# Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents inRN

Authors: Zhiying Deng; Yisheng Huang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper deals with a class of quasilinear elliptic systems involving
singular potentials and critical Sobolev exponents inRN. By using the symmetric
criticality principle of Palais and variational methods, we prove several existence and
multiplicity results of G-symmetric solutions under certain appropriate hypotheses on
the potentials and parameters.


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## Body

## 1. Introduction

In this work, we investigate the existence and multiplicity of nontrivial solutions for the following quasilinear elliptic system:(1)Lp,μu=Qxup*-2u+ςαp*uα-2uvβ+λh1(x)uq-2u,inRN,Lp,μv=Qxvp*-2v+ςβp*uαvβ-2v+λh2(x)vq-2v,inRN,ux,vx⟶0,asx⟶+∞,
where Lp,μ≜-div⁡(|∇·|p-2∇·)-μ|·|p-2·/|x|p is a quasilinear elliptic operator,  1<q<p<N, 0<ς<+∞, 0≤μ<μ¯ with μ¯=(N-p/p)p, λ≥0, and α,β>1 satisfy α+β=p*, p*≜Np/N-p denotes the critical Sobolev exponent, and Q∈C(RN)∩L∞(RN) and hi∈Lθ(RN)(i=1,2) with θ=Np/Np-q(N-p) are G-symmetric functions (see Section 2 for details) with respect to a closed subgroup G of O(N).In recent years, considerable attention has been paid to the scalar singular elliptic problem:(2)Lp,μu=Q(x)|u|p*-2u+λh(x)|u|q-2u,inΩ,u=0,on⁡∂Ω,
where Ω⊂RN is a smooth domain (bounded or unbounded) containing the origin. The study of this type of equation is motivated by its definite physics background and various applications, including celestial mechanics, fluid mechanics, and flow through porous media (see [1]). The mathematical interest lies in the fact that these problems like (2) are doubly critical due to the presence of the Sobolev embedding and the singularities. For this reason, many existence, nonexistence, and multiplicity results of nontrivial solutions for the single equations like (2) have been established with different assumptions on the potentials Q(x), h(x) and the parameters μ, λ, and q; we refer to [2–8] and the references therein.In a recent paper, Deng and Jin [9] considered the following single semilinear elliptic problem:
(3)-Δu-μux2=kxxsu2*s-1,u>0inRN,
where N>2, 0≤μ<(N-2/2)2, 0≤s<2, 2*(s)=2(N-s)/N-2, and k satisfies some symmetry conditions with respect to G⊂O(N). By using analytic techniques and variational arguments, the authors proved the existence and multiplicity of G-symmetric solutions to (3) under certain hypotheses on k. Subsequently, Waliullah [10] improved the results in [9] by using the minimizing sequence and the concentration-compactness principle. Recently, Deng and Huang [11] extended the results in [9, 10] to the scalar weighted elliptic problems in a bounded G-symmetric domain. Besides these, when μ=s=0 and the right-hand side term |x|-su2*(s)-1 is replaced by a term f(u) of the pure power, such as f(u)=ur-1 with 1<r<2N/N-2 or r=2N/N-2, there are many interesting results on the existence and multiplicity of G-symmetric solutions of (3), which can be found in [12–14] and the references therein.On the other hand, there have been many papers concerned with the existence and multiplicity of nontrivial solutions for elliptic systems. In [15], Wu considered the following semilinear elliptic system:
(4)-Δu=2αα+βQxuα-2uvβ+λfxuq-2u,inΩ,-Δv=2βα+βQxuαvβ-2v+δhxvq-2v,inΩ,u=v=0,on⁡∂Ω,
where Ω⊂RN(N≥3) is a smooth bounded domain, 1<q<2, α, β>1, α+β<2*, and the weight functions Q, f, h fulfill certain suitable conditions. Via the analytic techniques of Nehari manifold and variational methods, the author proved that the system (4) admits at least two nontrivial nonnegative solutions if the pair of parameters (λ,δ) belongs to a certain subset of R2. Very recently, Nyamoradi [16], Lü and Xiao [17], and Li and Gao [18] generalized the corresponding results of [15] to the nonlinear singular elliptic systems involving critical Hardy-Sobolev exponents. Other results about existence and multiplicity of nontrivial solutions, also for related elliptic systems, can be seen in [19–23] and the references therein.However, as far as we know, the existence and multiplicity ofG-symmetric solutions for singular elliptic systems were seldom studied; we can only find some G-symmetric results for singular elliptic systems in [24] and, when G=O(N), some radial and nonradial results for nonsingular elliptic systems in [25]. Inspired by [9, 12, 25], in this paper we are concerned with the existence and multiplicity of positive G-symmetric solutions for system (1). The main difficulties lie in the fact that there are not only the nonlinear perturbations λh1(x)|u|q-2u, λh2(x)|v|q-2v and the Hardy singular terms |u|p-2u/|x|p, |v|p-2v/|x|p in (1), but also four nonlinear terms with the critical Sobolev exponents in RN. Compared with (3) and (4), the singular quasilinear elliptic system (1) becomes more complicated to deal with. Moreover, the approach involving the Nehari manifold requires that the corresponding nonlinearity is second order derivative about u and v. Hence, in order to obtain the multiple G-symmetric solutions of system (1), the Nehari manifold techniques in the literature mentioned above are invalid and we need to look for other methods. To our knowledge, even in the particular case λ=0 and p=2, there are no results on the existence and multiplicity of G-symmetric solutions for system (1). It is therefore meaningful for us to investigate system (1) deeply. Let Q~>0 be a constant. Note that, here, we will try to treat both the cases of λ=0, Q(x)≡Q~ and λ>0, Q(x)≡Q~.This paper is schemed as follows. In Section2, we establish the appropriate Sobolev space which is applicable to the study of the elliptic system (1) and state the main results of this paper. In Section 3, we detail the proofs of several existence and multiplicity results for the case λ=0 and Q(x)≡Q~ in (1). In Section 4, we will present the proofs of multiplicity results for the case λ>0 and Q(x)≡Q~ in (1). Our methods in this paper are mainly based upon the symmetric criticality principle of Palais (see [26]) and variational arguments.


## 2. Preliminaries and Main Results

LetO(N) be the group of orthogonal linear transformations in RN and let G⊂O(N) be a closed subgroup. For x≠0 we denote the cardinality of Gx={gx;g∈G} by |Gx| and set |G|=inf⁡0≠x∈RN|Gx|. Note that, here, |G| may be +∞. We say that f:RN→R is G-symmetric (or G-invariant) if f(gx)=f(x) for every g∈G and x∈RN and in the context of Sobolev spaces this equality is understood a.e. on RN. In particular, if f is radially symmetric, then the corresponding group G is O(N) and |G|=+∞. We call Ω a G-symmetric subset of RN; if x∈Ω, then gx∈Ω for all g∈G.LetD1,p(RN) denote the closure of C0∞(RN) functions with respect to the norm (∫RN‍|∇u|pdx)1/p. We recall that the well-known Hardy inequality (see [2, 3]) holds:
(5)∫RN‍|u|p|x|pdx≤1μ¯∫RN‍|∇u|pdx,∀u∈D1,p(RN),
where μ¯=(N-p/p)p. For μ∈[0,μ¯), we employ the following norm in D1,p(RN):
(6)uμ=∫RN‍∇up-μupxpdx1/p.
