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

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  • 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.

    Mathematics Subject Classification: 35B25, 35J65.

    Citation:

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