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On Brezis' First Open Problem: A Complete Solution

Liming Sun, Juncheng Wei, Wen Yang

TL;DR

The work resolves Brezis' Open Problem 1.1 in three dimensions by constructing infinitely many sign-changing solutions to the BN problem in the unit ball for any $\lambda>0$, employing a novel necklace of inverted crown bubbles rooted in Yamabe sign-changing solutions. Central to the approach are precise properties of the crown solution (non-degenerate, smooth nodal set) and a sophisticated inner–outer gluing scheme that translates an infinite-dimensional PDE problem into a finite-dimensional energy minimization over a carefully balanced parameter space. The authors develop an intricate energy expansion, manage long-range bubble interactions via contour-integral techniques for specific series, and perform a detailed multi-parameter analysis to locate an interior minimum, yielding a nontrivial, sign-changing solution with high Morse index. Their framework also provides a generalized Brezis–Open Problem result for strictly star-shaped 3D domains and offers a robust blueprint for tackling other critical-exponent elliptic problems using sign-changing building blocks and gluing methods.

Abstract

In 2023, H.\,Brezis published a list of his ``favorite open problems", which he described as challenges he had ``raised throughout his career and has resisted so far". We provide a complete resolution to the first one--Open Problem 1.1--in Brezis's favorite open problems list: the existence of solutions to the long-standing Brezis-Nirenberg problem on a three-dimensional ball. Furthermore, using the building blocks of Del Pino-Musso-Pacard-Pistoia sign-changing solutions to the Yamabe problem, we establish the existence of infinitely many sign-changing, nonradial solutions for the full range of the parameter.

On Brezis' First Open Problem: A Complete Solution

TL;DR

The work resolves Brezis' Open Problem 1.1 in three dimensions by constructing infinitely many sign-changing solutions to the BN problem in the unit ball for any , employing a novel necklace of inverted crown bubbles rooted in Yamabe sign-changing solutions. Central to the approach are precise properties of the crown solution (non-degenerate, smooth nodal set) and a sophisticated inner–outer gluing scheme that translates an infinite-dimensional PDE problem into a finite-dimensional energy minimization over a carefully balanced parameter space. The authors develop an intricate energy expansion, manage long-range bubble interactions via contour-integral techniques for specific series, and perform a detailed multi-parameter analysis to locate an interior minimum, yielding a nontrivial, sign-changing solution with high Morse index. Their framework also provides a generalized Brezis–Open Problem result for strictly star-shaped 3D domains and offers a robust blueprint for tackling other critical-exponent elliptic problems using sign-changing building blocks and gluing methods.

Abstract

In 2023, H.\,Brezis published a list of his ``favorite open problems", which he described as challenges he had ``raised throughout his career and has resisted so far". We provide a complete resolution to the first one--Open Problem 1.1--in Brezis's favorite open problems list: the existence of solutions to the long-standing Brezis-Nirenberg problem on a three-dimensional ball. Furthermore, using the building blocks of Del Pino-Musso-Pacard-Pistoia sign-changing solutions to the Yamabe problem, we establish the existence of infinitely many sign-changing, nonradial solutions for the full range of the parameter.

Paper Structure

This paper contains 18 sections, 33 theorems, 486 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Nodal solutions to Yamabe problem and the non-degeneracy
  4. Sketch of Proofs.
  5. On the Del Pino-Musso-Pacard-Pistoia solution
  6. Estimates of $\psi$
  7. On the nodal set of Del Pino-Musso-Pacard-Pistoia's solution
  8. Non-degeneracy
  9. Preliminary on Approximate Solutions
  10. Energy expansion
  11. Gluing Procedure
  12. The linear and nonlinear problem
  13. The finite-dimensional reduction and the critical point
  14. Critical computations
  15. Estimates on Type I
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume that Then there are infinitely many (sign-changing) solutions to (main-eqn).

Figures (3)

  • Figure 1: Left picture is the region $\Sigma_K$ and its two boundaries inside $B_1$. Right one is the necklace solution.
  • Figure 2: The left one is the nodal sets of $U_*$ inside the cube $[-5/2,5/2]^3$ when $m=16$ and $d_m=\sqrt{2}m\log m\left(\sum_{j=1}^{m-1}\csc \frac{j\pi}{m}\right)^{-1}$. The right one is the intersection of nodal sets of $q_m$ with $z_1z_2$-plane.
  • Figure 3: The illustration of the integration contour in Lemma \ref{['S-integral']}.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['th2.qm-nodal']}.
  • ...and 58 more