Multiplicity of Solutions for Nonlocal Elliptic System of (𝑝,𝑞)-Kirchhoff Type

Abstract

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type −[𝑀₁(∫_Ω|∇𝑢|^𝑝)]^𝑝−1Δ_𝑝𝑢=𝜆𝐹_𝑢(𝑥,𝑢,𝑣), in Ω, −[𝑀₂(∫_Ω|∇𝑣|^𝑞)]^𝑞−1Δ_𝑞𝑣=𝜆𝐹_𝑣(𝑥,𝑢,𝑣), in Ω, 𝑢=𝑣=0, on 𝜕Ω. Under bounded condition on M and some novel and periodic condition on F, some new results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by means of Bonanno's multiple critical points theorems without the Palais-Smale condition and Ricceri's three critical points theorem, respectively.

1. Introduction and Preliminaries

We are concerned with the following nonlocal elliptic system of (𝑝,𝑞)-Kirchhoff type: −𝑀₁_Ω||∇𝑢||^𝑝^𝑝−1−𝑀Δ_𝑝𝑢=𝜆𝐹_𝑢(𝑥,𝑢,𝑣),inΩ,₂_Ω||∇𝑣||^𝑞^𝑞−1𝑢=𝑣=Δ0_𝑞;𝑣=on𝜆𝐹𝜕_𝑣Ω(,𝑥,𝑢,𝑣),inΩ,(1.1) where Ω⊂𝑅^𝑁(𝑁≥1) is a bounded smooth domain, 𝜆∈(0,+∞), 𝑝>𝑁,𝑞>𝑁, Δ_𝑝 is the 𝑝-Laplacian operator Δ_𝑝𝑢=div||∇𝑢||^𝑝−2∇𝑢,(1.2) and 𝑀_𝑖∶𝑅⁺→𝑅,𝑖=1,2, are continuous functions with bounded conditions.(𝑀)There are two positive constants 𝑚₀,𝑚₁ such that 𝑚₀≤𝑀_𝑖(𝑡)≤𝑚₁,∀𝑡≥0,𝑖=1,2.(1.3) Furthermore, 𝐹∶Ω×𝑅×𝑅→𝑅 is a function such that 𝐹(𝑥,𝑠,𝑡) is measurable in 𝑥 for all (𝑠,𝑡)∈𝑅×𝑅 and 𝐹(𝑥,𝑠,𝑡) is 𝐶¹ in (𝑠,𝑡) for a.e. 𝑥∈Ω, and 𝐹_𝑢 denotes the partial derivative of 𝐹 with respect to 𝑢. Moreover, 𝐹(𝑥,𝑠,𝑡) satisfies the following.(𝐹1)𝐹(𝑥,0,0)=0 for a.e. 𝑥∈Ω.(𝐹2) There exist two positive constants 𝛾<𝑝,𝛽<𝑞 and a positive real function 𝛼(𝑥)∈𝐿^∞(Ω) such that ||𝐹(𝑥,𝑠,𝑡)||≤𝛼(𝑥)1+|𝑠|^𝛾+|𝑡|^𝛽,fora.e.𝑥∈Ωandall(𝑠,𝑡)∈𝑅×𝑅.(1.4)

The system (1.1) is related to a model given by the equation of elastic strings 𝜌𝜕²𝑢𝜕𝑡²−𝑃⁰ℎ+2𝐸𝐿^𝐿0|||𝜕𝑢𝜕𝑥|||²𝑑𝑥𝜕²𝑢𝜕𝑥²=0(1.5) which was proposed by Kirchhoff [1] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings, where the parameters in (1.5) have the following meanings: 𝜌 is the mass density, 𝑃₀ is the initial tension, ℎ is the area of the cross-section, 𝐸 is the Young modulus of the material, and 𝐿 is the length of the string. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations.

