# Multiple Solutions for a Singular Quasilinear Elliptic System

Authors: Lin Chen; Caisheng Chen; Zonghu Xiu
Journal: The Scientific World Journal
 (2013)

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## Abstract

We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system-div(|x|-ap|∇u|p-2∇u)+f1(x)|u|p-2u=(α/(α+β))g(x)|u|α-2u|v|β+λh1(x)|u|γ-2u+l1(x), -div(|x|-ap|∇v|p-2∇v)+f2(x)|v|p-2v=(β/(α+β))g(x)|v|β-2v|u|α+μh2(x)|v|γ-2v+l2(x), u(x)>0, v(x)>0, x∈ℝN, where λ,μ>0, 1<p<N, 1<γ<p<α+β<p*=Np/(N-pd), 0≤a<(N-p)/p, a≤b<a+1, d=a+1-b>0. The functions f1(x), f2(x), g(x), h1(x), h2(x), l1(x), and
l2(x) satisfy some suitable conditions. We will prove that the problem has at least two nontrivial solutions by using Mountain Pass Theorem and Ekeland's variational principle.


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## Body

## 1. Introduction

In this paper, we consider the multiplicity of solutions to a class of quasilinear elliptic system(1)-div(|x|-ap|∇u|p-2∇u)+f1(x)|u|p-2u=αα+βg(x)|u|α-2u|v|β+λh1(x)|u|γ-2u+l1(x),-div(|x|-ap|∇v|p-2∇v)+f2(x)|v|p-2v=βα+βg(x)|v|β-2v|u|α+μh2(x)|v|γ-2v+l2(x),u(x)>0,v(x)>0,x∈ℝN,
where λ,μ>0, 1<p<N, 1<γ<p<α+β<p*=Np/(N-pd), 0≤a<(N-p)/p, a≤b<a+1, d=a+1-b>0.In recent 20 years, existence and multiplicity for a quasilinear elliptic equation and equations have been studied widely. In [1], Yu has studied the existence of a nontrivial weak solution to the quasilinear elliptic problem
(2)-div(a(x)|∇u|p-2∇u)+b(x)|u|p-2u=g(x)|u|r-2u,inΩ,u=0,on∂Ω,
where Ω is a smooth exterior domain in ℝN, 1<p<N, a(x)∈L∞(Ω)∩C0,δ(Ω-) is a positive function, and b∈L∞(Ω)∩C(Ω) is a nonnegative function. Problems of this type are motivated by mathematical physics, since certain stationary waves in nonlinear Klein-Gordon or Schrödinger equations can be reduced to this form (see [2, 3]). Problem (2) began to attract more and more attention after the work of Yu [1]; we refer the readers to [4–8].In [9], Hsu studied the following elliptic system:
(3)-div(|∇u|p-2∇u)=λ|u|q-2u+2αα+β|u|α-2u|v|β,x∈Ω,-div(|∇v|p-2∇v)=μ|v|q-2v+2βα+β|v|β-2v|u|α,x∈Ω,u=v=0,x∈∂Ω,
where Ω⊂ℝN is a bounded domain, λ,μ>0, 1<p<q<N, and α>1, β>1 satisfy α+β=p*. The problem (3) is the special case of problem (1), while a=0, f1(x)=f2(x)=0. By applying Nehari manifold method, the author proved the problem (3) has at least one positive solution when 0<λp/(p-q)+μp/(p-q)<λ1 and has at least two positive solutions when 0<λp/(p-q)+μp/(p-q)<λ2.This paper is motivated by the results of [9–12]. Replacing the Nehari manifold methods, we use the Mountain Pass Theory and Ekeland’s variational principle to study the existence of multiple solutions for problem (1). It seems difficult to get the same result by Nehari manifold methods.Throughout this paper, we make the following hypotheses:(H1)