By inequality (5), we see that the above norm is equivalent to the usual norm (∫RN‍|∇u|pdx)1/p. The elliptic operator Lp,μ=-div⁡(|∇·|p-2∇·)-μ|·|p-2·/|x|p is positive in D1,p(RN) if μ∈[0,μ¯). Moreover, we define the product space D1,p(RN)×D1,p(RN) endowed with the norm
(7)222222(u,v)μ=(uμp+vμp)1/p,∀u,v∈D1,pRN×D1,pRN.
The natural functional space to study system (1) is the Banach space DG1,p(RN)×DG1,p(RN), which is the subspace of D1,p(RN)×D1,p(RN) consisting of all G-symmetric functions. Now in this paper, we are concerned with the following elliptic problems:
(8)(PλQ)Lp,μu=Qxup*-2u+ςαp*uα-2uvβvvvvvvvvvvv+λh1xuq-2u,inRN,Lp,μv=Qxvp*-2v+ςβp*uαvβ-2vvvvvv+λh2(x)|v|q-2v,inRN,(u,v)∈DG1,p(RN)×DG1,p(RN),u>0,v>0,inRN.
To mention our main results, we need to introduce two notations Aμ and yϵ(x), which are, respectively, defined by
(9)Aμ≜inf⁡u∈D1,p(RN)∖0∫RN‍∇up-μup/xpdx∫RNup*dxp/p*,(10)yϵ(x)≜Cϵ-ϑUμxϵ,
where ϵ>0, ϑ≜N-p/p, and the constant C=C(N,p,μ)>0, depending only on N, p, and μ. From Kang [4], we see that yϵ(x) satisfies the equations
(11)yϵμp=∫RN‍∇yϵp-μyϵpxpdx=1,(12)∫RN‍yϵp*dx=∫RN‍yϵα+βdx=Aμ-α+β/p.
The function Uμ>0 in (10) is radially symmetric. Moreover, the following asymptotic properties at the origin and infinity for Uμ(r) and Uμ′(r) hold [4]:
(13)lim⁡r→0rl1Uμr=C1>0,lim⁡r→0rl1+1Uμ′r=C1l1>0,(14)lim⁡r→+∞rl2Uμ(r)=C2>0,lim⁡r→+∞rl2+1Uμ′r=C2l2>0,
where C1, C2 are positive constants and l1=l1(N,p,μ) and l2=l2(N,p,μ) are the zeroes of the function
(15)L(t)=(p-1)tp-(N-p)tp-1+μ,t≥0,0≤μ<μ¯,
satisfying
(16)0≤l1<ϑ<l2≤N-pp-1,ϑ=N-pp.We suppose that the functionsQ, h1, and h2 verify the following hypotheses.(q.1)
Q∈C(RN)∩L∞(RN), and Q is G-symmetric.(q.2)
Q+≡0, where Q+=max⁡{0,Q}.(h.1)
h1 and h2 are G-symmetric.(h.2)
hi(x)≥0, hi(x)≡0, and hi(x)∈Lθ(RN) with θ=Np/Np-q(N-p), where i=1,2.The main results of this paper are summarized in the following.Theorem 1.
Suppose that(q.1) and(q.2) hold. If
(17)∫RN‍Q(x)yϵα+βdx≥max⁡Q+0Aμα+β/p,Q+∞Aμα+β/p,,Q+∞Gα+β-p/pA0α+β/p>0,
for some ϵ>0, where Q+(∞)=limsup⁡|x|→∞Q+(x), then problem (P0Q) has at least one positive solution in DG1,p(RN)×DG1,p(RN).Corollary 2.
Suppose that(q.1) and(q.2) hold. Then we have the following statements. (1)
Problem(P0Q)has a positive solution if(18)Q(0)>0,Q(0)≥max⁡Q+∞,Gp-α+β/pA0Aμ-α+β/pQ+∞,and either (i)Q(x)≥Q(0)+Λ0|x|(α+β)(l2-ϑ)for someΛ0>0and|x|small or (ii)|Q(x)-Q(0)|≤Λ1|x|κfor some constantΛ1>0,κ>(α+β)(l2-ϑ)and|x|small and(19)∫RN‍(Q(x)-Q(0))|x|-N-(α+β)(l2-ϑ)dx>0.(2)
Problem(P0Q)admits at least one positive solution iflim|x|→∞Q(x)=Q(∞)exists and is positive,(20)Q∞≥max⁡Q+0,Gp-α+β/pA0Aμ-α+β/pQ+∞,and either (i)Q(x)≥Q(∞)+Λ2|x|-(α+β)(ϑ-l1)for someΛ2>0and large|x|or (ii)|Q(x)-Q(∞)|≤Λ3|x|-ιfor some constantsΛ3>0,ι>(α+β)(ϑ-l1)and large|x|and(21)∫RN‍(Q(x)-Q(∞))|x|-N+(α+β)(ϑ-l1)dx>0.(3)
IfQ(x)≥Q(∞)=Q(0)>0onRNand(22)Q(∞)=Q(0)≥|G|p-(α+β)/pA0Aμ-α+β/pQ+∞,then problem(P0Q)has at least one positive solution.Theorem 3.
Suppose thatQ+(0)=Q+(∞)=0 and |G|=+∞. Then problem (P0Q) has infinitely many G-symmetric solutions.Corollary 4.
IfQ is a radially symmetric function such that Q+(0)=Q+(∞)=0, then problem (P0Q) has infinitely many solutions which are radially symmetric.Theorem 5.
LetQ~>0 be a constant. Suppose that Q(x)≡Q~ and(h.1),(h.2) hold. Then there exists λ*>0 such that, for any λ∈(0,λ*), problem (PλQ~) possesses at least two positive solutions in DG1,p(RN)×DG1,p(RN).Remark 6.
The main results of this paper generalize, extend, and complement some results of the aforementioned papers [9–12, 24, 25].In the sequel, we denote byDG1,p(RN)×DG1,p(RN) the subspace of D1,p(RN)×D1,p(RN) consisting of all G-symmetric functions. The dual space of DG1,p(RN)×DG1,p(RN) (D1,p(RN)×D1,p(RN), resp.) is denoted by DG-1,p′(RN)×DG-1,p′(RN) (D-1,p′(RN)×D-1,p′(RN), resp.), where 1/p+1/p′=1. The ball of center x and radius r is denoted by Br(x). We employ C,C1,C2,… to denote (possibly different) positive constants and denote by “→” convergence in norm in a given Banach space X and by “⇀” weak convergence. Hereafter, on(1) denotes a datum which tends to 0 as n→∞. Lq(RN,h(x)) denotes the weighted Lq(RN) space with the norm (∫RN‍h(x)|u|qdx)1/q. A functional J∈C1(X,R) is said to satisfy the (PS)c condition if each sequence {un} in X satisfying J(un)→c, J′(un)→0 in X* has a subsequence which strongly converges to some element in X.


## 3. Existence and Multiplicity Results for Problem(P0Q)

The corresponding energy functional of problem(P0Q) is defined in DG1,p(RN)×DG1,p(RN) by
(23)Eu,v=1pu,vμp-1p*∫RN‍Qxup*+vp*+ςuαvβdx.
