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 $a(x)$.