hi∈L∞(ℝN), hi(x)|x|bγ∈Lθ(ℝN), θ=p*/(p*-γ), i=1,2.(H2)
g(x)|x|b(α+β)∈Lδ(ℝN)∩L∞(ℝN), δ=p*/(p*-α-β).(H3)
l1(x)|x|b, l2(x)|x|b∈Lσ(Ω), σ=p*/(p*-1), l1(x)>0, l2(x)>0.In this paper, we letLbp(ℝN) the completion of the space C0∞(ℝN) with respect to the norm ∥u∥Lbp=(∫ℝN‍|x|-bp|u|pdx)1/p, and let Wa1,p(ℝN) the completion of C0∞(ℝN) with respect to the norm ∥u∥Wa1,p=(∫ℝN‍|x|-ap|∇u|pdx)1/p.We define(4)h1θ=(∫ℝN‍(|h1(x)||x|bγ)θdx)1/θ,h2θ=(∫ℝN‍(|h2(x)||x|bγ)θdx)1/θ,l1σ=(∫ℝN‍|x|bσ|l1(x)|σdx)1/σ,l2σ=(∫ℝN‍|x|bσ|l2(x)|σdx)1/σ,
where θ=p*/(p*-γ), σ=p*/(p*-1).The following inequality is called Caffarelli-Kohn-Nirenberg inequality [13]. There is a constant S>0 such that
(5)(∫ℝN‍|x|-bp*|u|p*dx)1/p*≤S(∫ℝN‍|x|-ap|∇u|pdx)1/p,
where -∞<a<(N-p)/p, a≤b<a+1, d=a+1-b, and p*=pN/(N-pd).LetX=Wa1,p(ℝN)×Wa1,p(ℝN) denote the usual Sobolev space with the norm
(6)∥(u,v)∥=(∫ℝN|x|-ap|∇u|p+f1(x)|u|p)dx+∫ℝN(|x|-ap|∇v|p+f2(x)|v|pdx)1/p.
Then X is a reflexive and separable Banach space endowed with the norm ∥(·,·)∥.Definition 1.
We say that(u,v)∈X is a weak solution of problem (1) if for all (φ,ψ)∈X, there holds
(7)∫ℝN(|x|-ap|∇u|p-2∇u∇φ+f1(x)|u|p-2uφ)dx+∫ℝN(|x|-ap|∇v|p-2∇v∇ψ+f2(x)|v|p-2vψ)dx-∫ℝN‍l1(x)φdx-∫ℝN‍l2(x)ψdx-αα+β∫ℝN(g(x)|u|α-2u|v|β+λh1(x)|u|γ-2u)φdx-βα+β∫ℝN(g(x)|v|β-2v|u|α+μh2(x)|v|γ-2v)ψdx=0.
It is clear that problem (1) has a variational structure. Let Jλ,μ:X→ℝ1 be the energy functional corresponding to problem (1) which is defined by
(8)Jλ,μ(u,v)=1p∥(u,v)∥p-1α+β∫ℝNg(x)|u|α|v|βdx-1γGλ,μ(u,v)-L(u,v),
where
(9)Gλ,μ(u,v)=∫ℝNλh1(x)|u|γdx+∫ℝNμh2(x)|v|γdx,L(u,v)=∫ℝN(l1(x)u+l2(x)v)dx.
Then, we see that functional Jλ,μ∈C1(X,ℝ1) and for all (φ,ψ)∈X, there holds
(10)〈Jλ,μ′(u,v),(φ,ψ)〉=∫ℝN(|x|-ap|∇u|p-2∇u∇φ+f1(x)|u|p-2uφ)dx+∫ℝN(|x|-ap|∇v|p-2∇v∇ψ+f2(x)|v|p-2vψ)dx-αα+β∫ℝN(g(x)|u|α-2u|v|β+λh1(x)|u|γ-2u)φdx-βα+β∫ℝN(g(x)|v|β-2v|u|α+μh2(x)|v|γ-2v)ψdx-∫ℝNl1(x)φdx-∫ℝNl2(x)ψdx.
In particular, it follows from (10) that
(11)〈Jλ,μ′(u,v),(u,v)〉=∥(u,v)∥p-∫ℝNg(x)|u|α|v|βdx-Gλ,μ(u,v)-L(u,v).
It is clear that the critical points of the energy functional Jλ,μ(u,v) correspond to the weak solutions of problem (1). Thus, to prove the existence of weak solutions for the problem (1), it is sufficient to show that Jλ,μ(u,v) admits a sequence of critical points in X. Our main result is the following.Theorem 2.
