International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 727398 | 15 pages |

Existence Results for the p(x)-Laplacian with Nonlinear Boundary Condition

Academic Editor: Y. Tsompanakis
Received28 Mar 2012
Accepted08 May 2012
Published24 Jul 2012


By using the variational method, under appropriate assumptions on the perturbation terms ๐‘“(๐‘ฅ,๐‘ข),๐‘”(๐‘ฅ,๐‘ข) such that the associated functional satisfies the global minimizer condition and the fountain theorem, respectively, the existence and multiple results for the ๐‘(๐‘ฅ)-Laplacian with nonlinear boundary condition in bounded domain โ„ฆ were studied. The discussion is based on variable exponent Lebesgue and Sobolev spaces.

1. Introduction

In recent years, increasing attention has been paid to the study of differential and partial differential equations involving variable exponent conditions. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, or calculus of variations. For more information on modeling physical phenomena by equations involving ๐‘(๐‘ฅ)-growth condition we refer to [1โ€“3]. The appearance of such physical models was facilitated by the development of variable exponent Lebesgue and Sobolev spaces, ๐ฟ๐‘(๐‘ฅ) and ๐‘Š1,๐‘(๐‘ฅ), where ๐‘(๐‘ฅ) is a real-valued function. Variable exponent Lebesgue spaces appeared for the first time in the literature as early as 1931 in an article by Orlicz [4]. The spaces ๐ฟ๐‘(๐‘ฅ) are special cases of Orlicz spaces ๐ฟ๐œ‘ originated by Nakano [5] and developed by Musielak and Orlicz [6, 7], where ๐‘“โˆˆ๐ฟ๐œ‘ if and only if โˆซ๐œ‘(๐‘ฅ,|๐‘“(๐‘ฅ)|)๐‘‘๐‘ฅ<โˆž for a suitable ๐œ‘. Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers. In that context we refer to the studies of Tsenov [8], Sharapudinov [9], and Zhikov [10, 11].

In this paper, we consider the following nonlinear elliptic boundary value problem: ๎‚€||||โˆ’div๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)โˆ’2๎‚โˆ‡๐‘ข+๐‘(๐‘ฅ)|๐‘ข|๐‘(๐‘ฅ)โˆ’2||||๐‘ข=๐œ†๐‘“(๐‘ฅ,๐‘ข),๐‘ฅโˆˆฮฉ,๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)โˆ’2๐œ•๐‘ข๐œ•๐œˆ=๐‘(๐‘ฅ)|๐‘ข|๐‘ž(๐‘ฅ)โˆ’2๐‘ข+๐œ‡๐‘”(๐‘ฅ,๐‘ข),๐‘ฅโˆˆ๐œ•ฮฉ,(1.1) where ฮฉโŠ‚โ„๐‘› is a bounded domain with Lipschitz boundary ๐œ•ฮฉ,๐œ•/๐œ•๐œˆ is outer unit normal derivative, ๐‘(๐‘ฅ)โˆˆ๐ถ(ฮฉ),๐‘ž(๐‘ฅ)โˆˆ๐ถ(๐œ•ฮฉ),๐‘(๐‘ฅ),๐‘ž(๐‘ฅ)>1, and ๐‘(๐‘ฅ)โ‰ ๐‘ž(๐‘ฆ) for any ๐‘ฅโˆˆฮฉ,๐‘ฆโˆˆ๐œ•ฮฉ;๐œ†,๐œ‡โˆˆโ„;๐‘“โˆถฮฉร—โ„โ†’โ„, and ๐‘”โˆถ๐œ•ฮฉร—โ„โ†’โ„ are Carathรฉdory functions. Throughout this paper, we assume that ๐‘Ž(๐‘ฅ),๐‘(๐‘ฅ), and ๐‘(๐‘ฅ) satisfy 0<๐‘Ž1โ‰ค๐‘Ž(๐‘ฅ)โ‰ค๐‘Ž2,0<๐‘1โ‰ค๐‘(๐‘ฅ)โ‰ค๐‘2, and 0โ‰ค๐‘1โ‰ค๐‘(๐‘ฅ)โ‰ค๐‘2.

The operator โˆ’ฮ”๐‘(๐‘ฅ)๐‘ขโˆถ=โˆ’div(|โˆ‡๐‘ข|๐‘(๐‘ฅ)โˆ’2โˆ‡๐‘ข) is called ๐‘(๐‘ฅ)-Laplacian, which is a natural extension of the ๐‘-Laplace operator, with ๐‘ being a positive constant. However, such generalizations are not trivial since the ๐‘(๐‘ฅ)-Laplace operator possesses a more complicated structure than the ๐‘-Laplace operator, for example, it is inhomogeneous. For related results involving the Laplace operator, see [12, 13].

In the past decade, many people have studied the nonlinear boundary value problems involving ๐‘-Laplacian. For example, if ๐œ†=๐œ‡=1,๐‘Ž(๐‘ฅ)=๐‘(๐‘ฅ)=๐‘(๐‘ฅ)โ‰ก1,๐‘(๐‘ฅ)โ‰ก๐‘, and ๐‘ž(๐‘ฅ)โ‰ก๐‘ž (a constant), then problem (1.1) becomes ๎‚€||||โˆ’divโˆ‡๐‘ข๐‘โˆ’2๎‚โˆ‡๐‘ข+|๐‘ข|๐‘โˆ’2||||๐‘ข=๐‘“(๐‘ฅ,๐‘ข),๐‘ฅโˆˆฮฉ,โˆ‡๐‘ข๐‘โˆ’2๐œ•๐‘ข๐œ•๐œˆ=|๐‘ข|๐‘žโˆ’2๐‘ข+๐‘”(๐‘ฅ,๐‘ข),๐‘ฅโˆˆ๐œ•ฮฉ.(1.2) Bonder and Rossi [14] considered the existence of nontrivial solutions of problem (1.2) when ๐‘“(๐‘ฅ,๐‘ข)โ‰ก0 and discussed different cases when ๐‘”(๐‘ฅ,๐‘ข) is subcritical, critical, and supercritical with respect to ๐‘ข. We also mention that Martรญnez and Rossi [15] studied the existence of solutions when ๐‘=๐‘ž and the perturbation terms ๐‘“(๐‘ฅ,๐‘ข) and ๐‘”(๐‘ฅ,๐‘ข) satisfy the Landesman-Lazer-type conditions. Recently, J.-H. Zhao and P.-H. Zhao [16] studied the nonlinear boundary value problem, assumed that ๐‘“(๐‘ฅ,๐‘ข) and ๐‘”(๐‘ฅ,๐‘ข) satisfy the Ambrosetti-Rabinowitz-type condition, and got the multiple results.

If ๐œ†=๐œ‡=1,๐‘(๐‘ฅ)โ‰ก๐‘, and ๐‘ž(๐‘ฅ)โ‰ก๐‘ž (a constant), then problem (1.1) becomes ๎‚€||||โˆ’div๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘โˆ’2๎‚โˆ‡๐‘ข+๐‘(๐‘ฅ)|๐‘ข|๐‘โˆ’2||||๐‘ข=๐‘“(๐‘ฅ,๐‘ข),๐‘ฅโˆˆฮฉ,๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘โˆ’2๐œ•๐‘ข๐œ•๐œˆ=๐‘(๐‘ฅ)|๐‘ข|๐‘žโˆ’2๐‘ข+๐‘”(๐‘ฅ,๐‘ข),๐‘ฅโˆˆ๐œ•ฮฉ.(1.3) There are also many people who studied the ๐‘-Laplacian nonlinear boundary value problems involving (1.3). For example, Cรฎrstea and RวŽdulescu [17] used the weighted Sobolev space to discuss the existence and nonexistence results and assumed that ๐‘“(๐‘ฅ,๐‘ข) is a special case in the problem (1.3), where ฮฉ is an unbounded domain. Pflรผger [18], by using the same technique, considered the existence and multiplicity of solutions when ๐‘(๐‘ฅ)โ‰ก0. The author showed the existence result when ๐‘“(๐‘ฅ,๐‘ข) and ๐‘”(๐‘ฅ,๐‘ข) are superlinear and satisfy the Ambrosetti-Rabinowitz-type condition and got the multiplicity of solutions when one of ๐‘“(๐‘ฅ,๐‘ข) and ๐‘”(๐‘ฅ,๐‘ข) is sublinear and the other one is superlinear.