Note that (q.1) and (5) imply that E∈C1(DG1,p(RN)×DG1,p(RN),R). It is well known that there exists a one-to-one correspondence between the weak solutions of problem (P0Q) and the critical points of E. More precisely, any weak solution of (P0Q) is exactly the critical point of E by the following symmetric principle (see Lemma 7); namely, (u,v)∈DG1,p(RN)×DG1,p(RN) satisfies (P0Q) if and only if for all (φ1,φ2)∈D1,p(RN)×D1,p(RN), there holds
(24)0=∫RN‍|u|p-2uφ1+|v|p-2vφ2|x|p|∇u|p-2∇u∇φ1+|∇v|p-2∇v∇φ2vvvvvv-μ|u|p-2uφ1+|v|p-2vφ2|x|pdx-∫RN‍Q(x)up*-2u+ςαp*uα-2uvβφ1vvvvvvvvvvv+vp*-2v+ςβp*uαvβ-2vφ2dx.Lemma 7.
E
′
(
u
,
v
)
=
0 in DG-1,p′(RN)×DG-1,p′(RN) implies E′(u,v)=0 in D-1,p′(RN)×D-1,p′(RN).Proof.
See the proof of [12, Lemma 1] (see also [25, Proposition 2.8]).Now, for any0≤μ<μ¯, 0<ς<+∞, α, β>1, and α+β=p*, we define
(25)Aμ,ς(α,β)≜inf⁡u,v∈D1,p(RN)∖{0}∫RN‍∇up+∇vp-μup+vp/xpdx∫RN‍up*+vp*+ςuαvβdxp/α+β,(26)K(τ)≜1+τp1+ςτβ+τα+βp/α+β,τ≥0,(27)K(τmin⁡)≜min⁡τ≥0K(τ)>0,
where τmin⁡>0 is a minimal point of K(τ) and therefore a root of the equation
(28)(α+β)τα+β-p+ςβτβ-p-ςατβ-(α+β)=0,τ≥0.Lemma 8.
Suppose that1<p<N, 0<ς<+∞, and 0≤μ<μ¯. Then Aμ,ς(α,β)=K(τmin⁡)Aμ, and Aμ,ς(α,β) has the minimizer (yϵ(x),τmin⁡yϵ(x)), ∀ϵ>0, where yϵ(x) is the extremal function of Aμ defined as in (10).Proof.
The proof is similar to the proof in Nyamoradi [16, Theorem 2].Lemma 9.
Let{(un,vn)} be a weakly convergent sequence to (u,v) in DG1,p(RN)×DG1,p(RN) such that |∇un|p⇀η(1), |∇vn|p⇀η(2), |un|p*⇀σ(1), |vn|p*⇀σ(2), |un|α|vn|β⇀ν, |x|-p|un|p⇀γ(1), and |x|-p|vn|p⇀γ(2) in the sense of measures. Then there exists some at most countable set J, {ηj(1)≥0}j∈J∪{0}, {ηj(2)≥0}j∈J∪{0}, {σj(1)≥0}j∈J∪{0}, {σj(2)≥0}j∈J∪{0}, {νj≥0}j∈J∪{0}, γ0(1)≥0, γ0(2)≥0, and {xj}j∈J⊂RN∖{0} such that
(a)
η
(
1
)
≥
|
∇
u
|
p
+
∑
j
∈
J
η
j
(
1
)
δ
x
j
+
η
0
(
1
)
δ
0, η(2)≥|∇v|p+∑j∈Jηj(2)δxj+η0(2)δ0,
(b)
σ
(
1
)
=
|
u
|
p
*
+
∑
j
∈
J
σ
j
(
1
)
δ
x
j
+
σ
0
(
1
)
δ
0, σ(2)=|v|p*+∑j∈Jσj(2)δxj+σ0(2)δ0,
(c)
ν
=
|
u
|
α
|
v
|
β
+
∑
j
∈
J
ν
j
δ
x
j
+
ν
0
δ
0, γ(1)=|u|p/|x|p+γ0(1)δ0, γ(2)=|v|p/|x|p+γ0(2)δ0,
(d)
A
0
,
ς
(
α
,
β
)
(
σ
j
(
1
)
+
σ
j
(
2
)
+
ς
ν
j
)
p
/
(
α
+
β
)
≤
η
j
(
1
)
+
η
j
(
2
), A0(σj(1))p/p*≤ηj(1), A0(σj(2))p/p*≤ηj(2),
(e)
A
μ
,
ς
(
α
,
β
)
(
σ
0
(
1
)
+
σ
0
(
2
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+
ς
ν
0
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p
/
(
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+
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)
≤
η
0
(
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)
+
η
0
(
2
)
-
μ
(
γ
0
(
1
)
+
γ
0
(
2
)
), Aμ(σ0(1))p/p*≤η0(1)-μγ0(1), Aμ(σ0(2))p/p*≤η0(2)-μγ0(2), where δxj, j∈J∪0, is the Dirac-mass of 1 concentrated at xj∈RN.Proof.
The proof is similar to that of the concentration compactness principle in [27, 28] (see also [20, Lemma 2.2]) and is omitted here.In order to find critical points ofE, we need the following local (PS)c condition.Lemma 10.
Suppose that(q.1) and(q.2) hold. Then the (PS)c condition in DG1,p(RN)×DG1,p(RN) holds for E if
(29)c<c0*≜1N×min⁡Aμ,ςα,βN/pQ+0N-p/p,Aμ,ςα,βN/pQ+∞N-p/p,GA0,ςα,βN/pQ+∞N-p/p.Proof.
The proof is similar to that in [12, Proposition 2]. We sketch the argument here for completeness. Suppose {(un,vn)}⊂DG1,p(RN)×DG1,p(RN) satisfies E(un,vn)→c and E′(un,vn)→0 with c<c0*. It is easy to show that {(un,vn)} is bounded in DG1,p(RN)×DG1,p(RN) and then (un,vn)⇀(u,v) up to a subsequence. Moreover, we know from Lemma 9 that there exist measures η(1), η(2), σ(1), σ(2), ν, γ(1), and γ(2) such that relations (a)–(e) of this lemma hold. Let xj≠0 be a singular point of measures η(1), η(2), and ν. As in [20], we can choose two functions ϕ1,ϕ2∈C1(RN) such that 0≤ϕ1, ϕ2≤1, ϕ1=ϕ2=1 for |x-xj|≤ϵ/2, ϕ1=ϕ2=0 for |x-xj|≥ϵ and |∇ϕ1|≤4/ϵ, |∇ϕ2|≤4/ϵ. By Lemma 7, lim⁡n→∞〈E′(un,vn),(unϕ1,vnϕ2)〉=0, and, hence, using the Sobolev inequality and the Hölder inequality, we have
(30)∫RN‍ςQxp*(ϕ1dη1+ϕ2dη2)-Q(x)vvvvv×ϕ1dσ1+ϕ2dσ2-ςQxp*(αϕ1+βϕ2)dν-∫RN‍μϕ1dγ1+ϕ2dγ2≤limsup⁡n→∞∫RN‍un∇unp-1∇ϕ1+vn∇vnp-1∇ϕ2dx≤sup⁡n≥1∫RN‍∇unpdxp-1/p×limsup⁡n→∞∫RN‍un|p∇ϕ1|pdx1/p+sup⁡n≥1∫RN‍∇vnpdxp-1/p×limsup⁡n→∞∫RN‍vn|p∇ϕ2|pdx1/p≤C∫RN‍u|p∇ϕ1|pdx1/p+∫RN‍v|p∇ϕ2|pdx1/p≤C∫Bϵxj‍up*dx1/p*∫RN‍∇ϕ1N1/Nvvv+∫Bϵxj‍vp*dx1/p*∫RN‍∇ϕ2N1/N≤C∫Bϵxj‍∇updx1/p+∫Bϵxj‍∇vpdx1/p.
