American Institute of Mathematical Sciences

February & 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 & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
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