Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 526026 | 13 pages | https://doi.org/10.1155/2011/526026

Multiplicity of Solutions for Nonlocal Elliptic System of (𝑝,ğ‘ž)-Kirchhoff Type

Academic Editor: Josip E. Pečarić
Received16 May 2011
Accepted23 Jun 2011
Published11 Aug 2011

Abstract

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type −[𝑀1(∫Ω|∇𝑢|𝑝)]𝑝−1Δ𝑝𝑢=𝜆𝐹𝑢(𝑥,𝑢,𝑣), in Ω, −[𝑀2(∫Ω|∇𝑣|ğ‘ž)]ğ‘žâˆ’1Î”ğ‘žğ‘£=𝜆𝐹𝑣(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,ğ‘ž)-Kirchhoff type: −𝑀1Ω||||∇𝑢𝑝𝑝−1Δ𝑝𝑢=𝜆𝐹𝑢(𝑥,𝑢,𝑣),in−𝑀Ω,2Ω||||âˆ‡ğ‘£ğ‘žî‚¶î‚¹ğ‘žâˆ’1Î”ğ‘žğ‘£=𝜆𝐹𝑣(𝑥,𝑢,𝑣),inΩ,𝑢=𝑣=0;on𝜕Ω,(1.1) where Ω⊂𝑅𝑁(𝑁≥1) is a bounded smooth domain, 𝜆∈(0,+∞), 𝑝>𝑁,ğ‘ž>𝑁, Δ𝑝 is the 𝑝-Laplacian operator Δ𝑝||||𝑢=div∇𝑢𝑝−2,∇𝑢(1.2) and 𝑀𝑖∶𝑅+→𝑅,𝑖=1,2, are continuous functions with bounded conditions.(𝑀)There are two positive constants 𝑚0,𝑚1 such that 𝑚0≤𝑀𝑖(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 𝐹𝑢 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<ğ‘ž and a positive real function 𝛼(𝑥)âˆˆğ¿âˆž(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+𝐸2𝐿𝐿0|||𝜕𝑢|||𝜕𝑥2𝜕𝑑𝑥2𝑢𝜕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 𝑢𝑡𝑡−𝑀Ω|∇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 −𝑀Ω|∇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||||∇𝑢𝑝𝑝−1Δ𝑝𝑢=𝑓(𝑢,𝑣)+𝜌1(𝑥),in−𝑀Ω,2Ω||||∇𝑣𝑝𝑝−1Δ𝑝𝑣=𝑔(𝑢,𝑣)+𝜌2(𝑥),inΩ,𝜕𝑢=𝜕𝜂𝜕𝑣𝜕𝜂=0,on𝜕Ω,(𝑃) where 𝜂 is the unit exterior vector on 𝜕Ω, and 𝑀𝑖,𝜌𝑖(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,ğ‘ž)-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=𝑊01,𝑝(Ω)×𝑊01,ğ‘ž(Ω) be the Cartesian product of two Sobolev spaces, which is a reflexive real Banach space endowed with the norm ‖(𝑢,𝑣)‖=‖𝑢‖𝑝+â€–ğ‘£â€–ğ‘ž,(1.8) where ‖⋅‖𝑝 and â€–â‹…â€–ğ‘ž denote the norms of 𝑊01,𝑝(Ω) and 𝑊01,ğ‘ž(Ω), respectively. That is, ‖𝑢‖𝑝=Ω|∇𝑢|𝑝1/𝑝,â€–ğ‘£â€–ğ‘ž=Ω|∇𝑣|ğ‘žî‚¶1/ğ‘ž(1.9) for all 𝑢∈𝑊01,𝑝(Ω) and 𝑣∈𝑊01,ğ‘ž(Ω).

