Elliptic equations with critical growth and a large set of boundary singularities

Abstract:

We solve variationally certain equations of stellar dynamics of the form $-\sum _i\partial _{ii} u(x) =\frac {|u|^{p-2}u(x)}{\textrm {dist} (x,{\mathcal A} )^s}$ in a domain $\Omega$ of $\mathbb {R}^n$, where ${\mathcal A}$ is a proper linear subspace of $\mathbb {R}^n$. Existence problems are related to the question of attainability of the best constant in the following inequality due to Maz’ya (1985): \[ 0<\mu _{s,\mathcal {P}}(\Omega ) =\inf \left \{\int _{\Omega }|\nabla u|^2 dx\; \left |\; u\in H_{1,0}^2(\Omega ) \;\mathrm { and }\; \int _{\Omega }\frac {|u(x)|^{2^{\star }(s)}}{|\pi (x)|^s} dx=1\right .\right \},\] where $0<s<2$, $2^{\star }(s) =\frac {2(n-s)}{n-2}$ and where $\pi$ is the orthogonal projection on a linear space $\mathcal {P}$, where $\operatorname {dim}_{\mathbb {R}}\mathcal {P}\geq 2$ (see also Badiale-Tarantello (2002)). We investigate this question and how it depends on the relative position of the subspace ${\mathcal P}^{\bot }$, the orthogonal of $\mathcal P$, with respect to the domain $\Omega$, as well as on the curvature of the boundary $\partial \Omega$ at its points of intersection with ${\mathcal P}^{\bot }$.