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

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

2005-08-18

arxiv.org/abs/math/0508348v1

Mathematics

Analysis of PDEs

27 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif/

Scientific paper

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

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