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A note on sign-changing solutions to supercritical Yamabe-type equations

Jurgen Julio-Batalla

Abstract

On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type equation $-Δ_g u + λu = λ|u|^{q-1}u$, where $λ\in \mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$ with focal submanifolds of positive dimension. If $k>0$ is the minimum of the dimensions of the focal submanifolds of $f$, we let $q^* =\frac{n-k+2}{n-k-2}$. We prove the existence of infinite $f$-invariant sign-changing solutions to the equation when $1<q<q^*$.

A note on sign-changing solutions to supercritical Yamabe-type equations

Abstract

On a closed Riemannian manifold , we consider the Yamabe-type equation , where and . We assume that admits a proper isoparametric function with focal submanifolds of positive dimension. If is the minimum of the dimensions of the focal submanifolds of , we let . We prove the existence of infinite -invariant sign-changing solutions to the equation when .
Paper Structure (3 sections, 5 theorems, 44 equations)

This paper contains 3 sections, 5 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on isoparametric functions
  3. Mountain pass critical points

Key Result

Theorem 1.1

Consider a proper isoparametric function $f$ on a closed Riemannian manifold $(M^n,g)$. Assume that the minimum dimension of level sets of $f$, $k$, is strictly positive. Assume that $1<q<\frac{n-k+2}{n-k-2}$. Then there exist infinite solutions of Equation (YT) with arbitrarily large energy.

Theorems & Definitions (8)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof