Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 526026, 13 pages
http://dx.doi.org/10.1155/2011/526026
Research Article

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

Bitao Cheng,1Β Xian Wu,2Β and Jun Liu1

1College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China
2Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China

Received 16 May 2011; Accepted 23 June 2011

Academic Editor: Josip E.Β PečariΔ‡

Copyright Β© 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.

Abstract

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type βˆ’[𝑀1(∫Ω|βˆ‡π‘’|𝑝)]π‘βˆ’1Δ𝑝𝑒=πœ†πΉπ‘’(π‘₯,𝑒,𝑣), in Ξ©, βˆ’[𝑀2(∫Ω|βˆ‡π‘£|π‘ž)]π‘žβˆ’1Ξ”π‘žπ‘£=πœ†πΉπ‘£(π‘₯,𝑒,𝑣), in Ξ©, 𝑒=𝑣=0, on πœ•Ξ©. 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,π‘ž)-Kirchhoff type: βˆ’ξ‚Έπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Άξ‚Ήπ‘βˆ’1Δ𝑝𝑒=πœ†πΉπ‘’(π‘₯,𝑒,𝑣),inβˆ’ξ‚Έπ‘€Ξ©,2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άξ‚Ήπ‘žβˆ’1Ξ”π‘žπ‘£=πœ†πΉπ‘£(π‘₯,𝑒,𝑣),inΞ©,𝑒=𝑣=0;onπœ•Ξ©,(1.1) where Ξ©βŠ‚π‘…π‘(𝑁β‰₯1) is a bounded smooth domain, πœ†βˆˆ(0,+∞), 𝑝>𝑁,π‘ž>𝑁, Δ𝑝 is the 𝑝-Laplacian operator Δ𝑝||||𝑒=divβˆ‡π‘’π‘βˆ’2,βˆ‡π‘’(1.2) and π‘€π‘–βˆΆπ‘…+→𝑅,𝑖=1,2, are continuous functions with bounded conditions.(𝑀)There are two positive constants π‘š0,π‘š1 such that π‘š0≀𝑀𝑖(𝑑)β‰€π‘š1,βˆ€π‘‘β‰₯0,𝑖=1,2.(1.3) Furthermore, πΉβˆΆΞ©Γ—π‘…Γ—π‘…β†’π‘… is a function such that 𝐹(π‘₯,𝑠,𝑑) is measurable in π‘₯ for all (𝑠,𝑑)βˆˆπ‘…Γ—π‘… and 𝐹(π‘₯,𝑠,𝑑) is 𝐢1 in (𝑠,𝑑) for a.e. π‘₯∈Ω, and 𝐹𝑒 denotes the partial derivative of 𝐹 with respect to 𝑒. Moreover, 𝐹(π‘₯,𝑠,𝑑) satisfies the following.(𝐹1)𝐹(π‘₯,0,0)=0 for a.e. π‘₯∈Ω.(𝐹2) There exist two positive constants 𝛾<𝑝,𝛽<π‘ž and a positive real function 𝛼(π‘₯)∈𝐿∞(Ξ©) such that ||||𝐹(π‘₯,𝑠,𝑑)≀𝛼(π‘₯)1+|𝑠|𝛾+|𝑑|𝛽,fora.e.π‘₯∈Ωandall(𝑠,𝑑)βˆˆπ‘…Γ—π‘….(1.4)

The system (1.1) is related to a model given by the equation of elastic strings πœŒπœ•2π‘’πœ•π‘‘2βˆ’ξ‚΅π‘ƒ0β„Ž+πΈξ€œ2𝐿𝐿0|||πœ•π‘’|||πœ•π‘₯2ξ‚Άπœ•π‘‘π‘₯2π‘’πœ•π‘₯2=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, β„Ž 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 π‘’π‘‘π‘‘ξ‚΅ξ€œβˆ’π‘€Ξ©|βˆ‡π‘’|2Δ𝑒=𝑓(π‘₯,𝑒)inΞ©,(1.6) where π‘€βˆΆπ‘…+→𝑅 is a given function. After that, many people studied the nonlocal elliptic boundary value problem ξ‚΅ξ€œβˆ’π‘€Ξ©|βˆ‡π‘’|2Δ𝑒=𝑓(π‘₯,𝑒)inΞ©,𝑒=0onπœ•Ξ©,(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 𝑒 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(π‘₯,𝑑) satisfies condition 𝐴𝑅, that is, for some 𝜈>2 and 𝑅>0 such that 0<𝜈𝐹(π‘₯,𝑑)≀𝑓(π‘₯,𝑑)𝑑,βˆ€|𝑑|β‰₯𝑅,π‘₯∈Ω,(𝐴𝑅) where ∫𝐹(π‘₯,𝑑)=𝑑0𝑓(π‘₯,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, CorrΓͺa and Nascimento in [13] studied a nonlocal elliptic system of 𝑝-Kirchhoff type βˆ’ξ‚Έπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Άξ‚Ήπ‘βˆ’1Δ𝑝𝑒=𝑓(𝑒,𝑣)+𝜌1(π‘₯),inβˆ’ξ‚Έπ‘€Ξ©,2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘ξ‚Άξ‚Ήπ‘βˆ’1Δ𝑝𝑣=𝑔(𝑒,𝑣)+𝜌2(π‘₯),inΞ©,πœ•π‘’=πœ•πœ‚πœ•π‘£πœ•πœ‚=0,onπœ•Ξ©,(𝑃) where πœ‚ is the unit exterior vector on πœ•Ξ©, and 𝑀𝑖,πœŒπ‘–(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,π‘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=π‘Š01,𝑝(Ξ©)Γ—π‘Š01,π‘ž(Ξ©) be the Cartesian product of two Sobolev spaces, which is a reflexive real Banach space endowed with the norm β€–(𝑒,𝑣)β€–=‖𝑒‖𝑝+β€–π‘£β€–π‘ž,(1.8) where ‖⋅‖𝑝 and β€–β‹…β€–π‘ž denote the norms of π‘Š01,𝑝(Ξ©) and π‘Š01,π‘ž(Ξ©), respectively. That is, ‖𝑒‖𝑝=ξ‚΅ξ€œΞ©|βˆ‡π‘’|𝑝1/𝑝,β€–π‘£β€–π‘ž=ξ‚΅ξ€œΞ©|βˆ‡π‘£|π‘žξ‚Ά1/π‘ž(1.9) for all π‘’βˆˆπ‘Š01,𝑝(Ξ©) and π‘£βˆˆπ‘Š01,π‘ž(Ξ©).