Later, (1.5) was developed to the form 𝑢_𝑡𝑡−𝑀_Ω|∇𝑢|²Δ𝑢=𝑓(𝑥,𝑢)inΩ,(1.6) where 𝑀∶𝑅⁺→𝑅 is a given function. After that, many people studied the nonlocal elliptic boundary value problem −𝑀_Ω|∇𝑢|²Δ𝑢=𝑓(𝑥,𝑢)inΩ,𝑢=0on𝜕Ω,(1.7) which is the stationary counterpart of (1.6). It is pointed out in [2] that (1.7) models several physical and biological systems, where 𝑢 describes a process which depends on the average of itself (e.g., population density). By using the methods of sub and supersolutions, variational methods, and other techniques, many results of (1.7) were obtained, we can refer to [2–12] and the references therein. In particular, Alves et al. [2, Theorem  4] supposes that 𝑀 satisfies bounded condition (𝑀) and 𝑓(𝑥,𝑡) satisfies condition 𝐴𝑅, that is, for some 𝜈>2 and 𝑅>0 such that 0<𝜈𝐹(𝑥,𝑡)≤𝑓(𝑥,𝑡)𝑡,∀|𝑡|≥𝑅,𝑥∈Ω,(𝐴𝑅) where 𝐹(𝑥,𝑡)=∫^𝑡0𝑓(𝑥,𝑠)𝑑𝑠; one positive solutions for (1.7) was obtained. It is well known that condition 𝐴𝑅 plays an important role for showing the boundedness of Palais-Smale sequences. More recently, Corrêa and Nascimento in [13] studied a nonlocal elliptic system of 𝑝-Kirchhoff type −𝑀₁_Ω||∇𝑢||^𝑝^𝑝−1−𝑀Δ_𝑝𝑢=𝑓(𝑢,𝑣)+𝜌₁(𝑥),inΩ,₂_Ω||∇𝑣||^𝑝^𝑝−1𝜕𝑢Δ_𝑝𝑣=𝑔(𝑢,𝑣)+𝜌₂(𝑥),inΩ,𝜕𝜂=𝜕𝜕𝑣𝜂=0,on𝜕Ω,(𝑃) where 𝜂 is the unit exterior vector on 𝜕Ω, and 𝑀_𝑖,𝜌_𝑖(𝑖=1,2),𝑓,𝑔 satisfy suitable assumptions, where a special and important condition: periodic condition on nonlinearity was assumed. They obtained the existence of a weak solution for the nonlocal elliptic system of 𝑝-Kirchhoff type 𝑃 under Neumann boundary condition via Ekeland's Variational Principle.

In the present paper, our objective is to consider the nonlocal elliptic system of (𝑝,𝑞)-Kirchhoff-type (1.1), instead of the nonlocal elliptic system of 𝑝-Kirchhoff type and single Kirchhoff type equation. Under bounded condition on 𝑀 and some novel conditions without 𝑃𝑆 condition and periodic condition on 𝐹, we will prove the existence of two solutions and three solutions of system (1.1) by means of one multiple critical points theorem without the Palais-Smale condition of Bonanno in [14] and an equivalent formulation [15, Theorem  2.3] of Ricceri's three critical points theorem [16, Theorem  1], respectively.

In order to state our main results, we need the following preliminaries.

Let 𝑋=𝑊^01,𝑝(Ω)×𝑊^01,𝑞(Ω) be the Cartesian product of two Sobolev spaces, which is a reflexive real Banach space endowed with the norm ‖(𝑢,𝑣)‖=‖𝑢‖_𝑝+‖𝑣‖_𝑞,(1.8) where ‖⋅‖_𝑝 and ‖⋅‖_𝑞 denote the norms of 𝑊^01,𝑝(Ω) and 𝑊^01,𝑞(Ω), respectively. That is, ‖𝑢‖_𝑝=_Ω|∇𝑢|^𝑝^1/𝑝,‖𝑣‖_𝑞=_Ω|∇𝑣|^𝑞^1/𝑞(1.9) for all 𝑢∈𝑊^01,𝑝(Ω) and 𝑣∈𝑊^01,𝑞(Ω).

Since 𝑝>𝑁 and 𝑞>𝑁, 𝑊^01,𝑝(Ω) and 𝑊^01,𝑞(Ω) are compactly embedded in 𝐶⁰(Ω). Let 𝐶=max⎧⎪⎨⎪⎩sup_{𝑢∈𝑊01,𝑝(Ω)⧵{0}}max_𝑥∈Ω||𝑢(𝑥)||^𝑝‖𝑢‖^𝑝𝑝,sup_{𝑣∈𝑊01,𝑞(Ω)⧵{0}}max_𝑥∈Ω||𝑣(𝑥)||^𝑞‖𝑣‖^𝑞𝑞⎫⎪⎬⎪⎭;(1.10) then we have 𝐶<+∞. Furthermore, it is known from [17] that sup_{𝑢∈𝑊01,𝑝(Ω)⧵{0}}max_𝑥∈Ω||𝑢(𝑥)||^𝑝‖𝑢‖_𝑝≤𝑁^−1/𝑃√𝜋Γ1+𝑁2^1/𝑁𝑝𝑝−−𝑁1^1−1/𝑃||Ω||^{(1/𝑁)−(1/𝑃)},(1.11) where Γ denotes the Gamma function and |Ω| is the Lebesgue measure of Ω. Additionally, (1.11) is an equality when Ω is a ball.

