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

 

Isolated singularities for the exponential type semilinear elliptic equation in $ \mathbb{R}^2$


Authors: R. Dhanya, J. Giacomoni and S. Prashanth
Journal: Proc. Amer. Math. Soc. 137 (2009), 4099-4107
MSC (2000): Primary 35B32, 35B65, 35J25, 35J60
Published electronically: July 15, 2009
MathSciNet review: 2538571
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Abstract: In this article we study positive solutions of the equation $ -\Delta u= f(u)$ in a punctured domain $ \Omega'=\Omega\setminus\{0\}$ in $ \mathbb{R}^2$ and show sharp conditions on the nonlinearity $ f(t)$ that enables us to extend such a solution to the whole domain $ \Omega$ and also preserve its regularity. We also show, using the framework of bifurcation theory, the existence of at least two solutions for certain classes of exponential type nonlinearities.


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Additional Information

R. Dhanya
Affiliation: Tata Institute of Fundamental Research, Center for Applicable Mathematics, P.B. No. 6503, Sharadanagar, Chikkabommasandra, Bangalore 560065, India
Email: dhanya@math.tifrbng.res.in

J. Giacomoni
Affiliation: Laboratoire de Mathématiques Appliquées, Université de Pau et des Pays de l’Adour, B.P. 576, 64012 Pau cedex, France
Email: jgiacomo@univ-pau.fr

S. Prashanth
Affiliation: Tata Institute of Fundamental Research, Center for Applicable Mathematics, P.B. No. 6503, Sharadanagar, Chikkabommasandra, Bangalore 560065, India
Email: pras@math.tifrbng.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09988-2
PII: S 0002-9939(09)09988-2
Keywords: Isolated singularity, blow-up, Laplace equation.
Received by editor(s): September 30, 2008
Published electronically: July 15, 2009
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2009 American Mathematical Society