Nonradial sign changing solutions to Lane Emden equation

TL;DR

This paper proves the existence of multiple nonradial solutions with various symmetries for the Lane-Emden equation, revealing a complex solution structure beyond radial solutions.

Contribution

It introduces new methods to find nonradial solutions with prescribed nodal zones and symmetries, expanding understanding of the solution space.

Findings

Existence of infinitely many nonradial solutions

Solutions with prescribed number of nodal zones

Rich and complex structure of solution set

Abstract

In this paper we prove the existence of continua of nonradial solutions for the Lane-Emden equation. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane Emden problem has a very rich and complex structure.