# American Institute of Mathematical Sciences

November  2016, 15(6): 1975-2005. doi: 10.3934/cpaa.2016024

## Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities

 1 Instituto de Investigaciones Matemáticas Luis Santaló and CONICET, Facultad de Ciencias Exactas y Naturales, Pabellón I, Ciudad Universitaria, 1428, Argentina

Received  November 2012 Revised  May 2013 Published  September 2016

In this paper we analyze the following elliptic problem related to some Caffarelli-Kohn-Nirenberg inequalities: \begin{eqnarray} -div(|x|^{-2\gamma}\nabla u)-\lambda \frac{u}{|x|^{2(\gamma+1)}}=|\nabla u|^p|x|^{-\gamma p}+cf,\; u>0\; \mbox{ in }\; \Omega, \qquad u_{|\partial \Omega}\equiv0, \end{eqnarray} where $\Omega \subset R^N$ is a domain such that $0\in\Omega$, $N\geq 3$, and $c, \lambda, \gamma, p$ are positive constants verifying $0 < \lambda \leq \Lambda_{N,\gamma}=\left(\frac{N-2(\gamma+1)}{2}\right)^{2}$, $-\infty<\gamma<\frac{N-2} 2$ and $p>0$. Our study concerns to existence of solutions to the former problem. More precisely, first we determine a critical thereshold for the power $p$, in the sense that, beyond this value it does not exist any positive supersolution to our problem, not even in a very weak sense. In addition, we show existence of solutions for all the values $p>0$ below this threshold, with the restriction $\gamma>-\frac{N(1-p)+2}{2}$, whenever the righthand side verifies $f(x)\leq |x|^{-2(\gamma+1)}$ if $\gamma>-1$. When $-\frac{N(1-p)+2}{2}<\gamma\leq -1$ it suffices that $f\in L^{2/p}(\Omega)$. The existence of solutions for $0 < p < 1$ and $\gamma\leq -\frac{N(1-p)+2}{2}$ is an open question.
Citation: Mayte Pérez-Llanos. Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 1975-2005. doi: 10.3934/cpaa.2016024
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