Nontrivial Solutions for Dirichlet Boundary Value Systems with the -Laplacian

Using critical point theory due to Bonanno (2012), we prove the existence of at least one nontrivial solution for Dirichlet boundary value systems with the -Laplacian. 1. Introduction In this note, we prove the existence of at least one nontrivial solution for Dirichlet boundary value systems with the -Laplacian as follows: for , where is an integer, where denotes the -Laplacian differential operator and for , , is a nonempty bounded open set with smooth boundary , is a function such that is measurable in for all , is in for every and for every , and denotes the partial derivative of with respect to for . Due to importance of second-order Dirichlet and Neumann problems in describing a large class of physic phenomena, many authors have studied the existence and multiplicity of solutions for such a problem; we refer the reader to [1–13] and references therein. Some authors also study the system case; see [14–20]. In [6], the authors, employing a three-critical-point theorem due to Bonanno [3], determined an exact open interval of the parameter for which system (1) in the case admits non nontrivial weak solution. The aim of this paper is to prove the existence of at least one nontrivial weak solution for (1) for appropriate values of the parameter belonging to a precise real interval, which extend the results in [7]. For basic notation and definitions and also for a thorough account on the subject, we refer the reader to [21]. 2. Preliminaries and Basic Notation First, we recall for the reader’s convenience [22, Theorem 2.5] as given in [23, Theorem 2.1] (see also [3, Proposition 2.1]) which is our main tool to transfer the question of existence of at least one weak solution of (1) to the existence of a critical point of the Euler functional as follows. For a given nonempty set and two functionals , we define the following two functions as follows: for all , . Theorem 1 ([3, Theorem 5.1]). Let be a reflexive real Banach space and let be a sequentially weakly lower semicontinuous, coercive, and continuously Gateaux differentiable functional whose Gateaux derivative admits a continuous inverse on and let be a continuously Gateaux differentiable functional whose Gateaux derivative is compact. Put and assume that there are , , such that Then, for each , there is such that and . Let us introduce notation that will be used later. Let be the Sobolev space with the usual norm given by , where . Let with the norm where . Let Since for , one has . In addition, it is known [24, formula (6b)] that for , where denotes the Gamma function and is the Lebesgue measure of