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Stability and instability results for sign-changing solutions to second-order critical elliptic equations

Bruno Premoselli, Jérôme Vétois

Abstract

On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge3$, we consider the stationary Schrödinger equation $Δ_gu+h_0u=\left|u\right|^{2^*-2}u$, where $Δ_g:=-\text{div}_g\nabla$, $h_0\in C^1\left(M\right)$ and $2^* :=\frac{2n}{n-2}$. We prove that, up to perturbations of the potential function $h_0$ in $C^1\left(M\right)$, the sets of sign-changing solutions that are bounded in $H^1\left(M\right)$ are precompact in the $C^2$ topology. We obtain this result under the assumptions that $\left(M,g\right)$ is locally conformally flat, $n\ge7$ and $h_0\ne\frac{n-2}{4\left(n-1\right)}\text{Scal}_g$ at all points in $M$, where $\text{Scal}_g$ is the scalar curvature of the manifold. We then provide counterexamples in every dimension $n\ge3$ showing the optimality of these assumptions.

Stability and instability results for sign-changing solutions to second-order critical elliptic equations

Abstract

On a smooth, closed Riemannian manifold of dimension , we consider the stationary Schrödinger equation , where , and . We prove that, up to perturbations of the potential function in , the sets of sign-changing solutions that are bounded in are precompact in the topology. We obtain this result under the assumptions that is locally conformally flat, and at all points in , where is the scalar curvature of the manifold. We then provide counterexamples in every dimension showing the optimality of these assumptions.
Paper Structure (9 sections, 9 theorems, 23 equations)

This paper contains 9 sections, 9 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. A stability result for locally conformally flat manifolds
  3. Counterexamples: the case of dimensions $n\ge6$
  4. Counterexamples: the case of dimensions $n\in\left\{3,4,5\right\}$
  5. Organization of the article
  6. Additional results
  7. The case of dimensions $n \ge 6$.
  8. The case of dimensions $n \in \{3,4,5\}$.
  9. Proof of Theorem \ref{['Th1']}

Key Result

Theorem 1.1

Let $(M,g)$ be a smooth, closed, locally conformally flat Riemannian manifold of dimension $n\ge7$ and $$h_ɛ$_{0\le\varepsilon\ll1}$ be a family of functions in $C^1$M$$ such that $h_\varepsilon\to h_0$ in $C^1$M$$ and Then every family of solutions $$u_ɛ$_{0<\varepsilon\ll1}$ to IntroEq2 that is bounded in $H^1$M$$ converges up to a subsequence in $C^2$M$$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Corollary 2.5