Since 𝑝>𝑁 and π‘ž>𝑁, π‘Š01,𝑝(Ξ©) and π‘Š01,π‘ž(Ξ©) are compactly embedded in 𝐢0(Ξ©). Let ⎧βŽͺ⎨βŽͺ⎩𝐢=maxsupπ‘’βˆˆπ‘Š01,𝑝(Ξ©)⧡{0}maxπ‘₯βˆˆΞ©ξ€½||||𝑒(π‘₯)𝑝‖𝑒‖𝑝𝑝,supπ‘£βˆˆπ‘Š01,π‘ž(Ξ©)⧡{0}maxπ‘₯βˆˆΞ©ξ€½||||𝑣(π‘₯)π‘žξ€Ύβ€–π‘£β€–π‘žπ‘žβŽ«βŽͺ⎬βŽͺ⎭;(1.10) then we have 𝐢<+∞. Furthermore, it is known from [17] that supπ‘’βˆˆπ‘Š01,𝑝(Ξ©)⧡{0}maxπ‘₯βˆˆΞ©ξ€½||𝑒||(π‘₯)π‘ξ€Ύβ€–π‘’β€–π‘β‰€π‘βˆ’1/π‘ƒβˆšπœ‹ξ‚€Ξ“ξ‚€π‘1+21/π‘ξ‚΅π‘βˆ’1ξ‚Άπ‘βˆ’π‘1βˆ’1/𝑃||Ξ©||(1/𝑁)βˆ’(1/𝑃),(1.11) where Ξ“ denotes the Gamma function and |Ξ©| is the Lebesgue measure of Ξ©. Additionally, (1.11) is an equality when Ξ© is a ball.

Recall that (𝑒,𝑣)βˆˆπ‘‹ is called a weak solution of system (1.1) if 𝑀1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Άξ‚Ήπ‘βˆ’1ξ€œΞ©||||βˆ‡π‘’π‘βˆ’2ξ‚Έπ‘€βˆ‡π‘’βˆ‡πœ‘+2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άξ‚Ήπ‘žβˆ’1ξ€œΞ©||||βˆ‡π‘£π‘žβˆ’2ξ€œβˆ‡π‘£βˆ‡πœ“βˆ’πœ†Ξ©πΉπ‘’ξ€œ(π‘₯,𝑒,𝑣)πœ‘(π‘₯)𝑑π‘₯βˆ’πœ†Ξ©πΉπ‘£(π‘₯,𝑒,𝑣)πœ“(π‘₯)𝑑π‘₯=0,(1.12) for all (πœ‘,πœ“)βˆˆπ‘‹. Define the functional πΌβˆΆπ‘‹β†’π‘… given by 1𝐼(𝑒,𝑣)=π‘ξ‚Šπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Ά+1π‘žξ‚Šπ‘€2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άξ€œβˆ’πœ†Ξ©πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯(1.13) for all (𝑒,𝑣)βˆˆπ‘‹, and where ξ‚Šπ‘€1(ξ€œπ‘‘)=𝑑0𝑀1(𝑠)π‘βˆ’1ξ‚Šπ‘€π‘‘π‘ ,2(ξ€œπ‘‘)=𝑑0𝑀2(𝑠)π‘žβˆ’1𝑑𝑠,βˆ€π‘‘β‰₯0.(1.14) By the conditions (𝑀) and (𝐹2), it is easy to see that 𝐼∈𝐢1(𝑋,𝑅) and a critical point of 𝐼 corresponds to a weak solution of the system (1.1).