Since 𝑝>𝑁 and ğ‘ž>𝑁, 𝑊01,𝑝(Ω) and 𝑊01,ğ‘ž(Ω) are compactly embedded in 𝐶0(Ω). Let ⎧⎪⎨⎪⎩𝐶=maxsup𝑢∈𝑊01,𝑝(Ω)⧵{0}max𝑥∈Ω||||𝑢(𝑥)𝑝‖𝑢‖𝑝𝑝,sup𝑣∈𝑊01,ğ‘ž(Ω)⧵{0}max𝑥∈Ω||||𝑣(𝑥)ğ‘žî€¾â€–ğ‘£â€–ğ‘žğ‘žâŽ«âŽªâŽ¬âŽªâŽ­;(1.10) then we have 𝐶<+∞. Furthermore, it is known from [17] that sup𝑢∈𝑊01,𝑝(Ω)⧵{0}max𝑥∈Ω||𝑢||(𝑥)𝑝‖𝑢‖𝑝≤𝑁−1/𝑃√𝜋Γ𝑁1+21/𝑁𝑝−1𝑝−𝑁1−1/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||||∇𝑢𝑝𝑝−1Ω||||∇𝑢𝑝−2𝑀∇𝑢∇𝜑+2Ω||||âˆ‡ğ‘£ğ‘žî‚¶î‚¹ğ‘žâˆ’1Ω||||âˆ‡ğ‘£ğ‘žâˆ’2∇𝑣∇𝜓−𝜆Ω𝐹𝑢(𝑥,𝑢,𝑣)𝜑(𝑥)𝑑𝑥−𝜆Ω𝐹𝑣(𝑥,𝑢,𝑣)𝜓(𝑥)𝑑𝑥=0,(1.12) for all (𝜑,𝜓)∈𝑋. Define the functional 𝐼∶𝑋→𝑅 given by 1𝐼(𝑢,𝑣)=𝑝𝑀1Ω||||∇𝑢𝑝+1ğ‘žî‚Šğ‘€2Ω||||âˆ‡ğ‘£ğ‘žî‚¶î€œâˆ’ğœ†Î©ğ¹(𝑥,𝑢,𝑣)𝑑𝑥(1.13) for all (𝑢,𝑣)∈𝑋, and where 𝑀1(𝑡)=𝑡0𝑀1(𝑠)𝑝−1𝑀𝑑𝑠,2(𝑡)=𝑡0𝑀2(𝑠)ğ‘žâˆ’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∈𝑅𝑁∶|𝑦−𝑥|<𝑅}. Next we give some notations.𝛼1=𝛼1𝑁,𝑝,𝑅1,𝑅2=𝐶1/𝑃𝑅𝑁2−𝑅𝑁11/𝑃𝑅2−𝑅1𝜋𝑁/2Γ(1+𝑁/2)1/𝑃,𝛼2=𝛼2𝑁,ğ‘ž,𝑅1,𝑅2=𝐶1/ğ‘žî€·ğ‘…ğ‘2−𝑅𝑁11/ğ‘žğ‘…2−𝑅1𝜋𝑁/2Γ(1+𝑁/2)1/ğ‘ž.(1.15) Moreover, let ğ‘Ž,𝑐 be positive constants, denoteğ‘Žğ‘¦(𝑥)=𝑅2−𝑅1âŽ›âŽœâŽœâŽğ‘…2−𝑁𝑖=1𝑥𝑖−𝑥𝑖021/2âŽžâŽŸâŽŸâŽ î€·ğ‘¥,∀𝑥∈𝐵0,𝑅2𝑥⧵𝐵0,𝑅1,𝐴(𝑐)={(𝑠,𝑡)∈𝑅×𝑅∶|𝑠|𝑝+|𝑡|ğ‘žî€œâ‰¤ğ‘},𝑔(𝑐)=Ωsup(𝑠,𝑡)∈𝐴(𝑐)𝐹(𝑥,𝑠,𝑡)𝑑𝑥,𝑘(ğ‘Ž)=𝐵(𝑥0,𝑅2)⧵𝐵(𝑥0,𝑅1)𝐹(𝑥,𝑦(𝑥),𝑦(𝑥))𝑑𝑥+𝐵(𝑥0,𝑅1)â„Žğ¹(𝑥,ğ‘Ž,ğ‘Ž)𝑑𝑥,(𝑐,ğ‘Ž)=𝑘(ğ‘Ž)−𝑔(𝑐),𝑀+𝑚=max1𝑝−1𝑝,𝑚1ğ‘žâˆ’1ğ‘žîƒ°,𝑀−𝑚=min0𝑝−1𝑝,𝑚0ğ‘žâˆ’1ğ‘žîƒ°.(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 ğ‘Ž,𝑐1,𝑐2 with 𝑐1<(ğ‘Žğ›¼1)𝑝+(ğ‘Žğ›¼2)ğ‘žâ‰¤1<𝑐2 such that 𝑀+𝑔𝑐1<ğ‘€âˆ’â„Žî€·ğ‘1,ğ‘Ž,𝑀+𝑔𝑐2<ğ‘€âˆ’â„Žî€·ğ‘1.,ğ‘Ž(1.17) Then, for each 𝑀𝜆∈+î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»î€·ğ‘ğ¶â„Ž1,𝑀,ğ‘Žâˆ’ğ‘1î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»ğ¶îƒ¯1min𝑔𝑐1,1𝑔𝑐2,(1.18) there exists a positive real number 𝜌 such that the system (1.1) has at least two weak solutions (𝑢𝑖,𝑣𝑖)∈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 ğ‘Ž,𝑏, with (ğ‘Žğ›¼1)𝑝+(ğ‘Žğ›¼2)ğ‘ž>𝑏𝑀+/𝑀− such that
