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

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 \mathbb {R}^n, where {\mathcal A} is a proper linear subspace of \mathbb {R}^n. Existence problems are related to the ques...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2009-09, Vol.361 (9), p.4843-4870
Main Authors: Ghoussoub, Nassif, Robert, Frédéric
Format: Article
Language:English
Subjects:
Coordinate systems
Curvature
Elliptic equations
Exact sciences and technology
Functional analysis
Geometry
Greens function
Mathematical analysis
Mathematical constants
Mathematical growth
Mathematical inequalities
Mathematical vectors
Mathematics
Maximum principle
Normal vectors
Partial differential equations
Sciences and techniques of general use
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Summary: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 \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
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-09-04655-8