Assume that(H1)–(H3) hold. There exist c0,c1>0 such that if λ,μ>0 satisfy 0<λh1θ+μh2θ<c0 and 0≤l1σp/(p-1)+l2σp/(p-1)≤c1, then the problem (1) has at least two nontrivial solutions.


## 2. The Proof of Theorem2

In this section, we first introduce a compact embedding theory which is an extension of the classical Rellich-Kondrachov compactness theorem; see Xuan [13].Lemma 3.
Assume thatΩ∈ℝN is an open bounded domain with C1 boundary and 0∈Ω, N≥3, -∞<a<pN/(N-p). Then the embedding Wa1,p(Ω)↪Lαγ(Ω) is continuous if 1<γ≤Np/(N-p) and 0≤α≤(1+a)γ+N(1-γ/p) and is compact if 1≤γ<Np/(N-p) and 0≤α<(1+a)γ+N(1-γ/p).The following lemma is called the Mountain Pass Theorem, see [14] (also see [15]).Lemma 4 (Mountain Pass Theorem).
LetX be a real Banach space and J∈C1(X,ℝ1) with J(0,0)=0. Suppose that J(u,v) satisfies (PS) condition and(A1)
there areρ,α0>0, such that J(u,v)≥α0 when ∥(u,v)∥X=ρ;(A2)
there is(e1,e2)∈X, ∥(e1,e2)∥X>ρ such that J(e1,e2)<0.
Define(12)Γ={r∈C1([0,1],X)∣r(0)=(0,0),r(1)=(e1,e2)}.
Then
(13)c=infr∈Γmax0≤t≤1J(r(t))≥α0
is a critical value of  J(u,v).Lemma 5.
Assume that(H1)–(H3) hold. There exist c0,c1>0 such that if the parameters λ, μ satisfy 0<λh1θ+μh2θ<c0, and 0≤l1σp/(p-1)+l2σp/(p-1)≤c1, then Jλ,μ(u,v) satisfies the assumptions (A1)–(A2) in Lemma 4.Proof.
Let(u,v)∈X. By (8),
(14)Jλ,μ(u,v)=1p∥(u,v)∥p-1α+β∫ℝNg(x)|u|α|v|βdx-1γGλ,μ(u,v)-L(u,v).
By (H2) and the Hölder inequality, we have
(15)∫ℝNg(x)|u|α+βdx≤(∫ℝN(g(x)|x|b(α+β))δdx)1/δ×(∫ℝN|x|-bp*|u|p*dx)(α+β)/p*≤gδSα+β∥(u,v)∥α+β,
where gδ=(∫ℝN‍(g(x)|x|b(α+β))δdx)1/δ, δ=p*/(p*-α-β).
Similarly,(16)∫ℝNg(x)|v|α+βdx≤gδSα+β∥(u,v)∥α+β.
Hence, it follows from (15) and (16) that
(17)∫ℝNg(x)|u|α|v|βdx≤2gδSα+β∥(u,v)∥α+β.
Since h1(x)|x|bγ∈Lθ(ℝN)∩L∞(ℝN), we obtain from the Hölder inequality and (5) that
(18)∫ℝNλh1(x)|u|γdx≤λ(∫ℝN(h1(x)|x|bγ)θdx)1/θ×(∫ℝN‍|x|-bp*|u|p*dx)γ/p*≤λSγh1θ∥(u,v)∥γ.
Similarly,
(19)∫ℝNμh2(x)|v|γdx≤μSγh2θ∥(u,v)∥γ.
Then,
(20)Gλ,μ(u,v)≤(λh1θ+μh2θ)Sγ∥(u,v)∥γ.
Using the Hölder inequality, we get
(21)∫ℝN‍|l1(x)||u|dx≤(∫ℝN|x|-bp*|u|p*dx)1/p*×(∫ℝN|x|bσ|l1(x)|σdx)1/σ≤ε∥(u,v)∥p+Cε(Sl1σ)p/(p-1),
where ε>0, Cε>0.
Similarly, we have(22)∫ℝN|l2(x)||v|dx≤Sl2σ∥(u,v)∥≤ε∥(u,v)∥p+Cε(Sl2σ)p/(p-1).
