A refined blow-up analysis of the Brezis-Nirenberg equation and its application: The one-bubble case for $N\geq4$

Abstract: In this paper, we consider the famous Brezis-Nirenberg equation \begin{eqnarray*} \left{ \aligned &-Δu=λu+|u|^{\frac{4}{N-2}}u,\quad&\mbox{in}\,\, Ω,\ &u=0,\quad&\mbox{on}\,\, \partialΩ, \endaligned \right. \end{eqnarray*} where $N\geq3$ is the dimension, $Ω\subset\mathbb{R}<sup>N$ is a bounded domain with smooth boundary $\partialΩ$ and $λ&gt;0$ is a parameter. By developing a refined blow-up analysis based on the inverse reduction argument developed in \cite{WW2019,WW2019-2}, we classify, for the fist time, the Struwe decomposition of the Brezis-Nirenberg equation in the one-bubble case as the parameter $λ$ varies for $N\geq4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $λ\inσ(-Δ)$ in general bounded domains, where $σ(-Δ)$ is the spectrum of $-Δ$ in $H<sup>1_0(Ω)$. Our result completes the existence theory of the Brezis-Nirenberg equation for $N\geq4$ in \cite{AP2025,CFP1985,CFS,CSS1986,CW2005,CSZ2012,SWW2009,TYZ2022} since 1984.

• The paper presents a comprehensive classification of one-bubble blow-up phenomena for sign-changing solutions as the spectral parameter approaches critical values.
• The refined inverse reduction and orthogonal decomposition yield explicit asymptotic rates and energy estimates, particularly distinguishing cases in N=4.
• Methodological innovations directly prove the existence of least-energy solutions at spectral values, resolving longstanding open problems in critical Sobolev settings.

Refined Blow-Up Analysis and Struwe Decomposition for the Brezis-Nirenberg Equation ($N \geq 4$): The One-Bubble Case

Introduction and Context

This work presents a detailed and technically sophisticated blow-up analysis of the Brezis-Nirenberg equation in bounded smooth domains for dimension $N \geq 4$, with a focus on the "one-bubble" regime. It leverages and refines the method of inverse reduction to offer a complete classification of the formation of singularities (blow-up phenomena), specifically the so-called Struwe decomposition, in the setting where the nontrivial solution's mass concentrates at a single point as the parameter $\lambda$ approaches critical values. The analysis resolves several longstanding open questions in the study of sign-changing solutions, and in particular, delivers the first comprehensive existence result for least-energy sign-changing solutions at eigenvalues of $-\Delta$ in dimension $N=4$.

Theoretical Framework and Problem Statement

The Brezis-Nirenberg equation under analysis is

$$-\Delta u = \lambda u + |u|^{\frac{4}{N-2}}u \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega,$$

where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, $N \geq 4$, and $\lambda > 0$. The equation constitutes a paradigmatic critical exponent problem closely related to the Yamabe problem, and exhibits lack of compactness in the embedding $H_0^1(\Omega) \subset L^{2^*}(\Omega)$ ($N \geq 4$0).

Decades of work have delineated the scenarios for positive solutions, but the existence, multiplicity, and blow-up behavior of sign-changing solutions—especially for values of $N \geq 4$1 at the spectrum of $N \geq 4$2—remained incomplete, particularly in non-generic domains.

Methodological Innovations

Central to this work is a multi-layered, refined asymptotic expansion—substantially advancing beyond the classical Struwe decomposition—tailored to isolate and classify the potential blow-up scenarios for sequences of solutions $N \geq 4$3 as $N \geq 4$4. The major analytical tools are:

• Inverse Reduction Framework: Extends the finite-dimensional reduction by isolating the kernel components as $N \geq 4$5 approaches spectral values.
• Precise Orthogonal Decomposition: Decomposes the solution into three regimes: a bubble profile $N \geq 4$6 centered at $N \geq 4$7 matching the Aubin-Talenti solution, an eigenfunction component, and a refined remainder orthogonal to both.
• Successive Expansion of Orthogonality Conditions: By projecting on both the bubble's kernel and the involved eigenfunctions, higher-order interaction terms are computed, enabling a full asymptotic description up to second and, in degenerate cases, third order terms.

