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

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 $-{\Delta }_{p}u=\lambda u(x{)}^{q-1}(1+|\nabla u(x){|}^p)$ in $\Omega$ and give examples of super-linear nonlinearities which are also handled by our method.

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