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

Authors:
Nassif Ghoussoub and Frédéric Robert

Journal:
Trans. Amer. Math. Soc. **361** (2009), 4843-4870

MSC (2000):
Primary 35J35; Secondary 35B40

DOI:
https://doi.org/10.1090/S0002-9947-09-04655-8

Published electronically:
April 17, 2009

MathSciNet review:
2506429

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ${\mathcal P}^{\bot }$.

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Additional Information

**Nassif Ghoussoub**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada

MR Author ID:
73130

Email:
nassif@math.ubc.ca

**Frédéric Robert**

Affiliation:
Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France

Email:
frobert@math.unice.fr

Received by editor(s):
February 28, 2006

Received by editor(s) in revised form:
October 2, 2007

Published electronically:
April 17, 2009

Additional Notes:
This research was partially supported by the Natural Sciences and Engineering Research Council of Canada. The first author gratefully acknowledges the hospitality and support of the Université de Nice where this work was initiated.

The second author gratefully acknowledges the hospitality and support of the University of British Columbia.

Article copyright:
© Copyright 2009
American Mathematical Society