# Multiple solutions for resonant problems of the Robin $p$-Laplacian plus an indefinite potential

## TL;DR

This paper investigates a nonlinear Robin boundary value problem involving the p-Laplacian and an indefinite potential, demonstrating the existence of multiple solutions using advanced variational and topological methods.

## Contribution

It establishes the existence of at least three solutions for the problem, including two with fixed signs, under asymptotic resonance conditions.

## Key findings

- At least three nontrivial solutions exist.
- Two solutions have fixed signs.
- Solutions are obtained via variational, Morse theory, and truncation-perturbation techniques.

## Abstract

We study a nonlinear boundary value problem driven by the $p$-Laplacian plus an indefinite potential with Robin boundary condition. The reaction term is a Carath\'eodory function which is asymptotically resonant at $\pm\infty$ with respect to a nonprincipal Ljusternik-Schnirelmann eigenvalue. Using variational methods, together with Morse theory and truncation-perturbation techniques, we show that the problem has at least three nontrivial smooth solutions, two of which have a fixed sign.