Positive solutions for the Robin $p$-Laplacian plus an indefinite potential

TL;DR

This paper investigates positive solutions of a nonlinear elliptic equation involving the Robin p-Laplacian with an indefinite potential, analyzing how solutions change with the parameter and establishing bifurcation and monotonicity properties.

Contribution

It provides a bifurcation analysis for large parameters and characterizes the smallest positive solutions, including their monotonicity and continuity with respect to the parameter.

Findings

Existence of a smallest positive solution for all admissible parameters.

Bifurcation results for large parameter values.

Monotonicity and continuity of the solution map with respect to the parameter.

Abstract

We consider a nonlinear elliptic equation driven by the Robin $p$-Laplacian plus an indefinite potential. In the reaction we have the competing effects of a strictly $(p-1)$-sublinear parametric term and of a $(p-1)$-linear and nonuniformly nonresonant term. We study the set of positive solutions as the parameter $\lambda>0$ varies. We prove a bifurcation-type result for large values of the positive parameter $\lambda$. Also, we show that for all admissible $\lambda>0$, the problem has a smallest positive solution $\overline{u}_\lambda$ and we study the monotonicity and continuity properties of the map $\lambda\mapsto\overline{u}_\lambda$.