More recently, the study on the nonlinear boundary value problems with variable exponent has received considerable attention. For example, Deng [19] studied the eigenvalue of ๐‘(๐‘ฅ)-Laplacian Steklov problem, and discussed the properties of the eigenvalue sequence under different conditions. Fan [20] discussed the boundary trace embedding theorems for variable exponent Sobolev spaces and some applications. Yao [21] constrained the two nonlinear perturbation terms ๐‘“(๐‘ฅ,๐‘ข) and ๐‘”(๐‘ฅ,๐‘ข) in appropriate conditions and got a number of results for the existence and multiplicity of solutions. Motivated by Yao and problem (1.3), we consider the more general form of the variable exponent boundary value problem (1.1). Under appropriate assumptions on the perturbation terms ๐‘“(๐‘ฅ,๐‘ข) and ๐‘”(๐‘ฅ,๐‘ข), by using the global minimizer method and fountain theorem, respectively, the existence and multiplicity of solutions of (1.1) were obtained. These results extend some of the results in [21] and the classical results for the ๐‘-Laplacian in [14, 16, 22โ€“24].

2. Preliminaries

In order to discuss problem (1.1), we need some results for the spaces ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ), which we call variable exponent Sobolev spaces. We state some basic properties of the spaces ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ), which will be used later (for more details, see [25, 26]). Let ฮฉ be a bounded domain of โ„๐‘›, and denote ๐ถ+๎‚€ฮฉ๎‚=๎‚†๎‚€๐‘(๐‘ฅ)โˆฃ๐‘(๐‘ฅ)โˆˆ๐ถฮฉ๎‚;๐‘(๐‘ฅ)>1,โˆ€๐‘ฅโˆˆฮฉ๎‚‡.(2.1) For ๐‘(๐‘ฅ)โˆˆ๐ถ+(ฮฉ) write ๐‘+=max๐‘ฅโˆˆฮฉ๐‘(๐‘ฅ),๐‘โˆ’=min๐‘ฅโˆˆฮฉ๐‘(๐‘ฅ).(2.2) We can also denote ๐ถ+(๐œ•ฮฉ) and ๐‘ž+,๐‘žโˆ’ for any ๐‘ž(๐‘ฅ)โˆˆ๐ถ(๐œ•ฮฉ), and define ๐ฟ๐‘(๐‘ฅ)๎‚ป๎€œ(ฮฉ)=๐‘ขโˆฃ๐‘ขisameasurablereal-valuedfunction,ฮฉ||||๐‘ข(๐‘ฅ)๐‘(๐‘ฅ)๎‚ผ,๐ฟ๐‘‘๐‘ฅ<โˆž๐‘(๐‘ฅ)๎‚ป๎€œ(๐œ•ฮฉ)=๐‘ขโˆฃ๐‘ขโˆถ๐œ•ฮฉโŸถโ„isameasurablereal-valuedfunction,๐œ•ฮฉ||||๐‘ข(๐‘ฅ)๐‘(๐‘ฅ)๎‚ผ,๐‘‘๐œŽ<โˆž(2.3) with norms on ๐ฟ๐‘(๐‘ฅ)(ฮฉ) and ๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ) defined by|๐‘ข|๐ฟ๐‘(๐‘ฅ)(ฮฉ)=|๐‘ข|๐‘(๐‘ฅ)๎‚ป๎€œ=inf๐œ†>0โˆถฮฉ|||๐‘ข(๐‘ฅ)๐œ†|||๐‘(๐‘ฅ)๎‚ผ,๐‘‘๐‘ฅโ‰ค1|๐‘ข|๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ)๎‚ป๎€œ=inf๐œ>0โˆถ๐œ•ฮฉ|||๐‘ข(๐‘ฅ)๐œ|||๐‘(๐‘ฅ)๎‚ผ,๐‘‘๐œŽโ‰ค1(2.4) where ๐‘‘๐œŽ is the surface measure on ๐œ•ฮฉ. Then, (๐ฟ๐‘(๐‘ฅ)(ฮฉ),|โ‹…|๐‘(๐‘ฅ)) and (๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ),|โ‹…|๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ)) become Banach spaces, which we call variable exponent Lebesgue spaces. Let us define the space ๐‘Š1,๐‘(๐‘ฅ)๎€ฝ(ฮฉ)=๐‘ขโˆˆ๐ฟ๐‘(๐‘ฅ)||||(ฮฉ)โˆถโˆ‡๐‘ขโˆˆ๐ฟ๐‘(๐‘ฅ)๎€พ,(ฮฉ)(2.5) equipped with the norm ๎‚ป๎€œโ€–๐‘ขโ€–=inf๐œ†>0โˆถฮฉ๎‚ต|||โˆ‡๐‘ข(๐‘ฅ)๐œ†|||๐‘(๐‘ฅ)+|||๐‘ข(๐‘ฅ)๐œ†|||๐‘(๐‘ฅ)๎‚ถ๎‚ผ.๐‘‘๐‘ฅโ‰ค1(2.6) For ๐‘ขโˆˆ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ), if we define โ€–๐‘ขโ€–๎…ž๎‚ป๎€œ=inf๐œ†>0โˆถฮฉ๎‚ต|||๐‘Ž(๐‘ฅ)โˆ‡๐‘ข(๐‘ฅ)๐œ†|||๐‘(๐‘ฅ)|||+๐‘(๐‘ฅ)๐‘ข(๐‘ฅ)๐œ†|||๐‘(๐‘ฅ)๎‚ถ๎‚ผ,๐‘‘๐‘ฅโ‰ค1(2.7) then, from the assumptions of ๐‘Ž(๐‘ฅ) and ๐‘(๐‘ฅ), it is easy to check that โ€–๐‘ขโ€–๎…ž is an equivalent norm on ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ). For simplicity, we denote ๎€œฮ“(๐‘ข)=ฮฉ๎‚€||||๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)+|๐‘ข|๐‘(๐‘ฅ)๎‚๐‘‘๐‘ฅ.(2.8)

Hence, we have (see [27])(i)if ฮ“(๐‘ข)โ‰ฅ1, then ๐œ‰1โ€–๐‘ขโ€–๐‘โˆ’โ‰คฮ“(๐‘ข)โ‰ค๐œ‰2โ€–๐‘ขโ€–๐‘+,(ii)if ฮ“(๐‘ข)โ‰ค1, then ๐œ1โ€–๐‘ขโ€–๐‘+โ‰คฮ“(๐‘ข)โ‰ค๐œ2โ€–๐‘ขโ€–๐‘โˆ’,where ๐œ‰1,๐œ‰2 and ๐œ1,๐œ2 are positive constants independent of ๐‘ข.

Denote by ๐‘Š01,๐‘(๐‘ฅ)(ฮฉ) the closure of ๐ถโˆž0(ฮฉ) in ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ).

