Existence results for a borderline case of a class of p-Laplacian problems

Abstract

The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically “linear” problem $\left(\begin{array}{c}-div\left(\left(A_0\left(x\right)+A\left(x\right){\left|u\right|}^{ps}\right){|\nabla u|}^{p-2}\nabla u\right)+sA\left(x\right){\left|u\right|}^{ps-2}u{|\nabla u|}^p\\ =\mu {\left|u\right|}^{p(s+1)-2}u+g(x,u)& \text{in}\Omega \text{,}\\ u=0& \text{on}\partial \Omega \text{,}\end{array}\right)$where $\Omega$ is a bounded domain in $R^N$, $N\ge 2$, $1<p<N$, $s>1/p$, both the coefficients $A_0\left(x\right)$ and $A\left(x\right)$ are in $L^{\infty }\left(\Omega \right)$ and far away from 0, $\mu \in R$, and the “perturbation” term $g(x,t)$ is a Carathéodory function on $\Omega \times R$ which grows as ${\left|t\right|}^{r-1}$ with $1\le r<p(s+1)$ and is such that $g(x,t)\approx \nu {\left|t\right|}^{p-2}t$ as $t\to 0$.

By introducing suitable thresholds for the parameters $\nu$ and $\mu$, which are related to the coefficients $A_0\left(x\right)$, respectively $A\left(x\right)$, under suitable hypotheses on $g(x,t)$, the existence of a nontrivial weak solution is proved if either $\nu$ is large enough with $\mu$ small enough or $\nu$ is small enough with $\mu$ large enough.

Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used.