Thus,
(23)Jλ,μ(u,v)≥1p∥(u,v)∥p-1α+β·2gδSα+β∥(u,v)∥α+β-1γ(λh1θ+μh2θ)Sγ∥(u,v)∥γ-2ε∥(u,v)∥p-Cε(Sl1σ)p/(p-1)-Cε(Sl2σ)p/(p-1)≥12p∥(u,v)∥p-2gδα+βSα+β∥(u,v)∥α+β-1γ(λh1θ+μh2θ)Sγ∥(u,v)∥γ-Cε(Sl1σ)p/(p-1)-Cε(Sl2σ)p/(p-1)
with 0<ε≤1/4p.
Let(24)g(z)=2gδα+βSα+βzα+β-p+1γ(λh1θ+μh2θ)Sγzγ-p,z>0.
To verify (A1) in Lemma 4, it suffices to show that g(z1)<1/2p for some z1=∥(u,v)∥>0.
Note thatg(z)→+∞ where z→0+ or z→+∞. Then g(z) has a minimum at z1>0. In order to find z1, we have
(25)g′(z)=2gδ(α+β-p)α+βSα+βzα+β-p-1+γ-pγ(λh1θ+μh2θ)Sγzγ-p-1.
So that g′(z1)=0 and
(26)z1=((α+β)(p-γ)2γgδ(α+β-p)Sα+β-γ)1/(α+β-γ)×(λh1θ+μh2θ)1/(α+β-γ)=c2(λh1θ+μh2θ)1/(α+β-γ)>0.
Moreover, g(z1)<1/2p implies that
(27)g(z1)=(2gδα+βSα+βc2α+β-p+1γSγc2γ-p)×(λh1θ+μh2θ)(α+β-p)/(α+β-γ)=c3(λh1θ+μh2θ)(α+β-p)/(α+β-γ)<12p,
where c3=(2gδ/(α+β))Sα+βc2α+β-p+(1/γ)Sγc2γ-p.
Thus, it follows from (23) and (27) that there exist c0,c1,α0>0 such that Jλ,μ(u,v)≥α0 with 0<λh1θ+μh2θ<c0 and 0≤l1σp/(p-1)+l2σp/(p-1)≤c1. Then Jλ,μ(u,v) satisfies the assumption (A1) in Lemma 4.
We now verify(A2) in Lemma 4. Choose (φ1,φ2)∈C0∞(ℝN)  ×  C0∞(ℝN) such that
(28)∫ℝNg(x)|φ1|α|φ2|βdx>0.
Then
(29)Jλ,μ(tφ1,tφ2)=tpp∥(φ1,φ2)∥p-tα+βα+β∫ℝNg(x)|φ1|α|φ2|βdx-tγγGλ,μ(φ1,φ2)-tL(φ1,φ2)
and Jλ,μ(tφ1,tφ2)→-∞ (t→+∞), since α+β>p>γ. Therefore, there exists t large enough, such that Jλ,μ(tφ1,tφ2)<0. Then we take e=(tφ1,tφ2)∈X and Jλ,μ(e)<0. The condition (A2) of Lemma 4 is true. This completes the proof of Lemma 5.Lemma 6.
Let(H1)-(H2) hold.
If{un} and {vn} are bounded sequences in Wa1,p(ℝN), then there exist subsequences, still denoted {un} and {vn}, and u∈Wa1,p(ℝN), v∈Wa1,p(ℝN) such that as n→∞(30)∫ℝNg(x)|un|αdx⟶∫ℝNg(x)|u|αdx,∫ℝNh1(x)|un|γdx⟶∫ℝNh1(x)|u|γdx,∫ℝNg(x)|vn|αdx⟶∫ℝNg(x)|v|αdx,∫ℝNh2(x)|vn|γdx⟶∫ℝNh2(x)|u|γdx.Proof.
LetUk={x∈ℝN∣|x|<k}, k=1,2,….
Since{un} is bounded in Wa1,p(ℝN), {un} is bounded in Wa1,p(Uk) for for all k≥1. By Lemma 3, {un} has a subsequence {un(1)} which converges u-1 in Lγ(U1). Likewise, the subsequence {un(1)} is bounded in Wa1,p(U2). So that it has a subsequence {un(2)} which converges u-2 in Lγ(U2). Since {un(2)} is a subsequence of {un(1)}, u-2=u-1 in U¯1. Continuing this process, we obtain a sequence {u-k} with the following properties:
(31)u-k∈Lγ(Uk),u-k(x)=u-1(x),a.einU¯1,u-k(x)=u-2(x),a.einU¯2,⋮u-k(x)=u-k-1(x),a.einU¯k-1,k=1,2,….
