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

TL;DR

This paper establishes the existence of nontrivial solutions for a class of p-Laplacian problems with borderline asymptotic linearity, using variational methods and threshold parameters for coefficients.

Contribution

It introduces new existence results for a borderline p-Laplacian problem with asymptotically linear growth, employing variational techniques and parameter thresholds.

Findings

Existence of solutions when  large and  small.

Existence when  small and  large.

Application of Mountain Pass and minimization methods.

Abstract

The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically ``linear'' problem \[ \left\{ \begin{array}{ll} - {\rm div} \left[\left(A_0(x) + A(x) |u|^{ps}\right) |\nabla u|^{p-2} \nabla u\right] + s\ A(x) |u|^{ps-2} u\ |\nabla u|^p &\\ \qquad\qquad\qquad =\ \mu |u|^{p (s + 1) -2} u + g(x,u) & \hbox{in $\Omega$,}\\ u = 0 & \hbox{on $\partial{\Omega}$,} \end{array}\right.\] where $\Omega$ 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 $L^\infty(\Omega)$ and far away from 0, $\mu \in \mathbb{R}$, and the ``perturbation'' term $g(x,t)$ is a Carath\'{e}odory function on $\Omega \times \mathbb{R}$ which grows as $|t|^{r-1}$ with $1\le r < p (s + 1)$ and is such that $g(x,t) \approx \nu |t|^{p-2} t$ 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.