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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms


Authors: Norimichi Hirano and Naoki Shioji
Journal: Proc. Amer. Math. Soc. 134 (2006), 2585-2592
MSC (2000): Primary 35J65, 35J20
Published electronically: February 17, 2006
MathSciNet review: 2213736
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Abstract: Let $ N\geq4$, let $ 2^{\ast}=2N/(N-2)$ and let $ \Omega$ $ \subset\mathbb{R}^{N}$ be a bounded domain with a smooth boundary $ \partial\Omega$. Our purpose in this paper is to consider the existence of solutions of the problem:

\begin{displaymath} \left\{ \begin{array}[c]{rlll} -\Delta u-\mu\frac{u}{\left\v... ...& 0 & \qquad\qquad\text{on }\partial\Omega, \end{array}\right. \end{displaymath}

where $ 0<\mu<(\frac{N-2}{2})^{2}.$


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Additional Information

Norimichi Hirano
Affiliation: Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama, Japan
Email: hirano@math.sci.ynu.ac.jp

Naoki Shioji
Affiliation: Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama, Japan

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08405-X
PII: S 0002-9939(06)08405-X
Keywords: Critical Sobolev, Hardy inequality, semilinear elliptic problem
Received by editor(s): August 31, 2004
Received by editor(s) in revised form: March 21, 2005
Published electronically: February 17, 2006
Additional Notes: This work was partially supported by the Heisei16 joint research project fund in the Graduate School of Environment and Information Sciences of Yokohama National University
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2006 American Mathematical Society