Taking limits as ϵ→0 in (30), we obtain from Lemma 9 and the fact that α+β=p* that
(31)Q(xj)(σj(1)+σj(2)+ςνj)≥ηj(1)+ηj(2).
The above inequality implies that the concentration of the measures σ(1), σ(2), and ν cannot occur at points where Q(xj)≤0; that is, if Q(xj)≤0 then σj(1)=σj(2)=νj=ηj(1)=ηj(2)=0. Combining (31) and (d) of Lemma 9 we infer that either (i) σj(1)=σj(2)=νj=0 or (ii) σj(1)+σj(2)+ςνj≥(A0,ς(α,β)/∥Q+∥∞)N/p. For the point x=0, similarly as in the case xj≠0, we get
(32)η0(1)+η0(2)-μ(γ0(1)+γ0(2))-Q(0)(σ0(1)+σ0(2)+ςν0)≤0.
This, combined with (e) of Lemma 9, implies that either (iii) σ0(1)=σ0(2)=ν0=0 or (iv) σ0(1)+σ0(2)+ςν0≥(Aμ,ς(α,β)/Q+(0))N/p. To study the concentration at infinity of the sequence we need to consider the following quantities: (1)
η∞(1)=lim⁡R→∞limsup⁡n→∞∫|x|>R‍|∇un|pdx, η∞(2)=lim⁡R→∞limsup⁡n→∞∫|x|>R‍|∇vn|pdx,(2)
σ∞(1)=lim⁡R→∞limsup⁡n→∞∫|x|>R‍|un|p*dx, σ∞(2)=lim⁡R→∞limsup⁡n→∞∫|x|>R‍|vn|p*dx,(3)
ν∞=lim⁡R→∞limsup⁡n→∞∫|x|>R‍|un|α|vn|βdx,(4)
γ∞(1)=lim⁡R→∞limsup⁡n→∞∫|x|>R‍|un|p/|x|pdx, γ∞(2)=lim⁡R→∞limsup⁡n→∞∫|x|>R‍|vn|p/|x|pdx. Obviously, η∞(1), η∞(2), ν∞, γ∞(1), and γ∞(2) exist and are finite. For R>1, let ψR(1) and ψR(2) be two regular functions such that 0≤ψR(1), ψR(2)≤1, ψR(1)=ψR(2)=1 for |x|>R+1, ψR(1)=ψR(2)=0 for |x|<R and |∇ψR(1)|≤4/R, |∇ψR(2)|≤4/R. Since the sequence {(unψR(1),vnψR(2))} is bounded in D1,p(RN)×D1,p(RN), we get from (23) that
(33)0=lim⁡n→∞E′un,vn,unψR1,vnψR2=lim⁡n→∞∫RN‍unpψR1+vnpψR2xp|∇un|pψR1+|∇vn|pψR2vvvvvvvvvvvvvvvv-μunpψR1+vnpψR2xpdxvvvvvvvvvv+∫RN‍un∇unp-2∇un∇ψR1vvvvvvvvvvvvvvvvcv+vn∇vnp-2∇vn∇ψR2dxvvvvvvvvvv-∫RN‍Q(x)|un|p*ψR(1)+|vn|p*ψR(2)vvvvvvvvvvvvvvvvvvvvvvvvv+ςp*unαvnβvvvvvvvvvvvvvvvvvvvvvvvvv×αψR1+βψR2|un|p*ψR(1)+|vn|p*ψR(2)dx∫RN‍unpψR1+vnpψR2xp|∇un|pψR1+|∇vn|pψR2.
We now observe that un→u in Lp(R<|x|<R+1). Therefore, using the Sobolev inequality and the Hölder inequality we can easily check that
(34)lim⁡R→∞limsup⁡n→∞∫RN‍un∇unp-2∇un∇ψR1dx=0,lim⁡R→∞limsup⁡n→∞∫RN‍vn∇vnp-2∇vn∇ψR2dx=0.
Consequently, taking into account the definitions (1)–(4) of η∞(1), η∞(2), σ∞(1), σ∞(2), ν∞, γ∞(1), and γ∞(2), we deduce from (33) and (34) that
(35)Q(∞)(σ∞(1)+σ∞(2)+ςν∞)≥η∞(1)+η∞(2)-μ(γ∞(1)+γ∞(2)).
On the other hand, by (5) and the definition (25) of Aμ,ς(α,β) we easily see that μ¯γ∞(1)≤η∞(1), μ¯γ∞(2)≤η∞(2) and
(36)Aμ,ς(α,β)(σ∞1+σ∞(2)+ςν∞)p/α+β≤η∞(1)+η∞(2)-μ(γ∞(1)+γ∞(2)).
This, combined with (35), implies that either (v) σ∞(1)=σ∞(2)=ν∞=0 or (vi) σ∞(1)+σ∞(2)+ςν∞≥(Aμ,ς(α,β)/Q+(∞))N/p. We now rule out the cases (ii), (iv), and (vi). For every continuous nonnegative function ψ such that 0≤ψ(x)≤1 on RN, we obtain from (23) and (24) that
(37)c=lim⁡n→∞Eun,vn-1p*E′un,vn,un,vn=1Nlim⁡n→∞∫RN‍(|∇un|p+|∇vn|p-μ|un|p+|vn|p|x|p)dx≥1Nlimsup⁡n→∞∫RN‍|un|p+|vn|p|x|p|∇un|p+|∇vn|pvvvvvvvvvvvv-μ|un|p+|vn|p|x|pψ(x)dx.
If (ii) occurs, then the set J must be finite because the measures σ(1), σ(2), and ν are bounded. Since functions (un,vn) are G-symmetric, the measures σ(1), σ(2), and ν must be G-invariant. This means that if xj≠0 is a singular point of σ(1), σ(2), and ν, so is gxj for each g∈G, and the mass of σ(1), σ(2), and ν concentrated at gxj is the same for each g∈G. If we assume the existence of j∈J with xj≠0 such that (ii) holds, then we choose ψ with compact support so that ψ(gxj)=1 for each g∈G and we obtain
(38)c≥1NG(ηj(1)+ηj(2))≥1NGA0,ςα,β(σj1+σj2+ςνj)p/α+β≥1NG(A0,ς(α,β))N/pQ+∞-N-p/p,
which contradicts (29). Similarly, if (iv) holds for x=0, we choose ψ with compact support, so that ψ(0)=1 and we have
(39)c≥1N(η0(1)+η0(2)-μγ0(1)-μγ0(2))≥1NAμ,ςα,βσ01+σ02+ςν0p/α+β≥1NAμ,ςα,βN/p(Q+(0))-N-p/p,
which is impossible. Finally, if (vi) occurs at ∞, we take ψ=ψR(1)=ψR(2) to get
(40)c≥1N(η∞(1)+η∞(2)-μγ∞(1)-μγ∞(2))≥1NAμ,ς(α,β)σ∞1+σ∞2+ςν∞p/α+β≥1NAμ,ςα,βN/pQ+∞-N-p/p,
a contradiction with (29). Consequently, σj(1)=σj(2)=νj=0 for all j∈J∪{0,∞} and this implies that
(41)lim⁡n→∞∫RN‍(|un|p*+|vn|p*+ς|un|α|vn|β)dx=∫RN‍(|u|p*+|v|p*+ς|u|α|v|β)dx.
