Elliptic Equations with Critical Growth and a Large Set of Boundary Singularities

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