On problems with weighted elliptic operator and general growth nonlinearities

TL;DR

This paper investigates existence, non-existence, and Liouville-type theorems for weighted elliptic equations with general nonlinear growth, extending classical results and applying to systems related to Hardy-Sobolev inequalities.

Contribution

It introduces new non-existence and Liouville theorems for weighted elliptic equations with general growth nonlinearities, including systems, under various conditions.

Findings

Non-existence of solutions in bounded star-shaped domains with supercritical growth.

Existence of positive entire solutions for equations with similar growth.

Liouville-type theorem asserting no positive solutions in the entire space for subcritical growth.

Abstract

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form $$-div (|x|^{a} D u ) = f(x,u), ~ u > 0,\, \mbox{ in } \Omega,$$ where $N \geq 3$, $\Omega$ is an open domain in $\mathbb{R}^N$ containing the origin, $N-2+a > 0$ and $f$ satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided $f$ exhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the equation for $f$ exhibiting the same growth. A Liouville-type theorem is then established, which asserts no positive solution of the equation in $\Omega = \mathbb{R}^N$ exists provided the growth of $f$ is subcritical. The results are then extended to systems of the form $$-div (|x|^{a} D u_1) \!=\! f_{1}(x,u_1,u_2), -div (|x|^{a} D u_2) \!=\! f_{2}(x,u_1,u_2), u_1, u_2 \!>\! 0,\, \mbox{ in } \Omega,$$ but after overcoming additional obstacles not present in the single equation. Specific cases of our results recover classical ones for a renowned problem connected with finding best constants in Hardy-Sobolev and Caffarelli-Kohn-Nirenberg inequalities as well as existence results for well-known elliptic systems.