We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ ii} u(x) =\frac{ |u|^{ p-2}u(x)}{ { \rm dist} (x,{ \mathcal A} )^s}$ in a domain $\Omega$ of $\rn$, where ${ \mathcal A} $ is a proper linear subspace of $\rn$. Existence problems are related to the question of attainability of the best constant in the following recent inequality of Badiale-Tarantello [1]: $$0<\mu_{ s,\P}(\Omega)=\inf{ \int_{ \Omega}|\nabla u|^2 dx; u\in \huno \hbox{ and}\int_{ \Omega}\frac{ |u(x)|^{ \crit(s)}}{ |\pi(x)|^s} dx=1}$$ where $0<s<2$, $\crit(s)=\frac{ 2(n-s)}{ n-2}$ and where $\pi$ is the orthogonal projection on a linear space $\P$, where $\hbox{ dim}_{ \rr}\P \geq 2$. We investigate this question and how it depends on the relative position of the subspace $\Porth$, the orthogonal of $\P$, with respect to the domain $\Omega$ as well as on the curvature of the boundary $\partial\Omega$ at its points of intersection with $\Porth $.
Categories: math.AP