Recall that (𝑢,𝑣)∈𝑋 is called a weak solution of system (1.1) if 𝑀¹_Ω||∇𝑢||^𝑝^𝑝−1_Ω||∇𝑢||^𝑝−2∇𝑢∇𝜑+𝑀₂_Ω||∇𝑣||^𝑞^𝑞−1_Ω−𝜆_Ω||∇𝑣||^𝑞−2∇𝑣∇𝜓𝐹_𝑢(𝑥,𝑢,𝑣)𝜑(𝑥)𝑑𝑥−𝜆_Ω𝐹_𝑣(𝑥,𝑢,𝑣)𝜓(𝑥)𝑑𝑥=0,(1.12) for all (𝜑,𝜓)∈𝑋. Define the functional 𝐼∶𝑋→𝑅 given by 𝐼(𝑢,𝑣)=1𝑝𝑀¹_Ω||∇𝑢||^𝑝+1𝑞𝑀²_Ω||∇𝑣||^𝑞−𝜆_Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥(1.13) for all (𝑢,𝑣)∈𝑋, and where 𝑀₁(𝑡)=^𝑡0𝑀₁(𝑠)^𝑝−1𝑑𝑠,𝑀₂(𝑡)=^𝑡0𝑀₂(𝑠)^𝑞−1𝑑𝑠,∀𝑡≥0.(1.14) By the conditions (𝑀) and (𝐹2), it is easy to see that 𝐼∈𝐶¹(𝑋,𝑅) and a critical point of 𝐼 corresponds to a weak solution of the system (1.1).

Now, giving 𝑥₀∈Ω and choosing 𝑅₂>𝑅₁>0 such that 𝐵(𝑥₀,𝑅₂)⊆Ω, where 𝐵(𝑥,𝑅)={𝑦∈𝑅^𝑁∶|𝑦−𝑥|<𝑅}. Next we give some notations.𝛼₁=𝛼₁𝑁,𝑝,𝑅₁,𝑅₂=𝐶^1/𝑃𝑅^𝑁2−𝑅^𝑁1^1/𝑃𝑅₂−𝑅₁Γ(1𝜋+𝑁^𝑁/2/2)^1/𝑃𝛼₂,=𝛼₂𝑁,𝑞,𝑅₁,𝑅₂=𝐶^1/𝑞𝑅^𝑁2−𝑅^𝑁1^1/𝑞𝑅₂−𝑅₁Γ(1𝜋+𝑁^𝑁/2/2)^1/𝑞.(1.15) Moreover, let 𝑎,𝑐 be positive constants, denote𝑦(𝑥)=𝑎𝑅₂−𝑅₁⎛⎜⎜⎝𝑅₂−^𝑁_𝑖=1𝑥^𝑖−𝑥^𝑖0²^1/2⎞⎟⎟⎠,∀𝑥∈𝐵𝑥₀𝐴(𝑐)={(𝑠,𝑡)∈𝑅×𝑅∶|𝑠|_𝑝,𝑅₂⧵𝐵𝑥₀,𝑅₁,𝑘(𝑎)=_{𝐵(𝑥0,𝑅2)⧵𝐵(𝑥0,𝑅1)}+|𝑡|^𝑞≤𝑐},𝑔(𝑐)=_Ωsup_{(𝑠,𝑡)∈𝐴(𝑐)}𝐹(𝑥,𝑠,𝑡)𝑑𝑥,ℎ(𝑐,𝑎)=𝑘(𝑎)−𝑔(𝑐),𝑀₊𝐹(𝑥,𝑦(𝑥),𝑦(𝑥))𝑑𝑥+_{𝐵(𝑥0,𝑅1)}𝐹(𝑥,𝑎,𝑎)𝑑𝑥,=max𝑚^1𝑝−1𝑝,𝑚𝑞^1𝑞−1,𝑀⁻=min𝑚^0𝑝−1𝑝,𝑚𝑞^0𝑞−1.(1.16)

Now we are ready to state our main results for the system (1.1)

Theorem 1.1. Assume that (𝐹1)-(𝐹2) hold and there are three positive constants 𝑎,𝑐₁,𝑐₂ with 𝑐₁<(𝑎𝛼₁)^𝑝+(𝑎𝛼₂)^𝑞≤1<𝑐₂ such that 𝑀⁺𝑔𝑐₁<𝑀₋ℎ𝑐₁,𝑎,𝑀⁺𝑔𝑐₂<𝑀₋ℎ𝑐₁,𝑎.(1.17) Then, for each 𝜆∈𝑀⁺𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞𝐶ℎ𝑐₁,𝑎,𝑀⁻𝑐₁𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞𝐶min𝑔1𝑐₁,𝑔1𝑐₂,(1.18) there exists a positive real number 𝜌 such that the system (1.1) has at least two weak solutions (𝑢_𝑖,𝑣_𝑖)∈𝑋(𝑖=1,2) whose norms in 𝐶⁰(Ω) are less than some positive constant 𝜌.

