Electron. J. Diff. Eqns., Vol. 2000(2000), No. 69, pp. 1-40.

Behaviour of symmetric solutions of a nonlinear elliptic field equation in the semi-classical limit: Concentration around a circle

Teresa D'Aprile

Abstract:
In this paper we study the existence of concentrated solutions of the nonlinear field equation
$-h^{2}\Delta v+V(x)v-h^{p}\Delta_{p}v+ W'(v)=0$,
where $v:{\mathbb R}^{N}\to{\mathbb R}^{N+1}$, $N\geq 3$, , the potential $V$ is positive and radial, and $W$ is an appropriate singular function satisfying a suitable symmetric property. Provided that $h$ is sufficiently small, we are able to find solutions with a certain spherical symmetry which exhibit a concentration behaviour near a circle centered at zero as $h\to 0^{+}$. Such solutions are obtained as critical points for the associated energy functional; the proofs of the results are variational and the arguments rely on topological tools. Furthermore a penalization-type method is developed for the identification of the desired solutions.

Submitted May 15, 2000. Published November 16, 2000.
Math Subject Classifications: 35J20, 35J60.
Key Words: nonlinear Schrodinger equations, topological charge, existence, concentration.

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Teresa D'Aprile
Scuola Normale Superiore
Piazza dei Cavalieri 7, 56126 Pisa, Italy
e-mail: aprilet@cibs.sns.it

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