Proposition 2.1 (see [21, 28]). (1) The space (๐ฟ๐‘(๐‘ฅ)(ฮฉ), |โ‹…|๐‘(๐‘ฅ)) is a separable, uniformly convex Banach space, and its conjugate space is ๐ฟ๐‘ž(๐‘ฅ)(ฮฉ), where 1/๐‘ž(๐‘ฅ)+1/๐‘(๐‘ฅ)=1. For any ๐‘ขโˆˆ๐ฟ๐‘(๐‘ฅ)(ฮฉ) and ๐‘ฃโˆˆ๐ฟ๐‘ž(๐‘ฅ)(ฮฉ), one has ||||๎€œฮฉ||||โ‰ค๎‚ต1๐‘ข๐‘ฃd๐‘ฅ๐‘โˆ’+1๐‘žโˆ’๎‚ถ|๐‘ข|๐‘(๐‘ฅ)|๐‘ฃ|๐‘ž(๐‘ฅ).(2.9)

(2) If ๐‘1,๐‘2โˆˆ๐ถ+(ฮฉ), ๐‘1(๐‘ฅ)โ‰ค๐‘2(๐‘ฅ), for any ๐‘ฅโˆˆฮฉ, then ๐ฟ๐‘2(๐‘ฅ)(ฮฉ)โ†ช๐ฟ๐‘1(๐‘ฅ)(ฮฉ) and the imbedding is continuous.

Proposition 2.2 (see [20, 21, 28]). (1) โ€‰โ€‰๐‘Š1,๐‘(๐‘ฅ)(ฮฉ),๐‘Š01,๐‘(๐‘ฅ)(ฮฉ) are separable reflexive Banach spaces.
(2) If ๐‘ž(๐‘ฅ)โˆˆ๐ถ+(ฮฉ) and ๐‘ž(๐‘ฅ)<๐‘โˆ—(๐‘ฅ) for any ๐‘ฅโˆˆฮฉ, then the embedding from ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ) into ๐ฟ๐‘ž(๐‘ฅ)(ฮฉ) is compact and continuous, where ๐‘โˆ—(โŽงโŽชโŽจโŽชโŽฉ๐‘ฅ)=๐‘›๐‘(๐‘ฅ)๐‘›โˆ’๐‘(๐‘ฅ),if๐‘(๐‘ฅ)<๐‘›,โˆž,if๐‘(๐‘ฅ)โ‰ฅ๐‘›.(2.10)
(3) If ๐‘ž(๐‘ฅ)โˆˆ๐ถ+(๐œ•ฮฉ) and ๐‘ž(๐‘ฅ)<๐‘โˆ—(๐‘ฅ) for any ๐‘ฅโˆˆ๐œ•ฮฉ, then the trace imbedding from ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ) into ๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ) is compact and continuous, where ๐‘โˆ—(โŽงโŽชโŽจโŽชโŽฉ๐‘ฅ)=(๐‘›โˆ’1)๐‘(๐‘ฅ)๐‘›โˆ’๐‘(๐‘ฅ),if๐‘(๐‘ฅ)<๐‘›,โˆž,if๐‘(๐‘ฅ)โ‰ฅ๐‘›.(2.11)
(4) (Poincarรฉ inequality) There is a constant ๐ถ>0, such that |๐‘ข|๐‘(๐‘ฅ)||||โ‰ค๐ถโˆ‡๐‘ข๐‘(๐‘ฅ)โˆ€๐‘ขโˆˆ๐‘Š01,๐‘(๐‘ฅ)(ฮฉ).(2.12)

Proposition 2.3 (see [21, 28, 29]). If ๐‘“โˆถฮฉร—โ„โ†’โ„ is a Carathรฉodory function and satisfies ||||๐‘“(๐‘ฅ,๐‘ )โ‰ค๐‘Ž(๐‘ฅ)+๐‘|๐‘ |๐‘1(๐‘ฅ)/๐‘2(๐‘ฅ),forany๐‘ฅโˆˆฮฉ,๐‘ โˆˆโ„,(2.13) where ๐‘1(๐‘ฅ), ๐‘2(๐‘ฅ)โˆˆ๐ถ+(ฮฉ), ๐‘Ž(๐‘ฅ)โˆˆ๐ฟ๐‘2(๐‘ฅ)(ฮฉ), ๐‘Ž(๐‘ฅ)โ‰ฅ0, and ๐‘โ‰ฅ0 is a constant, then the Nemytsky operator from ๐ฟ๐‘1(๐‘ฅ)(ฮฉ) to ๐ฟ๐‘2(๐‘ฅ)(ฮฉ) defined by (๐‘๐‘“(๐‘ข))(๐‘ฅ)=๐‘“(๐‘ฅ,๐‘ข(๐‘ฅ)) is a continuous and bounded operator.

Proposition 2.4 (see [21, 28, 30]). Denote ๎€œ๐œŒ(๐‘ข)=ฮฉ|๐‘ข|๐‘(๐‘ฅ)d๐‘ฅ,โˆ€๐‘ขโˆˆ๐ฟ๐‘(๐‘ฅ)(ฮฉ).(2.14) Then,
(1)|๐‘ข|๐‘(๐‘ฅ)<1(=1;>1) if and only if ๐œŒ(๐‘ข)<1(=1;>1),
(2)โ€‰โ€‰|๐‘ข|๐‘(๐‘ฅ)>1 implies |๐‘ข|๐‘โˆ’๐‘(๐‘ฅ)โ‰ค๐œŒ(๐‘ข)โ‰ค|๐‘ข|๐‘+๐‘(๐‘ฅ) and |๐‘ข|๐‘(๐‘ฅ)<1 implies |๐‘ข|๐‘โˆ’๐‘(๐‘ฅ)โ‰ฅ๐œŒ(๐‘ข)โ‰ฅ|๐‘ข|๐‘+๐‘(๐‘ฅ),
(3)|๐‘ข|๐‘(๐‘ฅ)โ†’0 if and only if ๐œŒ(๐‘ข)โ†’0 and |๐‘ข(๐‘ฅ)|๐‘(๐‘ฅ)โ†’โˆž if and only if ๐œŒ(๐‘ข)โ†’โˆž.

Proposition 2.5 (see [19]). Denote ๎€œ๐œŒ(๐‘ข)=๐œ•ฮฉ|๐‘ข|๐‘(๐‘ฅ)d๐œŽ,โˆ€๐‘ขโˆˆ๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ).(2.15) Then, (1)|๐‘ข|๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ)>1 implies |๐‘ข|๐‘โˆ’๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ)โ‰ค๐œŒ(๐‘ข)โ‰ค|๐‘ข|๐‘+๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ),(2)|๐‘ข|๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ)<1 implies |๐‘ข|๐‘โˆ’๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ)โ‰ฅ๐œŒ(๐‘ข)โ‰ฅ|๐‘ข|๐‘+๐ฟ๐‘(๐‘ฅ)(๐œ•ฮฉ).

3. Assumptions and Statement of Main Results

In the following, let ๐‘‹ denote the generalized Sobolev space ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ), โ€‰๐‘‹โˆ— denote the dual space of ๐‘Š1,๐‘(๐‘ฅ)(ฮฉ), โŸจโ‹…โŸฉ denote the dual pair, and let โ†’ represent strong convergence, โ‡€ represent weak convergence, ๐ถ, ๐ถ๐‘– represent the generic positive constants.

