# Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth

## TL;DR

This paper studies a class of elliptic equations involving the p-Laplacian and gradient-dependent terms, transforming the problem to a variational form to establish existence and multiplicity results under various conditions.

## Contribution

It introduces a change of variables to handle gradient terms, enabling the use of variational methods for a class of nonlinear elliptic equations involving the p-Laplacian.

## Key findings

- Existence of solutions under certain growth conditions.
- Multiplicity results depending on the behavior of g and f.
- Analysis of the Ambrosetti-Rabinowitz condition in this context.

## Abstract

We investigate the problem $$ \left\{ \begin{array}{ll} -\Delta_p u = g(u)|\nabla u|^p + f(x,u) \ & \mbox{in} \ \ \Omega, \ \ \\ u>0 \ &\mbox{in} \ \ \Omega, \ \ u = 0 \ &\mbox{on} \ \ \partial\Omega, \end{array} \right. \leqno{(P)} $$ in a bounded smooth domain $\Omega \subset \mathbb{R}^N$. Using a Kazdan-Kramer change of variable we reduce this problem to a quasilinear one without gradient term and therefore approachable by variational methods. In this way we come to some new and interesting problems for quasilinear elliptic equations which are motivated by the need to solve $(P)$. Among other results, we investigate the validity of the Ambrosetti-Rabinowitz condition according to the behavior of $g$ and $f$. Existence and multiplicity results for $(P)$ are established in several situations.