Finally, observe that E′(u,v)=0 and, hence, by lim⁡n→∞〈E′(un,vn)-E′(u,v),(un-u,vn-v)〉=0, we obtain (un,vn)→(u,v) as n→∞ in D1,p(RN)×D1,p(RN). The assertion follows.As an immediate consequence of Lemma10 we obtain the following result.Corollary 11.
IfQ+(0)=Q+(∞)=0 and |G|=+∞, then the functional E satisfies the (PS)c condition for every c∈R.Proof of Theorem1.
Firstly, we chooseϵ>0 such that the condition (17) holds, where yϵ is the extremal function satisfying (10), (11), and (12). By (q.1), (23), and (25), we have
(42)Eu,v=1p(u,v)μp-1p*∫RN‍Q(x)(up*+vp*+ςuαvβ)dx≥1p(u,v)μp-1p*Q∞Aμ,ςα,β-α+β/p(u,v)μα+β.
Hence there exist constants α0>0 and ρ>0 such that E(u,v)≥α0 for all ∥(u,v)∥μ=ρ. Furthermore, if we set u=yϵ, v=τmin⁡yϵ and
(43)Φt=E(tu,tv)=E(tyϵ,tτmin⁡yϵ)=tpp∫RN‍1+τmin⁡p∇yϵp-μyϵpxpdx-tp*p*∫RN‍1+ςτmin⁡β+τmin⁡α+βQxyϵα+βdx,
with t≥0, then we can check that Φ(t) has a unique maximum at some t¯>0. A simple computation gives us the value
(44)t¯=1+τmin⁡p∫RN‍∇yϵp-μyϵp/xpdx1+ςτmin⁡β+τmin⁡α+β∫RN‍Q(x)yϵα+βdx1/α+β-p.
Consequently, we obtain from (26) and (27) that
(45)max⁡t≥0Φt=E(t¯yϵ,t¯τmin⁡yϵ)=1N×Kτmin⁡∫RN‍∇yϵp-μyϵp/xpdx∫RN‍Qxyϵα+βdxp/α+βα+β/α+β-p.
Since Φ(t)→-∞ as t→∞, we can choose t0>0 such that E(t0yϵ,t0τmin⁡yϵ)<0 and ∥(t0yϵ,t0τmin⁡yϵ)∥μ>ρ and set
(46)c0=inf⁡γ∈Γmax⁡t∈[0,1]E(γ(t)),
where
(47)Γ=γ∈C0,1,DG1,pRN×DG1,pRN;vvvγ0=0,0,γ1=t0yϵ,t0τmin⁡yϵDG1,p(RN).
From (11), (17), (29), (45), (46), and Lemma 8, we obtain that(48)c0≤E(t¯yϵ,t¯τmin⁡yϵ)=1NKτmin⁡∫RN‍∇yϵp-μyϵp/xpdx∫RN‍Qxyϵα+βdxp/α+βα+β/α+β-p≤1NKτmin⁡∫RN‍∇yϵp-μ|yϵ|p/|x|pdxmax⁡⁡Q+0/Aμα+β/p,Q+∞/Aμα+β/p,∥Q+∥∞/|G|α+β-p/pA0α+β/pp/α+βα+β/α+β-p=1Nmin⁡Aμ,ςα,βN/pQ+0N-p/p,Aμ,ςα,βN/pQ+∞N-p/p,|G|A0,ςα,βN/p∥Q+∥∞N-p/p=c0*.If c0<c0*, then we conclude from Lemma 10 that the (PS)c condition holds and the conclusion follows by the mountain pass theorem in [29] (see also [30]). If c0=c0*, then γ(t)=(tt0yϵ,tt0τmin⁡yϵ), with 0≤t≤1, is a path in Γ such that max⁡t∈[0,1]E(γ(t))=c0. Thus, either Φ′(t¯)=E′(t¯yϵ,t¯τmin⁡yϵ)=0 and we are done, or γ can be deformed to a path γ~∈Γ with max⁡t∈[0,1]E(γ~(t))<c0, which is impossible. Hence we have a nontrivial solution (u0,v0)∈DG1,p(RN)×DG1,p(RN) to problem (P0Q). In the following, we have just to show that the solution (u0,v0) can be chosen to be positive on RN. Consider the Nehari manifold
(49)Mμ≜u,v∈DG1,pRN×DG1,pRN∖0,0;vvE′u,v,u,v=0.
Writing an arbitrary element (u,v)∈Mμ as (u,v)=t(u~,v~)(t>0), with ∥(u~,v~))∥μ=1, we deduce from (24), (25), and the fact that〈E′(tu~,tv~),(tu~,tv~)〉=0 that
(50)0=1-tp*-p∫RN‍Qxu~p*+v~p*+ςu~αv~βdx≥1-Aμ,ςα,β-α+β/pQ∞u~,v~μα+β/ptp*-p≥1-Ctp*-p.
This implies that t≥ξ0, with a constant ξ0>0 independent of (u,v). Thus we conclude that the set Mμ is bounded away from 0 and inf⁡(u,v)∈MμE(u,v)>0. Set
(51)Gu,v=〈E′(u,v),(u,v)〉=u,vμp-∫RN‍Qxup*+vp*+ςuαvβdx.
Then 〈G′(u,v),(u,v)〉=(p-p*)∥(u,v)∥μp≠0, and, hence, Mμ is a C1-manifold. Notice that (u0,v0)∈Mμ and set c¯0=inf⁡(u,v)∈MμE(u,v). We now claim c¯0=c0. Indeed, if c¯0<c0, then we can find (u0*,v0*)∈Mμ such that E(u0*,v0*)<c0. Consequently, we deduce that
(52)∫RN‍Q(x)(u0*p*+|v0*|p*+ς|u0*|α|v0*|β)dx=(u0*,v0*)μp>0.
By a straightforward calculation, we get
(53)sup⁡t≥0Etu0*,tv0*=E(u0*,v0*)=1N∫RN‍Qxu0*p*+v0*p*+ςu0*αv0*βdx<c0.
Setting γ(t)=(tt0*u0*,tt0*v0*) for t∈[0,1], with t0* such that ∥(t0*u0*,t0*v0*)∥μ>ρ, we obtain that E(γ(t))=E(tt0*u0*,tt0*v0*)<c0 on [0,1], a contradiction with the definition (46) of c0. Hence we have c¯0=c0. Finally, by the strong maximum principle, we obtain u0>0 and v0>0 on RN. This, combined with (24) and Lemma 7, implies that (u0,v0) is a positive G-symmetric solution of (P0Q).Proof of Corollary2.
