Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth
Djairo Guedes De Figueiredo, Jean–Pierre Gossez, Humberto Ramos Quoirin, Pedro Ubilla · Revista Matemática Iberoamericana · 2019
We investigate the problem \left\{ \begin{array}{ll} -\Delta_p u = g(u)| abla u|^p + f(x,u) & \text{in } \Omega, \\ u>0 &\text{in } \Omega, \\ u = 0 &\text{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.