Existence and Multiplicity of Weak Solutions for a Class of Degenerate Nonlinear Elliptic Equations

The goal of this paper is to study a nonlinear elliptic equation in which the divergence form operator −div(a(x,∇u)) is involved. Such operators appear in many nonlinear diffusion problems, in particular in the mathematical modeling of non-Newtonian fluids (see [5] for a discussion of some physical background). Particularly, the p-Laplacian operator −div(|∇u|p−2∇u) is a special case of the operator −div(a(x,∇u)). Problems involving the p-Laplacian operator have been intensively studied in the last decades.We just remember the work on that topic of JoãoMarcos B. do Ó [7], Pflüger [12], Rădulescu and Smets [14] and the references therein. In the case of more general types of operators we point out the papers of João Marcos B. do Ó [6] and Nápoli and Mariani [4]. On the other hand, when the operator −div(a(x,∇u)) is of degenerate type we refer to Cı̂rstea and Rădulescu [15] and Motreanu and Rădulescu [11]. In this paper we study the existence and multiplicity of non-trivial weak solutions to equations of the type