Existence of Many Nonradial Positive Solutions of the Hénon Equation in R ³

Let

be the open unit ball in

\mathbb{R}^3 R 3

and let

2<p<6 2 < p < 6

. We show that for each

m\in \mathbb{N} m ∈ N

, there exists

\alpha_0>0 α 0 > 0

such that for each

\alpha\geq \alpha_0 α ≥ α 0

, there exist at least

m m

nonradial positive solutions of

-\Delta u = |x|^\alpha |u(x)|^{p-2}u(x) \quad\text{in $B_1$,}\qquad u = 0 \quad\text{on $\partial B_1$,} − Δ u = ∣ x ∣ α ∣ u ( x ) ∣ p − 2 u ( x ) in  B 1 , u = 0 on  ∂ B 1 ,

which are mutually nonequivalent if

m\geq 2 m ≥ 2

.