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

D’Aprile, Teresa

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 219-259.

We consider the problemwhere $\Omega \subset {ℝ}^3$ is a smooth and bounded domain, $\epsilon ,{\gamma }_1,{\gamma }_2>0,$ $v,V:\Omega \to ℝ$, $f:ℝ\to ℝ$. 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 $\Omega$.

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