We consider the problemwhere $\Omega \subset {\mathbb{R}}^{3}$ is a smooth and bounded domain, $\epsilon ,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{1},\phantom{\rule{0.166667em}{0ex}}{\gamma}_{2}>0,$ $v,\phantom{\rule{0.166667em}{0ex}}V:\Omega \to \mathbb{R}$, $f:\mathbb{R}\to \mathbb{R}$. 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 $.

@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://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, Serie 5, Volume 5 (2006) no. 2, pp. 219-259. http://www.numdam.org/item/ASNSP_2006_5_5_2_219_0/

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