Gelfand问题的扰动奇异解
Perturbing singular solutions of the Gelfand problem
摘要：
在单位球$B \subseteq \R^N$中提出的方程$-Δu = λe^u$，满足齐次Dirichlet条件$u|_{\partial B} = 0$，当$λ= 2(N-2)$时，有奇异解$U=\log\frac1{|x|^2}$.如果$N\ge 4$，我们表明，在小变形的球有一个奇异的解决方案$(u,λ)$接近$(U,2(N-2))$。在维度$N\ge 11$中，它对应于极值解--与存在的最大$λ$相关联的解。与此相反，我们证明，如果变形是足够大的，那么即使当$N\ge 10$，极值解在许多情况下仍然有界。
英文摘要：
he equation $-Δu = λe^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $λ= 2(N-2)$. If $N\ge 4$ we show that under small deformations of the ball there is a singular solution $(u,λ)$ close to $(U,2(N-2))$. In dimension $N\ge 11$ it corresponds to the extremal solution -- the one associated to the largest $λ$ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when $N\ge 10$, the extremal solution remains bounded in many cases.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
