Semilinear degenerate elliptic boundary value problems via Morse theory

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 and Robin problems. By making use of the Morse and Ljusternik–Schnirelman theories of critical points, we prove existence theorems of non-trivial solutions of our problem. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of semilinear elliptic boundary value problems with degenerate boundary conditions. The results here extend earlier theorems due to Ambrosetti–Lupo and Struwe to the degenerate case.