Implicit equations involving the $p$-Laplace operator

Abstract

In this work we study the existence of solutions $u\in W_0^{1,p}(\Omega )$ to the implicit elliptic problem $f(x,u,\nabla u,{\Delta }_{p}u)=0$ in $\Omega$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, $N\ge 2$, with smooth boundary $\partial \Omega$, $1<p<+\infty$, and $f:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$. We choose the particular case when the function $f$ can be expressed in the form $f(x,z,w,y)=\phi (x,z,w)-\psi (y)$, where the function $\psi$ depends only on the $p$-Laplacian ${\Delta }_{p}u$. We also present some applications of our results.

15 pages; comments are welcome