Now we state the assumptions on perturbation terms ๐‘“(๐‘ฅ,๐‘ข) and ๐‘”(๐‘ฅ,๐‘ข) for problem (1.1) as follows: (๐‘“0)โ€‰๐‘“โˆถฮฉร—โ„โ†’โ„ satisfies Carathรฉodory condition and there exist two constants ๐‘1โ‰ฅ0,๐‘2>0 such that ||||๐‘“(๐‘ฅ,๐‘ข)โ‰ค๐‘1+๐‘2|๐‘ข|๐›ผ(๐‘ฅ)โˆ’1,โˆ€(๐‘ฅ,๐‘ข)โˆˆฮฉร—โ„,(3.1) where ๐›ผ(๐‘ฅ)โˆˆ๐ถ+(ฮฉ) and ๐›ผ(๐‘ฅ)<๐‘โˆ—(๐‘ฅ) for any ๐‘ฅโˆˆฮฉ.(๐‘“1) There exist ๐‘€1>0,๐œƒ1>๐‘+ such that 0<๐œƒ1๐น(๐‘ฅ,๐‘ข)โ‰ค๐‘“(๐‘ฅ,๐‘ข)๐‘ข,|๐‘ข|โ‰ฅ๐‘€1,โˆ€๐‘ฅโˆˆฮฉ.(3.2)(๐‘“2)๐‘“(๐‘ฅ,โˆ’๐‘ข)=โˆ’๐‘“(๐‘ฅ,๐‘ข),forall๐‘ฅโˆˆฮฉ,๐‘ขโˆˆโ„.(๐‘”0)๐‘”โˆถ๐œ•ฮฉร—โ„โ†’โ„ satisfies Carathรฉodory condition and there exist two constants ๐‘๎…ž1โ‰ฅ0,๐‘๎…ž2>0 such that ||||๐‘”(๐‘ฅ,๐‘ข)โ‰ค๐‘๎…ž1+๐‘๎…ž2|๐‘ข|๐›ฝ(๐‘ฅ)โˆ’1,โˆ€(๐‘ฅ,๐‘ข)โˆˆ๐œ•ฮฉร—โ„,(3.3) where ๐›ฝ(๐‘ฅ)โˆˆ๐ถ+(๐œ•ฮฉ) and ๐›ฝ(๐‘ฅ)<๐‘โˆ—(๐‘ฅ) for any ๐‘ฅโˆˆ๐œ•ฮฉ.(๐‘”1) There exist ๐‘€2>0,๐œƒ2>๐‘+ such that 0<๐œƒ2๐บ(๐‘ฅ,๐‘ข)โ‰ค๐‘”(๐‘ฅ,๐‘ข)๐‘ข,|๐‘ข|โ‰ฅ๐‘€2,โˆ€๐‘ฅโˆˆ๐œ•ฮฉ.(3.4)(๐‘”2)๐‘”(๐‘ฅ,โˆ’๐‘ข)=โˆ’๐‘”(๐‘ฅ,๐‘ข),forall๐‘ฅโˆˆ๐œ•ฮฉ,๐‘ขโˆˆโ„.

The functional associated with problem (1.1) is ๎€œ๐œ‘(๐‘ข)=ฮฉ||||๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)+๐‘(๐‘ฅ)|๐‘ข|๐‘(๐‘ฅ)๎€œ๐‘(๐‘ฅ)d๐‘ฅโˆ’๐œ†ฮฉโˆ’๎€œ๐น(๐‘ฅ,๐‘ข)๐‘‘๐‘ฅ๐œ•ฮฉ๐‘(๐‘ฅ)|๐‘ž(๐‘ฅ)๐‘ข|๐‘ž(๐‘ฅ)๎€œd๐œŽโˆ’๐œ‡๐œ•ฮฉ๐บ(๐‘ฅ,๐‘ข)๐‘‘๐œŽ,(3.5) where ๐น(๐‘ฅ,๐‘ข) and ๐บ(๐‘ฅ,๐‘ข) are denoted by ๎€œ๐น(๐‘ฅ,๐‘ข)=๐‘ข0๎€œ๐‘“(๐‘ฅ,๐‘ )d๐‘ ,๐บ(๐‘ฅ,๐‘ข)=๐‘ข0๐‘”(๐‘ฅ,๐‘ )๐‘‘๐‘ .(3.6) By Propositions 3.1 and 3.2, and assumptions (๐‘“0), (๐‘”0), it is easy to see that the functional ๐œ‘โˆˆ๐ถ1(๐‘‹,โ„); moreover, ๐œ‘ is even if (๐‘“2) and (๐‘”3) hold. Then, ๎ซ๐œ‘๎…ž(๎ฌ=๎€œ๐‘ข),๐‘ฃฮฉ๎‚€||||๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)โˆ’2โˆ‡๐‘ขโˆ‡๐‘ฃ+๐‘(๐‘ฅ)|๐‘ข|๐‘(๐‘ฅ)โˆ’2๎‚๎€œ๐‘ข๐‘ฃ๐‘‘๐‘ฅโˆ’๐œ†ฮฉโˆ’๎€œ๐‘“(๐‘ฅ,๐‘ข)๐‘ฃ๐‘‘๐‘ฅ๐œ•ฮฉ๐‘(๐‘ฅ)|๐‘ข|๐‘ž(๐‘ฅ)โˆ’2๎€œ๐‘ข๐‘ฃ๐‘‘๐œŽโˆ’๐œ‡๐œ•ฮฉ๐‘”(๐‘ฅ,๐‘ข)๐‘ฃ๐‘‘๐œŽ,(3.7) so the weak solution of (1.1) corresponds to the critical point of the functional ๐œ‘.

Before giving our main results, we first give several propositions that will be used later.

Proposition 3.1 (see [31]). If one denotes ๎€œ๐ผ(๐‘ข)=ฮฉ||||๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)+๐‘(๐‘ฅ)|๐‘ข|๐‘(๐‘ฅ)๐‘(๐‘ฅ)d๐‘ฅ,โˆ€๐‘ขโˆˆ๐‘‹,(3.8) then ๐ผโˆˆ๐ถ1(๐‘‹,โ„) and the derivative operator of ๐ผ, denoted by ๐ผโ€ฒ, is ๎ซ๐ผ๎…ž(๎ฌ=๎€œ๐‘ข),๐‘ฃฮฉ๎‚€||||๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)โˆ’2โˆ‡๐‘ขโˆ‡๐‘ฃ+๐‘(๐‘ฅ)|๐‘ข|๐‘(๐‘ฅ)โˆ’2๎‚๐‘ข๐‘ฃd๐‘ฅ,โˆ€๐‘ข,๐‘ฃโˆˆ๐‘‹,(3.9) and one has:(i)๐ผโ€ฒโˆถ๐‘‹โ†’๐‘‹โˆ— is a continuous, bounded, and strictly monotone operator,(ii)๐ผโ€ฒ is a mapping of (S+) type, that is, if ๐‘ข๐‘›โ‡€๐‘ข in ๐‘‹ and limsup๐‘›โ†’โˆžโŸจ๐ผโ€ฒ(๐‘ข๐‘›)โˆ’๐ผโ€ฒ(๐‘ข),๐‘ข๐‘›โˆ’๐‘ขโŸฉโ‰ค0, then ๐‘ข๐‘›โ†’๐‘ข in ๐‘‹,(iii)๐ผโ€ฒโˆถ๐‘‹โ†’๐‘‹โˆ— is a homeomorphism.

Proposition 3.2 (see [19]). If one denotes ๎€œ๐ฝ(๐‘ข)=๐œ•ฮฉ๐‘(๐‘ฅ)|๐‘ž(๐‘ฅ)๐‘ข|๐‘ž(๐‘ฅ)d๐œŽ,โˆ€๐‘ขโˆˆ๐‘‹,(3.10) where ๐‘ž(๐‘ฅ)โˆˆ๐ถ+(๐œ•ฮฉ) and ๐‘ž(๐‘ฅ)<๐‘โˆ—(๐‘ฅ) for any ๐‘ฅโˆˆ๐œ•ฮฉ, then ๐ฝโˆˆ๐ถ1(๐‘‹,โ„) and the derivative operator ๐ฝโ€ฒ of ๐ฝ is ๎ซ๐ฝ๎…ž(๎ฌ=๎€œ๐‘ข),๐‘ฃ๐œ•ฮฉ๐‘(๐‘ฅ)|๐‘ข|๐‘ž(๐‘ฅ)โˆ’2๐‘ข๐‘ฃd๐œŽ,โˆ€๐‘ข,๐‘ฃโˆˆ๐‘‹,(3.11) and one has that ๐ฝโˆถ๐‘‹โ†’โ„ and ๐ฝโ€ฒโˆถ๐‘‹โ†’๐‘‹โˆ— are sequentially weakly-strongly continuous, namely, ๐‘ข๐‘›โ‡€๐‘ข in ๐‘‹ implies ๐ฝโ€ฒ(๐‘ข๐‘›)โ†’๐ฝโ€ฒ(๐‘ข).