(𝐹3)𝐹(𝑥,𝑠,𝑡)≥0 for a.e. 𝑥∈Ω⧵𝐵(𝑥0,𝑅1) and all (𝑠,𝑡)∈[0,ğ‘Ž]×[0,ğ‘Ž];
(𝐹4)[(ğ‘Žğ›¼1)𝑝+(ğ‘Žğ›¼2)ğ‘ž]|Ω|sup(𝑥,𝑠,𝑡)∈Ω×𝐴(𝑏𝑀+/𝑀−)∫𝐹(𝑥,𝑠,𝑡)<𝑏𝐵(𝑥0,𝑅1)𝐹(𝑥,ğ‘Ž,ğ‘Ž)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 𝑤𝑖=(𝑢𝑖,𝑣𝑖)∈𝑋(𝑖=1,2,3) whose norms ‖𝑤𝑖‖ 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 limâ€–ğ‘¢â€–â†’âˆžÎ¨(𝑢)=+∞. Assume also that there exist two constants 𝑟1 and 𝑟2 such that (𝑗)inf𝑋Ψ<𝑟1<𝑟2; (𝑗𝑗)𝜑1(𝑟1)<𝜑2(𝑟1,𝑟2);(𝑗𝑗𝑗)𝜑1(𝑟2)<𝜑2(𝑟1,𝑟2); where 𝜑1(𝑟)=inf𝑢∈Ψ−1(−∞,𝑟)Φ(𝑢)−inf𝑢∈Ψ−1(−∞,𝑟)𝑤Φ(𝑢),𝜑𝑟−Ψ(𝑢)2𝑟1,𝑟2=inf𝑢∈Ψ−1(−∞,𝑟1)sup𝑣∈Ψ−1[𝑟1,𝑟2)Φ(𝑢)−Φ(𝑣).Ψ(𝑣)−Ψ(𝑢)(2.1) Then, for each 1𝜆∈𝜑2𝑟1,𝑟21,min𝜑1𝑟1,1𝜑2𝑟2,(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)limâ€–ğ‘¢â€–â†’âˆž(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)inf𝑢∈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)+ğœ†Î¦î…ž(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 1Ψ(𝑢,𝑣)=𝑝𝑀1Ω||||∇𝑢𝑝+1ğ‘žî‚Šğ‘€2Ω||||âˆ‡ğ‘£ğ‘žî‚¶î€œ,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,ğ‘ž>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 (𝑢𝑛,𝑣𝑛)∈𝑋 with (𝑢𝑛,𝑣𝑛)⇀(𝑢,𝑣) in 𝑋, then 𝑢𝑛⇀𝑢 in 𝑊01,𝑝(Ω) and 𝑣𝑛⇀𝑣 in 𝑊01,ğ‘ž(Ω). Therefore, liminfğ‘›â†’âˆžâ€–â€–ğ‘¢ğ‘›â€–â€–ğ‘â‰¥â€–ğ‘¢â€–ğ‘,liminfğ‘›â†’âˆžâ€–â€–ğ‘£ğ‘›â€–â€–ğ‘žâ‰¥â€–ğ‘£â€–ğ‘ž(2.4) due to the weakly lower semicontinuity of norm. Hence by virtue of the continuity and monotonicity of 𝑀1 and 𝑀2, we conclude that 𝑀1Ω||||∇𝑢𝑝≤𝑀1liminfğ‘›â†’âˆžî€œÎ©||∇𝑢𝑛||𝑝≤liminfğ‘›â†’âˆžî‚Šğ‘€1Ω||∇𝑢𝑛||𝑝,𝑀2Ω||||âˆ‡ğ‘£ğ‘žî‚¶â‰¤î‚Šğ‘€2liminfğ‘›â†’âˆžî€œÎ©||∇𝑣𝑛||ğ‘žî‚¶â‰¤liminfğ‘›â†’âˆžî‚Šğ‘€2Ω||∇𝑣𝑛||ğ‘žî‚¶,(2.5) Consequently, Ψ is a sequentially weakly lower semicontinuous functional.

