Implicit equations involving the $p$-Laplace operator

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.