Let ๐‘‹ be a reflexive and separable Banach space. There exist ๐‘’๐‘–โˆˆ๐‘‹ and ๐‘’โˆ—๐‘—โˆˆ๐‘‹โˆ— such that ๐‘‹=๎€ฝ๐‘’span๐‘–๎€พโˆถ๐‘–=1,2,โ€ฆ,๐‘‹โˆ—=๎‚†๐‘’spanโˆ—๐‘—๎‚‡,๎ซ๐‘’โˆถ๐‘—=1,2,โ€ฆ๐‘–,๐‘’โˆ—๐‘—๎ฌ=๎‚ป1,๐‘–=๐‘—,0,๐‘–โ‰ ๐‘—.(3.12) For ๐‘˜=1,2,โ€ฆ, denote ๐‘‹๐‘˜๎€ฝ๐‘’=span๐‘˜๎€พ,๐‘Œ๐‘˜=๐‘˜๎ถ๐‘–=1๐‘‹๐‘–,๐‘๐‘˜=๎ถ๐‘–โ‰ฅ๐‘˜๐‘‹๐‘–.(3.13)

One important aspect of applying the standard methods of variational theory is to show that the functional ๐œ‘ satisfies the ๐‘ƒ๐‘Ž๐‘™๐‘Ž๐‘–๐‘ -๐‘†๐‘š๐‘Ž๐‘™๐‘’ condition, which is introduced by the following definition.

Definition 3.3. Let ๐œ‘โˆˆ๐ถ1(๐‘‹,โ„) and ๐‘โˆˆโ„. Then, functional ๐œ‘ satisfies the (PS)๐‘ condition if any sequence {๐‘ข๐‘›}โŠ‚๐‘‹ such that ๐œ‘๎€ท๐‘ข๐‘›๎€ธโŸถ๐‘,๐œ‘๎…ž๎€ท๐‘ข๐‘›๎€ธโŸถ0in๐‘‹โˆ—,as๐‘›โŸถโˆž(3.14) contains a subsequence converging to a critical point of ๐œ‘.
In what follows we write the (PS)๐‘ condition simply as the (PS) condition if it holds for every level ๐‘โˆˆโ„ for the ๐‘ƒ๐‘Ž๐‘™๐‘Ž๐‘–๐‘ -๐‘†๐‘š๐‘Ž๐‘™๐‘’ condition at level ๐‘.

Proposition 3.4 (Fountain theorem, see [23, 32]). Assume that(A1)๐‘‹ is a Banach space, ๐œ‘โˆˆ๐ถ1(๐‘‹,โ„) is an even functional, the subspaces ๐‘‹๐‘˜,๐‘Œ๐‘˜ and ๐‘๐‘˜ are defined by (3.13).Suppose that, for every ๐‘˜โˆˆ๐, there exist ๐œŒ๐‘˜>๐›พ๐‘˜>0 such that(A2)inf๐‘ขโˆˆ๐‘๐‘˜,โ€–๐‘ขโ€–=๐›พ๐‘˜๐œ‘(๐‘ข)โ†’โˆž as ๐‘˜โ†’โˆž,(A3)max๐‘ขโˆˆ๐‘Œ๐‘˜,โ€–๐‘ขโ€–=๐œŒ๐‘˜๐œ‘(๐‘ข)โ‰ค0,(A4)๐œ‘ satisfies (PS)๐‘ condition for every ๐‘>0.

Then, ๐œ‘ has a sequence of critical values tending to +โˆž.

Proposition 3.5 (see [21]). Suppose that hypotheses ๐›ผ(๐‘ฅ)โˆˆ๐ถ+(ฮฉ),๐›ผ(๐‘ฅ)<๐‘โˆ—(๐‘ฅ),forall๐‘ฅโˆˆฮฉ, and if ๐‘ž(๐‘ฅ)โˆˆ๐ถ+(๐œ•ฮฉ),๐‘ž(๐‘ฅ)<๐‘โˆ—(๐‘ฅ),forall๐‘ฅโˆˆ๐œ•ฮฉ, denote ๐›ผ๐‘˜๎€ฝ=sup|๐‘ข|๐ฟ๐›ผ(๐‘ฅ)(ฮฉ)โˆถโ€–๐‘ขโ€–=1,๐‘ขโˆˆ๐‘๐‘˜๎€พ;๐‘ž๐‘˜๎€ฝ=sup|๐‘ข|๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)โˆถโ€–๐‘ขโ€–=1,๐‘ขโˆˆ๐‘๐‘˜๎€พ,(3.15) then lim๐‘˜โ†’โˆž๐›ผ๐‘˜=0,lim๐‘˜โ†’โˆž๐‘ž๐‘˜=0.
Let us introduce the following lemma that will be useful in the proof of our main result.

Lemma 3.6. Let ๐œ†,๐œ‡โ‰ฅ0,๐‘žโˆ’>๐œƒ1,๐œƒ2, and assume that (๐‘“0),(๐‘“1),(๐‘”0),and(๐‘”1) are satisfied, then ๐œ‘ satisfies (PS) condition.

