Separating solutions of nonlinear problems using nonlinear generalized Rayleigh quotients

Abstract

This paper deals with nonlinear elliptic boundary value problems with complicated geometry of nonlinearities. A new method for obtaining multiple solutions based on a recursive use of the nonlinear generalized Rayleigh quotients to the split Nehari manifold into subsets without degeneracies is introduced. The method is applied to prove the multiplicity result for nonnegative solutions, as well as to find a ground state of elliptic boundary value problems with nonlinearities of polynomial type.