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




D'Aprile, Teresa

Journal Title

Journal ISSN

Volume Title


Southwest Texas State University, Department of Mathematics


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.



Nonlinear Schrodinger equations, Topological charge, Existence, Concentration


D'Aprile, T. (2000). Behaviour of symmetric solutions of a nonlinear elliptic field equation in the semi-classical limit: Concentration around a circle. <i>Electronic Journal of Differential Equations, 2000</i>(69), pp. 1-40.


Attribution 4.0 International

Rights Holder

Rights License