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

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 $v:{mathbb R}^{N}o{mathbb R}^{N+1}$, $Ngeq 3$, $p>N$, 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 $ho 0^{+}$. 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.