Multiplicity of Solutions for A Semilinear Elliptic Problem Via Generalized Nonlinear Rayleigh Quotient

Abstract

It is established existence and multiplicity of solutions for semilinear elliptic problems defined in the whole space \(\mathbb {R}^N\) considering subcritical nonlinearities with some parameters. Here we emphasize that our nonlinearities can be sign-changing functions. The main difficulty is proving the existence of nontrivial solutions by using the Nehari method, taking into account that the Lagrange multipliers theorem cannot be directly applied in our setting. In fact, we consider the case where the fibering map admits inflection points. In other words, we consider the case where the Nehari set admits degenerate critical points. Hence our main contribution is to consider a huge class of semilinear elliptic problems where the standard Nehari method cannot be applied. Using some fine estimates and recovering some compactness results together with the nonlinear Rayleigh quotient, we prove that our main problem admits at least three nontrivial solutions depending on the parameters.