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Tohoku Mathematical Journal
SECOND SERIES VOL. 64, NO. 1
|Tohoku Math. J.|
64 (2012), 79-103
MINIMIZING PROBLEMS FOR THE HARDY-SOBOLEV TYPE INEQUALITY WITH THE SINGULARITY ON THE BOUNDARY
Chang-Shou Lin and Hidemitsu Wadade
(Received June 4, 2010, revised March 31, 2011)
In this paper, we consider the existence of minimizers of the Hardy-Sobolev type variational problem. Recently, Ghoussoub and Robert [9, 10] proved that the Hardy-Sobolev best constant admits its minimizers provided the bounded smooth domain has the negative mean curvature at the origin on the boundary. We generalize their results by using the idea of Brézis and Nirenberg, and as a consequence, we shall prove the existence of positive solutions to the elliptic equation involving two different kinds of Hardy-Sobolev critical exponents.
2000 Mathematics Subject Classification.
Primary 35J60; Secondary 35B33.
Key words and phrases.
Minimizing problem, Hardy-Sobolev inequality, negative mean curvature.
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