These enable the extraction of sharp asymptotics for the blow-up rate $N \geq 4$8, the eigenmode amplitudes, and the remainder, with dimension-dependent distinctions ($N \geq 4$9, $\lambda$0, $\lambda$1).

Core Analytical Results

The main analytical achievements can be summarized as follows:

1. Classification of One-Bubble Blow-Up: The paper establishes a complete taxonomy for the possible limiting behaviors of sign-changing solutions as $\lambda$2, an eigenvalue of the Dirichlet Laplacian:

• For $\lambda$3, blow-up can occur, but only at points where eigenfunctions exhibit a singularity (i.e., a critical point with both function and gradient vanishing).
• For $\lambda$4, one-bubble blow-up is proved to be impossible.

1. Sharp Asymptotics: In $\lambda$5, two distinct asymptotic regimes are distinguished depending on whether the concentration occurs at an interior point or near the boundary. For both cases, explicit rates for $\lambda$6 and the amplitude of the sign-changing component are derived.
2. Resolution of Degenerate Cases: By further refining the inverse reduction, the analysis covers degenerate expansions where leading-order terms cancel. The resulting criteria involve nondegeneracy of certain matrices (Hessians of associated functionals restricted to the eigenspaces), leading to a condition equivalent to the non-degeneracy of the blow-up profile in the nonlinear setting.
3. Existence at Spectral Values — Completion of the Existence Theory: The refined blow-up analysis leads to a positive resolution of the open problem left since \cite{CFS}, \cite{CSS1986}: for $\lambda$7, least energy sign-changing solutions exist for all $\lambda$8, regardless of multiplicity or domain topology. The existence at $\lambda$9 was previously only guaranteed by indirect arguments; the present work gives a direct, variational-analytic proof.

Numerical and Structural Highlights

• Strong Monotonicity: The least energy for the sign-changing solution branch is shown to be strictly decreasing as a function of $-\Delta$0, with precise asymptotic values at spectrum endpoints.
• Sharp Lower Bound: Energy estimates confirm that least energy solutions at eigenvalues satisfy $-\Delta$1 as $-\Delta$2, matching the critical Sobolev constant.

Implications and Future Research Directions

Theoretical Implications

This refined classification of blow-up not only answers a specific existence question but provides the backbone for further study of compactness in critical exponent problems. The analytical infrastructure developed is adaptable to other critical equations, e.g., Yamabe-type equations on manifolds, Schrödinger equations with critical nonlinearity, and more.

Practical Implications

From an applied perspective, the existence of sign-changing least-energy states at all spectral values underpins the diversity of solution landscapes in nonlinear elliptic PDEs, impacting computational studies and bifurcation analyses in geometric analysis, physics, and engineering domains reliant on critical point theory.

Prospective Lines of Inquiry

• Extension to Systems and Higher Multiplicity Blow-Up: The methods appear adaptable to multi-bubble configurations, including systems of equations and settings with more complex domain/topology (including Riemannian manifolds).
• Stability and Uniqueness: Exploration of the stability, Morse index, and possibly uniqueness (modulo symmetries) of the least energy sign-changing states constructed at spectral values.
• Bifurcation Structure and Dynamics: Insights into how solution branches extend and turn as $-\Delta$3 moves along and through the spectrum, potentially informing bifurcation and blow-up diagrams.

Conclusion

This work provides a definitive analysis of one-bubble sign-changing blow-up for the Brezis-Nirenberg equation in dimensions $-\Delta$4, and, in doing so, settles open existence problems for $-\Delta$5 at spectral values of $-\Delta$6. The technical approach, fusing refined asymptotics, blow-up analysis, and variational characterization, establishes a new standard for the depth of singularity classification in critical Sobolev problems and opens several promising avenues for future research on nonlinear elliptic equations.