Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Abstract

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}$, and ${D_a^{1,2}(\Omega)}$ be the completion of ${C_0^\infty(\Omega)}$ with respect to the norm:

$$||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.$$

The Caffarelli–Kohn–Nirenberg inequalities state that there is a constant C > 0 such that

$$\begin{array}{ll}(\int_\Omega |x|^{-bq}|u|^q{d}x)^{\frac{2}{q}}\leqq C\int_\Omega|x|^{-2a}|\nabla u|^2{d}x \end{array}\quad\quad(0.1)$$

for ${u\in D_a^{1,2}(\Omega)}$ and

$$-\infty< a <\frac{N-2}{2},\quad 0\leqq b-a\leqq 1,\quad q=\frac{2N}{N-2+2(b-a)}.$$

We prove the best constant for (0.1)

$$S(a,b;\Omega)=\inf\limits_{u\in D_a^{1,2}\backslash\{0\}} \frac{\int_\Omega |x|^{-2a}|\nabla u|^2{d}x}{(\int_\Omega |x|^{-bq}|u|^q {d}x)^\frac{2}{q}}$$

is always achieved in ${D_a^{1,2}(\Omega)}$ provided that ${0\in\partial\Omega}$ and the mean curvature H(0) < 0, where a, b satisfies

$$(i)\,a<\, b<\,a+1\,{\rm and}\,N\geqq 3,{\,{\rm or}\,}(ii)\,b=a >0 {\,{\rm and}\, }N\geqq 4.$$

If a = 0 and 1 > b > 0, then the result was proved by Ghoussoub and Robert [12].