Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 69, pp. 1-40.
Title: Behaviour of symmetric solutions of a nonlinear
elliptic field equation in the semi-classical limit:
Concentration around a circle
Author: Teresa D'Aprile (Scuola Normale Superiore, Pisa, Italy)
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 $v:{\mathbb R}^{N}\to{\mathbb R}^{N+1}$, $N\geq 3$, $p>N$, 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.