Implicit equations involving the $p$-Laplace operator

In this work we study the existence of solutions $u \in W^{1,p}_0(Ω)$ to the implicit elliptic problem $f(x, u, \nabla u, Δ_p u)= 0$ in $Ω$, where $Ω$ is a bounded domain in $\mathbb R^N$, $N \ge 2$, with smooth boundary $\partial Ω$, $1< p< +\infty$, and $f\colon Ω\times \mathbb R \times \mathbb R^N \times \R \to \R$. We choose the particular case when the function $f$ can be expressed in the form $f(x, z, w, y)= φ(x, z, w)- ψ(y)$, where the function $ψ$ depends only on the $p$-Laplacian $Δ_p u$. We also present some applications of our results.

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