Implicit equations involving the $p$-Laplace operator

Abstract

In this work we study the existence of solutions $u \in W^{1,p}_0(\Omega)$ 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\colon \Omega \times \mathbb R \times \mathbb R^N \times \R \to \R $. We choose the particular case when the function $ f $ can be expressed in the form $ f(x, z, w, y)= \varphi(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.