Proof. By Propositions 2.2 and 2.3, we know that if we denote ๎€œฮฆ(๐‘ข)=๐œ†ฮฉ๎€œ๐น(๐‘ฅ,๐‘ข)๐‘‘๐‘ฅ+๐œ‡๐œ•ฮฉ๐บ(๐‘ฅ,๐‘ข)๐‘‘๐œŽ,(3.16) then ฮฆ is weakly continuous and its derivative operator, denoted by ฮฆโ€ฒ, is compact. By Propositions 3.1 and 3.2, we deduce that ๐œ‘โ€ฒ=๐ผโ€ฒโˆ’๐ฝโ€ฒโˆ’ฮฆโ€ฒ is also of (S+) type. To verify that ๐œ‘ satisfies (PS) condition on ๐‘‹, it is enough to verify that any (PS) sequence is bounded. Suppose that {๐‘ข๐‘›}โŠ‚๐‘‹ such that ๐œ‘๎€ท๐‘ข๐‘›๎€ธโŸถ๐‘,๐œ‘๎…ž๎€ท๐‘ข๐‘›๎€ธโŸถ0,in๐‘‹โˆ—,as๐‘›โŸถโˆž.(3.17) Then, for ๐‘› large enough, we can find ๐‘€3>0 such that ||๐œ‘๎€ท๐‘ข๐‘›๎€ธ||โ‰ค๐‘€3.(3.18) Since ๐œ‘โ€ฒ(๐‘ข๐‘›)โ†’0, we have โŸจ๐œ‘โ€ฒ(๐‘ข๐‘›),๐‘ข๐‘›โŸฉโ†’0. In particular, {โŸจ๐œ‘โ€ฒ(๐‘ข๐‘›),๐‘ข๐‘›โŸฉ} is bounded. Thus, there exists ๐‘€4>0 such that ||๎ซ๐œ‘๎…ž๎€ท๐‘ข๐‘›๎€ธ,๐‘ข๐‘›๎ฌ||โ‰ค๐‘€4.(3.19) We claim that the sequence {๐‘ข๐‘›} is bounded. If it is not true, by passing a subsequence if necessary, we may assume that โ€–๐‘ข๐‘›โ€–โ†’+โˆž. Without loss of generality, we assume that โ€–๐‘ข๐‘›โ€–โ‰ฅ1 appropriately large such that ๐œ‰1โ€–๐‘ขโ€–๐‘โˆ’<๐œ1โ€–๐‘ขโ€–๐‘+ for any ๐‘ฅโˆˆฮฉ. From (3.18) and (3.19) and letting ๐œƒ=min{๐œƒ1,๐œƒ2}, then ๐œƒ<๐‘žโˆ’, we have ๐‘€3๎€ท๐‘ขโ‰ฅ๐œ‘๐‘›๎€ธ๎€ท๐‘ข=๐ผ๐‘›๎€ธ๎€ท๐‘ขโˆ’๐ฝ๐‘›๎€ธ๎€ท๐‘ขโˆ’ฮฆ๐‘›๎€ธโ‰ฅ1๐‘+ฮ“๎€ท๐‘ข๐‘›๎€ธโˆ’1๐‘žโˆ’๎€œ๐œ•ฮฉ๐‘||๐‘ข(๐‘ฅ)๐‘›||๐‘ž(๐‘ฅ)๎€ท๐‘ข๐‘‘๐œŽโˆ’ฮฆ๐‘›๎€ธ,โ‰ฅ1๐‘+ฮ“๎€ท๐‘ข๐‘›๎€ธโˆ’1๐œƒ๎€œ๐œ•ฮฉ๐‘||๐‘ข(๐‘ฅ)๐‘›||๐‘ž(๐‘ฅ)๎€ท๐‘ข๐‘‘๐œŽโˆ’ฮฆ๐‘›๎€ธ,๐‘€(3.20)4๎ซ๐œ‘โ‰ฅโˆ’๎…ž๎€ท๐‘ข๐‘›๎€ธ,๐‘ข๐‘›๎ฌ๎€ท๐‘ข=โˆ’ฮ“๐‘›๎€ธ+๎€œ๐œ•ฮฉ๐‘||๐‘ข(๐‘ฅ)๐‘›||๐‘ž(๐‘ฅ)๎ซฮฆ๐‘‘๐œŽ+๎…ž๎€ท๐‘ข๐‘›๎€ธ,๐‘ข๐‘›๎ฌ.(3.21) By virtue of assumptions (๐‘“1) and (๐‘”1) and combining (3.20) and (3.21), we have ๐œƒ๐‘€3+๐‘€4โ‰ฅ๎‚ต๐œƒ๐‘+๎‚ถฮ“๎€ท๐‘ขโˆ’1๐‘›๎€ธ๎€ท๐‘ขโˆ’๐œƒฮฆ๐‘›๎€ธ+๎ซฮฆ๎…ž๎€ท๐‘ข๐‘›๎€ธ,๐‘ข๐‘›๎ฌโ‰ฅ๎‚ต๐œƒ๐‘+๎‚ถ๐œ‰โˆ’11โ€–โ€–๐‘ข๐‘›โ€–โ€–๐‘โˆ’๎€œ+๐œ†ฮฉ๎€ท๐‘“๎€ท๐‘ฅ,๐‘ข๐‘›๎€ธ๐‘ข๐‘›๎€ทโˆ’๐œƒ๐น๐‘ฅ,๐‘ข๐‘›๎€œ๎€ธ๎€ธ๐‘‘๐‘ฅ+๐œ‡๐œ•ฮฉ๎€ท๐‘”๎€ท๐‘ฅ,๐‘ข๐‘›๎€ธ๐‘ข๐‘›๎€ทโˆ’๐œƒ๐บ๐‘ฅ,๐‘ข๐‘›โ‰ฅ๎‚ต๐œƒ๎€ธ๎€ธ๐‘‘๐œŽ๐‘+๎‚ถ๐œ‰โˆ’11โ€–โ€–๐‘ข๐‘›โ€–โ€–๐‘โˆ’โˆ’๐ถ.(3.22) Note that ๐œƒ=min{๐œƒ1,๐œƒ2}>๐‘+, let ๐‘›โ†’โˆž we obtian a contradiction. It follows that the sequence {๐‘ข๐‘›} is bounded in ๐‘‹. Therefore, ๐œ‘ satisfies (PS) condition.

Under appropriate assumptions on the perturbation terms ๐‘“(๐‘ฅ,๐‘ข),๐‘”(๐‘ฅ,๐‘ข), a sequence of weak solutions with energy values tending to +โˆž was obtained. The main result of the paper reads as follows.

Theorem 3.7. Let ๐›ผโˆ’,๐›ฝโˆ’>๐‘+,๐‘žโˆ’>๐œƒ1,๐œƒ2, and ๐œ†,๐œ‡โ‰ฅ0, and assumed that (๐‘“0)โˆ’(๐‘“2),(๐‘”0)โˆ’(๐‘”2) are satisfied; then ๐œ‘ has a sequence of critical points {ยฑ๐‘ข๐‘›} such that ๐œ‘(ยฑ๐‘ข๐‘›)โ†’โˆž as ๐‘›โ†’โˆž.