Theorem 1.2. Assume that (𝐹1)-(𝐹2) hold and there are two positive constants 𝑎,𝑏, with (𝑎𝛼₁)^𝑝+(𝑎𝛼₂)^𝑞>𝑏𝑀⁺/𝑀₋ such that
(𝐹3)𝐹(𝑥,𝑠,𝑡)≥0 for a.e. 𝑥∈Ω⧵𝐵(𝑥₀,𝑅₁) and all (𝑠,𝑡)∈[0,𝑎]×[0,𝑎];
(𝐹4)[(𝑎𝛼₁)^𝑝+(𝑎𝛼₂)^𝑞]|Ω|sup_{(𝑥,𝑠,𝑡)∈Ω×𝐴(𝑏𝑀}⁺_/𝑀−)𝐹(𝑥,𝑠,𝑡)<𝑏∫_{𝐵(𝑥0,𝑅1)}𝐹(𝑥,𝑎,𝑎)𝑑𝑥.
Then there exist an open interval Λ⊆[0,+∞] and a positive real number 𝜌 such that, for each 𝜆∈Λ, the system (1.1) has at least three weak solutions 𝑤_𝑖=(𝑢_𝑖,𝑣_𝑖)∈𝑋(𝑖=1,2,3) whose norms ‖𝑤_𝑖‖ are less than 𝜌.

2. Proofs of Main Results

Before proving the results, we state one multiple critical points theorem without the Palais-Smale condition of Bonanno in [13] and an equivalent formulation [14, Theorem  2.3] of Ricceri's three critical points theorem [15, Theorem  1], which are our main tools.

Theorem 2.1 (see [14, Theorem  2.1]). Let 𝑋 be a reflexive real Banach space, and let Ψ,Φ∶𝑋→𝑅 be two sequentially weakly lower semicontinuous functions. Assume that Ψ is (strongly) continuous and satisfies lim_{‖𝑢‖→∞}Ψ(𝑢)=+∞. Assume also that there exist two constants 𝑟₁ and 𝑟₂ such that (𝑗)inf_𝑋Ψ<𝑟₁<𝑟₂; (𝑗𝑗)𝜑₁(𝑟₁)<𝜑₂(𝑟₁,𝑟₂);(𝑗𝑗𝑗)𝜑₁(𝑟₂)<𝜑₂(𝑟₁,𝑟₂); where 𝜑₁(𝑟)=inf_{𝑢∈Ψ−1(−∞,𝑟)}Φ(𝑢)−inf_{𝑢∈Ψ−1(−∞,𝑟)𝑤}Φ(𝑢)𝜑𝑟−Ψ(𝑢)₂,𝑟₁,𝑟₂=inf_{𝑢∈Ψ−1(−∞,𝑟1)}sup_{𝑣∈Ψ−1[𝑟1,𝑟2)}Φ(𝑢)−Φ(𝑣)Ψ(𝑣)−Ψ(𝑢).(2.1) Then, for each 𝜆∈1𝜑₂𝑟₁,𝑟₂,min𝜑₁1𝑟₁,𝜑₂1𝑟₂,(2.2) the functional Ψ+𝜆Φ has two local minima which lie in Ψ⁻¹(−∞,𝑟₁) and Ψ⁻¹[𝑟₁,𝑟₂), respectively.

Theorem 2.2 (see [15, Theorem  2.3]). Let 𝑋 be a separable and reflexive real Banach space. Ψ∶𝑋→𝐑 is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on 𝑋^∗; Φ∶𝑋→𝑅 is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Suppose that (i)lim_{‖𝑢‖→∞}(Ψ(𝑢)+𝜆Φ(𝑢))=+∞ for each 𝜆>0;(ii)There are a real number 𝑟, and 𝑢₀,𝑢₁∈𝑋 such that Ψ(𝑢₀)<𝑟<Ψ(𝑢₁);(iii)inf_𝑢∈Ψ⁻¹_{(−∞,𝑟]}Φ(𝑢)>((Ψ(𝑢₁)−𝑟)Φ(𝑢₀)+(𝑟−Ψ(𝑢₀))Φ(𝑢₁))/(Ψ(𝑢₁)−Ψ(𝑢₀)).
Then there exist an open interval Λ⊆[0,+∞] and a positive real number 𝜌 such that, for each 𝜆∈Λ, the equation Ψ^(𝑢)+𝜆Φ^(𝑢)=0 has at least three weak solutions whose norms in 𝑋 are less than 𝜌.

