Multiplicity Results for a Perturbed Elliptic Neumann Problem

The existence of three solutions for elliptic Neumann problems with a perturbed nonlinear term depending on two real parameters is investigated. Our approach is based on variational methods. 1. Introduction Here and in the sequel, is a bounded open set, with a boundary of class , with , ; and are -Carathéodory functions. The aim of this paper is to study the following perturbed boundary value problem with Neumann conditions: where is the -Laplacian, is the outer unit normal to , and are positive real parameters. Nonlinear boundary value problems involving the -Laplacian operator (with ) arise from a variety of physical problems. They are used in non-Newtonian fluids, reaction-diffusion problems, flow through porous media, and petroleum extraction (see, e.g., [1, 2]). In the last years, several researchers have studied nonlinear problems of this type through different approaches. In [1], the authors have obtained results on the existence of a solution for the problem by using the perturbation result on sums of ranges of nonlinear accretive operators. Subsequently, Wei and Agarwal, in [2], have studied the same problem by developing some new techniques in the wake of [1]. Problem ( ), when , and does not depend on , has been studied in [3]. In this paper, the authors have obtained the existence of at least three solutions for small , by using Implicit Function Theorem and Morse Theory. By using variational methods and in particular critical point results given by Ricceri in [4], Faraci, in her nice paper [5], has dealt with a Neumann Problem involving the -Laplacian (for any ) of type In particular, [5, Theorems , ] assure the existence of three solutions for the problem given above. In the present paper, we establish some results (Theorems 3.1, 3.2), which assure the existence of at least three weak solutions for the problem ( ). In particular the following result is a consequence of Theorem 3.2. Theorem 1.1. Let be a nonnegative continuous function such that Then, for every and for every positive continuous function there exists such that, for each , the problem has at least three nonzero classical solutions. With respect to [3, 5], we stress that our results hold under different assumptions (see Remarks 3.4 and 3.5). In particular, in Theorem 1.1, no asymptotic condition at infinity is required on the nonlinear term. We also point out that in Theorems 3.1 and 3.2, precise estimates of parameters and are given. 2. Preliminaries and Basic Notations Our main tools are three critical point theorems that we recall here in a convenient form. The first has