemma 2.2 and (3.21) we may assume zλ,μ2 is a nontrivial nonnegative solution of System (Sλf,μg,h). Finally, by using the same arguments as in the proof of Theorem 3.3, for all 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, we have that zλ,μ2 is a positive solution of System (Sλf,μg,h).Now, we complete the proofs of  Theorems1.1 and 1.2: Theorem 1.1 follows from Theorem 3.3. By Theorems 3.3 and 3.4, we obtain that for all 1≤q<p<N,p<α+β<p*, λ,μ>0, and 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1. System (Sλf,μg,h) has two positive solutions zλ,μ1, zλ,μ2 with zλ,μ1∈𝒩λ,μ+, zλ,μ2∈𝒩λ,μ-. Since 𝒩λ,μ+∩𝒩λ,μ-=∅, this implies that zλ,μ1 and zλ,μ2 are distinct. This completes the proofs of Theorems 1.1 and 1.2.


## 4. Proof of Theorem1.3

For the existence of a second positive solution of System (Sλf,μg,h) in the critical case α+β=p*, we will however need here a stronger restriction on h(x), namely, h(x)>0 in Ω but f(x) and g(x) may be also allowed to change sign in Ω. Now, we will establish the existence of a local minimum for Iλ,μ on 𝒩λ,μ- in the critical case α+β=p* to obtain a second positive solution of System (Sλf,μg,h).Lemma 4.1.
If{zn}⊂W is a (PS)c-sequence for Iλ,μ, then {zn} is bounded in W.Proof.
Letzn=(un,vn). We argue by contradiction. Assume that ∥zn∥→∞. Let
(4.1)ẑn=(ûn,v̂n)=zn‖zn‖=(un‖zn‖,vn‖zn‖).
We may assume that ẑn⇀ẑ=(û,v̂) in W. This implies that ûn→û,v̂n→v̂ strongly in Ls(Ω) for all 1≤s<p* and
(4.2)Kλ,μ(ẑn)=Kλ,μ(ẑ)+on(1).
Since {zn} is a (PS)c-sequence for Iλ,μ and ∥zn∥→∞, there hold
(4.3)1p‖ẑn‖p-‖zn‖q-pqKλ,μ(ẑn)-‖zn‖α+β-pα+βJ(ẑn)=on(1),(4.4)‖ẑn‖p-‖zn‖q-pKλ,μ(ẑn)-‖zn‖α+β-pJ(ẑn)=on(1).
From (4.2)–(4.4), we can deduce that
(4.5)‖ẑn‖p=p(α+β-q)q(α+β-p)‖zn‖q-pKλ,μ(ẑn)+on(1).
Since 1≤q<p and ∥zn∥→∞, (4.5) implies
(4.6)‖ẑn‖p⟶0,asn⟶∞,
which is contrary to the fact ∥ẑn∥=1.Denote(4.7)Sα,β=inf⁡u,v∈W01,p(Ω)∖{0}∫Ω(|∇u|p+|∇v|p)dx(∫Ω|u|α|v|βdx)p/(α+β).
Modifying the proof of Alves et al. [11, Theorem 5], we can easily deduce that(4.8)Sα,β=((αβ)β/(α+β)+(βα)α/(α+β))S,
where S is the best constant defined by(4.9)S=inf⁡u∈W01,p(Ω)∖{0}∫Ω|∇u|pdx(∫Ω|u|α+βdx)p/(α+β).
When α+β=p*, it is well known that S is achieved if and only if Ω=ℝN by the function(4.10)Uɛ(x)=CN(ɛ1/pɛ+|x|p/(p-1))(N-p)/p,ɛ>0,
where CN=[((N-p)/(p-1))p-1N](N-p)/p2.Lemma 4.2.
Letc∞=(1/N)Sα,βN/p|h|∞-(N-p)/p. If {zn}⊂W is a (PS)c-sequence for Iλ,μ with c∈(0,c∞), then there exists a subsequence of {zn} converging weakly to a nontrivial solution of System (Sλf,μg,h).Proof.
