Positive solutions for nonvariational Robin problems

TL;DR

This paper investigates the existence of positive smooth solutions for a nonlinear Robin boundary value problem involving the p-Laplacian and a gradient-dependent reaction term, using monotone operator theory and asymptotic analysis.

Contribution

It introduces a novel approach combining monotone operator theory and asymptotic analysis to establish positive solutions for nonvariational Robin problems with gradient dependence.

Findings

Existence of positive smooth solutions proven.

Application of monotone operator theory to nonlinear PDEs.

Asymptotic analysis used to handle gradient-dependent reactions.

Abstract

We study a nonlinear Robin problem driven by the $p$-Laplacian and with a reaction term depending on the gradient (the convection term). Using the theory of nonlinear operators of monotone-type and the asymptotic analysis of a suitable perturbation of the original equation, we show the existence of a positive smooth solution.