A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space

We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem in , on , in a suitable weighted Sobolev space, where is a bounded domain and is a continuous bounded nonlinearity. 1. Introduction Let be a bounded domain with boundary . Let be an operator in divergence form: with coefficients which are symmetric and satisfy the degenerate ellipticity condition: for all , and is an -weight . Let and and let be a real valued continuous function defined on . In this paper, we study the existence of weak solution of the BVP: under suitable hypotheses on the functions , , and . The present work is inspired by a semilinear problem in bounded domain given in the book by Zeidler [1]. In general, the Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic PDEs. For degenerate problems with various types of singularities in the coefficients it is natural to look for solutions in weighted Sobolev spaces; for example, see [2–8]. Section 2 deals with preliminaries and some basic results. Section 3 contains the main result and is about the existence of a weak solution to (3) in a suitable weighted Sobolev space. 2. Preliminaries We need the following preliminaries for the ensuing study. Let be a bounded domain (open connected set). Let be a locally integrable function with a.e. We say that belongs to the Muckenhoupt class , , or that is an -weight, if there is a constant such that for all balls in , where denotes the -dimensional Lebesgue measure in . We assume that , . We will denote by the usual Banach space of measurable real valued functions, , defined in for which For and a positive integer , the weighted Sobolev space is defined by with the associated norm In order to avoid too many suffices, at each step, a generic constant is denoted by or . We need the following result. Proposition 1 (the weighted Sobolev inequality). Let be a bounded domain and let . Then, there exist positive constants and such that, for all and all satisfying , A proof of the above statement can be found in [5, Theorem 1.3]. For and in the above inequality, we have where Further, we use function space which is defined as the closure of with respect to the norm (correctness of definition of this norm follows from inequality (9)). We also note that and are Hilbert spaces. For more details on -weight and weighted Sobolev spaces, we refer to [5, 7, 9–11]. Proposition 2. Let , , and let be a bounded open set in . If in , then there exist a subsequence and a function such that (i) as , -a.e. on ;(ii) , -a.e. on .