Proof of Theorem 1.1. Let 1Ψ(𝑢,𝑣)=𝑝𝑀1Ω||||∇𝑢𝑝+1ğ‘žî‚Šğ‘€2Ω||||âˆ‡ğ‘£ğ‘žî‚¶î€œ,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+â€–ğ‘£â€–ğ‘žğ‘žî€¸â‰¤Î¨(𝑢,𝑣)≤𝑀+‖𝑢‖𝑝𝑝+â€–ğ‘£â€–ğ‘žğ‘žî€¸.(2.7)
Therefore, (2.7) implies that lim‖(𝑢,𝑣)‖→+∞Ψ(𝑢,𝑣)=+∞.(2.8)
Put 𝑟1=𝑀−𝑐1ğ¶î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»,𝑟2=𝑀−𝑐2ğ¶î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€».(2.9)
Denote 𝜑1(𝑟)=inf(𝑢,𝑣)∈Ψ−1(−∞,𝑟)Φ(𝑢,𝑣)−inf(𝑢,𝑣)∈Ψ−1(−∞,𝑟)𝑤Φ(𝑢,𝑣),𝜑𝑟−Ψ(𝑢,𝑣)2𝑟1,𝑟2=inf(𝑢,𝑣)∈Ψ−1(−∞,𝑟1)sup(𝑢1,𝑣1)∈Ψ−1[𝑟1,𝑟2)𝑢Φ(𝑢,𝑣)−Φ1,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∈𝑥Ω⧵𝐵0,𝑅2,ğ‘Žğ‘…2−𝑅1âŽ›âŽœâŽœâŽğ‘…2−𝑁𝑖=1𝑥𝑖−𝑥𝑖01/2âŽžâŽŸâŽŸâŽ î€·ğ‘¥,𝑥∈𝐵0,𝑅2𝑥⧵𝐵0,𝑅1,î€·ğ‘¥ğ‘Ž,𝑥∈𝐵0,𝑅1.(2.11) Then (𝑢0,𝑣0)∈𝑋, where 𝑢0(𝑥)=𝑣0(𝑥)=𝑤0(𝑥) and ‖‖𝑢0‖‖𝑝𝑝=‖‖𝑤0‖‖𝑝𝑝=î€·ğ‘Žğ›¼1𝑝𝐶,‖‖𝑣0â€–â€–ğ‘žğ‘ž=‖‖𝑤0â€–â€–ğ‘žğ‘ž=î€·ğ‘Žğ›¼1î€¸ğ‘žğ¶.(2.12) Consequently, (2.7) and (2.12) imply that 𝑟1𝑢<Ψ0,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(𝑢,𝑣)−Φ1,𝑣1Ψ𝑢1,𝑣1−Ψ(𝑢,𝑣)≥inf(𝑢,𝑣)∈Ψ−1(−∞,𝑟1)𝑢Φ(𝑢,𝑣)−Φ0,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𝑐𝑑𝑥=𝑘(ğ‘Ž)>ℎ1>𝑀,ğ‘Ž+𝑀−𝑔𝑐1𝑐>𝑔1=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)ğ‘žî€·â€–â‰¤ğ¶ğ‘¢â€–ğ‘ğ‘+â€–ğ‘£â€–ğ‘žğ‘žî€¸â‰¤ğ¶ğ‘Ÿ1𝑀−=𝑐1î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»â‰¤ğ‘1.