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

The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically linear'' problem \[ \left\{ \begin{array}{ll} - {\rm div} \left[\left(A_0(x) + A(x) |u|^{ps}\right) |\nabla u|^{p-2} \nabla u\right] + s\ A(x) |u|^{ps-2} u\ |\nabla u|^p &\\ \qquad\qquad\qquad =\ μ|u|^{p (s + 1) -2} u + g(x,u) & \hbox{in $Ω$,}\\ u = 0 & \hbox{on $\partialΩ$,} \end{array}\right.\] where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N \ge 2$, $1 < p < N$, $s > 1/p$, both the coefficients $A_0(x)$ and $A(x)$ are in $L^\infty(Ω)$ and far away from 0, $μ\in \mathbb{R}$, and the perturbation’’ term $g(x,t)$ is a Carathéodory function on $\mathrm{\Omega }\times \mathbb{R}$ which grows as $\mathrm{\mid }t{\mathrm{\mid }}^{r-1}$ with $1\le r<p(s+1)$ and is such that $g(x,t)\approx \nu \mathrm{\mid }t{\mathrm{\mid }}^{p-2}t$ as $t\to 0$. By introducing suitable thresholds for the parameters $\nu$ and $\mu$, which are related to the coefficients $A_0(x)$, respectively $A(x)$, 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.