Now, giving π‘₯0∈Ω and choosing 𝑅2>𝑅1>0 such that 𝐡(π‘₯0,𝑅2)βŠ†Ξ©, where 𝐡(π‘₯,𝑅)={π‘¦βˆˆπ‘…π‘βˆΆ|π‘¦βˆ’π‘₯|<𝑅}. Next we give some notations.𝛼1=𝛼1𝑁,𝑝,𝑅1,𝑅2ξ€Έ=𝐢1/𝑃𝑅𝑁2βˆ’π‘…π‘1ξ€Έ1/𝑃𝑅2βˆ’π‘…1ξ‚΅πœ‹π‘/2ξ‚ΆΞ“(1+𝑁/2)1/𝑃,𝛼2=𝛼2𝑁,π‘ž,𝑅1,𝑅2ξ€Έ=𝐢1/π‘žξ€·π‘…π‘2βˆ’π‘…π‘1ξ€Έ1/π‘žπ‘…2βˆ’π‘…1ξ‚΅πœ‹π‘/2Ξ“ξ‚Ά(1+𝑁/2)1/π‘ž.(1.15) Moreover, let π‘Ž,𝑐 be positive constants, denoteπ‘Žπ‘¦(π‘₯)=𝑅2βˆ’π‘…1βŽ›βŽœβŽœβŽπ‘…2βˆ’ξƒ―π‘ξ“π‘–=1ξ€·π‘₯π‘–βˆ’π‘₯𝑖0ξ€Έ2ξƒ°1/2βŽžβŽŸβŽŸβŽ ξ€·π‘₯,βˆ€π‘₯∈𝐡0,𝑅2ξ€Έξ€·π‘₯⧡𝐡0,𝑅1ξ€Έ,𝐴(𝑐)={(𝑠,𝑑)βˆˆπ‘…Γ—π‘…βˆΆ|𝑠|𝑝+|𝑑|π‘žξ€œβ‰€π‘},𝑔(𝑐)=Ξ©sup(𝑠,𝑑)∈𝐴(𝑐)ξ€œπΉ(π‘₯,𝑠,𝑑)𝑑π‘₯,π‘˜(π‘Ž)=𝐡(π‘₯0,𝑅2)⧡𝐡(π‘₯0,𝑅1)ξ€œπΉ(π‘₯,𝑦(π‘₯),𝑦(π‘₯))𝑑π‘₯+𝐡(π‘₯0,𝑅1)β„ŽπΉ(π‘₯,π‘Ž,π‘Ž)𝑑π‘₯,(𝑐,π‘Ž)=π‘˜(π‘Ž)βˆ’π‘”(𝑐),𝑀+ξƒ―π‘š=max1π‘βˆ’1𝑝,π‘š1π‘žβˆ’1π‘žξƒ°,π‘€βˆ’ξƒ―π‘š=min0π‘βˆ’1𝑝,π‘š0π‘žβˆ’1π‘žξƒ°.(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 π‘Ž,𝑐1,𝑐2 with 𝑐1<(π‘Žπ›Ό1)𝑝+(π‘Žπ›Ό2)π‘žβ‰€1<𝑐2 such that 𝑀+𝑔𝑐1ξ€Έ<π‘€βˆ’β„Žξ€·π‘1ξ€Έ,π‘Ž,𝑀+𝑔𝑐2ξ€Έ<π‘€βˆ’β„Žξ€·π‘1ξ€Έ.,π‘Ž(1.17) Then, for each ξƒ©π‘€πœ†βˆˆ+ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»ξ€·π‘πΆβ„Ž1ξ€Έ,𝑀,π‘Žβˆ’π‘1ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»πΆξƒ―1min𝑔𝑐1ξ€Έ,1𝑔𝑐2ξ€Έ,ξƒ°ξƒͺ(1.18) there exists a positive real number 𝜌 such that the system (1.1) has at least two weak solutions (𝑒𝑖,𝑣𝑖)βˆˆπ‘‹(𝑖=1,2) whose norms in 𝐢0(Ξ©) are less than some positive constant 𝜌.

Theorem 1.2. Assume that (𝐹1)-(𝐹2) hold and there are two positive constants π‘Ž,𝑏, with (π‘Žπ›Ό1)𝑝+(π‘Žπ›Ό2)π‘ž>𝑏𝑀+/π‘€βˆ’ such that
(𝐹3)𝐹(π‘₯,𝑠,𝑑)β‰₯0 for a.e. π‘₯∈Ω⧡𝐡(π‘₯0,𝑅1) and all (𝑠,𝑑)∈[0,π‘Ž]Γ—[0,π‘Ž];
(𝐹4)[(π‘Žπ›Ό1)𝑝+(π‘Žπ›Ό2)π‘ž]|Ξ©|sup(π‘₯,𝑠,𝑑)βˆˆΞ©Γ—π΄(𝑏𝑀+/π‘€βˆ’)∫𝐹(π‘₯,𝑠,𝑑)<𝑏𝐡(π‘₯0,𝑅1)𝐹(π‘₯,π‘Ž,π‘Ž)𝑑π‘₯.
Then there exist an open interval Ξ›βŠ†[0,+∞] and a positive real number 𝜌 such that, for each πœ†βˆˆΞ›, the system (1.1) has at least three weak solutions 𝑀𝑖=(𝑒𝑖,𝑣𝑖)βˆˆπ‘‹(𝑖=1,2,3) whose norms ‖𝑀𝑖‖ 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 Ξ¨,Ξ¦βˆΆπ‘‹β†’π‘… be two sequentially weakly lower semicontinuous functions. Assume that Ξ¨ is (strongly) continuous and satisfies limβ€–π‘’β€–β†’βˆžΞ¨(𝑒)=+∞. Assume also that there exist two constants π‘Ÿ1 and π‘Ÿ2 such that (𝑗)inf𝑋Ψ<π‘Ÿ1<π‘Ÿ2; (𝑗𝑗)πœ‘1(π‘Ÿ1)<πœ‘2(π‘Ÿ1,π‘Ÿ2);(𝑗𝑗𝑗)πœ‘1(π‘Ÿ2)<πœ‘2(π‘Ÿ1,π‘Ÿ2); where πœ‘1(π‘Ÿ)=infπ‘’βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)Ξ¦(𝑒)βˆ’infπ‘’βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)𝑀Φ(𝑒),πœ‘π‘Ÿβˆ’Ξ¨(𝑒)2ξ€·π‘Ÿ1,π‘Ÿ2ξ€Έ=infπ‘’βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ1)supπ‘£βˆˆΞ¨βˆ’1[π‘Ÿ1,π‘Ÿ2)Ξ¦(𝑒)βˆ’Ξ¦(𝑣).Ξ¨(𝑣)βˆ’Ξ¨(𝑒)(2.1) Then, for each 1πœ†βˆˆπœ‘2ξ€·π‘Ÿ1,π‘Ÿ2ξ€Έξƒ―1,minπœ‘1ξ€·π‘Ÿ1ξ€Έ,1πœ‘2ξ€·π‘Ÿ2ξ€Έ,ξƒ°ξƒͺ(2.2) the functional Ξ¨+πœ†Ξ¦ has two local minima which lie in Ξ¨βˆ’1(βˆ’βˆž,π‘Ÿ1) and Ξ¨βˆ’1[π‘Ÿ1,π‘Ÿ2), respectively.

