AI 中文总结
本文针对半线性椭圆方程，在特定q范围完成非负解孤立奇点的分类并刻画奇异解渐近行为，改进了前人仅针对径向对称正解的结果，还推导了两类临界情形下解的渐近行为。
AI 中文摘要
本文研究如下半线性椭圆方程在Ω\{0}中的孤立奇点：-Δu + (1/2)x·∇u + (1/(q-1))u - u^q = 0，其中n≥3，Ω是ℝⁿ中的区域，0∈Ω且q>1。该方程源于半线性热方程的爆破剖面研究。当n/(n-2)<q<(n+2)/(n-2)时，本文对非负解的孤立奇点进行了完整分类，并刻画了奇异解的精确渐近行为。所得结果改进了Guedda与Kirane（《美国数学会汇刊》，1995：3595-3603）的成果，后者仅针对径向对称正解得到了类似结果。此外，本文还推导了Serrin临界情形q=n/(n-2)及超临界情形q>(n+2)/(n-2)下解的渐近行为。
英文摘要
In this paper, we study isolated singularities of the following semilinear elliptic equation $-Δu+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $Ω\setminus \{0\}$, where $n\ge 3$, $Ω\subset \mathbb{R}^n$ is a domain, $0 \in Ω$ and $q>1$. This equation arises in the study of blow-up profiles of semilinear heat equations. For $\frac{n}{n-2}< q < \frac{n+2}{n-2}$, we establish a complete classification of isolated singularities for nonnegative solutions and characterize the precise asymptotic behavior of singular solutions. Our results improve those of Guedda and Kirane (Trans. Amer. Math. Soc., 1995: 3595-3603), where analogous results were obtained only for radially symmetric positive solutions. In addition, we also derive the asymptotic behavior of solutions in the Serrin critical case $q=\frac{n}{n-2}$ and the supercritical case $q>\frac{n+2}{n-2}$.
Comments30 pages
