Positive Solutions for the p-Laplacian with Dependence on the Gradient

TL;DR

This paper establishes the existence of positive solutions for a class of p-Laplacian boundary value problems with gradient dependence, using geometric assumptions and fixed point methods, without requiring asymptotic conditions on the nonlinearities.

Contribution

The authors introduce a novel approach to prove positive solutions for p-Laplacian problems with gradient dependence, avoiding typical asymptotic assumptions and employing geometric conditions and fixed point techniques.

Findings

Existence of positive solutions under geometric conditions.

Application to nonlinearities with super-linear growth.

Construction of sub- and super-solutions for gradient-dependent problems.

Abstract

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on $f$, but simple geometric assumptions on a neighborhood of the first eigenvalue of the $p$-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder Fixed Point Theorem and this result is used to construct an ordered pair of sub- and super-solution, also valid for nonlinearities which are super-linear both at the origin and at $+\infty$. We apply our method to the Dirichlet problem $-\Delta_pu = \lambda u(x)^{q-1}(1+|\nabla u(x)|^p)$ in $\Omega$ and give examples of super-linear nonlinearities which are also handled by our method.