Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction

TL;DR

This paper proves the existence of multiple solutions to a Robin boundary value problem with indefinite potential and asymmetric nonlinearity using variational and Morse theory methods.

Contribution

It introduces new existence results for multiple solutions in Robin problems with indefinite weights and asymmetric nonlinearities, including conditions for a third solution.

Findings

Two nontrivial solutions established

Existence of a third solution under smoothness condition on f

Methods include variational, truncation, and Morse theory

Abstract

The existence of two nontrivial smooth solutions to a semilinear Robin problem with indefinite unbounded potential and asymmetric nonlinearity $f$ is established. Both crossing and resonance are allowed. A third nonzero solution exists provided $f$ is $C^1$. Proofs exploit variational methods, truncation techniques, and Morse theory.