Semilinear degenerate elliptic boundary value problems at resonance

Abstract

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems at resonance which include as particular cases the Dirichlet and Robin problems. The approach here is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the Lyapunov–Schmidt procedure and the global inversion theorem, we prove existence and uniqueness theorems for our problem. The results here extend an earlier theorem due to Landesman and Lazer to the degenerate case.