# Infinitely many nonradial solutions for the Hénon equation with critical growth

### Juncheng Wei

University of British Columbia, Vancouver, Canada### Shusen Yan

University of New England, Armidale, Australia

## Abstract

We consider the following Hénon equation with critical growth:

$(*) \begin{cases} - \Delta u = |y|^\alpha \, u^{\frac{N+2}{N-2}},\; u>0, & y\in B_1(0) , \\ u=0, &\text{on } \partial B_1(0), \end{cases}$

where $\alpha>0$ is a positive constant, $B_1(0)$ is the unit ball in $\mathbb{R}^N$, and $N\ge 4$. Ni [9] proved the existence of a radial solution and Serra [12] proved the existence of a nonradial solution for $\alpha$ *large* and $N \geq 4$. In this paper, we show the existence of a nonradial solution for *any* $\alpha>0$ and $N \geq 4$. Furthermore, we prove that equation (*) has *infinitely many nonradial* solutions, whose energy can be made arbitrarily large.

## Cite this article

Juncheng Wei, Shusen Yan, Infinitely many nonradial solutions for the Hénon equation with critical growth. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 997–1020

DOI 10.4171/RMI/747