    April  2021, 20(4): 1347-1361. doi: 10.3934/cpaa.2021023

## On problems with weighted elliptic operator and general growth nonlinearities

 University of Texas, Rio Grande Valley, Edinburg, TX 78539, USA

Received  August 2020 Revised  December 2020 Published  April 2021 Early access  April 2021

Fund Project: The author is partially supported by the Simons Foundation Collaboration Grants for Mathematicians 524335

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.
Citation: John Villavert. On problems with weighted elliptic operator and general growth nonlinearities. Communications on Pure &amp; Applied Analysis, 2021, 20 (4) : 1347-1361. doi: 10.3934/cpaa.2021023
##### References:
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##### References:
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