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Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

Tobias König, Paul Laurain

TL;DR

This work provides a complete multibubble blow-up analysis for the Brezis–Nirenberg-type problem in dimension three, establishing precise global and local asymptotics for sequences of blowing-up positive solutions and revealing how multiple concentration points interact through the Green's function and the interaction matrix M_a. The authors derive a sharp blow-up rate under nondegeneracy or analyticity assumptions on the Robin function, and they construct higher-order corrections to the bubble profile to obtain refined expansions. A key innovation is treating the vanishing Robin function scenario (ρ_a(x0)=0) for multiple bubbles, yielding an explicit blow-up formula that generalizes prior single-bubble results. The work also clarifies regularity requirements for φ_a and ρ_a, providing conditions under which the analytic structure of the concentration set can be leveraged in the asymptotic analysis.

Abstract

For a smooth bounded domain $Ω\subset \mathbb R^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_ε$ to $-Δu_ε+ (a+εV) u_ε= u_ε^5$ on $Ω$ with zero Dirichlet boundary conditions, which blow up as $ε\to 0$. We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up, thereby obtaining a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension $N=3$.

Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

TL;DR

This work provides a complete multibubble blow-up analysis for the Brezis–Nirenberg-type problem in dimension three, establishing precise global and local asymptotics for sequences of blowing-up positive solutions and revealing how multiple concentration points interact through the Green's function and the interaction matrix M_a. The authors derive a sharp blow-up rate under nondegeneracy or analyticity assumptions on the Robin function, and they construct higher-order corrections to the bubble profile to obtain refined expansions. A key innovation is treating the vanishing Robin function scenario (ρ_a(x0)=0) for multiple bubbles, yielding an explicit blow-up formula that generalizes prior single-bubble results. The work also clarifies regularity requirements for φ_a and ρ_a, providing conditions under which the analytic structure of the concentration set can be leveraged in the asymptotic analysis.

Abstract

For a smooth bounded domain and smooth functions and , we consider the asymptotic behavior of a sequence of positive solutions to on with zero Dirichlet boundary conditions, which blow up as . We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up, thereby obtaining a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension .
Paper Structure (14 sections, 19 theorems, 163 equations)

This paper contains 14 sections, 19 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper.
  4. Notation
  5. Asymptotic analysis of $u_\varepsilon$
  6. A first quantitative bound
  7. A refined expansion
  8. Proof of Theorem \ref{['theorem multibubble']}
  9. The main expansions
  10. Properties of the matrix $M_a(\bm{x})$ and the eigenvector $\rho_a(\bm{x})$
  11. Proof of Theorem \ref{['theorem multibubble']}
  12. Regularity of $\phi_a$ and $\rho_a$
  13. A Liouville type result for the solutions of the linearized equation
  14. Some computations

Key Result

Theorem 2.1

Let $(u_\varepsilon)$ be a sequence of solutions to brezis peletier additive with $\|u_\varepsilon\|_\infty \to \infty$. Then there exists $n \in \mathbb{N}$ and $n$ sequences of points $x_{1,\varepsilon},...,x_{n,\varepsilon} \in \Omega$ such that $\mu_{i,\varepsilon}:= u_\varepsilon(x_{i,\varepsil

Theorems & Definitions (38)

  • Theorem 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • Lemma 4.2
  • Remark 4.3
  • ...and 28 more