# Revista Matemática Iberoamericana

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**Volume 29, Issue 3, 2013, pp. 997–1020**

**DOI: 10.4171/RMI/747**

Published online: 2013-08-04

Infinitely many nonradial solutions for the Hénon equation with critical growth

Juncheng Wei^{[1]}and Shusen Yan

^{[2]}(1) University of British Columbia, Vancouver, Canada

(2) University of New England, Armidale, Australia

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.

*Keywords: *Hénon's equation, infinitely many solutions, critical Sobolev exponent, reduction method

Wei Juncheng, Yan Shusen: Infinitely many nonradial solutions for the Hénon equation with critical growth. *Rev. Mat. Iberoamericana* 29 (2013), 997-1020. doi: 10.4171/RMI/747