It is clear that u-k(x)→u(x) a.e in ℝ1, where
(32)u(x)=u-k(x),x∈U¯k,k=1,2,….
By a diagonal process, we take {um(m)} which is a subsequence of {un}. Thus, we have
(33)um(m)⟶u,stronglyinLγ(Uk),k=1,2,….
Without loss of generality, we assume that the subsequence {um(m)} is {un} itself. So
(34)un⟶u,inLγ(Uk),k=1,2,….
This implies that for n→∞,
(35)un(x)⟶u(x),a.einUk,k=1,2,….
Now, we prove
(36)limk→∞supu∈Wa1,p∖{0}∥u∥Lγ(Ukc,|h1|)∥u∥Wa1,p(ℝN)=0.
Indeed,
(37)∥u∥Lγ(Ukc,|h1|)γ=∫Ukc|h1||u|γdx≤(∫Ukc(|h1||x|bγ)θdx)1/θ×(∫Ukc|x|-bp*|u|p*dx)γ/p*≤(∫Ukc(|h1||x|bγ)θdx)1/θSγ∥u∥Wa1,pγ,
where θ=p*/(p*-γ).
The facth1(x)|x|bγ∈Lθ(ℝN) gives that limk→∞(∫Ukc(|h1||x|bγ)θdx)1/θ=0.
Then, (37) implies that
(38)∥u∥Lγ(Ukc,|h1|)∥u∥Wa1,p≤S(∫Ukc(|h1||x|bγ)θdx)1/γθ⟶0,hhhhhhhhhhhhhhhhask⟶∞.
This gives (36).
In the following, we show that(39)∫ℝN‍h1(x)|un|γdx⟶∫ℝNh1(x)|u|γdx,asn⟶∞.
Since the sequence {un} is bounded in Wa1,p(ℝN), we can assume (up to a subsequence) that un⇀u weakly in Wa1,p(ℝN) and ∥un∥Wa1,p≤C0 for some constant C0>0 and all n≥1.
By (36), we know that for any ε>0, there exists kε>0 so large that
(40)∥un∥Lγ(Ukεc,|h1|)≤C0-1ε∥un∥Wa1,p≤εforn=1,2,…,∥u∥Lγ(Ukεc,|h1|)≤ε.
Since the embeddding Wa1,p(Ukε)↪Lγ(Ukε) is compact (see Lemma 3), we have
(41)limn→∞∥un-u∥Lγ(Ukε)=0.
Thus, there exists N1>0, when n>N1,
(42)∥un-u∥Lγ(Ukε)<ε.
So
(43)∥un-u∥Lγ(ℝN,|h1|)≤∥h1∥L∞(ℝN)∥un-u∥Lγ(ℝN)≤∥h1∥L∞(ℝN)(∥un∥Lγ(Ukεc)+∥u∥Lγ(Ukεc)+∥un-u∥Lγ(Ukε)∥un∥Lγ(Ukεc)+∥u∥Lγ(Ukεc))≤3ε∥h1∥∞.
This shows that un→u in Lγ(ℝN,|h1|) as n→∞.
By Brezis-Lieb’s Lemma in [16], we obtain that (39) is true.
Similarly, we can prove(44)∫ℝNg(x)|un|αdx⟶∫ℝNg(x)|u|αdx,∫ℝNg(x)|vn|αdx⟶∫ℝNg(x)|v|αdx,∫ℝNh2(x)|vn|γdx⟶∫ℝNh2(x)|v|γdx.Remark 7.
By Brezis-Lieb’s Lemma, it follows from (30) that
(45)∫ℝN|g||un-u|αdx⟶0,∫ℝN|g||vn-v|αdx⟶0,∫ℝN|h1||un-u|γdx⟶0,∫ℝN|h2||vn-v|γdx⟶0,∫ℝN|l1||un-u|dx⟶0,∫ℝN|l2||vn-v|dx⟶0.Lemma 8.
Assume that(H1)–(H3) hold. Then Jλ,μ(u,v) defined by (8) satisfies (PS) condition on X.Proof.