Let{zn}⊂W be a (PS)c-sequence for Iλ,μ with c∈(0,c∞). Write zn=(un,vn). We know from Lemma 4.1 that {zn} is bounded in W, and then zn⇀z=(u,v) up to a subsequence, when z is a critical point of Iλ,μ. Furthermore, we may assume un⇀u, vn⇀v in W01,p(Ω) and un→u, vn→v in Ls(Ω) for all 1≤s<p* and un→u, vn→v a.e. on Ω. Hence we have that Iλ,μ′(z)=0 and
(4.11)Kλ,μ(zn)=Kλ,μ(z)+on(1).
Next we verify that u≢0 or v≢0. Arguing by contradiction, we assume u≡0 and v≡0. Set
(4.12)l=lim⁡n→∞J(zn).
Since Iλ,μ′(zn)=on(1) and {zn} is bounded in W, then by (4.11), we can deduce that
(4.13)0=〈lim⁡n→∞Iλ,μ′(zn),zn〉=lim⁡n→∞(‖zn‖p-J(zn))=lim⁡n→∞‖zn‖p-l,
that is,
(4.14)lim⁡n→∞‖zn‖p=l.
If l=0, then we get c=limn→∞Iλ,μ(zn)=0, which contradicts with c>0. Thus we conclude that l>0. By (4.7) and (4.12), we obtain
(4.15)Sα,βlp/p*=Sα,βlim⁡n→∞(∫Ωh(x)|un|α|vn|βdx)p/p*≤Sα,βlim⁡n→∞(∫Ω|h|∞|un|α|vn|βdx)p/p*≤|h|∞p/p*lim⁡n→∞‖zn‖p=|h|∞p/p*l,
which implies that
(4.16)l≥Sα,βN/p|h|∞-(N-p)/p.
Hence, from (4.11)–(4.16) we get
(4.17)c=lim⁡n→∞Iλ,μ(zn)=lim⁡n→∞(1p‖zn‖p-1qKλ,μ(zn)-1α+βJ(zn))=(1p-1α+β)l≥1NSα,βN/p|h|∞-(N-p)/p=c∞.
This is a contradiction to c<c∞. Therefore z is a nontrivial solution of System (Sλf,μg,h).Lemma 4.3.
Assume that(A1)–(A4) hold. Then for any λ,μ>0, there exist a nonnegative function zλ,μ∈W∖{0} such that
(4.18)sup⁡t≥0Iλ,μ(tzλ,μ)<c∞,
where c∞ is the constant given in Lemma 4.2.
In particular,θλ,μ-<c∞ for all 0<λp/(p-q)+μp/(p-q)<Λ1, where Λ1 is as in (1.6).Proof.
From(A4), we know that there exists 0<ρ0≤r0 such that, for all x∈B(0,2ρ0),
(4.19)h(x)=h(0)+o(|x|δ0)forsomeδ0>Np-1.
Now, we consider the functional Q:W→ℝ defined by
(4.20)Q(z)=1p‖z‖p-1α+βJ(z)∀z=(u,v)∈W
and define a cut-off function η(x)∈C0∞(Ω) such that η(x)=1 for |x|<ρ0,η(x)=0 for |x|>2ρ0,0≤η≤1, and |∇η|≤C. For ɛ>0, let
(4.21)uɛ(x)=ɛ(N-p)/p2η(x)(ɛ+|x|p/(p-1))(N-p)/p.
From Hsu [7, Lemma 4.3], we have(4.22)|∇uɛ|pp=∫RN|∇U|pdx+O(ɛ(N-p)/p)(4.23)∫Ω|∇uɛ|pdx(∫Ωh(x)|uɛ|p*)p/p*=S+O(ɛ(N-p)/p),
where U(x)=(1+|x|p/(p-1))-(N-p)/p.
