Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term

TL;DR

This paper establishes Liouville theorems for solutions to anisotropic p-Laplace equations with various types of nonlinear terms, providing new proofs and classifications for positive and sign-changing solutions.

Contribution

It introduces simplified proofs and extends Liouville theorems to anisotropic p-Laplace equations with sign-changing nonlinearities, including classifications of solutions.

Findings

Positive supersolutions and subsolutions are constant under certain conditions.

Nonexistence of positive solutions in the subcritical case.

Stable solutions or solutions stable outside compact sets are trivial.

Abstract

In this paper, we investigate Liouville theorems for solutions to the anisotropic $p$-Laplace equation $$-\Delta_p^H u=-\operatorname{div}(a(\nabla u))=f(u),\quad\text{in }\mathbb{R}^n,$$ where the semilinear term $f$ may be positive, negative, or sign-changing. When $f$ is positive (negative) and satisfies certain conditions, Serrin's technique is applied to show that every positive supersolution (subsolution) must be constant. For the subcritical case, we use the invariant tensor method to prove nonexistence results for positive solutions. In particular, by applying the differential identity established in the subcritical case to the critical case, we provide a simplified new proof of the classification of positive solutions to the critical case $f(u)=u^{p^*-1}$. For sign-changing solutions, every stable solution or solution that is stable outside a compact set is trivial under certain conditions on $f$.