We study positive solutions $u_p$ of the nonlinear Neumann elliptic problem $\mathrm{\Delta }u=u$ in Ω, $\partial u/\partial \nu =|u{|}^{p-1}u$ on ∂Ω, where Ω is a bounded open smooth domain in ${\mathbb{R}}^2$. We investigate the asymptotic behavior of families of solutions $u_p$ satisfying an energy bound condition when the exponent p is getting large. Inspired by the work of Davila-del Pino-Musso [8], we prove that $u_p$ is developing m peaks $x_i\in \partial \mathrm{\Omega }$, in the sense $u_p^p/{\int }_{\partial \mathrm{\Omega }}{u}_p^p$ approaches the sum of m Dirac masses at the boundary and we determine the localization of these concentration points.

中文翻译：

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具有大指数Neumann数据的平面椭圆问题的收敛性

我们研究积极的解决方案 ${ü}_p$ 诺伊曼椭圆问题的分析 $\mathrm{\Delta }ü=ü$ 以Ω为单位 $\partial ü/\partial \nu =|ü{|}^{p-\mathrm{1个}}ü$ 在∂Ω上，其中Ω是 ${\mathbb{[R}}^{\mathrm{2个}}$。我们调查解决方案族的渐近行为${ü}_p$当指数p变大时满足能量限制条件。受到Davila-del Pino-Musso [8]的启发，我们证明了${ü}_p$正在发展m个山峰$X_{\mathrm{一世}}\in \partial \mathrm{\Omega }$， 在这个意义上 ${ü}_p^p/{\int }_{\partial \mathrm{\Omega }}{ü}_p^p$接近边界处m个狄拉克质量的总和，我们确定了这些集中点的位置。