Theorem 2.2 (see [15, Theorem  2.3]). Let 𝑋 be a separable and reflexive real Banach space. Ξ¨βˆΆπ‘‹β†’π‘ is a continuously GΓ’teaux differentiable and sequentially weakly lower semicontinuous functional whose GΓ’teaux derivative admits a continuous inverse on π‘‹βˆ—; Ξ¦βˆΆπ‘‹β†’π‘… is a continuously GΓ’teaux differentiable functional whose GΓ’teaux derivative is compact. Suppose that (i)limβ€–π‘’β€–β†’βˆž(Ξ¨(𝑒)+πœ†Ξ¦(𝑒))=+∞ for each πœ†>0;(ii)There are a real number π‘Ÿ, and 𝑒0,𝑒1βˆˆπ‘‹ such that Ξ¨(𝑒0)<π‘Ÿ<Ξ¨(𝑒1);(iii)infπ‘’βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ]Ξ¦(𝑒)>((Ξ¨(𝑒1)βˆ’π‘Ÿ)Ξ¦(𝑒0)+(π‘Ÿβˆ’Ξ¨(𝑒0))Ξ¦(𝑒1))/(Ξ¨(𝑒1)βˆ’Ξ¨(𝑒0)).
Then there exist an open interval Ξ›βŠ†[0,+∞] and a positive real number 𝜌 such that, for each πœ†βˆˆΞ›, the equation Ξ¨ξ…ž(𝑒)+πœ†Ξ¦ξ…ž(𝑒)=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 1Ξ¨(𝑒,𝑣)=π‘ξ‚Šπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Ά+1π‘žξ‚Šπ‘€2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άξ€œ,Ξ¦(𝑒,𝑣)=βˆ’Ξ©πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯,(2.3) for all (𝑒,𝑣)βˆˆπ‘‹. Then Ξ¨ and Ξ¦ are continuously GΓ’teaux differentiable and sequentially weakly lower semicontinuous functionals. Moreover, the GΓ’teaux derivative of Ξ¨ admits a continuous inverse on π‘‹βˆ— and the GΓ’teaux derivative of Ξ¦ is compact.

Proof. By condition (𝑀), it is easy to see that Ξ¨ is continuously GΓ’teaux differentiable. Moreover, the GΓ’teaux derivative of Ξ¨ admits a continuous inverse on π‘‹βˆ—. Thanks to 𝑝>𝑁,π‘ž>𝑁, and (𝐹2), Ξ¦ is continuously GΓ’teaux differentiable and sequentially weakly lower semicontinuous functional whose GΓ’teaux derivative is compact. Next We will prove that Ξ¨ is a sequentially weakly lower semicontinuous functional. Indeed, for any (𝑒𝑛,𝑣𝑛)βˆˆπ‘‹ with (𝑒𝑛,𝑣𝑛)⇀(𝑒,𝑣) in 𝑋, then 𝑒𝑛⇀𝑒 in π‘Š01,𝑝(Ξ©) and 𝑣𝑛⇀𝑣 in π‘Š01,π‘ž(Ξ©). Therefore, liminfπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–π‘β‰₯‖𝑒‖𝑝,liminfπ‘›β†’βˆžβ€–β€–π‘£π‘›β€–β€–π‘žβ‰₯β€–π‘£β€–π‘ž(2.4) due to the weakly lower semicontinuity of norm. Hence by virtue of the continuity and monotonicity of ξ‚Šπ‘€1 and ξ‚Šπ‘€2, we conclude that ξ‚Šπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Άβ‰€ξ‚Šπ‘€1ξ‚΅liminfπ‘›β†’βˆžξ€œΞ©||βˆ‡π‘’π‘›||𝑝≀liminfπ‘›β†’βˆžξ‚Šπ‘€1ξ‚΅ξ€œΞ©||βˆ‡π‘’π‘›||𝑝,ξ‚Šπ‘€2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άβ‰€ξ‚Šπ‘€2ξ‚΅liminfπ‘›β†’βˆžξ€œΞ©||βˆ‡π‘£π‘›||π‘žξ‚Άβ‰€liminfπ‘›β†’βˆžξ‚Šπ‘€2ξ‚΅ξ€œΞ©||βˆ‡π‘£π‘›||π‘žξ‚Ά,(2.5) Consequently, Ξ¨ is a sequentially weakly lower semicontinuous functional.

