Nonradial solutions for the H\'enon equation close to the threshold

Abstract

We consider the H\'enon problem \begin{equation*} \left\{ \begin{array} - - \Delta u = |x|^{\alpha} u^{\frac{N+2+2\alpha}{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 $\alpha$ as a paramenter and prove the existence of a branch of nonradial solutions that bifurcates from the radial one when $\alpha$ is close to an even positive integer.