Behaviour of Symmetric Solutions of a Nonlinear Elliptic Field Equation in the Semi-classical Limit: Concentration Around a Circle

Abstract

In this paper we study the existence of concentrated solutions of the nonlinear field equation -h2 ∆v + V(x)v - hp ∆pv + W' (v) = 0, where v : ℝN → ℝN+1, N ≥ 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 → 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.