# Robin problems with indefinite linear part and competition phenomena

## TL;DR

This paper studies a parametric Robin boundary value problem with an indefinite potential and competing nonlinearities, establishing bifurcation results, existence of minimal solutions, and their properties as the parameter varies.

## Contribution

It introduces a bifurcation framework for Robin problems with indefinite linear parts and nonlinearities without the Ambrosetti-Rabinowitz condition, including the existence and properties of minimal solutions.

## Key findings

- Bifurcation-type theorem for positive solutions
- Existence of a minimal positive solution for each parameter
- Monotonicity and continuity of the solution map

## Abstract

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.