On problems with weighted elliptic operator and general growth nonlinearities

Abstract

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form \begin{document}$ \begin{equation*} -div (|x|^{a} D u ) = f(x,u), \; u > 0,\, \mbox{ in } \Omega, \end{equation*} $\end{document} where \begin{document}$ N \geq 3 $\end{document} , \begin{document}$ \Omega $\end{document} is an open domain in \begin{document}$ \mathbb{R}^N $\end{document} containing the origin, \begin{document}$ N-2+a > 0 $\end{document} and \begin{document}$ f $\end{document} satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided \begin{document}$ f $\end{document} exhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the equation for \begin{document}$ f $\end{document} exhibiting the same growth. A Liouville-type theorem is then established, which asserts no positive solution of the equation in \begin{document}$ \Omega = \mathbb{R}^N $\end{document} exists provided the growth of \begin{document}$ f $\end{document} is subcritical. The results are then extended to systems of the form \begin{document}$ \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*} $\end{document} 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.