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. The approach here 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 a variant of the Ljusternik-Schnirelman theory of critical points, we prove very exact results on the number of solutions of our problem. The results here extend earlier theorems due to Castro-Lazer to the degenerate case.
"Semilinear degenerate elliptic boundary value problems via critical point theory." Tsukuba J. Math. 36 (2) 311 - 365, December 2012. https://doi.org/10.21099/tkbjm/1358777003