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

We prove a result of existence of positive solutions of the Dirichlet problem for $-Δ_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $Δ_p$ 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 $-Δ_pu = λu(x)^{q-1}(1+|\nabla u(x)|^p)$ in $Ω$ and give examples of super-linear nonlinearities which are also handled by our method.

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