Let{(un,vn)} be a (PS)c sequence of Jλ,μ(u,v) in X; that is, Jλ,μ(un,vn)→c, Jλ,μ′(un,vn)→0, in X*. We first claim that {(un,vn)} is bounded in X. In fact, for large n, we obtain
(46)c+1+∥(un,vn)∥≥Jλ,μ(un,vn)-1α+β〈Jλ,μ′(un,vn),(un,vn)〉=(1p-1α+β)∥(un,vn)∥p-(1γ-1α+β)Gλ,μ(un,vn)-(1-1α+β)L(un,vn).
Since p>γ>1, we conclude that {(un,vn)} is bounded. Then, the {un} and {vn} are bounded in Wa1,p(ℝN), respectively. Thus, there exist {(u,v)}∈X and a subsequence of {(un,vn)}, still denoted by {(un,vn)}, such that
(47)un⇀u,vn⇀vweaklyinWa1,p(ℝN).
Denote
(48)Pn=〈Jλ,μ′(un,vn),(un-u,vn-v)〉=∫ℝN(|x|-ap|∇un|p-2∇un∇(un-u)+f1(x)|un|p-2un(un-u)+|x|-ap|∇vn|p-2∇vn∇(vn-v)+f2(x)|vn|p-2vn(vn-v))dx-αα+β∫ℝN‍(g(x)|un|α-2un|vn|β+λh1(x)|un|γ-2ung(x)|un|α-2un|vn|β)×(un-u)dx-βα+β×∫ℝN(g(x)|vn|β-2vn|un|α+μh2(x)|vn|γ-2vng(x)|vn|β-2vn|un|αg(x)|vn|β-2vn|un|α)(vn-v)dx-∫ℝN‍l1(x)(un-u)dx-∫ℝN‍l2(x)(vn-v)dx.
Then the fact Jλ,μ′(un,vn)→0 in X* implies Pn→0 as n→∞.
Since thatun⇀u,vn⇀v in Wa1,p(ℝN), we see that
(49)Qn=∫ℝN(|x|-ap|∇u|p-2∇u∇(un-u)+f1(x)|u|p-2u(un-u))dx+∫ℝN(|x|-ap|∇v|p-2∇v∇(vn-v)+f2(x)|v|p-2v(vn-v))dx⟶0,n⟶∞.
Let
(50)Tn=∫ℝN(|x|-ap|∇un|p-2∇un∇(un-u)+f1(x)|un|p-2un(un-u)+|x|-ap|∇vn|p-2∇vn∇(vn-v)+f2(x)|vn|p-2vn(vn-v))dx.
From dominated convergence theory and Lemma 6, we can conclude
(51)∫ℝN‍g(x)|un|α-2un|vn|β(un-u)dx⟶0,n⟶∞,(52)∫ℝN‍g(x)|vn|β-2vn|un|α(vn-v)dx⟶0,n⟶∞.
From Hölder inequality,
(53)∫ℝN|h1(x)||un|γ-1|un-u|dx≤(∫ℝN|h1||un-u|γdx)1/γ×(∫ℝN|h1||un|γdx)(γ-1)/γ⟶0,asn⟶∞.
Similarly, we have
(54)∫ℝN|h2||vn|γ-1|vn-v|dx⟶0,∫ℝN|l1(x)||un-u|dx⟶0,∫ℝN|l2(x)||vn-v|dx⟶0.
Then it follows from (48), (49) to (54) that Tn→0, as n→∞.
So(55)Tn-Qn=∫ℝN|x|-ap(|∇un|p-2∇un-|∇u|p-2∇u)∇(un-u)dx+∫ℝNf1(x)(|un|p-2un-|u|p-2u)(un-u)dx+∫ℝN|x|-ap(|∇vn|p-2∇vn-|∇v|p-2∇v)∇(vn-v)dx+∫ℝNf2(x)(|vn|p-2vn-|v|p-2v)×(vn-v)dx⟶0.
Using the standard inequality in ℝN given by
(56)〈|ξ|p-2ξ-|η|p-2η,ξ-η〉≥Cp|ξ-η|p,p≥2,〈|ξ|p-2ξ-|η|p-2η,ξ-η〉≥Cp|ξ-η|2(|ξ|+|η|)p-2,〈|ξ|p-2ξ|η|p-2η,hξ-η〉≥Cp|ξ-η|2≥11<p<2,
we have from (55) that ∥(un-u,vn-v)∥→0, as n→∞.
