# Asymmetric Robin problems with indefinite potential and concave terms

## TL;DR

This paper investigates a complex Robin boundary value problem involving an indefinite potential and asymmetric nonlinearities, establishing multiple solutions using advanced variational and topological methods.

## Contribution

It introduces new multiplicity results for Robin problems with indefinite potentials and asymmetric nonlinearities, employing variational, truncation, perturbation, and Morse theory techniques.

## Key findings

- Proved existence of four solutions for small positive parameters.
- Established five solutions under certain conditions.
- Demonstrated the effectiveness of variational and Morse theory methods.

## Abstract

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups) we prove two multiplicity theorems producing four and five respectively nontrivial smooth solutions when the parameter $\lambda>0$ is small.