First of all, we observe that, due to the identity (12), inequality (17) is equivalent to ∫RN‍(Q(x)-Q¯)yϵα+βdx≥0 for some ϵ>0, or equivalently
(54)ϵ-α+βϑ∫RN‍Qx-Q¯Uμxϵα+βdx≥0,
for some ϵ>0, where Q¯=max⁡{Q+(0),Q+(∞),|G|p-(α+β)/p(A0/Aμ)-α+β/p∥Q+∥∞} and ϑ=(N-p)/p. Part (1), case (i): according to (54), we need to show that
(55)ϵ-α+βl2∫RN‍Qx-Q0Uμxϵα+βdx≥0,
for some ϵ>0. We choose ϱ0>0 so that Q(x)≥Q(0)+Λ0|x|(α+β)(l2-ϑ) for |x|≤ϱ0. This, combined with (14) and (16) and the fact that -(α+β)ϑ=-N, implies that
(56)ϵ-(α+β)l2∫|x|≤ϱ0‍(Q(x)-Q(0))Uμxϵα+βdx≥Λ0∫x≤ϱ0‍x-α+βϑxϵl2Uμxϵα+βdx⟶+∞,
as ϵ→0. On the other hand, for all ϵ>0, we have
(57)ϵ-α+βl2∫x>ϱ0‍Qx-Q0Uμxϵα+βdx≤∫x>ϱ0‍Qx-Q0xα+βl2xϵl2Uμxϵα+βdx≤C1~,
for some constant C1~>0 independent of ϵ. Combining (56) and (57), we get (55) for ϵ sufficiently small.
Part (1), case (ii): we chooseϱ1>0 so that |Q(x)-Q(0)|≤Λ1|x|κ for |x|≤ϱ1. Since κ>(α+β)(l2-ϑ)>0, we deduce from (14) and the fact that (α+β)l2=N+(α+β)(l2-ϑ) that
(58)ϵ-α+βl2∫RN‍Qx-Q0Uμxϵα+βdx=∫RN‍Qx-Q0|x|(α+β)l2xϵl2Uμxϵα+βdx≤C∫RN‍Qx-Q0|x|N+(α+β)(l2-ϑ)dx≤CΛ1∫|x|≤ϱ1‍|x|κ-N-α+βl2-ϑdxvvv+∫|x|>ϱ1‍Qx-Q0x-N-α+βl2-ϑdx≤C.
So by (14), (16), (19), and the Lebesgue dominated convergence theorem we obtain that
(59)lim⁡ϵ→0ϵ-α+βl2∫RN‍Qx-Q0Uμxϵα+βdx=C∫RN‍Qx-Q0|x|-N-(α+β)(l2-ϑ)dx>0.
Hence, (55) holds for ϵ sufficiently small.
Part (2), case (i): from (54) it is sufficient to show that
(60)ϵ-α+βl1∫RN‍Qx-Q∞Uμxϵdxα+β≥0,
for some ϵ>0. We choose R1>0 such that Q(x)≥Q(∞)+Λ2|x|-(α+β)(ϑ-l1) for all |x|≥R1. Then
(61)ϵ-α+βl1∫x≥R1‍Qx-Q∞Uμxϵα+βdx=∫|x|≥R1‍Q(x)-Q(∞)|x|(α+β)l1xϵl1Uμxϵα+βdx≥Λ2∫x≥R1‍x-Nxϵl1Uμxϵα+βdx⟶+∞,
as ϵ→∞. Moreover, in view of (α+β)l1=N-(α+β)(ϑ-l1) and (α+β)(ϑ-l1)>0, we obtain
(62)ϵ-α+βl1∫x≤R1‍Qx-Q∞Uμxϵα+βdx=∫x≤R1‍Qx-Q∞|x|α+βl1xϵl1Uμxϵα+βdx≤C∫x≤R1‍Qx-Q∞x-N+α+βϑ-l1dx≤C2~,
for some constant C2~>0 independent of ϵ>0. These two estimates combined together give (60) for ϵ>0 large.
Part (2), case (ii): we chooseR2>0 such that |Q(x)-Q(∞)|≤Λ3|x|-ι for all |x|≥R2. Since ι>(α+β)(ϑ-l1)>0 and (α+β)l1=N-(α+β)(ϑ-l1), we get
(63)∫RN‍Qx-Q∞|x|-N+α+βϑ-l1dx≤Λ3∫x≥R2‍x-N-ι+α+βϑ-l1dx+∫x≤R2‍Qx-Q∞x-N+α+βϑ-l1dx<∞.
Consequently, by (13), (16), (21), and the Lebesgue dominated convergence theorem, we have
(64)lim⁡ϵ→∞ϵ-(α+β)l1∫RN‍(Q(x)-Q(∞))Uμxϵα+βdx=lim⁡ϵ→∞∫RN‍Qx-Q∞|x|α+βl1xϵl1Uμxϵα+βdx=C∫RN‍(Q(x)-Q(∞))x-N+α+βϑ-l1dx>0,
and (60) holds for ϵ>0  large. Similarly as above, we know part (3) holds.To prove Theorem3 we need the following version of the symmetric mountain pass theorem (cf. [31, Theorem 9.12]).Lemma 12.
LetX be an infinite dimensional Banach space and let E∈C1(X,R) be an even functional satisfying (PS)c condition for each c and E(0)=0. Furthermore, one supposes that (i)
there exist constantsα¯>0 and ρ>0 such that E(w)≥α¯ for all ∥w∥=ρ;(ii)
there exists an increasing sequence of subspaces{Xm} of X, with dim⁡Xm=m, such that for every m one can find a constant Rm>0 such that E(w)≤0 for all w∈Xm with ∥w∥≥Rm. Then E possesses a sequence of critical values {cm} tending to ∞ as m→∞.Proof of Theorem3.
We follow the arguments of [12]. Applying Lemma 12 with X=DG1,p(RN)×DG1,p(RN) and (u,v)=w∈X, we see from (q.1), (23), and (25) that
(65)Eu,v≥1p(u,v)μp-1p*Q∞(Aμ,ς(α,β))-α+β/p(u,v)μα+β.
Since α+β=p*>p, there exist constants α¯>0 and ρ>0 such that E(u,v)≥α¯ for all (u,v) with ∥(u,v)∥μ=ρ. To find a suitable sequence of finite dimensional subspaces of DG1,p(RN)×DG1,p(RN), we set Ω={x∈RN;Q(x)>0}. Obviously, the set Ω is G-symmetric and we can define DG1,p(Ω)×DG1,p(Ω), which is the subspace of G-symmetric functions of D1,p(Ω)×D1,p(Ω) (see Section 2). By extending functions in DG1,p(Ω)×DG1,p(Ω) by 0 outside Ω we can assume that DG1,p(Ω)×DG1,p(Ω)⊂DG1,p(RN)×DG1,p(RN). Let {Xm} be an increasing sequence of subspaces of DG1,p(Ω)×DG1,p(Ω) with dim⁡Xm=m for each m. Then we deduce that there exists a constant σ(m)>0 such that
(66)∫Ω‍Qxu~p*+v~p*+ςu~|αv~βdx≥σm,
for all (u~,v~)∈Xm, with ∥(u~,v~)∥μ=1. Consequently, if (u,v)∈Xm∖{(0,0)} then we write (u,v)=t(u~,v~), with t=∥(u,v)∥μ and ∥(u~,v~)∥μ=1. Hence we obtain
(67)Eu,v=1ptp-1p*tp*∫Ω‍Q(x)(u~p*+|v~|p*+ς|u~|α|v~|β)dx≤1ptp-σmp*tp*≤0,
for t large enough. Therefore we conclude from Lemma 12 and Corollary 11 that there exists a sequence of critical values cm→∞ and the results follow.Proof of Corollary4.
