Multiple Solutions for a Singular Quasilinear Elliptic System

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∈ℝ^𝑁, 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.

1. Introduction

In this paper, we consider the multiplicity of solutions to a class of quasilinear elliptic system −div|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2∇𝑢+𝑓₁(𝑥)|𝑢|^𝑝−2𝑢=𝛼𝑔(𝑥𝑢|𝛼+𝛽)|^𝛼−2𝑢|𝑣|^𝛽+𝜆ℎ₁(𝑥)|𝑢|^𝛾−2𝑢+𝑙₁(𝑥),−div|𝑥|^−𝑎𝑝|∇𝑣|^𝑝−2∇𝑣+𝑓₂(𝑥)|𝑣|^𝑝−2𝑣=𝛽𝑔(𝑥𝑣|𝛼+𝛽)|^𝛽−2𝑣|𝑢|^𝛼+𝜇ℎ₂(𝑥)|𝑣|^𝛾−2𝑣+𝑙₂(𝑥),𝑢(𝑥)>0,𝑣(𝑥)>0,𝑥∈ℝ^𝑁, (1) 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 −div𝑎(𝑥)|∇𝑢|^𝑝−2∇𝑢+𝑏(𝑥)|𝑢|^𝑝−2𝑢=𝑔(𝑥)|𝑢|^𝑟−2𝑢,inΩ,𝑢=0,on𝜕Ω, (2) where Ω is a smooth exterior domain in ℝ^𝑁, 1<𝑝<𝑁, 𝑎(𝑥)∈𝐿^∞(Ω)∩𝐶^0,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: −div|∇𝑢|^𝑝−2∇𝑢=𝜆|𝑢|^𝑞−2𝑢+2𝛼|𝑢|𝛼+𝛽^𝛼−2𝑢|𝑣|^𝛽||,𝑥∈Ω,−div∇𝑣^𝑝−2∇𝑣=𝜇|𝑣|^𝑞−2𝑣+2𝛽|𝑣|𝛼+𝛽^𝛽−2𝑣|𝑢|^𝛼,𝑥∈Ω,𝑢=𝑣=0,𝑥∈𝜕Ω, (3) where Ω⊂ℝ^𝑁 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<𝜆^{𝑝/(𝑝−𝑞)}+𝜇^{𝑝/(𝑝−𝑞)}<𝜆₁ and has at least two positive solutions when 0<𝜆^{𝑝/(𝑝−𝑞)}+𝜇^{𝑝/(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)ℎ_𝑖∈𝐿^∞(ℝ^𝑁), ℎ_𝑖(𝑥)|𝑥|^𝑏𝛾∈𝐿^𝜃(ℝ^𝑁), 𝜃=𝑝^∗/(𝑝^∗−𝛾), 𝑖=1,2.(𝐻₂)𝑔(𝑥)|𝑥|^{𝑏(𝛼+𝛽)}∈𝐿^𝛿(ℝ^𝑁)∩𝐿^∞(ℝ^𝑁), 𝛿=𝑝^∗/(𝑝^∗−𝛼−𝛽).(𝐻₃)𝑙₁(𝑥)|𝑥|^𝑏, 𝑙₂(𝑥)|𝑥|^𝑏∈𝐿^𝜎(Ω), 𝜎=𝑝^∗/(𝑝^∗−1), 𝑙₁(𝑥)>0, 𝑙₂(𝑥)>0.

In this paper, we let 𝐿^𝑝_𝑏(ℝ^𝑁) the completion of the space 𝐶^∞₀(ℝ^𝑁) with respect to the norm ‖𝑢‖_𝐿^𝑝_𝑏=(∫_ℝ𝑁|𝑥|^−𝑏𝑝|𝑢|^𝑝𝑑𝑥)^1/𝑝, and let 𝑊_𝑎^1,𝑝(ℝ^𝑁) the completion of 𝐶^∞₀(ℝ^𝑁) with respect to the norm ‖𝑢‖_𝑊𝑎^1,𝑝=(∫_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥)^1/𝑝.

We define ℎ_1𝜃=_ℝ𝑁ℎ₁(𝑥)|𝑥|^𝑏𝛾𝜃𝑑𝑥^1/𝜃,ℎ_2𝜃=_ℝ𝑁ℎ₂(𝑥)|𝑥|^𝑏𝛾𝜃𝑑𝑥^1/𝜃,𝑙_1𝜎=_ℝ𝑁|𝑥|^𝑏𝜎𝑙₁(𝑥)^𝜎𝑑𝑥^1/𝜎,𝑙_2𝜎=_ℝ𝑁|𝑥|^𝑏𝜎𝑙₂(𝑥)^𝜎𝑑𝑥^1/𝜎, (4) 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 _ℝ𝑁|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥^1/𝑝∗≤𝑆_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥^1/𝑝, (5) where −∞<𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏<𝑎+1, 𝑑=𝑎+1−𝑏, and 𝑝^∗=𝑝𝑁/(𝑁−𝑝𝑑).

