Nonradial Solutions for the Hénon Equation Close to the Threshold

Abstract

Abstract We consider the Hénon problem { - Δ ⁢ u = | x | α ⁢ u N + 2 + 2 ⁢ α N - 2 - ε in ⁢ B 1 , u > 0 in ⁢ B 1 , u = 0 on ⁢ ∂ ⁡ B 1 , \left\{\begin{aligned} &\displaystyle{-}\Delta u=\lvert x\rvert^{\alpha}u^{% \frac{N+2+2\alpha}{N-2}-\varepsilon}&&\displaystyle\phantom{}\text{in }B_{1},% \\ &\displaystyle u>0&&\displaystyle\phantom{}\text{in }B_{1},\\ &\displaystyle u=0&&\displaystyle\phantom{}\text{on }\partial B_{1},\end{% aligned}\right. where B 1 {B_{1}} is the unit ball in ℝ N {\mathbb{R}^{N}} and N ⩾ 3 {N\geqslant 3} . For ε > 0 {\varepsilon>0} small enough, we use α as a parameter and prove the existence of a branch of nonradial solutions that bifurcates from the radial one when α is close to an even positive integer.