SinceQ(x) is radially symmetric, that is, Q(x)=Q(|x|), we easily see that the corresponding group G=O(N) and |G|=+∞. According to Corollary 11, E satisfies the (PS)c condition for every c∈R. Hence, by applying the proof of Theorem 3 the conclusion follows.


## 4. Multiplicity Results for Problem(PλQ~)

Throughout this section we assume thatλ>0 and Q(x)≡Q~>0 is a constant. Since we are interested in positive G-symmetric solutions of problem (PλQ~), we define a functional Fλ:DG1,p(RN)×DG1,p(RN)→R given by
(68)Fλu,v=1p(u,v)μp-Q~p*∫RN‍u+p*+v+p*+ςu+αv+βdx-λq∫RN‍(h1(x)|u+|q+h2(x))|v+|qdx,
where 1<q<p<N, u+=max⁡{0,u} and v+=max⁡{0,v}. By (h.1), (h.2), and the Hölder inequality, we easily see that Fλ∈C1(DG1,p(RN)×DG1,p(RN),R) and there exists a one-to-one correspondence between the weak solutions of (PλQ~) and the critical points of Fλ. Furthermore, an analogously symmetric principle of Lemma 7 clearly holds; hence, the weak solutions of problem (PλQ~) are exactly the critical points of the functional Fλ.Lemma 13.
Suppose that(h.1) and(h.2) hold. Then there exists a positive constant M depending on N, p, q, Aμ, ∥h1∥θ, and ∥h2∥θ, such that any bounded sequence {(un,vn)}⊂DG1,p(RN)×DG1,p(RN), satisfying
(69)Fλ(un,vv)⟶c<1NQ~p-N/pAμ,ςα,βN/p-Mλp/p-q,Fλ′un,vv⟶0n⟶∞,
contains a convergent subsequence.Proof.
Since{(un,vn)} is bounded in DG1,p(RN)×DG1,p(RN), we can obtain a subsequence, still denoted by {(un,vn)}, satisfying
(70)un⇀u,vn⇀v,inDG1,p(RN),un⟶u,vn⟶v,a.e.inRN,∇un⟶∇u,∇vn⟶∇v,a.e.inRN.
Moreover, using (h.2) and the Hölder inequality and the Lebesgue dominated theorem, we may also assume
(71)∫RN‍h1(x)|un+|qdx⟶∫RN‍h1(x)|u+|qdx,∫RN‍h2xvn+qdx⟶∫RN‍h2xv+qdx,
as n→∞. By (71) and the standard argument, we easily show that (u,v) is a critical point of Fλ. Consequently, we deduce from (68), (9), (h.2), the Hölder inequality, and the fact that 1<q<p<p* that
(72)Fλu,v=1N(un,vn)μp-(1q-1p*)λ×∫RN‍(h1(x)|u+|q+h2(x)|v+|q)dx≥1Nu,vμp-p*-qqp*Aμ-q/pλ×h1θuμq+h2θvμq≥1N(uμp+vμp)-p*-qqp*Aμ-q/pλ×max⁡⁡h1θ,h2θuμq+vμq≥-2p-qpqNpq/p-q×p*-qqp*Aμ-q/pmax⁡h1θ,h2θp/p-q×λp/p-q≜-Mλp/p-q,
where M=2(p-q)/p)(qN/p)q/p-q[p*-q/qp*Aμ-q/pmax⁡{∥h1∥θ,∥h2∥θ}]p/p-q) is a positive constant. Now we set u¯n=un-u and v¯n=vn-v. Then by the Brezis-Lieb lemma [32] and arguing as in [33, Lemma 2.1] we get
(73)(u¯n,v¯n)μp=(un,vn)μp-(u,v)μp+on(1),∫RN‍|u¯n+|p*dx=∫RN‍|un+|p*dx-∫RN‍|u+|p*dx+on(1),∫RN‍|v¯n+|p*dx=∫RN‍|vn+|p*dx-∫RN‍|v+|p*dx+on(1),∫RN‍u¯n+αv¯n+βdx=∫RN‍|un+|α|vn+|βdx-∫RN‍|u+|α|v+|βdx+on(1).
Since Fλ(un,vn)=c+on(1) and Fλ′(un,vn)=on(1), we obtain from (68), (71), and (73) that
(74)1pu¯n,v¯nμp-Q~p*∫RN‍u¯n+p*+v¯n+p*+ςu¯n+αv¯n+βdx=c-Fλu,v+on1,u¯n,v¯nμp-Q~∫RN‍u¯n+p*+v¯n+p*+ςu¯n+αv¯n+βdx=on(1).
Hence, for a subsequence {(u¯n,v¯n)}, we have
(75)(u¯n,v¯n)μp⟶k≥0,Q~∫RN‍u¯n+p*+v¯n+p*+ςu¯n+|αv¯n+|βdx⟶k,
as n→∞. From the definition (25) of Aμ,ς(α,β) it follows that Aμ,ς(α,β)(k/Q~)p/α+β≤k, which implies either k=0 or k≥Q~p-N/p(Aμ,ς(α,β))N/p. If k≥Q~p-N/p(Aμ,ς(α,β))N/p, we obtain from (72) and (74) that
(76)c=Fλu,v+1p-1p*k≥1NQ~p-N/pAμ,ςα,βN/p-Mλp/p-q,
which contradicts (69). Consequently, we have ∥(u¯n,v¯n)∥μp→0 as n→∞, and, thus, (un,vn)→(u,v) in DG1,p(RN)×DG1,p(RN). The lemma is proved.Lemma 14.
Suppose that(h.1) and(h.2) hold. Then there exists λ1*>0 such that for any λ∈(0,λ1*) the following geometric conditions for Fλ(u,v) hold: (i)
Fλ(0,0)=0; there exist α~>0 and ρ>0 such that Fλ(u,v)≥α~ for all ∥(u,v)∥μ=ρ;(ii)
there exists(eu,ev)∈DG1,p(RN)×DG1,p(RN) such that ∥(eu,ev)∥μ>ρ and Fλ(eu,ev)<0.Proof.