First, we give one basic lemma.

Lemma 2.3. Assume that (𝑀) and (𝐹2) hold; let Ψ(𝑢,𝑣)=1𝑝𝑀¹_Ω||∇𝑢||^𝑝+1𝑞𝑀²_Ω||∇𝑣||^𝑞,Φ(𝑢,𝑣)=−_Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥,(2.3) for all (𝑢,𝑣)∈𝑋. Then Ψ and Φ are continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functionals. Moreover, the Gâteaux derivative of Ψ admits a continuous inverse on 𝑋^∗ and the Gâteaux derivative of Φ is compact.

Proof. By condition (𝑀), it is easy to see that Ψ is continuously Gâteaux differentiable. Moreover, the Gâteaux derivative of Ψ admits a continuous inverse on 𝑋^∗. Thanks to 𝑝>𝑁,𝑞>𝑁, and (𝐹2), Φ is continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative is compact. Next We will prove that Ψ is a sequentially weakly lower semicontinuous functional. Indeed, for any (𝑢_𝑛,𝑣_𝑛)∈𝑋 with (𝑢_𝑛,𝑣_𝑛)⇀(𝑢,𝑣) in 𝑋, then 𝑢_𝑛⇀𝑢 in 𝑊^01,𝑝(Ω) and 𝑣_𝑛⇀𝑣 in 𝑊^01,𝑞(Ω). Therefore, liminf_𝑛→∞‖‖𝑢_𝑛‖‖_𝑝≥‖𝑢‖_𝑝,liminf_𝑛→∞‖‖𝑣_𝑛‖‖_𝑞≥‖𝑣‖_𝑞(2.4) due to the weakly lower semicontinuity of norm. Hence by virtue of the continuity and monotonicity of 𝑀₁ and 𝑀₂, we conclude that 𝑀₁_Ω||∇𝑢||^𝑝≤𝑀¹liminf_𝑛→∞_Ω||∇𝑢_𝑛||^𝑝≤liminf_𝑛→∞𝑀₁_Ω||∇𝑢_𝑛||^𝑝𝑀₂,_Ω||∇𝑣||^𝑞≤𝑀²liminf_𝑛→∞_Ω||∇𝑣_𝑛||^𝑞≤liminf_𝑛→∞𝑀₂_Ω||∇𝑣_𝑛||^𝑞,(2.5) Consequently, Ψ is a sequentially weakly lower semicontinuous functional.

