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

TL;DR

This paper proves the existence of multiple solutions for a class of p-Laplacian elliptic problems with gradient terms, using variational methods and a nonquadraticity condition to overcome the lack of Ambrosetti-Rabinowitz condition.

Contribution

It establishes the existence of at least two solutions for a p-Laplacian problem with gradient terms under smallness conditions, employing a variational approach without the Ambrosetti-Rabinowitz condition.

Findings

Existence of at least two solutions under small norm conditions.

Application of Mountain Pass theorem to a non-AR nonlinear problem.

Use of nonquadraticity condition at infinity to handle nonlinearity.

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}.