Multiple solutions of semilinear degenerate elliptic boundary value problems

Abstract

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet problem and the Robin problem. The approach here is based on the super-sub-solution method in the degenerate case, and is distinguished by the extensive use of an Lp Schauder theory elaborated for second-order, elliptic differential operators with discontinuous zero-th order term. By using Schauder's fixed point theorem, we prove that the existence of an ordered pair of sub- and supersolutions of our problem implies the existence of a solution of the problem. The results extend an earlier theorem due to Kazdan and Warner to the degenerate case. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim