Infinitely many nonradial solutions for the Hénon equation with critical growth

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.