Implicit equations involving the $p$-Laplace operator

TL;DR

This paper investigates the existence of solutions to a class of implicit elliptic equations involving the p-Laplacian operator, focusing on cases where the nonlinear function separates into parts depending on the solution and its p-Laplacian, with applications provided.

Contribution

The work introduces new existence results for implicit p-Laplacian equations with a specific functional form, expanding understanding of such nonlinear elliptic problems.

Findings

Existence of solutions under certain conditions

Application to specific implicit elliptic equations

Extension of previous results to new functional forms

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.