边界爆破解：梯度渐近行为与唯一性
Boundary blow-up solutions: gradient asymptotics and uniqueness
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中文总结 AI 辅助
本文针对满足Keller-Osserman条件的f，研究Δu=f(u)的边界爆破解，确定归一化梯度Qᵤ为控制唯一性的量，分情况证明不同区域中边界爆破解的唯一性条件。
中文摘要 AI 辅助
设Ω是ℝⁿ中的有界区域，f为满足Keller-Osserman条件的非负非递减函数，本文研究Ω中Δu=f(u)的边界爆破解。其存在性是经典结论，但在上述假设下的唯一性仅对球形成立，对光滑凸域仍未解决。本文将归一化梯度Qᵤ=|∇u|²/(2F(u))（其中F'=f）确定为控制唯一性的量。在f的结构条件下，若limsupₓ→∂Ω Qᵤ(x)≤1，则边界爆破解u唯一，且无需对∂Ω施加任何正则性假设；对C¹,¹型区域，在f满足增长条件时，本文证明每个边界爆破解均满足Qᵤ(x)→1，结合结构条件可得唯一性；对凸域，本文证明最小边界爆破解满足Qᵤ≤1，当√F最终为凸函数时，无需额外边界正则性即可得到唯一性。
英文摘要
Let $Ω\subset\mathbb R^n$ be a bounded domain, and let $f$ be a nonnegative, nondecreasing function satisfying the Keller-Osserman condition. We study boundary blow-up solutions of $Δu=f(u)$ in $Ω$. Although existence is classical, uniqueness under these assumptions is known in balls but remains open even for smooth convex domains. We identify the normalized gradient $Q_u=|\nabla u|^2/(2F(u))$, $F'=f$, as a quantity governing uniqueness. Under a structural condition on $f$, a boundary blow-up solution $u$ is unique if $\limsup_{x\to\partialΩ}Q_u(x)\le 1$, without any regularity assumption on $\partial Ω$. For $C^{1,1}$ domains, assuming a growth condition on $f$, we prove $Q_u(x)\to 1$ for every boundary blow-up solution and hence obtain uniqueness under the structural condition. For convex domains, we prove $Q_u\le 1$ for the minimal boundary blow-up solution and obtain uniqueness when $\sqrt F$ is eventually convex, without imposing any additional boundary regularity.
