Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

Tobias König, Paul Laurain

TL;DR

The paper analyzes blow-up phenomena for a higher-dimensional Brezis–Nirenberg-type equation with a negative perturbation V on a bounded domain. It develops a comprehensive multi-bubble framework: solutions concentrating at finitely many points with bubbles modeled by Aubin–Talenti profiles, governed by a Green-function–based interaction matrix M and a Robin function φ. By deriving sharp a priori bounds for remainders and precise energy expansions, the authors reduce the problem to a finite-dimensional variational system for a reduced energy F(κ,x) (and its N=4 analogue), yielding existence, location, and speeds of concentration points, and characterizing degenerate cases where improved rates occur. The results extend the blow-up analysis to N≥4, complementing previous work in N=3, and provide a detailed description of the complete blow-up picture in the Brezis–Nirenberg setting.

Abstract

For a bounded set $Ω\subset \mathbb R^N$ and a perturbation $V \in C^1(\overlineΩ)$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -Δu_ε+ εV = N(N-2) u_ε^\frac{N+2}{N-2} \] for dimensions $N \geq 4$, which are non-critical in the sense of the Brezis--Nirenberg problem. For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of $-Δ$ on $Ω$. Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension $N = 3$, gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework.

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

TL;DR

The paper analyzes blow-up phenomena for a higher-dimensional Brezis–Nirenberg-type equation with a negative perturbation V on a bounded domain. It develops a comprehensive multi-bubble framework: solutions concentrating at finitely many points with bubbles modeled by Aubin–Talenti profiles, governed by a Green-function–based interaction matrix M and a Robin function φ. By deriving sharp a priori bounds for remainders and precise energy expansions, the authors reduce the problem to a finite-dimensional variational system for a reduced energy F(κ,x) (and its N=4 analogue), yielding existence, location, and speeds of concentration points, and characterizing degenerate cases where improved rates occur. The results extend the blow-up analysis to N≥4, complementing previous work in N=3, and provide a detailed description of the complete blow-up picture in the Brezis–Nirenberg setting.

Abstract

For a bounded set and a perturbation , we analyze the concentration behavior of a blow-up sequence of positive solutions to for dimensions , which are non-critical in the sense of the Brezis--Nirenberg problem. For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of on . Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension , gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework.
Paper Structure (16 sections, 19 theorems, 327 equations)

This paper contains 16 sections, 19 theorems, 327 equations.

Table of Contents

  1. Introduction and main results
  2. Main result
  3. Asymptotic analysis
  4. The bound on $r_{i,\varepsilon}$
  5. The bound on $q_{i,\varepsilon}$
  6. The bound on $p_{i,\varepsilon}$
  7. The main expansions
  8. Proof of Theorem \ref{['theorem multibubble']}
  9. Some computations
  10. Classical Asymptotic analysis
  11. Proof of Proposition \ref{['estim']}
  12. Proof of Proposition \ref{['proposition preliminaries multi bis']}
  13. Necessity of coercivity
  14. Harnack's inequality
  15. General Pohožaev's identities
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(u_\varepsilon)$ be a sequence of solutions to brezis peletier additive, with $V \in C^1(\overline{\Omega})$ and $V < 0$ on $\overline{\Omega}$, such that $\|u_\varepsilon\|_\infty \to \infty$. Then there exists $n \in \mathbb{N}$ and $n$ sequences of points $x_{1,\varepsilon},...,x_{n,\varepsi

Theorems & Definitions (40)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 30 more