Proof of Theorem 1.1. Let 1Ξ¨(𝑒,𝑣)=π‘ξ‚Šπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Ά+1π‘žξ‚Šπ‘€2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άξ€œ,Ξ¦(𝑒,𝑣)=βˆ’Ξ©πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯(2.6) for all (𝑒,𝑣)βˆˆπ‘‹. Under condition (𝑀), by a simple computation, we have π‘€βˆ’ξ€·β€–π‘’β€–π‘π‘+β€–π‘£β€–π‘žπ‘žξ€Έβ‰€Ξ¨(𝑒,𝑣)≀𝑀+‖𝑒‖𝑝𝑝+β€–π‘£β€–π‘žπ‘žξ€Έ.(2.7)
Therefore, (2.7) implies that limβ€–(𝑒,𝑣)β€–β†’+∞Ψ(𝑒,𝑣)=+∞.(2.8)
Put π‘Ÿ1=π‘€βˆ’π‘1πΆξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»,π‘Ÿ2=π‘€βˆ’π‘2πΆξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€».(2.9)
Denote πœ‘1(π‘Ÿ)=inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)Ξ¦(𝑒,𝑣)βˆ’inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)𝑀Φ(𝑒,𝑣),πœ‘π‘Ÿβˆ’Ξ¨(𝑒,𝑣)2ξ€·π‘Ÿ1,π‘Ÿ2ξ€Έ=inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ1)sup(𝑒1,𝑣1)βˆˆΞ¨βˆ’1[π‘Ÿ1,π‘Ÿ2)𝑒Φ(𝑒,𝑣)βˆ’Ξ¦1,𝑣1Ψ𝑒1,𝑣1ξ€Έ,βˆ’Ξ¨(𝑒,𝑣)(2.10) and Ξ¨βˆ’1(βˆ’βˆž,π‘Ÿ)𝑀 is the closure of Ξ¨βˆ’1(βˆ’βˆž,π‘Ÿ) in the weak topology.
Set 𝑀0⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩(π‘₯)=0,π‘₯βˆˆξ€·π‘₯Ω⧡𝐡0,𝑅2ξ€Έ,π‘Žπ‘…2βˆ’π‘…1βŽ›βŽœβŽœβŽπ‘…2βˆ’ξƒ―π‘ξ“π‘–=1ξ€·π‘₯π‘–βˆ’π‘₯𝑖0ξ€Έξƒ°1/2βŽžβŽŸβŽŸβŽ ξ€·π‘₯,π‘₯∈𝐡0,𝑅2ξ€Έξ€·π‘₯⧡𝐡0,𝑅1ξ€Έ,ξ€·π‘₯π‘Ž,π‘₯∈𝐡0,𝑅1ξ€Έ.(2.11) Then (𝑒0,𝑣0)βˆˆπ‘‹, where 𝑒0(π‘₯)=𝑣0(π‘₯)=𝑀0(π‘₯) and ‖‖𝑒0‖‖𝑝𝑝=‖‖𝑀0‖‖𝑝𝑝=ξ€·π‘Žπ›Ό1𝑝𝐢,‖‖𝑣0β€–β€–π‘žπ‘ž=‖‖𝑀0β€–β€–π‘žπ‘ž=ξ€·π‘Žπ›Ό1ξ€Έπ‘žπΆ.(2.12) Consequently, (2.7) and (2.12) imply that π‘Ÿ1𝑒<Ξ¨0,𝑣0ξ€Έ<π‘Ÿ2.(2.13) Furthermore, (2.13) implies that πœ‘2ξ€·π‘Ÿ1,π‘Ÿ2ξ€Έ=inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ1)sup(𝑒1,𝑣1)βˆˆΞ¨βˆ’1[π‘Ÿ1,π‘Ÿ2)Φ𝑒(𝑒,𝑣)βˆ’Ξ¦1,𝑣1Ψ𝑒1,𝑣1ξ€Έβˆ’Ξ¨(𝑒,𝑣)β‰₯inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ1)𝑒Φ(𝑒,𝑣)βˆ’Ξ¦0,𝑣0Ψ𝑒0,𝑣0ξ€Έ.βˆ’Ξ¨(𝑒,𝑣)(2.14) On the other hand, by (𝐹1), (1.17), and, (2.11), one has ξ€œΞ©πΉξ€·π‘₯,𝑒0,𝑣0𝑐𝑑π‘₯=π‘˜(π‘Ž)>β„Ž1ξ€Έ>𝑀,π‘Ž+π‘€βˆ’π‘”ξ€·π‘1𝑐>𝑔1ξ€Έ=ξ€œΞ©sup𝑐(𝑠,𝑑)∈𝐴1𝐹(π‘₯,𝑠,𝑑)𝑑π‘₯.(2.15) For each (𝑒,𝑣)βˆˆπ‘‹ with Ξ¨(𝑒,𝑣)β‰€π‘Ÿ1, and π‘₯∈Ω, by (2.7), we conclude ||||𝑒(π‘₯)𝑝+||||𝑣(π‘₯)π‘žξ€·β€–β‰€πΆπ‘’β€–π‘π‘+β€–π‘£β€–π‘žπ‘žξ€Έβ‰€πΆπ‘Ÿ1π‘€βˆ’=𝑐1ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»β‰€π‘1.