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

Abstract

We solve variationally certain equations of stellar dynamics of the form $-{\sum }_i{\partial }_{ii}u(x)=\frac{|u{|}^{p-2}u(x)}{\mathrm{dist}(x,\mathcal{A}{)}^s}$ 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]: $$0<μ_{s,¶}(Ω)=\inf{\int_Ω|\nabla u|^2 dx; u\in \huno \hbox{and}\int_Ω\frac{|u(x)|^{\crit(s)}}{|π(x)|^s} dx=1}$$ 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 $\hbox{dim}_{\rr}¶\geq 2$. 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 $.

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