Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem
Year: 2009
Type: article
Abstract: We consider the problemwhere ⊂ R 3 is a smooth and bounded domain, ε, γ 1 , γ 2 > 0, v, V : → R, f : R → R. We prove that this system has a least-energy solution v ε which develops, as ε → 0 + , a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schrödinger 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 solutions in powers of ε up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in .
Authors: Teresa D'Aprile, Juncheng Wei
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1889305042