Concentrating phenomena in some elliptic Neumann problem: asymptotic behavior of solutions (Q933152)

For a bounded smooth domain \(\Omega\subseteq\mathbb{R}^2\), smooth, positive \(a\colon\overline{\Omega}\to\mathbb{R}\), and small positive \(\varepsilon\) consider \[ \left\{ \begin{alignedat}{2} -\text{div}(a(x)\nabla u)+a(x)u &= 0,&&\qquad\text{in }\Omega,\\ \frac{\partial u}{\partial\nu}&=\varepsilon \text{e}^u, &&\qquad\text{in }\partial\Omega. \end{alignedat}\right.\tag{\(P_\varepsilon\)} \] The problem is a generalization of the one considered in [\textit{J. Dávila, M. del Pino, M. Musso}, J. Funct. Anal. 227, No. 2, 430--490 (2005; Zbl 1207.35158)], where \(a\equiv 1\). Suppose that \(\varepsilon_n\to0\) and that \(u_n\) is a solution of \((P_{\varepsilon_n})\), for each \(n\in\mathbb{N}\). If \(\varepsilon_n\int_{\partial\Omega}\text{e}^{u_n}\,\text{d}x\) remains bounded as \(n\to\infty\), it is shown that then, after passing to a subsequence, either \(u_n\) remains bounded in \(L^\infty(\Omega)\) or \(u_n\) blows up in a finite number of points on \(\partial\Omega\) that are critical points of \(a| _{\partial\Omega}\).