Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem

Abstract

We consider the problem e 2 �v − v − γ1V v + f (v) = 0 � V + γ2|v| 2 = 0 ,v = V = 0o n∂�, where � ⊂ R 3 is a smooth and bounded domain, e, γ1 ,γ 2 > 0 ,v ,V : � → R, f : R → R.W eprove that this system has a least-energy solution ve which develops, as e → 0 + ,a single spike layer located near the boundary, in striking contrast with the result in (37) for the single Schr¨ odinger equation. Moreover the unique peak approaches the most curved part of ∂� , i.e., where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem for the single-equation case. The technique employed is based on the so-called energy method, which consists in the derivation of an asymptotic expansion for the energy of the solu- tions in powers of e up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in � .