Existence of extremal functions for a family of Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper proves the existence of extremal functions for a broad class of Caffarelli-Kohn-Nirenberg inequalities on cones, establishes conditions for compact embeddings of related Sobolev spaces, and applies these results to certain nonlinear PDEs.

Contribution

It demonstrates the attainability of sharp constants in Caffarelli-Kohn-Nirenberg inequalities for large parameter ranges and identifies new conditions for compact Sobolev embeddings.

Findings

Sharp constants are achieved for large parameter spaces.

New sufficient conditions for compact embeddings of weighted Sobolev spaces.

Existence results for nonlinear PDEs involving weighted divergence operators.

Abstract

Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\it (Compositio Mathematica,1984):} $$\Big(\int_\Omega \frac{|u|^r}{|x|^s}dx\Big)^{\frac{1}{r}}\leq C(p,q,r,\mu,\sigma,s)\Big(\int_\Omega \frac{|\nabla u|^p}{|x|^\mu}dx\Big)^{\frac{a}{p}}\Big(\int_\Omega \frac{|u|^q}{|x|^\sigma}dx\Big)^{\frac{1-a}{q}},$$ where $\Omega \subset \R^N (N\geq 2)$ is an open set; $p, q, r, \mu, \sigma, s, a$ are some parameters satisfying some balanced conditions. When $\Omega$ is a cone in $\R^N$ (for example, $\Omega=\R^N)$, we prove the sharp constant $C(p,q,r,\mu,\sigma,s)$ can be achieved for a very large parameter space. Besides, we find some sufficient conditions which guarantee that the following Sobolev spaces $$W_{\mu}^{1,p}(\Omega),\; W_{\mu}^{1,p}(\Omega)\cap L^p(\Omega), \; H^{1,p}(\R^N) $$ are compactly embedded into $L^r(\R^N, \frac{dx}{|x|^s})$ for some new ranges of parameters, where $\displaystyle W_{\mu}^{1,p}(\Omega)$ is the completion of $C_0^\infty(\Omega)$ with respect to the norm $\displaystyle \Big(\int_\Omega \frac{ |\nabla u|^p}{|x|^\mu}dx\Big)^{\frac{1}{p}}. $ As applications, we also study the equation $$\displaystyle -div\Big(\frac{|\nabla u|^{p-2}\nabla u}{|x|^\mu}\Big)=\lambda V(x)|u|^{q-2}u, \;\;\; u\in W_{\mu}^{1,p}(\Omega)$$ under some proper conditions on $V(x)$.