Multiplicity of nodal solutions to the Yamabe problem

TL;DR

This paper proves the existence of multiple nodal solutions to a Yamabe-type equation on symmetric compact Riemannian manifolds, extending results to critical Sobolev exponents and scalar curvature conditions.

Contribution

It establishes new existence results for multiple nodal solutions to Yamabe-type equations under symmetry and curvature assumptions, including prescribed numbers of solutions.

Findings

Existence of positive and multiple nodal solutions under symmetry assumptions.

Conditions for prescribed numbers of solutions related to scalar curvature.

Extension of Yamabe problem solutions to critical Sobolev exponent cases.

Abstract

Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\nabla u)+bu=c|u|^{2^{\ast}-2}u\quad on\ M$$ where $a,b,c\in C^{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ is coercive, and $2^{\ast}=\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_{g}$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $$\Delta_{g}u+\frac{m-2}{4(m-1} R_{g}u=\kappa u^{2^{\ast}-2}\quad on\ M$$ admits a prescribed number of nodal solutions.