Global bifurcation for the H\'enon problem

TL;DR

This paper establishes the existence of nonradial solutions with multiple nodal zones for the Hénon problem in a ball, demonstrating how solutions branch off from radial solutions as parameters vary.

Contribution

It introduces a novel bifurcation analysis for the Hénon equation, including sign-changing solutions at , and details the structure and symmetry of solution branches.

Findings

Existence of nonradial solutions with arbitrary nodal zones.

Branches of solutions originate from radial solutions and increase with  and nodal zones.

Analysis of symmetry and intersection properties of solution branches.

Abstract

We prove the existence of nonradial solutions for the H\'enon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $\alpha$. For sign-changing solutions, the case $\alpha=0$ -- Lane-Emden equation -- is included. The obtained solutions form global continua which branch off from the curve of radial solutions $p\mapsto u_p$, and the number of branching points increases with both the number of nodal zones and the exponent $\alpha$. The proof technique relies on the index of fixed points in cones and provides information on the symmetry properties of the bifurcating solutions and the possible intersection and/or overlapping between different branches, thus allowing to separate them at least in some cases.