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

Abstract

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 $c$ and $ h$ belong to $L^{k}(\Omega)$ for some $k>\frac{N}{p}$. In this paper, we assume that $c\gneqq 0$ a.e. in $\Omega$ and $h$ without sign condition, then we prove the existence of at least two bounded solutions under the condition that $\|c\|_{k}$ and $\|h\|_{k}$ are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to $(P)$ with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the \textbf{nonquadraticity condition at infinity}.