(2.16) Therefore, the combination of (2.15) and (2.16) implies Φ𝑒(𝑒,𝑣)βˆ’Ξ¦0,𝑣0Ψ𝑒0,𝑣0ξ€Έ=βˆ«βˆ’Ξ¨(𝑒,𝑣)Ω𝐹π‘₯,𝑒0,𝑣0ξ€Έβˆ«π‘‘π‘₯βˆ’Ξ©πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯Ψ𝑒0,𝑣0ξ€Έβ‰₯βˆ«βˆ’Ξ¨(𝑒,𝑣)Ω𝐹π‘₯,𝑒0,𝑣0ξ€Έβˆ«π‘‘π‘₯βˆ’Ξ©sup|𝑒(π‘₯)|𝑝+|𝑣(π‘₯)|π‘žβ‰€π‘1𝐹(π‘₯,𝑒,𝑣)𝑑π‘₯Ψ𝑒0,𝑣0ξ€Έβ‰₯βˆ«βˆ’Ξ¨(𝑒,𝑣)Ω𝐹π‘₯,𝑒0,𝑣0ξ€Έβˆ«π‘‘π‘₯βˆ’Ξ©sup|𝑒(π‘₯)|𝑝+|𝑣(π‘₯)|π‘žβ‰€π‘1𝐹(π‘₯,𝑒,𝑣)𝑑π‘₯Ψ𝑒0,𝑣0ξ€Έβ‰₯βˆ«Ξ©πΉξ€·π‘₯,𝑒0,𝑣0ξ€Έβˆ«π‘‘π‘₯βˆ’Ξ©sup|𝑒(π‘₯)|𝑝+|𝑣(π‘₯)|π‘žβ‰€π‘1𝐹(π‘₯,𝑒,𝑣)𝑑π‘₯𝑀+‖‖𝑒0‖‖𝑝𝑝+‖‖𝑣0β€–β€–π‘žπ‘žξ€Έ=𝐢𝑀+ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»β„Žξ€·π‘1ξ€Έ.,π‘Ž(2.17) By (2.14) and (2.17), we have πœ‘2ξ€·π‘Ÿ1,π‘Ÿ2ξ€Έβ‰₯𝐢𝑀+ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»β„Žξ€·π‘1ξ€Έ.,π‘Ž(2.18) Similarly, for every (𝑒,𝑣)βˆˆπ‘‹ such that Ξ¨(𝑒,𝑣)β‰€π‘Ÿ, where π‘Ÿ is a positive real number, and π‘₯∈Ω, one has ||||𝑒(π‘₯)𝑝+||||𝑣(π‘₯)π‘žξ€·β‰€πΆβ€–π‘’β€–π‘π‘+β€–π‘£β€–π‘žπ‘žξ€Έβ‰€πΆπ‘Ÿπ‘€βˆ’.(2.19) By virtue of Ξ¨ being sequentially weakly lower semicontinuous, then Ξ¨βˆ’1(βˆ’βˆž,π‘Ÿ)𝑀=Ξ¨βˆ’1(βˆ’βˆž,π‘Ÿ]. Consequently, πœ‘1(π‘Ÿ)=inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)Ξ¦(𝑒,𝑣)βˆ’inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)𝑀Φ(𝑒,𝑣)β‰€π‘Ÿβˆ’Ξ¨(𝑒,𝑣)Ξ¦(0,0)βˆ’inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)𝑀Φ(𝑒,𝑣)β‰€π‘Ÿβˆ’Ξ¨(0,0)βˆ’inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)𝑀Φ(𝑒,𝑣)π‘Ÿβ‰€βˆ«Ξ©sup|𝑒(π‘₯)|𝑝+|𝑣(π‘₯)|π‘žβ‰€πΆπ‘Ÿ/π‘€βˆ’πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯π‘Ÿ.(2.20) It implies that πœ‘1ξ€·π‘Ÿ1≀𝑔𝑐1ξ€Έπ‘Ÿ1=πΆπ‘€βˆ’π‘1ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»π‘”ξ€·π‘1ξ€Έ<𝐢𝑀+ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»β„Žξ€·π‘1ξ€Έ,,π‘Ž(2.21)πœ‘1ξ€·π‘Ÿ2≀𝑔𝑐2ξ€Έπ‘Ÿ2=πΆπ‘€βˆ’π‘2ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»π‘”ξ€·π‘2ξ€Έ<𝐢𝑀+ξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»β„Žξ€·π‘1ξ€Έ.,π‘Ž(2.22) By (2.18)–(2.22), we conclude πœ‘1ξ€·π‘Ÿ1ξ€Έβ‰€πœ‘2ξ€·π‘Ÿ1,π‘Ÿ2ξ€Έ,πœ‘1ξ€·π‘Ÿ2ξ€Έβ‰€πœ‘2ξ€·π‘Ÿ1,π‘Ÿ2ξ€Έ.(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 Ξ¨+πœ†Ξ¦ has two local minima (𝑒1,𝑣1),(𝑒2,𝑣2)βˆˆπ‘‹, which lie in Ξ¨βˆ’1(βˆ’βˆž,π‘Ÿ1) and Ξ¨βˆ’1[π‘Ÿ1,π‘Ÿ2), respectively. Since 𝐼=Ξ¨+πœ†Ξ¦βˆˆπΆ1, (𝑒1,𝑣1),(𝑒2,𝑣2)βˆˆπ‘‹ are the solutions of the equation Ξ¨β€²(𝑒,𝑣)+πœ†Ξ¦ξ…ž(𝑒,𝑣)=0.(2.24) Then (𝑒1,𝑣1),(𝑒2,𝑣2)βˆˆπ‘‹ are the weak solutions of system (1.1).
Since Ξ¨(𝑒𝑖,𝑣𝑖)<π‘Ÿ2,𝑖=1,2, by (1.10) and (2.7), ||𝑒𝑖||(π‘₯)𝑝+||𝑣𝑖||(π‘₯)π‘žβ‰€πΆπ‘Ÿ2π‘€βˆ’β‰€π‘2,𝑖=1,2;(2.25) which implies there exists a positive real number 𝜌 such that the norms of (𝑒𝑖,𝑣𝑖)βˆˆπ‘‹(𝑖=1,2) in 𝐢0(Ξ©) are less than some positive constant 𝜌. This completes the proof of Theorem 1.1.

