Implicit equations involving the $p$-Laplace operator

In this work we study the existence of solutions $u\in W_0^{1,p}(\mathrm{\Omega })$ to the implicit elliptic problem $f(x,u,\mathrm{\nabla }u,{\mathrm{\Delta }}_{p}u)=0$ in $\mathrm{\Omega }$, where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$, $N\ge 2$, with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$, $1<p<+\mathrm{\infty }$, 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) $,wherethefunction$ \psi $ depends only on the $p$-Laplacian ${\mathrm{\Delta }}_{p}u$. We also present some applications of our results.