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

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.

Show me the PDF file (344K), TEX file, and other files for this article.

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

Return to the EJDE web page