A nonsmooth variational approach to semipositone quasilinear problems in $\mathbb{R}^N$

TL;DR

This paper establishes the existence of positive solutions for a class of semipositone quasilinear problems involving the p-Laplacian in al R^N, using nonsmooth critical point theory and comparison principles, with additional results like Hopf's Lemma and Liouville-type theorems.

Contribution

It introduces a nonsmooth variational approach to prove solution existence for semipositone problems with the p-Laplacian, extending classical methods to nonsmooth settings.

Findings

Existence of solutions for small parameter a.

Development of a nonsmooth critical points framework.

New versions of Hopf's Lemma and Liouville's theorem for the p-Laplacian.

Abstract

This paper concerns the existence of a solution for the following class of semipositone quasilinear problems \begin{equation*} \left \{ \begin{array}{rclcl} -\Delta_p u = h(x)(f(u)-a),\ & u > 0 & \mbox{in} & \mathbb{R}^N, \end{array} \right. \end{equation*} where $1<p<N$, $a>0$, $ f:[0,+\infty) \to [0,+\infty)$ is a function with subcritical growth and $f(0)=0$, while $h:\mathbb{R}^N \to (0,+\infty)$ is a continuous function that satisfies some technical conditions. We prove via nonsmooth critical points theory and comparison principle, that a solution exists for $a$ small enough. We also provide a version of Hopf's Lemma and a Liouville-type result for the $p$-Laplacian in the whole $\mathbb{R}^N$.