Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 219-259.

We consider the problemwhere Ω 3 is a smooth and bounded domain, ε,γ 1 ,γ 2 >0, v,V:Ω, f:. We prove that this system has a least-energy solution v ε which develops, as ε0 + , 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 Ω, 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 Ω.

Classification : 35B40, 35B45, 35J55, 92C15, 92C40
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D’Aprile, Teresa. Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 219-259. http://www.numdam.org/item/ASNSP_2006_5_5_2_219_0/

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