Setu0=αpuɛ,v0=βpuɛ, and z0=(u0,v0)∈W. Then, from (4.8), (4.23), and |h|∞=1, we conclude that(4.24)sup⁡t≥0Q(tz0)≤1N((α+β)∫Ω|∇uɛ|pdx(αα/pββ/p∫Ωh(x)|uɛ|p*)p/p*)N/p≤1N((αβ)β/(α+β)+(βα)α/(α+β))N/p(S+O(ɛ(N-p)/p))N/p=1N((αβ)α/(α+β)+(βα)α/(α+β))N/p(SN/p+O(ɛ(N-p)/p))=1NSα,βN/p+O(ɛ(N-p)/p),
where the following fact has been used:
(4.25)sup⁡t≥0(tppA-tα+βα+βB)=1NA(AB)(N-p)/p=1N(ABp/p*)N/p,A,B>0.
Using the definitions of Iλ,μ, z0 and by (A3) and (A4), we get
(4.26)Iλ,μ(tz0)≤tpp‖z0‖p=α+βptp|∇uɛ|pp∀t≥0,λ,μ>0.
Combining this with (4.22), let ɛ∈(0,1), then there exists t0∈(0,1) independent of ɛ such that
(4.27)sup⁡0≤t≤t0Iλ,μ(tz0)<c∞,∀λ,μ>0,∀ɛ∈(0,1).
Using the definitions of Iλ,μ,z0 and by α,β>1, (4.24) and (A3), we have
(4.28)sup⁡t≥t0Iλ,μ(tz0)=sup⁡t≥t0(Q(tz0)-tqqKλ,μ(tz0))≤1NSα,βN/p+O(ɛ(N-p)/p)-t0qq(a0αq/pλ+b0βq/pμ)∫B(0,ρ0)|uɛ|qdx≤1NSα,βN/p+O(ɛ(N-p)/p)-t0qqγ0(λ+μ)∫B(0,ρ0)|uɛ|qdx,
where γ0=min{a0αq/p,b0βq/p}.
Let0<ɛ≤ρ0p/(p-1); we have(4.29)∫B(0,ρ0)|uɛ|qdx=∫B(0,ρ0)ɛq(N-p)/p2(ɛ+|x|p/(p-1))((N-p)/p)qdx≥∫B(0,ρ0)ɛq(N-p)/p2(2ρ0p/(p-1))((N-p)/p)qdx=C1(N,p,q,ρ0)ɛq(N-p)/p2.
Combining with (4.28) and (4.29), for all ɛ∈(0,ρ0p/(p-1)), we get
(4.30)sup⁡t≥t0Iλ,μ(tz0)≤1NSα,βN/p+O(ɛ(N-p)/p)-t0qqC1γ0(λ+μ)ɛq(N-p)/p2.
Hence, for any λ,μ>0, we can choose small positive constant ɛλ,μ<min{1,ρ0p/(p-1)} such that
(4.31)O(ɛλ,μ(N-p)/p)-t0qqC1γ0(λ+μ)ɛλ,μq(N-p)/p2<0.
Now, we fix ɛλ,μ and let zλ,μ=(αpuɛλ,μ,βpuɛλ,μ). From (4.27), (4.30), (4.31), we can deduce that, for any λ,μ>0, there exists a nonnegative function zλ,μ∈W∖{0} such that
(4.32)sup⁡t≥0Iλ,μ(tzλ,μ)<c∞.
Finally, we prove that θλ,μ-<c∞ for all 0<λp/(p-q)+μp/(p-q)<Λ1. Recall that zλ,μ=(uλ,μ,vλ,μ)=(αpuɛλ,μ,βpuɛλ,μ). By (A3), (A4), and the definition of uɛλ,μ, we have
(4.33)J(zλ,μ)>0,Kλ,μ(zλ,μ)>0.
Combining this with Lemma 2.6 (ii), from the definition of θλ,μ- and (4.32), for all 0<λp/(p-q)+μp/(p-q)<Λ1, we obtain that there exists tλ,μ>0 such that tλ,μzλ,μ∈𝒩λ,μ- and
(4.34)θλ,μ-≤Iλ,μ(tλ,μzλ,μ)≤sup⁡t≥0Iλ,μ(tzλ,μ)<c∞.
This completes the proof.Theorem 4.4.