Proof. We will prove that ๐œ‘ satisfies the conditions of Proposition 3.4. Obviously, because of the assumptions of (๐‘“2) and (๐‘”2), ๐œ‘ is an even functional and satisfies (PS) condition (see Lemma 3.6). We will prove that if ๐‘˜ is large enough, then there exist ๐œŒ๐‘˜>๐›พ๐‘˜>0 such that (A2) and (A3) hold. By virtue of (๐‘“0), (๐‘”0), there exist two positive constants ๐ถ1,๐ถ2 such that ||||๐น(๐‘ฅ,๐‘ข)โ‰ค๐ถ1๎€ท1+|๐‘ข|๐›ผ(๐‘ฅ)๎€ธ||||,(๐‘ฅ,๐‘ข)โˆˆฮฉร—โ„;๐บ(๐‘ฅ,๐‘ข)โ‰ค๐ถ2๎€ท1+|๐‘ข|๐›ฝ(๐‘ฅ)๎€ธ,(๐‘ฅ,๐‘ข)โˆˆ๐œ•ฮฉร—โ„.(3.23) Letting ๐‘ขโˆˆ๐‘๐‘˜ with โ€–๐‘ขโ€–>1 appropriately large such that ๐œ‰1โ€–๐‘ขโ€–๐‘โˆ’<๐œ1โ€–๐‘ขโ€–๐‘+, we have โ‰ฅ1๐œ‘(๐‘ข)=๐ผ(๐‘ข)โˆ’๐ฝ(๐‘ข)โˆ’ฮฆ(๐‘ข)๐‘+๐‘ฮ“(๐‘ข)โˆ’2๐‘žโˆ’๎€œ๐œ•ฮฉ|๐‘ข|๐‘ž(๐‘ฅ)๎€œ๐‘‘๐œŽโˆ’๐œ†ฮฉ๐ถ1๎€ท1+|๐‘ข|๐›ผ(๐‘ฅ)๎€ธ๎€œ๐‘‘๐‘ฅโˆ’๐œ‡๐œ•ฮฉ๐ถ2๎€ท1+|๐‘ข|๐›ฝ(๐‘ฅ)๎€ธโ‰ฅ1๐‘‘๐œŽ๐‘+๎‚†๐œ‰min1โ€–๐‘ขโ€–๐‘โˆ’,๐œ1โ€–๐‘ขโ€–๐‘+๎‚‡โˆ’๐‘2๐‘žโˆ’๎‚†max|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)๎‚‡โˆ’๐œ†๐ถ1๎‚†max|๐‘ข|๐›ผ+๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),|๐‘ข|๐›ผโˆ’๐ฟ๐›ผ(๐‘ฅ)(ฮฉ)๎‚‡โˆ’๐œ‡๐ถ2๎‚†max|๐‘ข|๐›ฝ+๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐›ฝโˆ’๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ)๎‚‡โˆ’๐ถ3โ‰ฅ๐œ‰1๐‘+โ€–๐‘ขโ€–๐‘โˆ’โˆ’๐ถ(๐‘žโˆ’๎‚†,๐œ†,๐œ‡)max|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐›ผ+๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),|๐‘ข|๐›ผโˆ’๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),|๐‘ข|๐›ฝ+๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐›ฝโˆ’๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ)๎‚‡โˆ’๐ถ3.(3.24) If max{|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐›ผ+๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),|๐‘ข|๐›ผโˆ’๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),|๐‘ข|๐›ฝ+๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐›ฝโˆ’๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ)}=|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ), then by Proposition 3.5, we have ๐œ‘๎€ท๐‘ข๐‘›๎€ธโ‰ฅ๐œ‰1๐‘+โ€–๐‘ขโ€–๐‘โˆ’โˆ’๐ถ(๐‘žโˆ’,๐œ†,๐œ‡)|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)โˆ’๐ถ3โ‰ฅ๐œ‰1๐‘+โ€–๐‘ขโ€–๐‘โˆ’โˆ’๐ถ(๐‘žโˆ’,๐œ†,๐œ‡)๐‘ž๐‘ž+๐‘˜โ€–๐‘ขโ€–๐‘ž+โˆ’๐ถ3.(3.25) Choose ๐›พ๐‘˜=(๐‘ž+๐ถ(๐‘žโˆ’,๐œ†,๐œ‡)(๐‘ž_๐‘˜(๐‘ž+))/๐œ‰_1)1/(๐‘โˆ’โˆ’๐‘ž+). For ๐‘ขโˆˆ๐‘๐‘˜ with โ€–๐‘ขโ€–=๐›พ๐‘˜, we have ๐œ‘(๐‘ข)โ‰ฅ๐œ‰1๎‚ต1๐‘+โˆ’1๐‘ž+๎‚ถ๐›พ๐‘โˆ’๐‘˜โˆ’๐ถ3.(3.26) Since ๐‘ž๐‘˜โ†’0 as ๐‘˜โ†’โˆž and 1<๐‘โˆ’โ‰ค๐‘+<๐œƒ1,๐œƒ2<๐‘žโˆ’โ‰ค๐‘ž+, we have 1/๐‘+โˆ’1/๐‘ž+>0 and ๐›พ๐‘˜โ†’โˆž. Thus, for sufficiently large ๐‘˜, we have ๐œ‘(๐‘ข)โ†’โˆž with ๐‘ขโˆˆ๐‘๐‘˜ and โ€–๐‘ขโ€–=๐›พ๐‘˜ as ๐‘˜โ†’โˆž. In other cases, similarly, we can deduce ๐œ‘(๐‘ข)โŸถโˆž,since๐›ผ๐‘˜โŸถ0,๐‘ž๐‘˜=0,๐‘˜โŸถโˆž.(3.27) So (A2) holds.
By virtue of (๐‘“1) and (๐‘”1), there exist two positive constants ๐ถ4,๐ถ5 such that ๐น(๐‘ฅ,๐‘ข)โ‰ฅ๐ถ4๎€ท|๐‘ข|๐œƒ1๎€ธโˆ’1,โˆ€(๐‘ฅ,๐‘ข)โˆˆฮฉร—โ„;๐บ(๐‘ฅ,๐‘ข)โ‰ฅ๐ถ5๎€ท|๐‘ข|๐œƒ2๎€ธโˆ’1,โˆ€(๐‘ฅ,๐‘ข)โˆˆ๐œ•ฮฉร—โ„.(3.28) Letting ๐‘ขโˆˆ๐‘Œ๐‘˜, we have 1๐œ‘(๐‘ข)โ‰ค๐‘โˆ’๐‘ฮ“(๐‘ข)โˆ’1๐‘ž+๎€œ๐œ•ฮฉ|๐‘ข|๐‘ž(๐‘ฅ)๎€œ๐‘‘๐œŽโˆ’๐œ†ฮฉ๎€œ๐น(๐‘ฅ,๐‘ข)๐‘‘๐‘ฅโˆ’๐œ‡๐œ•ฮฉโ‰ค1๐บ(๐‘ฅ,๐‘ข)๐‘‘๐œŽ๐‘โˆ’๎‚†๐œ‰max2โ€–๐‘ขโ€–๐‘+,๐œ2โ€–๐‘ขโ€–๐‘โˆ’๎‚‡โˆ’๐‘1๐‘ž+๎‚†|min๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)๎‚‡โˆ’๐ถ4๐œ†๎€œฮฉ|๐‘ข|๐œƒ1๐‘‘๐‘ฅโˆ’๐ถ5๐œ‡๎€œ๐œ•ฮฉ|๐‘ข|๐œƒ2๐‘‘๐œŽ+๐ถ6.(3.29) If max{๐œ‰2โ€–๐‘ขโ€–๐‘+,๐œ2โ€–๐‘ขโ€–๐‘โˆ’}=๐œ‰2โ€–๐‘ขโ€–๐‘+,min{|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)}=|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ), then we have ๐œ‰๐œ‘(๐‘ข)โ‰ค2๐‘โˆ’โ€–๐‘ขโ€–๐‘+โˆ’๐‘1๐‘ž+|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)โˆ’๐ถ4๐œ†๎€œฮฉ|๐‘ข|๐œƒ1๐‘‘๐‘ฅโˆ’๐ถ5๐œ‡๎€œ๐œ•ฮฉ|๐‘ข|๐œƒ2๐‘‘๐œŽ+๐ถ6.(3.30) Since dim๐‘Œ๐‘˜<โˆž, all norms are equivalent in ๐‘Œ๐‘˜. So we get ๐œ‰๐œ‘(๐‘ข)โ‰ค2๐‘โˆ’โ€–๐‘ขโ€–๐‘+โˆ’๐‘1๐‘ž+๐ถ7โ€–๐‘ขโ€–๐‘žโˆ’โˆ’๐ถ8๐œ†โ€–๐‘ขโ€–๐œƒ1โˆ’๐ถ9๐œ‡โ€–๐‘ขโ€–๐œƒ2+๐ถ6.(3.31) Also, note that ๐‘žโˆ’>๐œƒ1,๐œƒ2>๐‘+, Then, we get ๐œ‘(๐‘ข)โ†’โˆ’โˆž as โ€–๐‘ขโ€–โ†’โˆž. For other cases, the proofs are similar and we omit them here. So (A3) holds. From the proof of (A2) and (A3), we can choose ๐œŒ๐‘˜>๐›พ๐‘˜>0. Thus, we complete the proof.

This time our idea is to show that ๐œ‘ possesses a nontrivial global minimum point in ๐‘‹.

Theorem 3.8. Let ๐›ผ+,๐›ฝ+,๐‘ž+<๐‘โˆ’, and assume (๐‘“0), (๐‘”0) are satisfied; then (1.1) has a weak solution.

