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

Abstract

Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\it (Compositio Mathematica,1984):} $$({\int }_{\Omega }\frac{|u{|}^r}{|x{|}^s}dx{)}^{\frac{1}{r}}\le C(p,q,r,\mu ,\sigma ,s)({\int }_{\Omega }\frac{|\nabla u{|}^p}{|x{|}^{\mu }}dx{)}^{\frac{a}{p}}({\int }_{\Omega }\frac{|u{|}^q}{|x{|}^{\sigma }}dx{)}^{\frac{1-a}{q}},$$ where $\Omega \subset {\mathbb{R}}^N(N\ge 2)$ is an open set; $p,q,r,\mu ,\sigma ,s,a$ are some parameters satisfying some balanced conditions. When $\Omega$ is a cone in ${\mathbb{R}}^N$ (for example, $\Omega ={\mathbb{R}}^N)$, 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 $$W_{\mu }^{1,p}(\Omega ),W_{\mu }^{1,p}(\Omega )\cap L^p(\Omega ),H^{1,p}({\mathbb{R}}^N)$$ are compactly embedded into $L^r({\mathbb{R}}^N,\frac{dx}{|x{|}^s})$ for some new ranges of parameters, where $W_{\mu }^{1,p}(\Omega )$ is the completion of $C_0^{\infty }(\Omega )$ with respect to the norm $({\int }_{\Omega }\frac{|\nabla u{|}^p}{|x{|}^{\mu }}dx{)}^{\frac{1}{p}}.$ As applications, we also study the equation $$-div\left(\frac{|\nabla u{|}^{p-2}\nabla u}{|x{|}^{\mu }}\right)=\lambda V(x)|u{|}^{q-2}u,u\in W_{\mu }^{1,p}(\Omega )$$ under some proper conditions on $V(x)$.

25 pages

Cited by in corpus (1)

• On Elliptic Equations and Systems involving critical Hardy-Sobolev exponents (non-limit case)