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, p. 219-259
We consider the problemwhere $\Omega \subset {ℝ}^{3}$ is a smooth and bounded domain, $\epsilon ,\phantom{\rule{0.166667em}{0ex}}{\gamma }_{1},\phantom{\rule{0.166667em}{0ex}}{\gamma }_{2}>0,$ $v,\phantom{\rule{0.166667em}{0ex}}V:\Omega \to ℝ$, $f:ℝ\to ℝ$. We prove that this system has a least-energy solution ${v}_{\epsilon }$ which develops, as $\epsilon \to {0}^{+}$, 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 $\partial \Omega$, 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 $\epsilon$ up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in $\Omega$.
Classification:  35B40,  35B45,  35J55,  92C15,  92C40
@article{ASNSP_2006_5_5_2_219_0,
author = {D'Aprile, Teresa},
title = {Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 5},
number = {2},
year = {2006},
pages = {219-259},
zbl = {1150.35006},
mrnumber = {2244699},
language = {en},
url = {http://http://www.numdam.org/item/ASNSP_2006_5_5_2_219_0}
}

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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