Let 𝑋=𝑊_𝑎^1,𝑝(ℝ^𝑁)×𝑊_𝑎^1,𝑝(ℝ^𝑁) denote the usual Sobolev space with the norm ‖(𝑢,𝑣)‖=_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝+𝑓₁(𝑥)|𝑢|^𝑝+𝑑𝑥_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑣|^𝑝+𝑓₂(𝑥)|𝑣|^𝑝𝑑𝑥^1/𝑝. (6) 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 _ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2∇𝑢∇𝜑+𝑓₁(𝑥)|𝑢|^𝑝−2+𝑢𝜑𝑑𝑥_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑣|^𝑝−2∇𝑣∇𝜓+𝑓₂(𝑥)|𝑣|^𝑝−2−𝑣𝜓𝑑𝑥_ℝ𝑁𝑙₁(𝑥)𝜑𝑑𝑥−_ℝ𝑁𝑙₂(𝑥)−𝛼𝜓𝑑𝑥𝛼+𝛽_ℝ^𝑁𝑔(𝑥)|𝑢|^𝛼−2𝑢|𝑣|^𝛽+𝜆ℎ₁(𝑥)|𝑢|^𝛾−2𝑢−𝛽𝜑𝑑𝑥𝛼+𝛽_ℝ^𝑁𝑔(𝑥)|𝑣|^𝛽−2𝑣|𝑢|^𝛼+𝜇ℎ₂(𝑥)|𝑣|^𝛾−2𝑣𝜓𝑑𝑥=0. (7)
It is clear that problem (1) has a variational structure. Let 𝐽_𝜆,𝜇∶𝑋→ℝ¹ be the energy functional corresponding to problem (1) which is defined by 𝐽_𝜆,𝜇()=1𝑢,𝑣𝑝‖(𝑢,𝑣)‖^𝑝−1𝛼+𝛽_ℝ^𝑁𝑔(𝑥)𝑢|^𝛼𝑣|^𝛽−1𝑑𝑥𝛾𝐺_𝜆,𝜇(𝑢,𝑣)−𝐿(𝑢,𝑣), (8) where 𝐺_𝜆,𝜇(𝑢,𝑣)=_ℝ𝑁𝜆ℎ₁(𝑥)|𝑢|^𝛾𝑑𝑥+_ℝ𝑁𝜇ℎ₂(𝑥)|𝑣|^𝛾𝐿()=𝑑𝑥,𝑢,𝑣_ℝ𝑁𝑙₁(𝑥)𝑢+𝑙₂(𝑥)𝑣𝑑𝑥. (9) Then, we see that functional 𝐽_𝜆,𝜇∈𝐶¹(𝑋,ℝ¹) and for all (𝜑,𝜓)∈𝑋, there holds 𝐽^′_𝜆,𝜇(𝑢,𝑣),𝜑,𝜓=_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2∇𝑢∇𝜑+𝑓₁(𝑥)|𝑢|^𝑝−2+𝑢𝜑𝑑𝑥_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑣|^𝑝−2∇𝑣∇𝜓+𝑓₂(𝑥)|𝑣|^𝑝−2−𝛼𝑣𝜓𝑑𝑥𝛼+𝛽_ℝ^𝑁𝑔(𝑥)|𝑢|^𝛼−2𝑢|𝑣|^𝛽+𝜆ℎ₁(𝑥)|𝑢|^𝛾−2𝑢−𝛽𝜑𝑑𝑥𝛼+𝛽_ℝ^𝑁𝑔(𝑥)|𝑣|^𝛽−2𝑣|𝑢|^𝛼+𝜇ℎ₂(𝑥)|𝑣|^𝛾−2𝑣−𝜓𝑑𝑥_ℝ𝑁𝑙₁(𝑥)𝜑𝑑𝑥−_ℝ𝑁𝑙₂(𝑥)𝜓𝑑𝑥. (10) In particular, it follows from (10) that 𝐽^′_𝜆,𝜇(𝑢,𝑣),(𝑢,𝑣)=‖(𝑢,𝑣)‖^𝑝−_ℝ𝑁𝑔(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥−𝐺_𝜆,𝜇(𝑢,𝑣)−𝐿(𝑢,𝑣). (11) It is clear that the critical points of the energy functional 𝐽_𝜆,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 𝐽_𝜆,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<𝜆ℎ_1𝜃+𝜇ℎ_2𝜃<𝑐₀ and 0≤𝑙^{𝑝/(𝑝−1)}_1𝜎+𝑙^{𝑝/(𝑝−1)}_2𝜎≤𝑐₁, then the problem (1) has at least two nontrivial solutions.

2. The Proof of Theorem 2

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 that Ω∈ℝ^𝑁 is an open bounded domain with 𝐶¹ boundary and 0∈Ω, 𝑁≥3, −∞<𝑎<𝑝𝑁/(𝑁−𝑝). Then the embedding 𝑊_𝑎^1,𝑝(Ω)↪𝐿^𝛾_𝛼(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). Let 𝑋 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 are 𝜌,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 Γ=𝑟∈𝐶¹([0,1],𝑋)∣𝑟(0)=(0,0),𝑟(1)=𝑒₁,𝑒₂. (12) Then 𝑐=inf_𝑟∈Γmax_0≤𝑡≤1𝐽(𝑟(𝑡))≥𝛼₀ (13) 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<𝜆ℎ_1𝜃+𝜇ℎ_2𝜃<𝑐₀, and 0≤𝑙^{𝑝/(𝑝−1)}_1𝜎+𝑙^{𝑝/(𝑝−1)}_2𝜎≤𝑐₁, then 𝐽_𝜆,𝜇(𝑢,𝑣) satisfies the assumptions (𝐴₁)--(𝐴₂) in Lemma 4.

Proof. Let (𝑢,𝑣)∈𝑋. By (8), 𝐽_𝜆,𝜇()=1𝑢,𝑣𝑝‖(𝑢,𝑣)‖^𝑝−1𝛼+𝛽_ℝ^𝑁𝑔(𝑥)|𝑢|^𝛼|𝑣|^𝛽−1𝑑𝑥𝛾𝐺_𝜆,𝜇(𝑢,𝑣)−𝐿(𝑢,𝑣). (14) By (𝐻₂) and the Hölder inequality, we have _ℝ𝑁𝑔(𝑥)|𝑢|^𝛼+𝛽𝑑𝑥≤_ℝ𝑁𝑔(𝑥)|𝑥|^{𝑏(𝛼+𝛽)𝛿}𝑑𝑥^1/𝛿×_ℝ𝑁|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥^{(𝛼+𝛽)/𝑝∗}≤𝑔_𝛿𝑆^𝛼+𝛽‖(𝑢,𝑣)‖^𝛼+𝛽, (15) where 𝑔_𝛿=(∫_ℝ𝑁(𝑔(𝑥)|𝑥|^{𝑏(𝛼+𝛽)})^𝛿𝑑𝑥)^1/𝛿, 𝛿=𝑝^∗/(𝑝^∗−𝛼−𝛽).
