Semiclassical states for a class of nonlinear elliptic field equations

Article type: Research Article

Authors: D'Aprile, Teresa

Affiliations: Dipartimento di Matematica, via E. Orabona 4, 70125 Bari, Italy E‐mail: daprile@dm.uniba.it

Abstract: In this paper we are concerned with the problem of finding solutions for the following nonlinear field equation −ħ2Δv+V(x)v−ħpΔpv+W′(v)=0, where v : $\mathbb{R} $N→$\mathbb{R} $N+1, N≥3, p>N, ħ>0, the potential V is positive and W is an appropriate singular function. Assuming that V has a mountain pass geometry, determined by the existence of two strict local minima, then for small values of the parameter ħ we are able to find a solution which concentrates at the related mountain pass point of V in the semiclassical limit (i.e., as ħ→0+). The proof of our results uses a variational approach and the solutions are captured as critical points for the associated energy functional.

Journal: Asymptotic Analysis, vol. 37, no. 2, pp. 109-141, 2004