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 $$(i)\,a 0 {\,{\rm and}\, }N\geqq 4.$$ If a = 0 and 1 > b > 0, then the result was proved by Ghoussoub and Robert [12].