According to (h.1) and (h.2), for all0<σ<1/p, we deduce from (9), (25), (68), the Young inequality, and the Hölder inequality that
(77)Fλ(u,v)≥1p(u,v)μp-1p*Q~Aμ,ςα,β-α+β/p(u,v)μα+β-λqAμ-q/pmax⁡{h1θ,h2θ}(uμq+vμq)≥1p(u,v)μp-1p*Q~Aμ,ςα,β-α+β/p×u,vμα+β-Cλu,vμq≥(1p-σ)(u,v)μp-1p*Q~Aμ,ςα,β-α+β/p×(u,v)μα+β-C(σ)λp/p-q,
where C(σ)>0 is a constant depending on σ>0. The last inequality and the fact α+β=p*>p imply that, for small σ, there exist constants α~>0, ρ>0, and λ1*>0 such that Fλ(u,v)≥α~>0 for all ∥(u,v)∥μ=ρ and 0<λ<λ1*. On the other hand, since ∫RN‍(h1(x)|u+|q+h2(x)|v+|q)dx≥0, we conclude from (68) that there exists (u~,v~)∈DG1,p(RN)×DG1,p(RN)∖{(0,0)} such that Fλ(tu~,tv~)→-∞ as t→+∞, which completes this proof.Lemma 15.
Suppose that(h.1) and(h.2) hold. Then there exists λ2*>0 such that
(78)sup⁡t≥0Fλ(tyϵ,tτmin⁡yϵ)<1NQ~p-N/p(Aμ,ς(α,β))N/p-Mλp/p-q
for any λ∈(0,λ2*) and small ϵ>0, where τmin⁡>0 satisfies (26)–(28) and M>0 is given in Lemma 13.Proof.
First, we define the functions(79)Ψt=Fλ(tyϵ,tτmin⁡yϵ)=tpp(1+τmin⁡p)×∫RN‍(|∇yϵ|p-μ|yϵ|p|x|p)dx-Q~tp*p*(1+ςτmin⁡β+τmin⁡α+β)∫RN‍|yϵ|p*dx-λtqq∫RN‍h1x+τmin⁡qh2xyϵqdx,(80)Ψ~t=tpp1+τmin⁡p∫RN‍∇yϵp-μyϵpxpdx-Q~tp*p*1+ςτmin⁡β+τmin⁡α+β∫RN‍yϵp*dx,
with t≥0. Note that Ψ~(0)=0, Ψ~(t)>0 for t→0+, and lim⁡t→+∞Ψ~(t)=-∞. Hence sup⁡t≥0Ψ~(t) can be achieved at some finite t~ϵ>0 at which Ψ~′(t) becomes zero. By direct calculation, we obtain from (11), (12), (26), (27), (80), and Lemma 8 that
(81)sup⁡t≥0Ψ~(t)=Ψ~(t~ϵ)=1N×1+τmin⁡p∫RN‍∇yϵp-μ|yϵ|p/|x|pdx(Q~(1+ςτmin⁡β+τmin⁡α+β)∫RN‍|yϵ|p*dx)p/p*p*/p*-p=1NQ~p-N/pKτmin⁡AμN/p=1NQ~p-N/pAμ,ςα,βN/p.
Let Λ¯>0 be such that 1/NQ~p-N/p(Aμ,ς(α,β))N/p-Mλp/p-q>0, ∀λ∈(0,Λ¯). Then from (h.1), (h.2), (11), and (79), we have
(82)Ψ(t)=Fλ(tyϵ,tτmin⁡yϵ)≤tpp(1+τmin⁡p),∀t≥0,λ>0,
and there exists T0∈(0,1) independent of ϵ such that
(83)sup⁡0≤t≤T0Ψt≤T0pp1+τmin⁡p<1NQ~p-N/p(Aμ,ς(α,β))N/p-Mλp/p-q,∀λ∈(0,Λ¯).
Moreover, we obtain from (79)–(81) that
(84)sup⁡t≥T0Ψ(t)≤sup⁡t≥0Ψ~(t)vvvvvvvvv-λqT0q∫RN‍(h1(x)+τmin⁡qh2(x))|yϵ|qdxvvvvvvv=1NQ~p-N/p(Aμ,ς(α,β))N/pvvvvvvvvv-λqT0q∫RN‍(h1(x)+τmin⁡qh2(x))|yϵ|qdx.
Now, taking λ>0 such that
(85)-λqT0q∫RN‍h1x+τmin⁡qh2x|yϵ|qdx<-Mλp/p-q,
that is,
(86)0<λ<T0qqM∫RN‍h1x+τmin⁡qh2xyϵqdxp-q/p≜Λ~,
we have
(87)sup⁡t≥T0Ψt<1NQ~p-N/pAμ,ςα,βN/p-Mλp/p-q,∀λ∈(0,Λ~).
Choosing λ2*=min⁡{Λ¯,Λ~}, we deduce from (83) and (87) that
(88)sup⁡t≥0Ψt<1NQ~p-N/pAμ,ςα,βN/p-Mλp/p-q,v∀λ∈(0,λ2*).
Therefore the result of this lemma follows.Proof of Theorem5.
Takingρ>0 and λ*=min⁡{λ1*,λ2*}, for 0<λ<λ*, given in the proofs of Lemmas 14 and 15, we define
(89)c1≜inf⁡B¯ρ(0)Fλ(u,v),
where B¯ρ(0)={(u,v)∈DG1,p(RN)×DG1,p(RN);∥(u,v)∥μ≤ρ}. Since the metric space B¯ρ(0) is complete, we deduce from the Ekeland variational principle [34] that there exists a sequence {(un,vn)}⊂B¯ρ(0) such that Fλ(un,vn)→c1 and Fλ′(un,vn)→0 as n→∞. Let φ0,ψ0∈C0∞(RN) be the G-symmetric functions such that φ0,ψ0>0. By (h.1) and (h.2), we have ∫RN‍(h1(x)φ0q+h2(x)ψ0q)dx>0. This, combined with the fact that 1<q<p<p*, implies that there exists t0=t0(φ0,ψ0)>0 sufficiently small such that
(90)Fλt0φ0,t0ψ0=t0pp(φ0,ψ0)μp-Q~p*t0p*∫RN‍φ0p*+ψ0p*+ςφ0αψ0βdx-λqt0q∫RN‍(h1(x)φ0q+h2(x)ψ0q)dx<0.
Therefore we obtain c1<0<1/NQ~p-N/p(Aμ,ς(α,β))N/p-Mλp/p-q for any λ∈(0,λ*). By Lemma 13, Fλ possesses a critical point (u1,v1) with Fλ(u1,v1)=c1<0. Taking (u1-,v1-) as a pair of test functions, where u1-=min⁡{0,u1} and v1-=min⁡{0,v1}, we deduce from (68) that 0=〈Fλ′(u1,v1),(u1-,v1-)〉=∥(u1-,v1-)∥μp, which implies u1≥0 and v1≥0 in RN. Consequently, by the strong maximum principle and the symmetric criticality principle, we conclude that (u1,v1) is a positive G-symmetric solution of problem (PλQ~).
On the other hand, we define(91)c2≜inf⁡γ∈Γmax⁡t∈[0,1]Fλ(γ(t)),
where Γ={γ∈C([0,1],DG1,p(RN)×DG1,p(RN));γ(0)=(0,0),γ(1)=(eu,ev)}. It follows from Lemmas 14 and 15 that
(92)0<α~≤c2<1NQ~p-N/pAμ,ςα,βN/p-Mλp/p-q,v∀λ∈(0,λ*).
Hence c2 is a critical value of Fλ by the mountain pass theorem. Similar to the arguments above, problem (PλQ~) admits another positive G-symmetric solution (u2,v2) with Fλ(u2,v2)=c2>0.



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*Source: 430976-2014-11-16.xml*