Proof. Firstly, we show that ๐œ‘ is coercive. For sufficiently large norm of ๐‘ข(โ€–๐‘ขโ€–โ‰ฅ1), and by virtue of (3.23), ๎€œ๐œ‘(๐‘ข)=ฮฉ||||๐‘Ž(๐‘ฅ)โˆ‡๐‘ข๐‘(๐‘ฅ)+๐‘(๐‘ฅ)|๐‘ข|๐‘(๐‘ฅ)๎€œ๐‘(๐‘ฅ)๐‘‘๐‘ฅโˆ’๐œ†ฮฉ๎€œ๐น(๐‘ฅ,๐‘ข)๐‘‘๐‘ฅโˆ’๐œ•ฮฉ๐‘(๐‘ฅ)๐‘ž(๐‘ฅ)|๐‘ข|๐‘ž(๐‘ฅ)๎€œ๐‘‘๐œŽโˆ’๐œ‡๐œ•ฮฉโ‰ฅ๐œ‰๐บ(๐‘ฅ,๐‘ข)๐‘‘๐œŽ1๐‘+โ€–๐‘ขโ€–๐‘โˆ’โˆ’||๐œ†||๎€œฮฉ๐ถ1๎€ท1+|๐‘ข|๐›ผ(๐‘ฅ)๎€ธ๐‘๐‘‘๐‘ฅโˆ’2๐‘žโˆ’๎€œ๐œ•ฮฉ|๐‘ข|๐‘ž(๐‘ฅ)||๐œ‡||๎€œ๐‘‘๐œŽโˆ’๐œ•ฮฉ๐ถ2๎€ท1+|๐‘ข|๐›ฝ(๐‘ฅ)๎€ธโ‰ฅ๐œ‰๐‘‘๐œŽ1๐‘+โ€–๐‘ขโ€–๐‘โˆ’โˆ’||๐œ†||๐ถ1๎‚†max|๐‘ข|๐›ผ+๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),|๐‘ข|๐›ผโˆ’๐ฟ๐›ผ(๐‘ฅ)(ฮฉ)๎‚‡โˆ’๐‘2๐‘žโˆ’๎‚†max|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)๎‚‡โˆ’||๐œ‡||๐ถ2๎‚†max|๐‘ข|๐›ฝ+๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐›ฝโˆ’๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ)๎‚‡โˆ’๐ถ10.(3.32) If ๎‚†max|๐‘ข|๐›ผ+๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),|๐‘ข|๐›ผโˆ’๐ฟ๐›ผ(๐‘ฅ)(ฮฉ)๎‚‡=|๐‘ข|๐›ผ+๐ฟ๐›ผ(๐‘ฅ)(ฮฉ),๎‚†max|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐‘žโˆ’๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ)๎‚‡=|๐‘ข|๐‘ž+๐ฟ๐‘ž(๐‘ฅ)(๐œ•ฮฉ),๎‚†max|๐‘ข|๐›ฝ+๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ),|๐‘ข|๐›ฝโˆ’๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ)๎‚‡=|๐‘ข|๐›ฝ+๐ฟ๐›ฝ(๐‘ฅ)(๐œ•ฮฉ),(3.33) then ๐œ‰๐œ‘(๐‘ข)โ‰ฅ1๐‘+โ€–๐‘ขโ€–๐‘โˆ’โˆ’๐ถ11||๐œ†||โ€–๐‘ขโ€–๐›ผ+โˆ’๐ถ12โ€–๐‘ขโ€–๐‘ž+โˆ’๐ถ13||๐œ‡||โ€–๐‘ขโ€–๐›ฝ+โˆ’๐ถ10โŸถโˆžasโ€–๐‘ขโ€–โŸถโˆž.(3.34) So ๐œ‘ is coercive since ๐›ผ+,๐›ฝ+,๐‘ž+<๐‘โˆ’. Secondly, by Proposition 2.2, it is easy to verify that ๐œ‘ is weakly lower semicontinuous. Thus, ๐œ‘ is bounded below and ๐œ‘ attains its infimum in ๐‘‹, that is, ๐œ‘(๐‘ข0)=inf๐‘ขโˆˆ๐‘‹๐œ‘(๐‘ข) and ๐‘ข0 is a critical point of ๐œ‘, which is a weak solution of (1.1).

In the Theorem 3.8, we cannot guarantee that ๐‘ข0 is nontrivial. In fact, under the assumptions on the above theorem, we can also get a nontrivial weak solution of ๐œ‘.

Corollary 3.9. Under the assumptions in Theorem 3.8, if one of the following conditions holds, (1.1) has a nontrivial weak solution.(1)If ๐œ†,๐œ‡โ‰ 0, there exist two positive constants ๐‘‘1,๐‘‘2<๐‘โˆ’ such that liminf๐‘ขโ†’0sgn(๐œ†)๐น(๐‘ฅ,๐‘ข)|๐‘ข|๐‘‘1>0,for๐‘ฅโˆˆฮฉuniformly,liminf๐‘ขโ†’0sgn(๐œ‡)๐บ(๐‘ฅ,๐‘ข)|๐‘ข|๐‘‘2>0,for๐‘ฅโˆˆ๐œ•ฮฉuniformly.(3.35)(2)If ๐œ†=0,๐œ‡โ‰ 0, there exist two positive constants ๐‘‘2<๐‘โˆ’ such that liminf๐‘ขโ†’0sgn(๐œ‡)๐บ(๐‘ฅ,๐‘ข)|๐‘ข|๐‘‘2>0,for๐‘ฅโˆˆ๐œ•ฮฉuniformly.(3.36)(3)If ๐œ†โ‰ 0,๐œ‡=0, there exist two positive constants ๐‘‘1<๐‘โˆ’ such that lim๐‘ขโ†’0infsgn(๐œ†)๐น(๐‘ฅ,๐‘ข)|๐‘ข|๐‘‘1>0,for๐‘ฅโˆˆฮฉuniformly.(3.37)

Proof. From Theorem 3.8, we know that ๐œ‘ has a global minimum point ๐‘ข0. We just need to show that ๐‘ข0 is nontrivial. We only consider the case ๐œ†,๐œ‡โ‰ 0 here. From (1), we know that for 0<๐‘ข<1 small enough, there exists two positive constants ๐ถ14,๐ถ15>0 such that sgn(๐œ†)๐น(๐‘ฅ,๐‘ข)โ‰ฅ๐ถ14|๐‘ข|๐‘‘1,sgn(๐œ‡)๐บ(๐‘ฅ,๐‘ข)โ‰ฅ๐ถ15|๐‘ข|๐‘‘2.(3.38) Choose ๐‘ขโ‰ก๐‘€>0; then ๐‘ขโˆˆ๐‘‹. For 0<๐‘ก<1 small enough, we have ๐œ‘๎€ท๐‘ก๐‘ข๎€ธโ‰ค๐‘2๐‘ก๐‘โˆ’๐‘โˆ’๎€œฮฉ||๐‘ข||๐‘(๐‘ฅ)||๐œ†||๎€œ๐‘‘๐‘ฅโˆ’ฮฉ๎€ทsgn(๐œ†)๐น๐‘ฅ,๐‘ก๐‘ข๎€ธ๐‘๐‘‘๐‘ฅโˆ’1๐‘ž+๎€œ๐œ•ฮฉ||๐‘ก๐‘ข||๐‘ž(๐‘ฅ)โˆ’||๐œ‡||๎€œ๐‘‘๐œŽ๐œ•ฮฉ๎€ทsgn(๐œ‡)๐บ๐‘ฅ,๐‘ก๐‘ข๎€ธโ‰ค๐‘๐‘‘๐œŽ2๐‘ก๐‘โˆ’๐‘โˆ’๎€œฮฉ||๐‘€||๐‘(๐‘ฅ)d๐‘ฅโˆ’๐ถ14||๐œ†||๐‘ก๐‘‘1๎€œฮฉ||๐‘€||๐‘‘1๐‘๐‘‘๐‘ฅโˆ’1๐‘ž+๐‘ก๐‘žโˆ’๎€œ๐œ•ฮฉ||๐‘€||๐‘ž(๐‘ฅ)๐‘‘๐œŽโˆ’๐ถ15||๐œ‡||๐‘ก๐‘‘2๎€œ๐œ•ฮฉ||๐‘€||๐‘‘2๐‘‘๐œŽโ‰ค๐ถ16๐‘ก๐‘โˆ’โˆ’๐ถ17||๐œ†||๐‘ก๐‘‘1โˆ’๐ถ18๐‘1๐‘ก๐‘žโˆ’โˆ’๐ถ19||๐œ‡||๐‘ก๐‘‘2.(3.39) Since ๐‘‘1,๐‘‘2,<๐‘โˆ’ and ๐‘žโˆ’โ‰ค๐‘ž+<๐‘โˆ’, there exists 0<๐‘ก0<1 small enough such that ๐œ‘(๐‘ก0๐‘ข)<0. So the global minimum point ๐‘ข0 of ๐œ‘ is nontrivial.

Remark 3.10. Suppose that ๐‘“(๐‘ฅ,๐‘ข)=sgn(๐œ†)|๐‘ข|๐›ผ(๐‘ฅ)โˆ’2๐‘ข,๐‘”(๐‘ฅ,๐‘ข)=sgn(๐œ‡)|๐‘ข|๐›ฝ(๐‘ฅ)โˆ’2๐‘ข and ๐‘โˆ’>๐›ผ+,๐›ฝ+,๐‘ž+; then the conditions in Corollary 3.9 can be fulfilled.


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Copyright © 2012 Zhiqiang Wei and Zigao Chen. 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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