Nonradial solutions for the H\'enon equation close to the threshold

TL;DR

This paper proves the existence of nonradial solutions for the Hénon equation near the critical exponent, using bifurcation theory when the parameter alpha is close to an even positive integer.

Contribution

It introduces a new bifurcation approach to find nonradial solutions for the Hénon problem near critical parameters, expanding understanding of solution symmetry.

Findings

Existence of nonradial solutions bifurcating from radial solutions.

Bifurcation occurs when alpha is near an even positive integer.

Solutions are constructed for small epsilon close to zero.

Abstract

We consider the H\'enon problem \begin{equation*} \left\{ \begin{array} - - \Delta u = |x|^{\alpha} u^{\frac{N+2+2\alpha}{N-2}-\varepsilon} & \ \ \text{in} \ B_1, \\ u > 0 & \ \ \text{in} \ B_1, \\ u=0 & \ \ \text{on} \ \partial B_1, \end{array} \right. \end{equation*} where $B_1$ is the unit ball in ${\mathbb R}^N$ and $N\geqslant 3$. For $\varepsilon > 0$ small enough, we use $\alpha$ as a paramenter and prove the existence of a branch of nonradial solutions that bifurcates from the radial one when $\alpha$ is close to an even positive integer.