Thus,Jλ,μ(u,v) satisfies (PS)c condition on X.Proof of Theorem2.
By Lemmas5 and 8, Jλ,μ(u,v) satisfies all assumptions in Lemma 4. Then there exists (u1,v1)∈X such that (u1,v1) is a solution of problem (1) by Lemma 4. Furthermore, Jλ,μ(u1,v1)≥α0>0.
We now seek a solution(u2,v2) of problem (1). Choose (φ1,φ2)∈C0∞(ℝN)×C0∞(ℝN) such that ∫ℝN‍l1(x)φ1dx+∫ℝN‍l2(x)φ2dx>0, and then
(57)Jλ,μ(tφ1,tφ2)=1ptp∥(φ1,φ2)∥p-1α+βtα+β∫ℝNg(x)|φ1|α|φ2|βdx-t∫ℝN(l1(x)φ1+l2(x)φ2)dx-1γtγGλ,μ(φ1,φ2)<0
for small t>0 and thus for any open ball Bτ⊂X such that
(58)-∞<cτ=infB¯τJλ,μ(u,v)<0.
Thus, exists ρ>0, such that
(59)cρ=inf(u,v)∈B¯ρJλ,μ(u,v)<0,inf(u,v)∈∂BρJλ,μ(u,v)>0.
Letting εn↓0 such that
(60)0<εn<inf(u,v)∈∂BρJλ,μ(u,v)-inf(u,v)∈BρJλ,μ(u,v).
Then, by Ekeland’s variational principle in [17], there exists {(un,vn)}⊂B¯ρ such that
(61)cρ≤Jλ,μ(un,vn)<cρ+εn,(62)Jλ,μ(un,vn)<Jλ,μ(u,v)+εn∥(un-u,vn-v)∥,∀(u,v)∈B¯ρB¯ρ∀(u,v)∈B¯ρ,u≠un,v≠vn.
Then it follows from (59) to (61) that
(63)Jλ,μ(un,vn)<cρ+εn≤inf(u,v)∈BρJλ,μ(u,v)+εn<inf(u,v)∈∂BρJλ,μ(u,v),
so that (un,vn)∈Bρ. We now consider the functional F:B¯ρ→ℝ given by
(64)F(u,v)=Jλ,μ(u,v)+εn∥(u-un,v-vn)∥,(u,v)∈B¯ρ.
Then (62) shows that F(un,vn)<F(u,v), (u,v)∈B¯ρ, u≠un, v≠vn, and thus (un,vn) is a strict local minimum of F. Moreover,
(65)t-1(F(un+tφ1,vn+tφ2)-F(un,vn))≥0,hhhhhhhforsmallt>0,∀(φ1,φ2)∈B1.
Hence,
(66)t-1(Jλ,μ(un+tφ1,vn+tφ2)-Jλ,μ(un,vn))+εn∥(φ1,φ2)∥≥0.
Let t→0+; then,
(67)〈Jλ,μ′(un,vn),(φ1,φ2)〉+εn∥(φ1,φ2)∥≥0,∀(φ1,φ2)∈B1∀∀∀∀∀∀∀∀∀∀(φ1,φ2)∈B1.
Replacing (φ1,φ2) in (67) by (-φ1,-φ2), we get
(68)-〈Jλ,μ′(un,vn),(φ1,φ2)〉+εn∥(φ1,φ2)∥≥0,∀(φ1,φ2)∈B1∀∀(φ1,φ2)∈∀∀(φ1,φ2)∈B1.
So that ∥Jλ,μ′(un,vn)∥≤εn.
Therefore, there is a sequence{(un,vn)}⊂Bρ such that Jλ,μ(un,vn)→cρ<0, and Jλ,μ′(un,vn)→0 in X* as n→∞. By Lemma 5, {(un,vn)} has a convergent subsequence in X, still denoted by {(un,vn)}, such that (un,vn)→(u2,v2) in X. Thus (u2,v2) is a solution of (1) with Jλ,μ(u2,v2)<0. Then the proof of Theorem 2 is complete.



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*Source: 278013-2013-10-24.xml*