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

 

On a semilinear Schrödinger equation with critical Sobolev exponent


Authors: Jan Chabrowski and Andrzej Szulkin
Journal: Proc. Amer. Math. Soc. 130 (2002), 85-93
MSC (2000): Primary 35B33, 35J65, 35Q55
Published electronically: May 22, 2001
MathSciNet review: 1855624
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Abstract:

We consider the semilinear Schrödinger equation $-\Delta u+V(x)u = K(x)\vert u\vert^{2^{*}-2}u+g(x,u)$, $u\in W^{1,2}(\mathbf{R}^{N})$, where $N\ge 4$, $V,K,g$ are periodic in $x_{j}$ for $1\le j\le N$, $K>0$, $g$ is of subcritical growth and 0 is in a gap of the spectrum of $-\Delta+V$. We show that under suitable hypotheses this equation has a solution $u\ne 0$. In particular, such a solution exists if $K\equiv 1$ and $g\equiv 0$.


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Additional Information

Jan Chabrowski
Affiliation: Department of Mathematics, University of Queensland, St. Lucia 4072, Queensland, Australia
Email: jhc@maths.uq.edu.au

Andrzej Szulkin
Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
Email: andrzejs@matematik.su.se

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06143-3
PII: S 0002-9939(01)06143-3
Keywords: Semilinear Schr\"odinger equation, critical Sobolev exponent, linking
Received by editor(s): May 20, 2000
Published electronically: May 22, 2001
Additional Notes: The second author was supported in part by the Swedish Natural Science Research Council
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society