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Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data. (English) Zbl 1207.35158

Summary: We consider the elliptic equation \(-\Delta u+u=0\) in a bounded, smooth domain \(\Omega\) in \(\mathbb R^2\) subject to the nonlinear Neumann boundary condition \(\frac{\partial u}{\partial v}= \varepsilon e^u\). Here \(\varepsilon>0\) is a small parameter. We prove that any family of solutions \(u_\varepsilon\) for which \(\varepsilon\int_{\partial\Omega} e^u\) is bounded, develops up to subsequences a finite number \(m\) of peaks \(\xi_i\in\partial\Omega\), in the sense that \(\varepsilon e^u\rightharpoonup 2\pi \sum_{k=1}^m \delta_{\xi_i}\) as \(\varepsilon\to 0\). Reciprocally, we establish that at least two such families indeed exist for any given \(m\geq 1\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
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