Similarly, _ℝ𝑁𝑔(𝑥)|𝑣|^𝛼+𝛽𝑑𝑥≤𝑔_𝛿𝑆^𝛼+𝛽‖(𝑢,𝑣)‖^𝛼+𝛽. (16) Hence, it follows from (15) and (16) that _ℝ𝑁𝑔(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥≤2𝑔_𝛿𝑆^𝛼+𝛽‖(𝑢,𝑣)‖^𝛼+𝛽. (17) Since ℎ₁(𝑥)|𝑥|^𝑏𝛾∈𝐿^𝜃(ℝ^𝑁)∩𝐿^∞(ℝ^𝑁), we obtain from the Hölder inequality and (5) that _ℝ𝑁𝜆ℎ₁(𝑥)|𝑢|^𝛾𝑑𝑥≤𝜆_ℝ𝑁ℎ₁(𝑥)|𝑥|^𝑏𝛾𝜃𝑑𝑥^1/𝜃×_ℝ𝑁|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥^𝛾/𝑝∗≤𝜆𝑆^𝛾ℎ_1𝜃‖(𝑢,𝑣)‖^𝛾. (18) Similarly, _ℝ𝑁𝜇ℎ₂(𝑥)|𝑣|^𝛾𝑑𝑥≤𝜇𝑆^𝛾ℎ_2𝜃‖(𝑢,𝑣)‖^𝛾. (19) Then, 𝐺_𝜆,𝜇(𝑢,𝑣)≤𝜆ℎ_1𝜃+𝜇ℎ_2𝜃𝑆^𝛾‖(𝑢,𝑣)‖^𝛾. (20) Using the Hölder inequality, we get _ℝ𝑁𝑙₁(𝑥)|𝑢|𝑑𝑥≤_ℝ𝑁|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥^1/𝑝∗×_ℝ𝑁|𝑥|^𝑏𝜎𝑙₁(𝑥)^𝜎𝑑𝑥^1/𝜎≤𝜀‖(𝑢,𝑣)‖^𝑝+𝐶_𝜀𝑆𝑙_1𝜎^{𝑝/(𝑝−1)}, (21) where 𝜀>0, 𝐶_𝜀>0.
Similarly, we have _ℝ𝑁𝑙₂(𝑥)|𝑣|𝑑𝑥≤𝑆𝑙_2𝜎‖(𝑢,𝑣)‖≤𝜀‖(𝑢,𝑣)‖^𝑝+𝐶_𝜀𝑆𝑙_2𝜎^{𝑝/(𝑝−1)}. (22) Thus, 𝐽_𝜆,𝜇()≥1𝑢,𝑣𝑝‖(𝑢,𝑣)‖^𝑝−1𝛼+𝛽⋅2𝑔_𝛿𝑆^𝛼+𝛽‖(𝑢,𝑣)‖^𝛼+𝛽−1𝛾𝜆ℎ_1𝜃+𝜇ℎ_2𝜃𝑆^𝛾‖(𝑢,𝑣)‖^𝛾−2𝜀‖(𝑢,𝑣)‖^𝑝−𝐶_𝜀𝑆𝑙_1𝜎^{𝑝/(𝑝−1)}−𝐶_𝜀𝑆𝑙_2𝜎^{𝑝/(𝑝−1)}≥12𝑝‖(𝑢,𝑣)‖^𝑝−2𝑔_𝛿𝑆𝛼+𝛽^𝛼+𝛽‖(𝑢,𝑣)‖^𝛼+𝛽−1𝛾𝜆ℎ_1𝜃+𝜇ℎ_2𝜃𝑆^𝛾‖(𝑢,𝑣)‖^𝛾−𝐶_𝜀𝑆𝑙_1𝜎^{𝑝/(𝑝−1)}−𝐶_𝜀𝑆𝑙_2𝜎^{𝑝/(𝑝−1)} (23) with 0<𝜀≤1/4𝑝.
Let 𝑔(𝑧)=2𝑔_𝛿𝑆𝛼+𝛽^𝛼+𝛽𝑧^{𝛼+𝛽−𝑝}+1𝛾𝜆ℎ_1𝜃+𝜇ℎ_2𝜃𝑆^𝛾𝑧^𝛾−𝑝,𝑧>0. (24) To verify (𝐴₁) in Lemma 4, it suffices to show that 𝑔(𝑧₁)<1/2𝑝 for some 𝑧₁=‖(𝑢,𝑣)‖>0.
