Effect of the boundary conditions in the behavior of the optimal constant of some Caffarelli-Kohn-Nirenberg inequalities. Application to some doubly critical nonlinear elliptic problems
Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral · Advances in Differential Equations · 2006
This work deals with the study of the optimal constants of Sobolev and Hardy-Sobolev inequalities with weights and their relations with the behavior of some mixed Dirichlet-Neumann boundary conditions. More precisely, we analyze the attainability of the Sobolev constant \begin{equation}\label{eq:sobb00} S^2_{\gamma}({\Omega},\Sigma_1)=\inf_{u\in {E_{\Sigma_1}^{2,\gamma}(\Omega)};u ot\equiv 0}\frac{{\displaystyle\int}_\Omega {{|x|^{-2\gamma}}} |{ abla} u|^2dx}{ \Big ({\displaystyle\int}_\Omega |x|^{-2^*\gamma}|u|^{2^*}dx \Big )^{\frac{2}{2^*}}}, \end{equation} and the Hardy-Sobolev constant \begin{equation}\label{HS00} {\Lambda}_{N,\gamma}(\Omega,\Sigma_1)=\inf_{u\in{E_{\Sigma_1}^{2,\gamma}({\Omega})} , u ot\equiv 0} \frac{{\displaystyle\int}_\Omega {{|x|^{-2\gamma}}} |{ abla} u|^2dx}{ {\displaystyle\int}_\Omega \frac{|u|^2}{|x|^{2(\gamma+1)}} dx} \end{equation} where $\Omega\subset{{\rm I\! R^{N}}}$, $N\ge 3$, is a smooth bounded domain such that $0\in\Omega$, $-\infty 0{\quad\mbox{in }}\Omega,\\ u & = & 0{\quad\mbox{on }}\Sigma_1,\\ {{|x|^{-2\gamma}}} \frac {{\partial} u}{{\partial} u} & = & 0{\quad\mbox{on }} \Sigma_2, \end{array}\right. \end{equation} where $q$ and $r$ are given real parameters under convenient hypotheses and $\overline \Sigma_1$, $\overline \Sigma_2$, is a smooth partition of $\partial\Omega$.