Hénon方程在阈值附近的非径向解
Nonradial solutions for the Hénon equation close to the threshold
摘要：
考虑Hénon问题\begin{equation*} \left\{ \begin{array} - - Δu = |x|^α u^{\frac{N+2+2α}{N-2}-\varepsilon} & \ \ \text{in} \ B_1, \\ u > 0 & \ \ \text{in} \ B_1, \\ u=0 & \ \ \text{on} \ \partial B_1, \end{array} \right. \end{equation*}，其中$B_1$是${\mathbb R}^N$和$N\geqslant 3$中的单位球.当$\varepsilon > 0$足够小时，我们以$α$为参数，证明了当$α$接近于一个偶数正整数时，非径向解分支的存在性.
英文摘要：
We consider the Hénon problem \begin{equation*} \left\{ \begin{array} - - Δu = |x|^α u^{\frac{N+2+2α}{N-2}-\varepsilon} & \ \ \text{in} \ B_1, \\ u > 0 & \ \ \text{in} \ B_1, \\ u=0 & \ \ \text{on} \ \partial B_1, \end{array} \right. \end{equation*} where $B_1$ is the unit ball in ${\mathbb R}^N$ and $N\geqslant 3$. For $\varepsilon > 0$ small enough, we use $α$ as a paramenter and prove the existence of a branch of nonradial solutions that bifurcates from the radial one when $α$ is close to an even positive integer.
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