Positive solutions for nonlinear nonhomogeneous parametric Robin problems

TL;DR

This paper investigates positive solutions for a nonlinear Robin boundary value problem driven by a nonhomogeneous differential operator, establishing bifurcation results, solution behavior as the parameter varies, and properties of minimal solutions.

Contribution

It introduces a bifurcation theorem for small parameters and analyzes the behavior of positive solutions as the parameter approaches zero, including the existence of minimal solutions.

Findings

Existence of positive solutions for small parameters.

Solutions can have arbitrarily large or small Sobolev norms as the parameter tends to zero.

Existence and properties of the minimal positive solution map.

Abstract

We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carath\'eodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter $\lambda>0$ approaches zero we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally we show that for every admissible parameter value there is a smallest positive solution $u^*_{\lambda}$ of the problem and we investigate the properties of the map $\lambda\mapsto u^*_{\lambda}$.