Multiplicity of solutions for a class of elliptic problem of $p$-Laplacian type with a $p$-Gradient term

Zakariya Chaouai, Soufiane Maatouk

Published 2017-10-10Version 1

We consider the following problem $$(P) \begin{cases} -\Delta_{p}u= c(x)|u|^{q-1}u+\mu |\nabla u|^{p}+h(x) & \ \ \mbox{ in }\Omega, u=0 & \ \ \mbox{ on } \partial\Omega, \end{cases}$$ where $\Omega$ is a bounded set in $\mathbb{R}^{N}$ ($N\geq 3$) with a smooth boundary, $1<p<N$, $q>0$, $\mu \in \mathbb{R}^{*}$ and $c, h$ belongs to $L^{k}(\Omega)$ for some $k>\frac{N}{p}$. In this paper, we assume that $c$ and $h$ are allowed to change sign such that $c^{+}\not \equiv 0$, then we prove the existence of at least two bounded solutions when $\|c\|_{k}$ and $\|h\|_{k}$ are suitably small. For this purpose, we use the Mountain Pass theorem, due to Ambrosetti and Rabinowitz, on a problem with variational structure reduced from $(P)$. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the Ambrosetti and Rabinowitz condition by the \textbf{nonquadraticity condition at infinity}.