On problems with weighted elliptic operator and general growth nonlinearities

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form

$ \begin{equation*} -div (|x|^{a} D u ) = f(x,u), \; u > 0,\, \mbox{ in } \Omega, \end{equation*} $

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

$ \begin{equation*} -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, \end{equation*} $

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.