Sharp existence and nonexistence results for an elliptic equation associated with Caffarelli-Kohn-Nirenberg inequalities

John Villavert

Published 2018-07-13Version 1

This article establishes sharp existence and Liouville type theorems for the following nonlinear elliptic equation with singular coefficients, \begin{equation*} -div (|x|^{a} D u ) = |x|^{b}u^{p}, ~ u > 0,\, \mbox{ in } \mathbb{R}^N, \end{equation*} where $N \geq 3$, $p > 1$, and $b > a - 2 > -N$. In certain cases, this recovers the Euler-Lagrange equations connected with finding the best constant in Sobolev and Caffarelli-Kohn-Nirenberg inequalities. The first main result indicates that regular solutions exist if and only if the exponent $p$ is either critical or supercritical, i.e., either \begin{equation*} p = \frac{N + 2 + 2b - a}{N - 2 + a} ~ \mbox{ or } ~ p > \frac{N + 2 + 2b - a}{N - 2 + a}, \mbox{ respectively}. \end{equation*} Similarly, the second main result provides necessary and sufficient conditions for the existence of finite energy solutions.