Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain

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September 2008, 22(3): 465-508. doi: 10.3934/dcds.2008.22.465

Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain

1.	Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
2.	Department of Mathematics, Shenzhen University, Shenzhen, China

Received July 2007 Revised March 2008 Published August 2008

Abstract
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We consider the equation $\varepsilon^2\Delta$ ũ-ũ+ũ$^p =0$ in a bounded, smooth domain $\Omega$ in $\R^2$ under homogeneous Neumann boundary conditions. Let $\Gamma$ be a segment contained in $\Omega$, connecting orthogonally the boundary, non-degenerate and non-minimal with respect to the curve length. For any given integer $N\ge 2$ and for small $\varepsilon$ away from certain critical numbers, we construct a solution exhibiting $N$ interior layers at mutual distances $O(\varepsilon|\ln\varepsilon|)$ whose center of mass collapse onto $\Gamma$ at speed $O(\varepsilon^{1+\mu})$ for small positive constant $\mu$ as $\varepsilon\to 0$. Asymptotic location of these layers is governed by a Toda system.

Keywords: Gap condition, Clustered layers, singularly perturbed Neumann problem., Toda system.

Mathematics Subject Classification: Primary: 35B25, 35J60; Secondary: 35K99, 92C4.

Citation: Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465

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