Note that 𝑔(𝑧)→+∞ where 𝑧→0⁺ or 𝑧→+∞. Then 𝑔(𝑧) has a minimum at 𝑧₁>0. In order to find 𝑧₁, we have 𝑔^′(𝑧)=2𝑔_𝛿𝛼+𝛽−𝑝𝑆𝛼+𝛽^𝛼+𝛽𝑧^{𝛼+𝛽−𝑝−1}+𝛾−𝑝𝛾𝜆ℎ_1𝜃+𝜇ℎ_2𝜃𝑆^𝛾𝑧^{𝛾−𝑝−1}. (25) So that 𝑔^′(𝑧₁)=0 and 𝑧₁=𝛼+𝛽𝑝−𝛾2𝛾𝑔_𝛿𝛼+𝛽−𝑝𝑆^{𝛼+𝛽−𝛾1/(𝛼+𝛽−𝛾)}×𝜆ℎ_1𝜃+𝜇ℎ_2𝜃^{1/(𝛼+𝛽−𝛾)}=𝑐₂𝜆ℎ_1𝜃+𝜇ℎ_2𝜃^{1/(𝛼+𝛽−𝛾)}>0. (26) Moreover, 𝑔(𝑧₁)<1/2𝑝 implies that 𝑔𝑧₁=2𝑔_𝛿𝑆𝛼+𝛽^𝛼+𝛽𝑐₂^{𝛼+𝛽−𝑝}+1𝛾𝑆^𝛾𝑐₂^𝛾−𝑝×𝜆ℎ_1𝜃+𝜇ℎ_2𝜃^{(𝛼+𝛽−𝑝)/(𝛼+𝛽−𝛾)}=𝑐₃𝜆ℎ_1𝜃+𝜇ℎ_2𝜃^{(𝛼+𝛽−𝑝)/(𝛼+𝛽−𝛾)}<1,2𝑝 (27) where 𝑐₃=(2𝑔_𝛿/(𝛼+𝛽))𝑆^𝛼+𝛽𝑐₂^{𝛼+𝛽−𝑝}+(1/𝛾)𝑆^𝛾𝑐₂^𝛾−𝑝.
Thus, it follows from (23) and (27) that there exist 𝑐₀,𝑐₁,𝛼₀>0 such that 𝐽_𝜆,𝜇(𝑢,𝑣)≥𝛼₀ with 0<𝜆ℎ_1𝜃+𝜇ℎ_2𝜃<𝑐₀ and 0≤𝑙^{𝑝/(𝑝−1)}_1𝜎+𝑙^{𝑝/(𝑝−1)}_2𝜎≤𝑐₁. Then 𝐽_𝜆,𝜇(𝑢,𝑣) satisfies the assumption (𝐴₁) in Lemma 4.
We now verify (𝐴₂) in Lemma 4. Choose (𝜑₁,𝜑₂)∈𝐶^∞₀(ℝ^𝑁)  ×  𝐶^∞₀(ℝ^𝑁) such that _ℝ𝑁𝑔(𝑥)𝜑₁^𝛼𝜑₂^𝛽𝑑𝑥>0. (28) Then 𝐽_𝜆,𝜇𝑡𝜑₁,𝑡𝜑₂=𝑡^𝑝𝑝𝜑₁,𝜑₂^𝑝−𝑡^𝛼+𝛽𝛼+𝛽_ℝ^𝑁𝑔(𝑥)𝜑₁^𝛼𝜑₂^𝛽−𝑡𝑑𝑥_𝛾𝛾𝐺_𝜆,𝜇𝜑₁,𝜑₂−𝑡𝐿𝜑₁,𝜑₂ (29) and 𝐽_𝜆,𝜇(𝑡𝜑₁,𝑡𝜑₂)→−∞ (𝑡→+∞), since 𝛼+𝛽>𝑝>𝛾. Therefore, there exists 𝑡 large enough, such that 𝐽_𝜆,𝜇(𝑡𝜑₁,𝑡𝜑₂)<0. Then we take 𝑒=(𝑡𝜑₁,𝑡𝜑₂)∈𝑋 and 𝐽_𝜆,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 {𝑢_𝑛} and {𝑣_𝑛} are bounded sequences in 𝑊_𝑎^1,𝑝(ℝ^𝑁), then there exist subsequences, still denoted {𝑢_𝑛} and {𝑣_𝑛}, and 𝑢∈𝑊_𝑎^1,𝑝(ℝ^𝑁), 𝑣∈𝑊_𝑎^1,𝑝(ℝ^𝑁) such that as 𝑛→∞_ℝ𝑁𝑔(𝑥)𝑢_𝑛^𝛼𝑑𝑥⟶_ℝ𝑁𝑔(𝑥)|𝑢|^𝛼𝑑𝑥,_ℝ𝑁ℎ₁(𝑥)𝑢_𝑛^𝛾𝑑𝑥⟶_ℝ𝑁ℎ₁(𝑥)|𝑢|^𝛾𝑑𝑥,_ℝ𝑁𝑔(𝑥)𝑣_𝑛^𝛼𝑑𝑥⟶_ℝ𝑁𝑔(𝑥)|𝑣|^𝛼𝑑𝑥,_ℝ𝑁ℎ₂(𝑥)𝑣_𝑛^𝛾𝑑𝑥⟶_ℝ𝑁ℎ₂(𝑥)|𝑢|^𝛾𝑑𝑥. (30)

Proof. Let 𝑈_𝑘={𝑥∈ℝ^𝑁∣|𝑥|<𝑘}, 𝑘=1,2,….