Proof of Theorem 1.2. Let 1Ξ¨(𝑒,𝑣)=π‘ξ‚Šπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Ά+1π‘žξ‚Šπ‘€2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άξ€œ,Ξ¦(𝑒,𝑣)=βˆ’Ξ©πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯(2.26) for all (𝑒,𝑣)βˆˆπ‘‹. By (𝐹2) and (2.7), we have 1Ξ¨(𝑒,𝑣)+πœ†Ξ¦(𝑒,𝑣)=π‘ξ‚Šπ‘€1ξ‚΅ξ€œΞ©||||βˆ‡π‘’π‘ξ‚Ά+1π‘žξ‚Šπ‘€2ξ‚΅ξ€œΞ©||||βˆ‡π‘£π‘žξ‚Άξ€œβˆ’πœ†Ξ©πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯β‰₯π‘€βˆ’ξ€·β€–π‘’β€–π‘π‘+β€–π‘£β€–π‘žπ‘žξ€Έξ€œβˆ’πœ†Ξ©ξ‚€||||𝛼(π‘₯)1+𝑒(π‘₯)𝛾+||||𝑣(π‘₯)𝛽𝑑π‘₯β‰₯π‘€βˆ’ξ€·β€–π‘’β€–π‘π‘+β€–π‘£β€–π‘žπ‘žξ€Έβˆ’πœ†β€–π›Όβ€–βˆžξ‚€||Ξ©||+π‘˜1‖𝑒‖𝛾𝑝+π‘˜2β€–π‘£β€–π›½π‘žξ‚,(2.27) where π‘˜1,π‘˜1 are positive constants. Since 𝛾<𝑝,𝛽<π‘ž, (2.27) implies that limβ€–(𝑒,𝑣)β€–β†’+∞(Ξ¨(𝑒,𝑣)+πœ†Ξ¦(𝑒,𝑣))=+∞.(2.28) The same as in (2.11), defining a function 𝑀0(π‘₯), and letting 𝑒0(π‘₯)=𝑣0(π‘₯)=𝑀0(π‘₯), then (2.12) is also satisfied. Choosing π‘Ÿ=𝑏𝑀+/𝐢, by (2.7), (2.12), and (π‘Žπ›Ό1)𝑝+(π‘Žπ›Ό2)π‘ž>𝑏𝑀+/π‘€βˆ’, we conclude Ψ𝑒0,𝑣0ξ€Έβ‰₯π‘€βˆ’ξ€·β€–β€–π‘’0‖‖𝑝𝑝+‖‖𝑣0β€–β€–π‘žπ‘žξ€Έ=π‘€βˆ’πΆξ€Ίξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€»>π‘€βˆ’πΆπ‘π‘€+π‘€βˆ’=π‘Ÿ.(2.29) By (𝐹3) and the definitions of 𝑒0 and 𝑣0, one has ||Ξ©||supξ€·(π‘₯,𝑠,𝑑)βˆˆΞ©Γ—π΄π‘π‘€+/π‘€βˆ’ξ€Έπ‘πΉ(π‘₯,𝑠,𝑑)<ξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€œπ΅(π‘₯0,𝑅1)=𝐹(π‘₯,π‘Ž,π‘Ž)𝑑π‘₯𝑏𝑀+𝐢∫𝐡(π‘₯0,𝑅1)𝐹(π‘₯,π‘Ž,π‘Ž)𝑑π‘₯𝑀+ξ€·ξ€·π‘Žπ›Ό1𝑝+ξ€·π‘Žπ›Ό2ξ€Έπ‘žξ€Έβ‰€/𝐢𝑏𝑀+𝐢∫Ω⧡𝐡(π‘₯0,𝑅1)𝐹π‘₯,𝑒0,𝑣0ξ€Έβˆ«π‘‘π‘₯+𝐡(π‘₯0,𝑅1)𝐹π‘₯,𝑒0,𝑣0𝑑π‘₯𝑀+‖‖𝑒0‖‖𝑝𝑝+‖‖𝑣0β€–β€–π‘žπ‘žξ€Έβ‰€π‘π‘€+πΆβˆ«Ξ©πΉξ€·π‘₯,𝑒0,𝑣0𝑑π‘₯Ψ𝑒0,𝑣0ξ€Έ.(2.30) For every (𝑒,𝑣)βˆˆπ‘‹ such that Ξ¨(𝑒,𝑣)β‰€π‘Ÿ, and π‘₯∈Ω, one has ||||𝑒(π‘₯)𝑝+||||𝑣(π‘₯)π‘žξ€·β‰€πΆβ€–π‘’β€–π‘π‘+β€–π‘£β€–π‘žπ‘žξ€Έβ‰€πΆπ‘Ÿπ‘€βˆ’=πΆπ‘€βˆ’π‘π‘€+𝐢=𝑏𝑀+π‘€βˆ’.(2.31) By the combination of (2.30) and (2.31), we have sup(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)(βˆ’Ξ¦(𝑒,𝑣))=sup{(𝑒,𝑣)∣Ψ(𝑒,𝑣)β‰€π‘Ÿ}ξ€œΞ©πΉ(π‘₯,𝑒,𝑣)𝑑π‘₯≀sup{(𝑒,𝑣)||𝑒(π‘₯)|𝑝+|𝑣(π‘₯)|π‘žβ‰€π‘π‘€+/π‘€βˆ’}ξ€œΞ©β‰€ξ€œπΉ(π‘₯,𝑒,𝑣)𝑑π‘₯Ξ©supξ€·(𝑠,𝑑)βˆˆπ΄π‘π‘€+/π‘€βˆ’ξ€Έβ‰€||Ξ©||𝐹(π‘₯,𝑠,𝑑)𝑑π‘₯supξ€·(π‘₯,𝑠,𝑑)βˆˆΞ©Γ—π΄π‘π‘€+/π‘€βˆ’ξ€Έβ‰€πΉ(π‘₯,𝑠,𝑑)𝑏𝑀+πΆβˆ«Ξ©πΉξ€·π‘₯,𝑒0,𝑣0𝑑π‘₯Ψ𝑒0,𝑣0𝑒=π‘Ÿβˆ’Ξ¦0,𝑣0Ψ𝑒0,𝑣0ξ€Έ.(2.32) Therefore, inf(𝑒,𝑣)βˆˆΞ¨βˆ’1(βˆ’βˆž,π‘Ÿ)Φ𝑒Φ(𝑒,𝑣)>π‘Ÿ0,𝑣0Ψ𝑒0,𝑣0ξ€Έ.(2.33) Note that Ξ¦(0,0)=Ξ¨(0,0)=0, we conclude that inf(𝑒,𝑣)βˆˆΞ¨βˆ’1](βˆ’βˆž,π‘Ÿξ€·Ξ¨ξ€·π‘’Ξ¦(𝑒,𝑣)>0,𝑣0ξ€Έξ€ΈΞ¦ξ€·π‘’βˆ’π‘Ÿ(0,0)+(π‘Ÿβˆ’Ξ¨(0,0))Ξ¦0,𝑣0Ψ𝑒0,𝑣0ξ€Έ.