Proof of Theorem 1.1. Let Ψ(𝑢,𝑣)=1𝑝𝑀¹_Ω||∇𝑢||^𝑝+1𝑞𝑀²_Ω||∇𝑣||^𝑞,Φ(𝑢,𝑣)=−_Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥(2.6) for all (𝑢,𝑣)∈𝑋. Under condition (𝑀), by a simple computation, we have 𝑀₋‖𝑢‖^𝑝𝑝+‖𝑣‖^𝑞𝑞≤Ψ(𝑢,𝑣)≤𝑀⁺‖𝑢‖^𝑝𝑝+‖𝑣‖^𝑞𝑞.(2.7)
Therefore, (2.7) implies that lim_{‖(𝑢,𝑣)‖→+∞}Ψ(𝑢,𝑣)=+∞.(2.8)
Put 𝑟₁=𝑀⁻𝑐₁𝐶𝑎𝛼¹^𝑝+𝑎𝛼₂^𝑞,𝑟₂=𝑀⁻𝑐₂𝐶𝑎𝛼¹^𝑝+𝑎𝛼₂^𝑞.(2.9)
Denote 𝜑₁(𝑟)=inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)}Φ(𝑢,𝑣)−inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)𝑤}Φ(𝑢,𝑣)𝜑𝑟−Ψ(𝑢,𝑣)₂,𝑟₁,𝑟₂=inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟1)}sup_{(𝑢1,𝑣1)∈Ψ−1[𝑟1,𝑟2)}Φ(𝑢,𝑣)−Φ𝑢₁,𝑣₁Ψ𝑢₁,𝑣₁−Ψ(𝑢,𝑣),(2.10) and Ψ⁻¹(−∞,𝑟)^𝑤 is the closure of Ψ⁻¹(−∞,𝑟) in the weak topology.
Set 𝑤₀(𝑥)=⎧⎪⎪⎪⎨⎪⎪⎪⎩0,𝑥∈𝑎Ω⧵𝐵𝑥₀,𝑅₂,𝑅₂−𝑅₁⎛⎜⎜⎝𝑅₂−^𝑁_𝑖=1𝑥^𝑖−𝑥^𝑖0^1/2⎞⎟⎟⎠,𝑥∈𝐵𝑥₀𝑎,𝑥∈𝐵𝑥₀,𝑅₂⧵𝐵𝑥₀,𝑅₁,,𝑅₁.(2.11) Then (𝑢₀,𝑣₀)∈𝑋, where 𝑢₀(𝑥)=𝑣₀(𝑥)=𝑤₀(𝑥) and ‖‖𝑢₀‖‖^𝑝𝑝=‖‖𝑤₀‖‖^𝑝𝑝=𝑎𝛼¹^𝑝𝐶,‖‖𝑣⁰‖‖^𝑞𝑞=‖‖𝑤₀‖‖^𝑞𝑞=𝑎𝛼¹^𝑞𝐶.(2.12) Consequently, (2.7) and (2.12) imply that 𝑟₁<Ψ𝑢₀,𝑣₀<𝑟₂.(2.13) Furthermore, (2.13) implies that 𝜑₂𝑟₁,𝑟₂=inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟1)}sup_{(𝑢1,𝑣1)∈Ψ−1[𝑟1,𝑟2)}Φ(𝑢,𝑣)−Φ𝑢₁,𝑣₁≥inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟1)}Ψ𝑢₁,𝑣₁−Ψ(𝑢,𝑣)Φ(𝑢,𝑣)−Φ𝑢₀,𝑣₀Ψ𝑢₀,𝑣₀−Ψ(𝑢,𝑣).(2.14) On the other hand, by (𝐹1), (1.17), and, (2.11), one has _Ω𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥=𝑘(𝑎)>ℎ𝑐₁,𝑎>𝑀⁺𝑀₋𝑔𝑐₁>𝑔𝑐₁=_Ωsup_{𝑐(𝑠,𝑡)∈𝐴1}𝐹(𝑥,𝑠,𝑡)𝑑𝑥.(2.15) For each (𝑢,𝑣)∈𝑋 with Ψ(𝑢,𝑣)≤𝑟₁, and 𝑥∈Ω, by (2.7), we conclude ||𝑢(𝑥)||^𝑝+||𝑣(𝑥)||^𝑞≤𝐶‖𝑢‖^𝑝𝑝+‖𝑣‖^𝑞𝑞≤𝐶𝑟¹𝑀₋=𝑐₁𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞≤𝑐₁.(2.16) Therefore, the combination of (2.15) and (2.16) implies Φ(𝑢,𝑣)−Φ𝑢₀,𝑣₀Ψ𝑢₀,𝑣₀−Ψ(𝑢,𝑣)=∫^Ω𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥−∫_Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥≥∫Ψ𝑢₀,𝑣₀−Ψ(𝑢,𝑣)_Ω𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥−∫_Ωsup_|𝑢(𝑥)|^𝑝_{+|𝑣(𝑥)|}^𝑞_≤𝑐1𝐹(𝑥,𝑢,𝑣)𝑑𝑥≥∫Ψ𝑢₀,𝑣₀−Ψ(𝑢,𝑣)_Ω𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥−∫_Ωsup_|𝑢(𝑥)|^𝑝_{+|𝑣(𝑥)|}^𝑞_≤𝑐1𝐹(𝑥,𝑢,𝑣)𝑑𝑥≥∫_ΩΨ𝑢₀,𝑣₀𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥−∫_Ωsup_|𝑢(𝑥)|^𝑝_{+|𝑣(𝑥)|}^𝑞_≤𝑐1𝐹(𝑥,𝑢,𝑣)𝑑𝑥=𝐶𝑀⁺‖‖𝑢₀‖‖^𝑝𝑝+‖‖𝑣₀‖‖^𝑞𝑞𝑀⁺𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞ℎ𝑐¹,𝑎.