(2.16) Therefore, the combination of (2.15) and (2.16) implies Φ𝑢(𝑢,𝑣)−Φ0,𝑣0Ψ𝑢0,𝑣0=∫−Ψ(𝑢,𝑣)Ω𝐹𝑥,𝑢0,𝑣0∫𝑑𝑥−Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥Ψ𝑢0,𝑣0≥∫−Ψ(𝑢,𝑣)Ω𝐹𝑥,𝑢0,𝑣0∫𝑑𝑥−Ωsup|𝑢(𝑥)|𝑝+|𝑣(𝑥)|ğ‘žâ‰¤ğ‘1𝐹(𝑥,𝑢,𝑣)𝑑𝑥Ψ𝑢0,𝑣0≥∫−Ψ(𝑢,𝑣)Ω𝐹𝑥,𝑢0,𝑣0∫𝑑𝑥−Ωsup|𝑢(𝑥)|𝑝+|𝑣(𝑥)|ğ‘žâ‰¤ğ‘1𝐹(𝑥,𝑢,𝑣)𝑑𝑥Ψ𝑢0,𝑣0≥∫Ω𝐹𝑥,𝑢0,𝑣0∫𝑑𝑥−Ωsup|𝑢(𝑥)|𝑝+|𝑣(𝑥)|ğ‘žâ‰¤ğ‘1𝐹(𝑥,𝑢,𝑣)𝑑𝑥𝑀+‖‖𝑢0‖‖𝑝𝑝+‖‖𝑣0â€–â€–ğ‘žğ‘žî€¸=𝐶𝑀+î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»â„Žî€·ğ‘1.,ğ‘Ž(2.17) By (2.14) and (2.17), we have 𝜑2𝑟1,𝑟2≥𝐶𝑀+î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»â„Žî€·ğ‘1.,ğ‘Ž(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)ğ‘žî€·â‰¤ğ¶â€–ğ‘¢â€–ğ‘ğ‘+â€–ğ‘£â€–ğ‘žğ‘žî€¸â‰¤ğ¶ğ‘Ÿğ‘€âˆ’.(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)𝑤Φ(𝑢,𝑣)≤𝑟−Ψ(𝑢,𝑣)Φ(0,0)−inf(𝑢,𝑣)∈Ψ−1(−∞,𝑟)𝑤Φ(𝑢,𝑣)≤𝑟−Ψ(0,0)−inf(𝑢,𝑣)∈Ψ−1(−∞,𝑟)𝑤Φ(𝑢,𝑣)𝑟≤∫Ωsup|𝑢(𝑥)|𝑝+|𝑣(𝑥)|ğ‘žâ‰¤ğ¶ğ‘Ÿ/𝑀−𝐹(𝑥,𝑢,𝑣)𝑑𝑥𝑟.(2.20) It implies that 𝜑1𝑟1≤𝑔𝑐1𝑟1=𝐶𝑀−𝑐1î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»ğ‘”î€·ğ‘1<𝐶𝑀+î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»â„Žî€·ğ‘1,,ğ‘Ž(2.21)𝜑1𝑟2≤𝑔𝑐2𝑟2=𝐶𝑀−𝑐2î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»ğ‘”î€·ğ‘2<𝐶𝑀+î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»â„Žî€·ğ‘1.,ğ‘Ž(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)+ğœ†Î¦î…ž(𝑢,𝑣)=0.(2.24) Then (𝑢1,𝑣1),(𝑢2,𝑣2)∈𝑋 are the weak solutions of system (1.1).
