# Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

## TL;DR

This paper investigates the existence and uniqueness of positive solutions for a perturbed Robin eigenvalue problem involving the Laplacian and an indefinite potential, analyzing sublinear and superlinear cases.

## Contribution

It provides new results on positive solutions for Robin eigenvalue problems with indefinite potentials, including existence, uniqueness, and properties of minimal solutions.

## Key findings

- Unique positive solution for sublinear perturbations when λ<λ̂₁
- Multiple positive solutions for superlinear perturbations when λ<λ̂₁
- No positive solutions for λ≥λ̂₁ in both cases

## Abstract

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $\lambda<\widehat{\lambda}_{1}$ ($\widehat{\lambda}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. In the superlinear case, for $\lambda<\widehat{\lambda}_{1}$ we have at least two positive solutions and for $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_{\lambda}$ and we investigate the properties of the map $\lambda\mapsto\bar{u}_{\lambda}$.