(2.17) By (2.14) and (2.17), we have 𝜑₂𝑟₁,𝑟₂≥𝐶𝑀⁺𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞ℎ𝑐¹,𝑎.(2.18) Similarly, for every (𝑢,𝑣)∈𝑋 such that Ψ(𝑢,𝑣)≤𝑟, where 𝑟 is a positive real number, and 𝑥∈Ω, one has ||𝑢(𝑥)||^𝑝+||𝑣(𝑥)||^𝑞≤𝐶‖𝑢‖^𝑝𝑝+‖𝑣‖^𝑞𝑞≤𝐶𝑟𝑀₋.(2.19) By virtue of Ψ being sequentially weakly lower semicontinuous, then Ψ⁻¹(−∞,𝑟)^𝑤=Ψ⁻¹(−∞,𝑟]. Consequently, 𝜑₁(𝑟)=inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)}Φ(𝑢,𝑣)−inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)𝑤}Φ(𝑢,𝑣)≤Φ(0,0)−inf𝑟_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)𝑤}−Ψ(𝑢,𝑣)Φ(𝑢,𝑣)≤−inf𝑟−Ψ(_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)𝑤}0,0)Φ(𝑢,𝑣)≤∫_Ω𝑟sup_|𝑢(𝑥)|^𝑝_{+|𝑣(𝑥)|}^𝑞_{≤𝐶𝑟/𝑀−}𝐹(𝑥,𝑢,𝑣)𝑑𝑥𝑟.(2.20) It implies that 𝜑₁𝑟₁≤𝑔𝑐¹𝑟₁=𝐶𝑀₋𝑐₁𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞𝑔𝑐¹<𝐶𝑀⁺𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞ℎ𝑐¹,𝑎,(2.21)𝜑₁𝑟₂≤𝑔𝑐²𝑟₂=𝐶𝑀₋𝑐₂𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞𝑔𝑐²<𝐶𝑀⁺𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞ℎ𝑐¹,𝑎.(2.22) By (2.18)–(2.22), we conclude 𝜑₁𝑟₁≤𝜑₂𝑟₁,𝑟₂,𝜑₁𝑟₂≤𝜑₂𝑟₁,𝑟₂.(2.23) Therefore, the conditions (𝑗), (𝑗𝑗), and (𝑗𝑗𝑗) in Theorem 2.1 are satisfied. Consequently, by Lemma 2.3 and above facts, the functional Ψ+𝜆Φ has two local minima (𝑢₁,𝑣₁),(𝑢₂,𝑣₂)∈𝑋, which lie in Ψ⁻¹(−∞,𝑟₁) and Ψ⁻¹[𝑟₁,𝑟₂), respectively. Since 𝐼=Ψ+𝜆Φ∈𝐶¹, (𝑢₁,𝑣₁),(𝑢₂,𝑣₂)∈𝑋 are the solutions of the equation Ψ′(𝑢,𝑣)+𝜆Φ^(𝑢,𝑣)=0.(2.24) Then (𝑢₁,𝑣₁),(𝑢₂,𝑣₂)∈𝑋 are the weak solutions of system (1.1).
Since Ψ(𝑢_𝑖,𝑣_𝑖)<𝑟₂,𝑖=1,2, by (1.10) and (2.7), ||𝑢_𝑖(𝑥)||^𝑝+||𝑣_𝑖(𝑥)||^𝑞≤𝐶𝑟²𝑀₋≤𝑐₂,𝑖=1,2;(2.25) which implies there exists a positive real number 𝜌 such that the norms of (𝑢_𝑖,𝑣_𝑖)∈𝑋(𝑖=1,2) in 𝐶⁰(Ω) are less than some positive constant 𝜌. This completes the proof of Theorem 1.1.

