Infinitely many nonradial solutions for the Hénon equation with critical growth

Abstract

We consider the following Hénon equation with critical growth:

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 $N\ge 4$. In this paper, we show the existence of a nonradial solution for any $\alpha >0$ and $N\ge 4$. Furthermore, we prove that equation (*) has infinitely many nonradial solutions, whose energy can be made arbitrarily large.