Since {𝑢_𝑛} is bounded in 𝑊_𝑎^1,𝑝(ℝ^𝑁), {𝑢_𝑛} is bounded in 𝑊_𝑎^1,𝑝(𝑈_𝑘) for for all 𝑘≥1. By Lemma 3, {𝑢_𝑛} has a subsequence {𝑢_𝑛⁽¹⁾} which converges 𝑢₁ in 𝐿^𝛾(𝑈₁). Likewise, the subsequence {𝑢_𝑛⁽¹⁾} is bounded in 𝑊_𝑎^1,𝑝(𝑈₂). So that it has a subsequence {𝑢_𝑛⁽²⁾} which converges 𝑢₂ in 𝐿^𝛾(𝑈₂). Since {𝑢_𝑛⁽²⁾} is a subsequence of {𝑢_𝑛⁽¹⁾}, 𝑢₂=𝑢₁ in 𝑈₁. Continuing this process, we obtain a sequence {𝑢_𝑘} with the following properties: 𝑢_𝑘∈𝐿^𝛾𝑈_𝑘,𝑢_𝑘(𝑥)=𝑢₁(𝑥),a.ein𝑈₁,𝑢_𝑘(𝑥)=𝑢₂(𝑥),a.ein𝑈₂,⋮𝑢_𝑘(𝑥)=𝑢_𝑘−1(𝑥),a.ein𝑈_𝑘−1,𝑘=1,2,…. (31) It is clear that 𝑢_𝑘(𝑥)→𝑢(𝑥) a.e in ℝ¹, where 𝑢(𝑥)=𝑢_𝑘(𝑥),𝑥∈𝑈_𝑘,𝑘=1,2,…. (32) By a diagonal process, we take {𝑢_𝑚^(𝑚)} which is a subsequence of {𝑢_𝑛}. Thus, we have 𝑢_𝑚^(𝑚)⟶𝑢,stronglyin𝐿^𝛾𝑈_𝑘,𝑘=1,2,…. (33) Without loss of generality, we assume that the subsequence {𝑢_𝑚^(𝑚)} is {𝑢_𝑛} itself. So 𝑢_𝑛⟶𝑢,in𝐿^𝛾𝑈_𝑘,𝑘=1,2,…. (34) This implies that for 𝑛→∞, 𝑢_𝑛(𝑥)⟶𝑢(𝑥),a.ein𝑈_𝑘,𝑘=1,2,…. (35) Now, we prove lim_𝑘→∞sup_{𝑢∈𝑊𝑎1,𝑝⧵{0}}‖𝑢‖_𝐿^𝛾_(𝑈^𝑐_𝑘,|ℎ1|)‖𝑢‖_𝑊𝑎^1,𝑝_(ℝ𝑁)=0. (36) Indeed, ‖𝑢‖^𝛾_{𝐿𝛾𝑈𝑐𝑘,ℎ1}=_𝑈𝑐𝑘ℎ₁|𝑢|^𝛾≤𝑑𝑥_𝑈𝑐𝑘ℎ₁|𝑥|^𝑏𝛾𝜃𝑑𝑥^1/𝜃×_𝑈𝑐𝑘|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥^𝛾/𝑝∗≤_𝑈𝑐𝑘ℎ₁|𝑥|^𝑏𝛾𝜃𝑑𝑥^1/𝜃𝑆^𝛾‖𝑢‖^𝛾_{𝑊𝑎1,𝑝}, (37) where 𝜃=𝑝^∗/(𝑝^∗−𝛾).
The fact ℎ₁(𝑥)|𝑥|^𝑏𝛾∈𝐿^𝜃(ℝ^𝑁) gives that lim_𝑘→∞(∫_𝑈𝑐𝑘(|ℎ₁||𝑥|^𝑏𝛾)^𝜃𝑑𝑥)^1/𝜃=0.
Then, (37) implies that ‖𝑢‖_𝐿^𝛾_(𝑈^𝑐_𝑘,|ℎ1|)‖𝑢‖_𝑊𝑎^1,𝑝≤𝑆_𝑈𝑐𝑘ℎ₁|𝑥|^𝑏𝛾𝜃𝑑𝑥^1/𝛾𝜃⟶0,as𝑘⟶∞. (38) This gives (36).
In the following, we show that _ℝ𝑁ℎ₁(𝑥)𝑢_𝑛^𝛾𝑑𝑥⟶_ℝ𝑁ℎ₁(𝑥)|𝑢|^𝛾𝑑𝑥,as𝑛⟶∞. (39) Since the sequence {𝑢_𝑛} is bounded in 𝑊_𝑎^1,𝑝(ℝ^𝑁), we can assume (up to a subsequence) that 𝑢_𝑛⇀𝑢 weakly in 𝑊_𝑎^1,𝑝(ℝ^𝑁) and ‖𝑢_𝑛‖_𝑊𝑎^1,𝑝≤𝐶₀ for some constant 𝐶₀>0 and all 𝑛≥1.
By (36), we know that for any 𝜀>0, there exists 𝑘_𝜀>0 so large that 𝑢_{𝑛𝐿𝛾(𝑈𝑐𝑘𝜀,|ℎ1|)}≤𝐶₀⁻¹𝜀𝑢_𝑛𝑊𝑎^1,𝑝‖𝑢‖≤𝜀for𝑛=1,2,…,_{𝐿𝛾(𝑈𝑐𝑘𝜀,|ℎ1|)}≤𝜀. (40) Since the embeddding 𝑊_𝑎^1,𝑝(𝑈_𝑘𝜀)↪𝐿^𝛾(𝑈_𝑘𝜀) is compact (see Lemma 3), we have lim_𝑛→∞𝑢_𝑛−𝑢_{𝐿𝛾(𝑈𝑘𝜀)}=0. (41) Thus, there exists 𝑁₁>0, when 𝑛>𝑁₁, 𝑢_𝑛−𝑢_{𝐿𝛾(𝑈𝑘𝜀)}<𝜀. (42) So 𝑢_𝑛−𝑢_{𝐿𝛾(ℝ𝑁,|ℎ1|)}≤ℎ_{1𝐿∞(ℝ𝑁)}𝑢_𝑛−𝑢_{𝐿𝛾(ℝ𝑁)}≤ℎ_{1𝐿∞(ℝ𝑁)}𝑢_{𝑛𝐿𝛾(𝑈𝑐𝑘𝜀)}+‖𝑢‖_𝐿^𝛾_(𝑈^𝑐_𝑘𝜀)+𝑢_𝑛−𝑢_{𝐿𝛾(𝑈𝑘𝜀)}≤3𝜀ℎ_1∞. (43) This shows that 𝑢_𝑛→𝑢 in 𝐿^𝛾(ℝ^𝑁,|ℎ₁|) as 𝑛→∞.