Proof of Theorem 1.2. Let Ψ(𝑢,𝑣)=1𝑝𝑀¹_Ω||∇𝑢||^𝑝+1𝑞𝑀²_Ω||∇𝑣||^𝑞,Φ(𝑢,𝑣)=−_Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥(2.26) for all (𝑢,𝑣)∈𝑋. By (𝐹2) and (2.7), we have Ψ(𝑢,𝑣)+𝜆Φ(𝑢,𝑣)=1𝑝𝑀¹_Ω||∇𝑢||^𝑝+1𝑞𝑀²_Ω||∇𝑣||^𝑞−𝜆_Ω≥𝑀₋𝐹(𝑥,𝑢,𝑣)𝑑𝑥≥𝑀₋‖𝑢‖^𝑝𝑝+‖𝑣‖^𝑞𝑞−𝜆_Ω𝛼(𝑥)1+||𝑢(𝑥)||^𝛾+||𝑣(𝑥)||^𝛽𝑑𝑥‖𝑢‖^𝑝𝑝+‖𝑣‖^𝑞𝑞−𝜆‖𝛼‖_∞||Ω||+𝑘¹‖𝑢‖^𝛾𝑝+𝑘₂‖𝑣‖^𝛽𝑞,(2.27) where 𝑘₁,𝑘₁ are positive constants. Since 𝛾<𝑝,𝛽<𝑞, (2.27) implies that lim_{‖(𝑢,𝑣)‖→+∞}(Ψ(𝑢,𝑣)+𝜆Φ(𝑢,𝑣))=+∞.(2.28) The same as in (2.11), defining a function 𝑤₀(𝑥), and letting 𝑢₀(𝑥)=𝑣₀(𝑥)=𝑤₀(𝑥), then (2.12) is also satisfied. Choosing 𝑟=𝑏𝑀⁺/𝐶, by (2.7), (2.12), and (𝑎𝛼₁)^𝑝+(𝑎𝛼₂)^𝑞>𝑏𝑀⁺/𝑀₋, we conclude Ψ𝑢₀,𝑣₀≥𝑀₋‖‖𝑢₀‖‖^𝑝𝑝+‖‖𝑣₀‖‖^𝑞𝑞=𝑀⁻𝐶𝑎𝛼¹^𝑝+𝑎𝛼₂^𝑞>𝑀⁻𝐶𝑏𝑀𝑀⁺₋=𝑟.(2.29) By (𝐹3) and the definitions of 𝑢₀ and 𝑣₀, one has ||Ω||sup_{(𝑥,𝑠,𝑡)∈Ω×𝐴𝑏𝑀+/𝑀−}𝐹(𝑥,𝑠,𝑡)<𝑏𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞_{𝐵(𝑥0,𝑅1)}=𝑏𝑀₊𝐹(𝑥,𝑎,𝑎)𝑑𝑥𝐶∫^{𝐵(𝑥0,𝑅1)}𝐹(𝑥,𝑎,𝑎)𝑑𝑥≤𝑏𝑀₊𝑀⁺𝑎𝛼₁^𝑝+𝑎𝛼₂^𝑞/𝐶𝐶∫^{Ω⧵𝐵(𝑥0,𝑅1)}𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥+∫_{𝐵(𝑥0,𝑅1)}𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥≤𝑏𝑀₊𝑀⁺‖‖𝑢₀‖‖^𝑝𝑝+‖‖𝑣₀‖‖^𝑞𝑞𝐶∫^Ω𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥Ψ𝑢₀,𝑣₀.(2.30) For every (𝑢,𝑣)∈𝑋 such that Ψ(𝑢,𝑣)≤𝑟, and 𝑥∈Ω, one has ||𝑢(𝑥)||^𝑝+||𝑣(𝑥)||^𝑞≤𝐶‖𝑢‖^𝑝𝑝+‖𝑣‖^𝑞𝑞≤𝐶𝑟𝑀₋=𝐶𝑀₋𝑏𝑀⁺𝐶=𝑏𝑀𝑀⁺₋.(2.31) By the combination of (2.30) and (2.31), we have sup_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)}(−Φ(𝑢,𝑣))=sup_{{(𝑢,𝑣)∣Ψ(𝑢,𝑣)≤𝑟}}_Ω≤sup𝐹(𝑥,𝑢,_{{(𝑢,𝑣)||𝑢(𝑥)|𝑝+|𝑣(𝑥)|𝑞≤𝑏𝑀+/𝑀−}}𝑣)𝑑𝑥_Ω≤𝐹(𝑥,𝑢,𝑣)𝑑𝑥_Ω≤||Ω||suspup_{(𝑠,𝑡)∈𝐴𝑏𝑀+/𝑀−}𝐹(𝑥_{(𝑥,𝑠,𝑡)∈Ω×𝐴𝑏𝑀+/𝑀−},𝑠,𝑡)𝑑𝑥≤𝑏𝑀₊𝐹(𝑥,𝑠,𝑡)𝐶∫^Ω𝐹𝑥,𝑢₀,𝑣₀𝑑𝑥=𝑟−Φ𝑢₀Ψ𝑢₀,𝑣₀,𝑣₀Ψ𝑢₀,𝑣₀.(2.32) Therefore, inf_{(𝑢,𝑣)∈Ψ−1(−∞,𝑟)}Φ(𝑢,𝑣)>𝑟Φ𝑢⁰,𝑣₀Ψ𝑢₀,𝑣₀.(2.33) Note that Φ(0,0)=Ψ(0,0)=0, we conclude that inf_{(𝑢,𝑣)∈Ψ−1](−∞,𝑟}Φ(𝑢,𝑣)>Ψ𝑢⁰,𝑣₀−𝑟Φ(0,0)+(𝑟−Ψ(0,0))Φ𝑢₀,𝑣₀Ψ𝑢₀,𝑣₀−Ψ(0,0).(2.34) Hence, by Lemma 2.3 and above facts, Ψ and Φ satisfy all conditions of Theorem 2.2; then the conclusion directly follows from Theorem 2.2.

Acknowledgments

The authors would like to thank the referee for the useful suggestions. This work is supported in part by the National Natural Science Foundation of China (10961028), Yunnan NSF Grant no. 2010CD086, and the Foundation of young teachers of Qujing Normal University (2009QN018).