Mathematical and Statistical Sciences Faculty Publications and Presentations

Title

On problems with weighted elliptic operator and general growth nonlinearities

Document Type

Article

Publication Date

2021

Abstract

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form −div(|x| aDu) = f(x, u), u > 0, in Ω, where N ≥ 3, Ω is an open domain in 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 Ω = R N exists provided the growth of f is subcritical. The results are then extended to systems of the form −div(|x| aDu1)=f1(x, u1, u2), −div(|x| aDu2)=f2(x, u1, u2), u1, u2 >0, in Ω, 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.

Comments

Original published source available at http://dx.doi.org/10.3934/cpaa.2021023

Publication Title

Communications on Pure and Applied Analysis

DOI

10.3934/cpaa.2021023

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