Since Ψ(𝑢𝑖,𝑣𝑖)<𝑟2,𝑖=1,2, by (1.10) and (2.7), ||𝑢𝑖||(𝑥)𝑝+||𝑣𝑖||(𝑥)ğ‘žâ‰¤ğ¶ğ‘Ÿ2𝑀−≤𝑐2,𝑖=1,2;(2.25) which implies there exists a positive real number 𝜌 such that the norms of (𝑢𝑖,𝑣𝑖)∈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 1Ψ(𝑢,𝑣)=𝑝𝑀1Ω||||∇𝑢𝑝+1ğ‘žî‚Šğ‘€2Ω||||âˆ‡ğ‘£ğ‘žî‚¶î€œ,Φ(𝑢,𝑣)=−Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥(2.26) for all (𝑢,𝑣)∈𝑋. By (𝐹2) and (2.7), we have 1Ψ(𝑢,𝑣)+𝜆Φ(𝑢,𝑣)=𝑝𝑀1Ω||||∇𝑢𝑝+1ğ‘žî‚Šğ‘€2Ω||||âˆ‡ğ‘£ğ‘žî‚¶î€œâˆ’ğœ†Î©ğ¹(𝑥,𝑢,𝑣)𝑑𝑥≥𝑀−‖𝑢‖𝑝𝑝+â€–ğ‘£â€–ğ‘žğ‘žî€¸î€œâˆ’ğœ†Î©î‚€||||𝛼(𝑥)1+𝑢(𝑥)𝛾+||||𝑣(𝑥)𝛽𝑑𝑥≥𝑀−‖𝑢‖𝑝𝑝+â€–ğ‘£â€–ğ‘žğ‘žî€¸âˆ’ğœ†â€–ğ›¼â€–âˆžî‚€||Ω||+𝑘1‖𝑢‖𝛾𝑝+𝑘2â€–ğ‘£â€–ğ›½ğ‘žî‚,(2.27) where 𝑘1,𝑘1 are positive constants. Since 𝛾<𝑝,𝛽<ğ‘ž, (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 (ğ‘Žğ›¼1)𝑝+(ğ‘Žğ›¼2)ğ‘ž>𝑏𝑀+/𝑀−, we conclude Ψ𝑢0,𝑣0≥𝑀−‖‖𝑢0‖‖𝑝𝑝+‖‖𝑣0â€–â€–ğ‘žğ‘žî€¸=ğ‘€âˆ’ğ¶î€ºî€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€»>𝑀−𝐶𝑏𝑀+𝑀−=𝑟.(2.29) By (𝐹3) and the definitions of 𝑢0 and 𝑣0, one has ||Ω||sup(𝑥,𝑠,𝑡)∈Ω×𝐴𝑏𝑀+/𝑀−𝑏𝐹(𝑥,𝑠,𝑡)<î€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€œğµ(𝑥0,𝑅1)=𝐹(𝑥,ğ‘Ž,ğ‘Ž)𝑑𝑥𝑏𝑀+𝐶∫𝐵(𝑥0,𝑅1)𝐹(𝑥,ğ‘Ž,ğ‘Ž)𝑑𝑥𝑀+î€·î€·ğ‘Žğ›¼1𝑝+î€·ğ‘Žğ›¼2î€¸ğ‘žî€¸â‰¤/𝐶𝑏𝑀+𝐶∫Ω⧵𝐵(𝑥0,𝑅1)𝐹𝑥,𝑢0,𝑣0∫𝑑𝑥+𝐵(𝑥0,𝑅1)𝐹𝑥,𝑢0,𝑣0𝑑𝑥𝑀+‖‖𝑢0‖‖𝑝𝑝+‖‖𝑣0â€–â€–ğ‘žğ‘žî€¸â‰¤ğ‘ğ‘€+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)ğ‘žî€·â‰¤ğ¶â€–ğ‘¢â€–ğ‘ğ‘+â€–ğ‘£â€–ğ‘žğ‘žî€¸â‰¤ğ¶ğ‘Ÿğ‘€âˆ’=𝐶𝑀−𝑏𝑀+𝐶=𝑏𝑀+𝑀−.(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}Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥≤sup{(𝑢,𝑣)||𝑢(𝑥)|𝑝+|𝑣(𝑥)|ğ‘žâ‰¤ğ‘ğ‘€+/𝑀−}Ω≤𝐹(𝑥,𝑢,𝑣)𝑑𝑥Ωsup(𝑠,𝑡)∈𝐴𝑏𝑀+/𝑀−≤||Ω||𝐹(𝑥,𝑠,𝑡)𝑑𝑥sup(𝑥,𝑠,𝑡)∈Ω×𝐴𝑏𝑀+/𝑀−≤𝐹(𝑥,𝑠,𝑡)𝑏𝑀+𝐶∫Ω𝐹𝑥,𝑢0,𝑣0𝑑𝑥Ψ𝑢0,𝑣0𝑢=𝑟−Φ0,𝑣0Ψ𝑢0,𝑣0.(2.32) Therefore, inf(𝑢,𝑣)∈Ψ−1(−∞,𝑟)Φ𝑢Φ(𝑢,𝑣)>𝑟0,𝑣0Ψ𝑢0,𝑣0.(2.33) Note that Φ(0,0)=Ψ(0,0)=0, we conclude that inf(𝑢,𝑣)∈Ψ−1](−∞,𝑟Ψ𝑢Φ(𝑢,𝑣)>0,𝑣0Φ𝑢−𝑟(0,0)+(𝑟−Ψ(0,0))Φ0,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).

