On a semilinear Schrödinger equation with critical Sobolev exponent
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- by Jan Chabrowski and Andrzej Szulkin PDF
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Abstract:
We consider the semilinear Schrödinger equation $-\Delta u+V(x)u = K(x)|u|^{2^{*}-2}u+g(x,u)$, $u\in W^{1,2}(\mathbf {R}^{N})$, 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 $g\equiv 0$.References
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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
- MR Author ID: 210814
- Email: andrzejs@matematik.su.se
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 85-93
- MSC (2000): Primary 35B33, 35J65, 35Q55
- DOI: https://doi.org/10.1090/S0002-9939-01-06143-3
- MathSciNet review: 1855624