## Concentrating phenomena in some elliptic Neumann problem: asymptotic behavior of solutions.(English)Zbl 1181.35106

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 [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 $$a| _{\partial\Omega}$$.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems

Zbl 1207.35158
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