On problems with weighted elliptic operator and general growth nonlinearities

2018/07/13 by Villavert, John
#35B53 #35J15 #35J47 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35B09 35B33 #Secondary: 35B38

paper · doi:10.48550/arxiv.1807.04940

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 gt; 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|^a D u₁) = f₁(x,u₁,u₂), -div (|x|^a D u₂) = f₂(x,u₁,u₂), u₁, u₂ gt; 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.

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