Non-radial solutions of the problem $-\Delta u = |u|^{4/(n-2)}u$ in $R^n$, $n\geq3$

TL;DR

This paper establishes the existence of infinitely many non-radial, nodal solutions to a critical nonlinear elliptic PDE in higher dimensions, expanding understanding of solution symmetry and multiplicity.

Contribution

It proves the existence of an infinite sequence of non-radial, G-invariant solutions to the critical elliptic problem, which was previously unknown.

Findings

Existence of infinitely many non-radial solutions

Solutions are G-invariant and nodal

Solutions exist for all dimensions n ≥ 3

Abstract

We prove the existence of an infinite sequence of distinct non-radial nodal $G-$invariant solutions for the following critical nonlinear elliptic problem: $({\mathrm{P}})\quad {*{20}c} {-\Delta u = |u|^{4/(n-2)}u},\quad u\in C^2(\mathbb{R}^n), \quad n\geq3}$