On the profile of solutions for an elliptic problem arising in nonlinear optics

Daomin Cao; Ezzat S. Noussair; Shusen Yan

doi:10.3934/dcds.2004.11.649

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2004, Volume 11, Issue 2&3: 649-666. Doi: 10.3934/dcds.2004.11.649

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On the profile of solutions for an elliptic problem arising in nonlinear optics

1.
Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100080
2.
School of Mathematics, The University of New South Wales, Sydney 2052
3.
School of Mathematics and Statistics, University of Sydney, NSW 2006

Received: April 2003

Revised: April 2004

Published: September 2004

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Abstract

We study the profile of solutions of

$-\Delta u + (\lambda - h(x)) u = g(x) (u^{p-1} + f(u))$ in $\ \mathbb R^N,$

$u > 0$ in $\mathbb R^N,$

$u \in H^1(\mathbb R^N),$

where $\lambda > 0$ is a parameter, $h$ and $g$ are nonnegative functions in $L^\infty(\mathbb R^N).$ We obtain the asymptotic behaviour of the least energy solutions or solutions obtained by the minimax principle. From the asymptotic behaviour we conclude that those solutions are asymmetric for $\lambda$ large even if $h$ and $g$ are radially symmetric.

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Mathematics Subject Classification: 35B25, 35J65.

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