By Brezis-Lieb’s Lemma in [16], we obtain that (39) is true.
Similarly, we can prove _ℝ𝑁𝑔(𝑥)𝑢_𝑛^𝛼𝑑𝑥⟶_ℝ𝑁𝑔(𝑥)|𝑢|^𝛼𝑑𝑥,_ℝ𝑁𝑔(𝑥)𝑣_𝑛^𝛼𝑑𝑥⟶_ℝ𝑁𝑔(𝑥)|𝑣|^𝛼𝑑𝑥,_ℝ𝑁ℎ₂(𝑥)𝑣_𝑛^𝛾𝑑𝑥⟶_ℝ𝑁ℎ₂(𝑥)|𝑣|^𝛾𝑑𝑥. (44)

Remark 7. By Brezis-Lieb’s Lemma, it follows from (30) that _ℝ𝑁𝑔𝑢_𝑛−𝑢^𝛼𝑑𝑥⟶0,_ℝ𝑁𝑔𝑣_𝑛−𝑣^𝛼𝑑𝑥⟶0,_ℝ𝑁ℎ₁𝑢_𝑛−𝑢^𝛾𝑑𝑥⟶0,_ℝ𝑁ℎ₂𝑣_𝑛−𝑣^𝛾𝑑𝑥⟶0,_ℝ𝑁𝑙₁𝑢_𝑛−𝑢𝑑𝑥⟶0,_ℝ𝑁𝑙₂𝑣_𝑛−𝑣𝑑𝑥⟶0. (45)

Lemma 8. Assume that (𝐻₁)--(𝐻₃) hold. Then 𝐽_𝜆,𝜇(𝑢,𝑣) defined by (8) satisfies (𝑃𝑆) condition on 𝑋.

Proof. Let {(𝑢_𝑛,𝑣_𝑛)} be a (𝑃𝑆)_𝑐 sequence of 𝐽_𝜆,𝜇(𝑢,𝑣) in 𝑋; that is, 𝐽_𝜆,𝜇(𝑢_𝑛,𝑣_𝑛)→𝑐, 𝐽^′_𝜆,𝜇(𝑢_𝑛,𝑣_𝑛)→0, in 𝑋^∗. We first claim that {(𝑢_𝑛,𝑣_𝑛)} is bounded in 𝑋. In fact, for large 𝑛, we obtain 𝑐+1+𝑢_𝑛,𝑣_𝑛≥𝐽_𝜆,𝜇𝑢_𝑛,𝑣_𝑛−1𝐽𝛼+𝛽^′𝜆,𝜇𝑢_𝑛,𝑣_𝑛,𝑢_𝑛,𝑣_𝑛=1𝑝−1𝑢𝛼+𝛽^𝑛,𝑣_𝑛^𝑝−1𝛾−1𝐺𝛼+𝛽^𝜆,𝜇𝑢_𝑛,𝑣_𝑛−11−𝐿𝑢𝛼+𝛽^𝑛,𝑣_𝑛. (46) Since 𝑝>𝛾>1, we conclude that {(𝑢_𝑛,𝑣_𝑛)} is bounded. Then, the {𝑢_𝑛} and {𝑣_𝑛} are bounded in 𝑊_𝑎^1,𝑝(ℝ^𝑁), respectively. Thus, there exist {(𝑢,𝑣)}∈𝑋 and a subsequence of {(𝑢_𝑛,𝑣_𝑛)}, still denoted by {(𝑢_𝑛,𝑣_𝑛)}, such that 𝑢_𝑛⇀𝑢,𝑣_𝑛⇀𝑣weaklyin𝑊_𝑎^1,𝑝ℝ^𝑁. (47) Denote 𝑃_𝑛=𝐽^′_𝜆,𝜇𝑢_𝑛,𝑣_𝑛,𝑢_𝑛−𝑢,𝑣_𝑛=−𝑣_ℝ𝑁|𝑥|^−𝑎𝑝∇𝑢_𝑛^𝑝−2∇𝑢_𝑛∇𝑢_𝑛−𝑢+𝑓₁(𝑥)𝑢_𝑛^𝑝−2𝑢_𝑛𝑢_𝑛+|𝑥|−𝑢_−𝑎𝑝∇𝑣_𝑛^𝑝−2∇𝑣_𝑛∇𝑣_𝑛−𝑣+𝑓₂(𝑥)𝑣_𝑛^𝑝−2𝑣_𝑛𝑣_𝑛−𝛼−𝑣𝑑𝑥𝛼+𝛽_ℝ^𝑁𝑔(𝑥)𝑢_𝑛^𝛼−2𝑢_𝑛𝑣_𝑛^𝛽+𝜆ℎ₁(𝑥)𝑢_𝑛^𝛾−2𝑢_𝑛×𝑢_𝑛𝛽−𝑢𝑑𝑥−×𝛼+𝛽_ℝ𝑁𝑔(𝑥)𝑣_𝑛^𝛽−2𝑣_𝑛𝑢_𝑛^𝛼+𝜇ℎ₂(𝑥)𝑣_𝑛^𝛾−2𝑣_𝑛𝑣_𝑛−−𝑣𝑑𝑥_ℝ𝑁𝑙₁(𝑥)𝑢_𝑛−−𝑢𝑑𝑥_ℝ𝑁𝑙₂(𝑥)𝑣_𝑛−𝑣𝑑𝑥. (48) Then the fact 𝐽^′_𝜆,𝜇(𝑢_𝑛,𝑣_𝑛)→0 in 𝑋^∗ implies 𝑃_𝑛→0 as 𝑛→∞.
