# Revista Matemática Iberoamericana

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**Volume 35, Issue 1, 2019, pp. 173–194**

**DOI: 10.4171/rmi/1052**

Published online: 2019-02-04

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

Djairo Guedes de Figueiredo^{[1]}, Jean-Pierre Gossez

^{[2]}, Humberto Ramos Quoirin

^{[3]}and Pedro Ubilla

^{[4]}(1) IMECC - UNICAMP, Campinas, Brazil

(2) Université Libre de Bruxelles, Belgium

(3) Universidad de Santiago de Chile, Chile

(4) Universidad de Santiago de Chile, Chile

We investigate the problem $$ \left\{ \begin{array}{ll} -\Delta_p u = g(u)|\nabla u|^p + f(x,u) & \mbox{in } \Omega, \\ u>0 &\mbox{in } \Omega, \\ u = 0 &\mbox{on } \partial\Omega, \end{array} \right. $$ 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.

*Keywords: *Quasilinear elliptic problem, natural growth in the gradient, variational methods, $p$-Laplacian

Guedes de Figueiredo Djairo, Gossez Jean-Pierre, Ramos Quoirin Humberto, Ubilla Pedro: Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth. *Rev. Mat. Iberoam.* 35 (2019), 173-194. doi: 10.4171/rmi/1052