On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations

TL;DR

This paper studies the antimaximum principle for the p-Laplacian with sublinear perturbations, identifying parameter ranges where solutions obey maximum principles, and extends results to low regularity cases and the linear case.

Contribution

It provides new conditions under which the antimaximum principle holds for the p-Laplacian, including low regularity assumptions and the linear case, expanding existing theoretical understanding.

Findings

Solutions satisfy the antimaximum principle near the first eigenvalue for certain weights.

New results on antimaximum principle validity under low regularity assumptions.

Existence results for solutions under specified conditions.

Abstract

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-\Delta_p u = \lambda m(x)|u|^{p-2}u + \eta a(x)|u|^{q-2}u + f(x)$ in a bounded domain $\Omega \subset \mathbb{R}^N$, where $q<p$. Under certain regularity and qualitative assumptions on the weights $m, a$ and the source function $f$, we identify ranges of parameters $\lambda$ and $\eta$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $\eta=0$, and among them there are results providing new information even in the linear case $p=2$. In particular, we show that for any $p>1$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the $p$-Laplacian provided $m,f \in L^\gamma(\Omega)$ with $\gamma>N$. For completeness, we also investigate the existence of solutions.