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One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations

Bruno Premoselli, Frédéric Robert

TL;DR

The paper analyzes sign-changing solutions to the asymptotically critical Schrödinger-type equation on smooth closed manifolds that blow up as a single bubble. It develops a sharp one-bubble blow-up framework, proving that the solutions are asymptotically described by a rescaled Yamabe bubble $V$ and a geometric constraint involving the Weyl tensor via the term $\mathrm{Weyl}_g\otimes B$, the scalar curvature, and the limit rate $\Lambda$ defined by the approach to criticality. In dimensions $n\ge5$ the new Weyl interaction governs possible blow-up and leads to precise necessary conditions; in dimensions $n=3,4$ the mass $m_h(x_0)$ and integrals of $V$ play analogous roles, reflecting different higher-order interactions. These results illuminate the delicate interplay between geometry, the limiting bubble, and the asymptotically critical nonlinearity, and they enable ruling out certain blow-up configurations through Pohozaev-type identities. The work also discusses stability implications for sign-changing solutions and provides a non-blow-up scenario to illustrate the utility of the derived conditions.

Abstract

We investigate in this work families $(u_ε)_{ε>0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type: $$Δ_g u_ε+ h_εu_ε= |u_ε|^{p_ε-2} u_ε$$ in a closed manifold $(M,g)$, where $h_ε$ converges to $h$ in $C^1(M)$. Assuming that $(u_ε)_{ε>0}$ blows-up as \emph{a single sign-changing bubble}, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between $h$, the geometry of $(M,g)$ and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of $u_ε$.

One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations

TL;DR

The paper analyzes sign-changing solutions to the asymptotically critical Schrödinger-type equation on smooth closed manifolds that blow up as a single bubble. It develops a sharp one-bubble blow-up framework, proving that the solutions are asymptotically described by a rescaled Yamabe bubble and a geometric constraint involving the Weyl tensor via the term , the scalar curvature, and the limit rate defined by the approach to criticality. In dimensions the new Weyl interaction governs possible blow-up and leads to precise necessary conditions; in dimensions the mass and integrals of play analogous roles, reflecting different higher-order interactions. These results illuminate the delicate interplay between geometry, the limiting bubble, and the asymptotically critical nonlinearity, and they enable ruling out certain blow-up configurations through Pohozaev-type identities. The work also discusses stability implications for sign-changing solutions and provides a non-blow-up scenario to illustrate the utility of the derived conditions.

Abstract

We investigate in this work families of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type: in a closed manifold , where converges to in . Assuming that blows-up as \emph{a single sign-changing bubble}, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between , the geometry of and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of .
Paper Structure (16 sections, 11 theorems, 189 equations)

This paper contains 16 sections, 11 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Consequences of Theorem \ref{['maintheo']} on the blow-up picture of \ref{['Intro:eq1']}
  4. Finite-energy nodal solutions of the Yamabe equation in $\mathbb{R}^n$
  5. Riemannian preliminaries
  6. Sign-changing families of bubbles in $M$
  7. The product $\hbox{Weyl}_g\otimes B$
  8. The mass of the Green's function in 3D
  9. A priori pointwise estimates for families of bubbles
  10. Statement of the results
  11. Proof of Proposition \ref{['prop:C0']}
  12. Proof of Theorems \ref{['maintheo']} and \ref{['maintheo2']}
  13. The case $n\geq 5$.
  14. The case $n=4$.
  15. The case $n=3$.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M^n,g), n \ge 5$ be a smooth, connected and closed manifold, let $h, (h_\varepsilon)_{0 < \varepsilon \le 1}\in C^1(M)$ such that $\lim_{\varepsilon\to 0}h_\varepsilon=h$ in $C^1(M)$ and $\Delta_g + h$ is coercive. Let $(p_\varepsilon)_{0< \varepsilon \le 1}\in (2, 2^\star]$ be such that $\lim We assume that $(u_\varepsilon)_{0< \varepsilon \le 1}$ satisfies for some $c>0$, where $B=(B_\var

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Proposition 3.1
  • ...and 18 more