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

Abstract

We solve variationally certain equations of stellar dynamics of the form − ∑ i ∂ i i u ( x ) = | u | p − 2 u ( x ) dist ( x , A ) s -\sum _i\partial _{ii} u(x) =\frac {|u|^{p-2}u(x)}{\textrm {dist} (x,{\mathcal A} )^s} in a domain Ω \Omega of R n \mathbb {R}^n , where A {\mathcal A} is a proper linear subspace of R n \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): \[ 0 > μ s , P ( Ω ) = inf { ∫ Ω | ∇ u | 2 d x | u ∈ H 1 , 0 2 ( Ω ) a n d ∫ Ω | u ( x ) | 2 ⋆ ( s ) | π ( x ) | s d x = 1 } , 0>\mu _{s,\mathcal {P}}(\Omega ) =\inf \left \{\int _{\Omega }|abla u|^2 dx\; \left |\; u\in H_{1,0}^2(\Omega ) \;\mathrm { and }\; \int _{\Omega }\frac {|u(x)|^{2^{\star }(s)}}{|\pi (x)|^s} dx=1\right .\right \}, \] where 0 > s > 2 0>s>2 , 2 ⋆ ( s ) = 2 ( n − s ) n − 2 2^{\star }(s) =\frac {2(n-s)}{n-2} and where π \pi is the orthogonal projection on a linear space P \mathcal {P} , where dim R ⁡ P ≥ 2 \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 P ⊥ {\mathcal P}^{\bot } , the orthogonal of P \mathcal P , with respect to the domain Ω \Omega , as well as on the curvature of the boundary ∂ Ω \partial \Omega at its points of intersection with P ⊥ {\mathcal P}^{\bot } .