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.