三维临界 Lin-Ni-Takagi 问题的内部 bubbling 解
Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3
AI总结：
研究三维有界光滑区域中带零Neumann边界条件的临界Lin-Ni-Takagi问题，通过分析Green函数，发现并刻画了正参数 $λ_*$，使得当 $λ$ 逼近该值时，问题存在在内部点发生 bubbling 现象的解。
摘要：
我们考虑在有界光滑区域 $Ω$（属于 $\mathbb{R}^3$）中寻找方程 $Δu - λu +u^5 = 0$ 正解的问题，并采用零 Neumann 边界条件。此处 $λ$ 为正数。我们将 $λ$ 视为参数，分析了 $-Δ+λ$ 的 Green 函数在解于域内某点呈现单一 bubbling 行为时所起的作用。作为结果的一个特例，我们找到并刻画了一个正值 $λ_*$，使得当 $λ-λ^*>0$ 足够小时，该问题可由解 $u_λ$ 求解，且当 $λ\downarrow λ_*$ 时，该解会在 $Ω$ 的某个内部点处通过 bubbling 发生 blow-up。
英文摘要：
We consider the problem of finding positive solutions of the problem $Δu - λu +u^5 = 0$ in a bounded, smooth domain $Ω$ in $\mathbb{R}^3$, under zero Neumann boundary conditions. Here $λ$ is a positive number. We analyze the role of Green's function of $-Δ+λ$ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when $λ$ is regarded as a parameter. As a special case of our results, we find and characterize a positive value $λ_*$ such that if $λ-λ^*>0$ is sufficiently small, then this problem is solvable by a solution $u_λ$ which blows-up by bubbling at a certain interior point of $Ω$ as $λ\downarrow λ_*$.
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