βˆ’Ξ¨(0,0)(2.34) Hence, by Lemma 2.3 and above facts, Ξ¨ and Ξ¦ 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. CorrΓͺa, and T. F. Ma, β€œPositive solutions for a quasilinear elliptic equation of Kirchhoff type,” Computers & Mathematics with Applications, vol. 49, no. 1, pp. 85–93, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. B. Cheng and X. Wu, β€œExistence results of positive solutions of Kirchhoff type problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4883–4892, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. B. Cheng, X. Wu, and J. Liu, β€œMultiplicity of nontrivial solutions for Kirchhoff type problems,” Boundary Value Problems, vol. 2010, Article ID 268946, 13 pages, 2010. View at Publisher Β· View at Google Scholar
  5. M. Chipot and B. Lovat, β€œSome remarks on nonlocal elliptic and parabolic problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 30, no. 7, pp. 4619–4627, 1997. View at Publisher Β· View at Google Scholar
  6. P. D'Ancona and S. Spagnolo, β€œGlobal solvability for the degenerate Kirchhoff equation with real analytic data,” Inventiones Mathematicae, vol. 108, no. 2, pp. 247–262, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. X. He and W. Zou, β€œInfinitely many positive solutions for Kirchhoff-type problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 3, pp. 1407–1414, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. T. Ma and J. E. MuΓ±oz Rivera, β€œPositive solutions for a nonlinear nonlocal elliptic transmission problem,” Applied Mathematics Letters, vol. 16, no. 2, pp. 243–248, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. T. Ma, β€œRemarks 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, β€œSign-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 Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. K. Perera and Z. Zhang, β€œNontrivial solutions of Kirchhoff-type problems via the Yang index,” Journal of Differential Equations, vol. 221, no. 1, pp. 246–255, 2006. View at Publisher Β· View at Google Scholar
  12. Z. Zhang and K. Perera, β€œSign 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 Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. F. J. S. A. CorrΓͺa and R. G. Nascimento, β€œOn 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 Β· View at Google Scholar
  14. G. Bonanno, β€œMultiple 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 Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. G. Bonanno, β€œA 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 Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. B. Ricceri, β€œOn a three critical points theorem,” Archiv der Mathematik, vol. 75, no. 3, pp. 220–226, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. G. Talenti, β€œSome inequalities of Sobolev type on two-dimensional spheres,” in General Inequalities, W. Walter, Ed., vol. 5 of Internat. Schriftenreihe Numer. Math., pp. 401–408, BirkhΓ€user, Basel, Germany, 1987. View at Google Scholar Β· View at Zentralblatt MATH