March  2004, 11(2&3): 649-666. doi: 10.3934/dcds.2004.11.649

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

1. 

Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100080, China

2. 

School of Mathematics, The University of New South Wales, Sydney 2052, Australia

3. 

School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Received  April 2003 Revised  April 2004 Published  June 2004

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.

Citation: Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
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