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July  2008, 7(4): 925-946. doi: 10.3934/cpaa.2008.7.925

Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions

Long Wei 1,

1. 

Department of Mathematics, East China Normal University, Shanghai 200062, China

Received  May 2007 Revised  November 2007 Published  April 2008

We investigate here the elliptic equation -div$(a(x)\nabla u)+a(x)u=0$ posed on a bounded smooth domain $\Omega$ in $\mathbb R^2$ with nonlinear Neumann boundary condition $\frac{\partial u}{\partial \nu}=\varepsilon e^u$, where $\varepsilon$ is a small parameter. We extend the work of Davila-del Pino-Musso [5] and show that if a family of solutions $u_\varepsilon$ for which $\varepsilon\int_{\partial Omega}e^{u_\varepsilon}$ is bounded, then it will develop up to subsequences a finite number of bubbles $\xi_i\in\partial Omega$, in the sense that $\varepsilon e^{u_\varepsilon}\to 2\pi\sum_{i=1}^k m_i\delta_{\xi_i}$ as $\varepsilon\rightarrow 0$ with $k, m_i \in \mathbb Z^+$. Location of blow-up points is characterized in terms of function $a(x)$.
Keywords: Exponential Neumann nonlinearity, concentration of solutions..
Mathematics Subject Classification: Primary: 35J65, 35J25; Secondary: 35J6.
Citation: Long Wei. Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions. Communications on Pure and Applied Analysis, 2008, 7 (4) : 925-946. doi: 10.3934/cpaa.2008.7.925
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