Assume that(A1)–(A4) hold. If 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then Iλ,μ satisfies the (PS)θλ,μ-condition. Moreover, Iλ,μ has a minimizer zλ,μ2 in 𝒩λ,μ- and satisfies  the following:(i)
Iλ,μ(zλ,μ2)=θλ,μ-;(ii)
zλ,μ2 is a positive solution of System (Sλf,μg,h), where Λ1 is the same as in (1.6).Proof.
If0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, then by Theorem 2.5(ii), Proposition 3.2(ii), and Lemma 4.3, there exists a (PS)θλ,μ--sequence {zn}⊂𝒩λ,μ- in W for Iλ,μ with θλ,μ-∈(0,c∞). From Lemma 4.2, there exist a subsequence still denoted by {zn} and a nontrivial solution zλ,μ2=(uλ,μ2,vλ,μ2)∈W of System (Sλf,μg,h)  such that zn⇀zλ,μ2 weakly in W. Now we prove that zn→zλ,μ2 strongly in W and Iλ,μ(zλ,μ2)=θλ,μ-. By (3.4), if z∈𝒩λ,μ, then
(4.35)Iλ,μ(z)=p*-pp*p‖z‖p-p*-qp*qKλ,μ(z).
First, we prove that zλ,μ2∈𝒩λ,μ-. On the contrary, if zλ,μ2∈𝒩λ,μ+, then by 𝒩λ,μ- closed in W, we have ∥zλ,μ2∥<limn→∞∥zn∥. By Lemma 2.3 (i) and (A4), we obtain that
(4.36)Kλ,μ(zλ,μ2)>0,J(zλ,μ2)>0.
By Lemma 2.7, there exists a unique t- such that t-zλ,μ2∈𝒩λ,μ-. Since zn∈𝒩λ,μ-, Iλ,μ(zn)≥Iλ,μ(tzn) for all t≥0 and by (4.35), we have
(4.37)θλ,μ-≤Iλ,μ(t-zλ,μ2)<lim⁡n→∞Iλ,μ(t-zn)≤lim⁡n→∞Iλ,μ(zn)=θλ,μ-,
and this is a contradiction.
In order to prove thatIλ,μ(zλ,μ2)=θλ,μ-, it suffices to recall that zn,zλ,μ2∈𝒩λ,μ- for all n, by (4.35). and apply Fatou's lemma to get (4.38)θλ,μ-≤Iλ,μ(zλ,μ2)=p*-pp*p‖zλ,μ2‖p-p*-qp*qKλ,μ(zλ,μ2)≤lim⁡inf⁡n→∞(p*-pp*p‖zn‖p-p*-qp*qKλ,μ(zn))≤lim⁡inf⁡n→∞Iλ,μ(zn)=θλ,μ-.
This implies that Iλ,μ(zλ,μ2)=θλ,μ- and limn→∞∥zn∥p=∥zλ,μ2∥p. Let z̃n=zn-zλ,μ2, then by Brézis and Lieb lemma [26] implies
(4.39)‖z̃n‖p=‖zn‖p-‖zλ,μ2‖p.
Therefore, zn→zλ,μ2 strongly in W.
SinceIλ,μ(zλ,μ2)=Iλ,μ(|zλ,μ2|) and |zλ,μ2|∈𝒩λ,μ-, by Lemmas 2.2, and 2.3 (iii), we may assume that zλ,μ2 is a nontrivial nonnegative solution of System (Sλf,μg,h). Finally, by using the same arguments as in the proof of Theorem 3.3, for all 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, we have that zλ,μ2 is a positive solution of System (Sλf,μg,h).Proof of Theorem1.3.
By Theorems3.3, and 4.4, we obtain that for all λ,μ>0 and 0<λp/(p-q)+μp/(p-q)<(q/p)p/(p-q)Λ1, (Sλf,μg,h) has two positive solutions zλ,μ1, zλ,μ2 with zλ,μ1∈𝒩λ,μ+, zλ,μ2∈𝒩λ,μ-. Since 𝒩λ,μ+∩𝒩λ,μ-=∅, this implies that zλ,μ1 and zλ,μ2 are distinct. This completes the proof of Theorem 1.3.



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