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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. Iberoam. 29 (2013), 997-1020. doi: 10.4171/RMI/747