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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
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Abstract: We solve variationally certain equations of stellar dynamics of the form $ -\sum_i\partial_{ii} u(x) =\frac{\vert u\vert^{p-2}u(x)}{{\rm 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):

$\displaystyle 0<\mu_{s,\mathcal{P}}(\Omega) =\inf\left\{\int_{\Omega}\vert\nabl... ...\frac{\vert u(x)\vert^{2^{\star}(s)}}{\vert\pi(x)\vert^s} dx=1\right.\right\},$

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 $ \hbox{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
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

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

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