References

  1. G. Kirchhoff, Mechanik, Teubner, leipzig, Germany, 1883.
  2. C. O. Alves, F. J. S. A. Corrêa, and T. F. Ma, “Positive solutions for a quasilinear elliptic equation of Kirchhoff type,” Computers & Mathematics with Applications, vol. 49, no. 1, pp. 85–93, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. B. Cheng and X. Wu, “Existence results of positive solutions of Kirchhoff type problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4883–4892, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. B. Cheng, X. Wu, and J. Liu, “Multiplicity of nontrivial solutions for Kirchhoff type problems,” Boundary Value Problems, vol. 2010, Article ID 268946, 13 pages, 2010. View at: Publisher Site | Google Scholar
  5. M. Chipot and B. Lovat, “Some remarks on nonlocal elliptic and parabolic problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 30, no. 7, pp. 4619–4627, 1997. View at: Publisher Site | Google Scholar
  6. P. D'Ancona and S. Spagnolo, “Global solvability for the degenerate Kirchhoff equation with real analytic data,” Inventiones Mathematicae, vol. 108, no. 2, pp. 247–262, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. X. He and W. Zou, “Infinitely many positive solutions for Kirchhoff-type problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 3, pp. 1407–1414, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. T. Ma and J. E. Muñoz Rivera, “Positive solutions for a nonlinear nonlocal elliptic transmission problem,” Applied Mathematics Letters, vol. 16, no. 2, pp. 243–248, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. T. Ma, “Remarks on an elliptic equation of Kirchhoff type,” Nonlinear Analysis. Theory, Methods & Applications, vol. 63, pp. 1967–1977, 2005. View at: Google Scholar
  10. A. Mao and Z. Zhang, “Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 3, pp. 1275–1287, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. K. Perera and Z. Zhang, “Nontrivial solutions of Kirchhoff-type problems via the Yang index,” Journal of Differential Equations, vol. 221, no. 1, pp. 246–255, 2006. View at: Publisher Site | Google Scholar
  12. Z. Zhang and K. Perera, “Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,” Journal of Mathematical Analysis and Applications, vol. 317, no. 2, pp. 456–463, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. F. J. S. A. Corrêa and R. G. Nascimento, “On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 598–604, 2009. View at: Publisher Site | Google Scholar
  14. G. Bonanno, “Multiple critical points theorems without the Palais-Smale condition,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 600–614, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. G. Bonanno, “A minimax inequality and its applications to ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 210–229, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. B. Ricceri, “On a three critical points theorem,” Archiv der Mathematik, vol. 75, no. 3, pp. 220–226, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. G. Talenti, “Some inequalities of Sobolev type on two-dimensional spheres,” in General Inequalities, W. Walter, Ed., vol. 5 of Internat. Schriftenreihe Numer. Math., pp. 401–408, Birkhäuser, Basel, Germany, 1987. View at: Google Scholar | Zentralblatt MATH

Copyright © 2011 Bitao Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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