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Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions
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 $a(x)$.