Since that 𝑢_𝑛⇀𝑢,𝑣_𝑛⇀𝑣 in 𝑊_𝑎^1,𝑝(ℝ^𝑁), we see that 𝑄_𝑛=_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2∇𝑢∇𝑢_𝑛−𝑢+𝑓₁(𝑥)|𝑢|^𝑝−2𝑢𝑢_𝑛+−𝑢𝑑𝑥_ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑣|^𝑝−2∇𝑣∇𝑣_𝑛−𝑣+𝑓₂(𝑥)|𝑣|^𝑝−2𝑣𝑣_𝑛−𝑣𝑑𝑥⟶0,𝑛⟶∞. (49) Let 𝑇_𝑛=_ℝ𝑁|𝑥|^−𝑎𝑝∇𝑢_𝑛^𝑝−2∇𝑢_𝑛∇𝑢_𝑛−𝑢+𝑓₁(𝑥)𝑢_𝑛^𝑝−2𝑢_𝑛𝑢_𝑛+|𝑥|−𝑢_−𝑎𝑝∇𝑣_𝑛^𝑝−2∇𝑣_𝑛∇𝑣_𝑛−𝑣+𝑓₂(𝑥)𝑣_𝑛^𝑝−2𝑣_𝑛𝑣_𝑛−𝑣𝑑𝑥. (50) From dominated convergence theory and Lemma 6, we can conclude _ℝ𝑁𝑔(𝑥)𝑢_𝑛^𝛼−2𝑢_𝑛𝑣_𝑛^𝛽𝑢_𝑛−𝑢𝑑𝑥⟶0,𝑛⟶∞,_ℝ𝑁𝑔(𝑥)𝑣_𝑛^𝛽−2𝑣_𝑛𝑢_𝑛^𝛼𝑣_𝑛−𝑣𝑑𝑥⟶0,𝑛⟶∞. (51)(52) From Hölder inequality, _ℝ𝑁ℎ₁(𝑥)𝑢_𝑛^𝛾−1𝑢_𝑛−𝑢𝑑𝑥≤_ℝ𝑁ℎ₁𝑢_𝑛−𝑢^𝛾𝑑𝑥^1/𝛾×_ℝ𝑁ℎ₁𝑢_𝑛^𝛾𝑑𝑥^{(𝛾−1)/𝛾}⟶0,as𝑛⟶∞. (53) Similarly, we have _ℝ𝑁ℎ₂𝑣_𝑛^𝛾−1𝑣_𝑛−𝑣𝑑𝑥⟶0,_ℝ𝑁𝑙₁(𝑥)𝑢_𝑛−𝑢𝑑𝑥⟶0,_ℝ𝑁𝑙₂(𝑥)𝑣_𝑛−𝑣𝑑𝑥⟶0. (54) Then it follows from (48), (49) to (54) that 𝑇_𝑛→0, as 𝑛→∞.
So 𝑇_𝑛−𝑄_𝑛=_ℝ𝑁|𝑥|^−𝑎𝑝∇𝑢_𝑛^𝑝−2∇𝑢_𝑛−|∇𝑢|^𝑝−2∇𝑢∇𝑢_𝑛+−𝑢𝑑𝑥_ℝ𝑁𝑓₁(𝑥)𝑢_𝑛^𝑝−2𝑢_𝑛−|𝑢|^𝑝−2𝑢𝑢_𝑛+−𝑢𝑑𝑥_ℝ𝑁|𝑥|^−𝑎𝑝∇𝑣_𝑛^𝑝−2∇𝑣_𝑛−|∇𝑣|^𝑝−2∇𝑣∇𝑣_𝑛+−𝑣𝑑𝑥_ℝ𝑁𝑓₂(𝑥)𝑣_𝑛^𝑝−2𝑣_𝑛−|𝑣|^𝑝−2𝑣×𝑣_𝑛−𝑣𝑑𝑥⟶0. (55) Using the standard inequality in ℝ^𝑁 given by 𝜉^𝑝−2𝜉−𝜂^𝑝−2𝜂,𝜉−𝜂≥𝐶_𝑝𝜉−𝜂^𝑝𝜉,𝑝≥2,_𝑝−2𝜉−𝜂^𝑝−2𝜂,𝜉−𝜂≥𝐶_𝑝𝜉−𝜂²𝜉+𝜂^𝑝−2,1<𝑝<2, (56) we have from (55) that ‖(𝑢_𝑛−𝑢,𝑣_𝑛−𝑣)‖→0, as 𝑛→∞.
Thus, 𝐽_𝜆,𝜇(𝑢,𝑣) satisfies (𝑃𝑆)_𝑐 condition on 𝑋.

Proof of Theorem 2. By Lemmas 5 and 8, 𝐽_𝜆,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, 𝐽_𝜆,𝜇(𝑢₁,𝑣₁)≥𝛼₀>0.
We now seek a solution (𝑢₂,𝑣₂) of problem (1). Choose (𝜑₁,𝜑₂)∈𝐶^∞₀(ℝ^𝑁)×𝐶^∞₀(ℝ^𝑁) such that ∫_ℝ𝑁𝑙₁(𝑥)𝜑₁𝑑𝑥+∫_ℝ𝑁𝑙₂(𝑥)𝜑₂𝑑𝑥>0, and then 𝐽_𝜆,𝜇𝑡𝜑₁,𝑡𝜑₂=1𝑝𝑡^𝑝𝜑₁,𝜑₂^𝑝−1𝑡𝛼+𝛽^𝛼+𝛽_ℝ^𝑁𝑔(𝑥)𝜑₁^𝛼𝜑₂^𝛽𝑑𝑥−𝑡_ℝ𝑁𝑙₁(𝑥)𝜑₁+𝑙₂(𝑥)𝜑₂−1𝑑𝑥𝛾𝑡^𝛾𝐺_𝜆,𝜇𝜑₁,𝜑₂<0 (57) for small 𝑡>0 and thus for any open ball 𝐵_𝜏⊂𝑋 such that −∞<𝑐_𝜏=inf_𝐵𝜏𝐽_𝜆,𝜇(𝑢,𝑣)<0. (58) Thus, exists 𝜌>0, such that 𝑐_𝜌=inf_{()∈𝑢,𝑣𝐵𝜌}𝐽_𝜆,𝜇(𝑢,𝑣)<0,inf_{()𝑢,𝑣∈𝜕𝐵𝜌}𝐽_𝜆,𝜇(𝑢,𝑣)>0. (59) Letting 𝜀_𝑛↓0 such that 0<𝜀_𝑛<inf_{()𝑢,𝑣∈𝜕𝐵𝜌}𝐽_𝜆,𝜇(𝑢,𝑣)−inf_{()𝑢,𝑣∈𝐵𝜌}𝐽_𝜆,𝜇(𝑢,𝑣). (60) Then, by Ekeland’s variational principle in [17], there exists {(𝑢_𝑛,𝑣_𝑛)}⊂𝐵_𝜌 such that 𝑐_𝜌≤𝐽_𝜆,𝜇𝑢_𝑛,𝑣_𝑛<𝑐_𝜌+𝜀_𝑛,𝐽_𝜆,𝜇𝑢_𝑛,𝑣_𝑛<𝐽_𝜆,𝜇(𝑢,𝑣)+𝜀_𝑛𝑢_𝑛−𝑢,𝑣_𝑛,∀()∈−𝑣𝑢,𝑣𝐵_𝜌,𝑢≠𝑢_𝑛,𝑣≠𝑣_𝑛. (61)(62) Then it follows from (59) to (61) that 𝐽_𝜆,𝜇𝑢_𝑛,𝑣_𝑛<𝑐_𝜌+𝜀_𝑛≤inf_{()𝑢,𝑣∈𝐵𝜌}𝐽_𝜆,𝜇(𝑢,𝑣)+𝜀_𝑛<inf_{()𝑢,𝑣∈𝜕𝐵𝜌}𝐽_𝜆,𝜇(𝑢,𝑣), (63) so that (𝑢_𝑛,𝑣_𝑛)∈𝐵_𝜌. We now consider the functional 𝐹∶𝐵_𝜌→ℝ given by 𝐹(𝑢,𝑣)=𝐽_𝜆,𝜇(𝑢,𝑣)+𝜀_𝑛𝑢−𝑢_𝑛,𝑣−𝑣_𝑛,(𝑢,𝑣)∈𝐵_𝜌. (64) Then (62) shows that 𝐹(𝑢_𝑛,𝑣_𝑛)<𝐹(𝑢,𝑣), (𝑢,𝑣)∈𝐵_𝜌, 𝑢≠𝑢_𝑛, 𝑣≠𝑣_𝑛, and thus (𝑢_𝑛,𝑣_𝑛) is a strict local minimum of 𝐹. Moreover, 𝑡⁻¹𝐹𝑢_𝑛+𝑡𝜑₁,𝑣_𝑛+𝑡𝜑₂−𝐹𝑢_𝑛,𝑣_𝑛𝜑≥0,forsmall𝑡>0,∀₁,𝜑₂∈𝐵₁. (65) Hence, 𝑡⁻¹𝐽_𝜆,𝜇𝑢_𝑛+𝑡𝜑₁,𝑣_𝑛+𝑡𝜑₂−𝐽_𝜆,𝜇𝑢_𝑛,𝑣_𝑛+𝜀_𝑛𝜑₁,𝜑₂≥0. (66) Let 𝑡→0⁺; then, 𝐽^′_𝜆,𝜇𝑢_𝑛,𝑣_𝑛,𝜑₁,𝜑₂+𝜀_𝑛𝜑₁,𝜑₂∀𝜑≥0,₁,𝜑₂∈𝐵₁. (67) Replacing (𝜑₁,𝜑₂) in (67) by (−𝜑₁,−𝜑₂), we get −𝐽^′_𝜆,𝜇𝑢_𝑛,𝑣_𝑛,𝜑₁,𝜑₂+𝜀_𝑛𝜑₁,𝜑₂∀𝜑≥0,₁,𝜑₂∈𝐵₁. (68) So that ‖𝐽^′_𝜆,𝜇(𝑢_𝑛,𝑣_𝑛)‖≤𝜀_𝑛.
Therefore, there is a sequence {(𝑢_𝑛,𝑣_𝑛)}⊂𝐵_𝜌 such that 𝐽_𝜆,𝜇(𝑢_𝑛,𝑣_𝑛)→𝑐_𝜌<0, and 𝐽^′_𝜆,𝜇(𝑢_𝑛,𝑣_𝑛)→0 in 𝑋^∗ as 𝑛→∞. By Lemma 5, {(𝑢_𝑛,𝑣_𝑛)} has a convergent subsequence in 𝑋, still denoted by {(𝑢_𝑛,𝑣_𝑛)}, such that (𝑢_𝑛,𝑣_𝑛)→(𝑢₂,𝑣₂) in 𝑋. Thus (𝑢₂,𝑣₂) is a solution of (1) with 𝐽_𝜆,𝜇(𝑢₂,𝑣₂)<0. Then the proof of Theorem 2 is complete.