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Existence and local uniqueness of multi-spike solutions for Br\'{e}zis-Nirenberg problem with prescribed mass

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Pith's one-line read The authors prove the existence and local uniqueness of multi-peak solutions with prescribed L2 mass for the critical Brézis-Nirenberg problem in dimensions at least six.

arxiv 2505.14168 v1 pith:3V3CTOPI submitted 2025-05-20 math.AP

Zongyan Lv , Xiaoyu Zeng , Huan-Song Zhou This is my paper · ORCID
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keywords solutionsomegaequationlocalmassproblemquadzis-nirenberg
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Many wave models in physics conserve the total amount of stuff, for example the number of atoms in a Bose-Einstein condensate or the power of a laser beam. When a solution has a prescribed total mass, the usual frequency in the equation becomes an unknown multiplier. This paper studies a critical nonlinear elliptic equation on a bounded container with such a mass constraint. The main question is whether the equation can have solutions that look like a sum of sharp peaks, called spikes, concentrating near selected points.

The authors show that if a geometric function built from the domain's Green's function has a nondegenerate critical point with a positive matrix, then for any small mass the problem admits a solution with k spikes. They also prove that among all solutions concentrating at the same limiting points, this k-spike solution is unique and, in favorable cases, can be counted exactly. The proof works by rescaling known fixed-frequency spike solutions so that their L2 norm matches the prescribed mass, and then carefully comparing two hypothetical normalized solutions through local Pohozaev identities and blow-up analysis. The new difficulty is that the two solutions may carry different multipliers, and the authors derive estimates that make the multiplier difference negligible.

The theorem is conditional on the existence of such a critical point, and the paper does not exhibit an explicit domain satisfying it.

Extended reading notes

Core claim

The central claim is Theorem 1.2: for N >= 6, if (a_k, mu_k) is a nondegenerate critical point of Psi_k with M_k(a_k) positive, then for all sufficiently small rho > 0 the normalized k-spike solution u_rho satisfying (1.6) and (1.7) is unique; moreover, if the nondegeneracy holds for every mu_k in Q_{a_k}, the number of solutions satisfying (1.5) is exactly the cardinality of Q_{a_k}. Theorem 1.1 establishes existence of such a solution under the same hypotheses.

Load-bearing premise

The uniqueness conclusion depends on Lemma 3.2, stated without proof, which asserts that for two normalized solutions concentrating at the same limiting points, the differences of their spike centers and scales are o(1/mu_bar^2). This rate is used in equations (3.9) and (3.10) to show that the two solutions agree up to o(1/mu_bar), and it feeds the limiting kernel analysis. If the rate were weaker, the expansion of the difference and the final contradiction argument would break. A second unresolved premise is that an admissible domain with a nondegenerate critical point of Psi_k actually exists; the paper never constructs one.

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Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof depends primarily on imported prior theorems and on a nondegeneracy hypothesis that is never instantiated. The two ad hoc assumptions, the unproved Lemma 3.2 and the admissibility condition, mark the points where a reader must trust either omitted details or an unexhibited domain.

assumptions (5)
  • domain assumption The fixed-frequency problem (1.8) has a k-spike solution branch u_lambda with the estimates and local uniqueness proved by Musso-Pistoia [32] and Cao-Luo-Peng [12].
    Used in Proposition 2.1 and throughout Section 2 to build the normalized solution by scaling; the present paper does not reprove these results.
  • ad hoc to paper The domain Omega is admissible: there exists (a_k, mu_k) in Q_{Omega,k} with M_k(a_k) positive and a nondegenerate critical point of Psi_k.
    This is Definition 1.1 and the hypothesis of Theorems 1.1 and 1.2. No explicit construction of such a domain is given in the paper.
  • ad hoc to paper Lemma 3.2: for two normalized solutions, |x^(1)_{j,rho}-x^(2)_{j,rho}| = o(1/mu_bar^2) and |mu^(1)_{j,rho}-mu^(2)_{j,rho}| = o(1/mu_bar^2).
    Stated without proof and asserted to follow as in [12, Props 3.2 and 3.3]. It is load-bearing for the difference estimates (3.9) and (3.10) and the subsequent kernel analysis.
  • standard math Standard elliptic estimates, Green's function asymptotics, and local Pohozaev identities for solutions of (1.1).
    Used in Sections 3 and 4; these are classical background facts in the field.
  • standard math The kernel of the linearized operator L(u) = -Delta u - ((N+2)/(N-2)) U^(4/(N-2)) u is spanned by the N+1 bubble derivatives psi_0, psi_1, ..., psi_N.
    Invoked in Lemmas 3.6 and 3.7 to classify the limiting difference profile.

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Pith. "Pith review of Existence and local uniqueness of multi-spike solutions for Br\'{e}zis-Nirenberg problem with prescribed mass." pith.science (2026). https://pith.science/paper/3V3CTOPI

@misc{pith2026250514168,
  author       = {Pith},
  title        = {Pith review of: Existence and local uniqueness of multi-spike solutions for Br\'ezis-Nirenberg problem with prescribed mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3V3CTOPI}},
  note         = {Machine review of arXiv:2505.14168}
}
abstract

In this paper, we consider the following Br\'{e}zis-Nirenberg problem with prescribed $ L^2$-norm (mass) constraint: \begin{equation*} \begin{cases} -\Delta u=|u|^{2^*-2} u +\lambda_\rho u\quad \text { in } \Omega, u>0, \quad u \in H_0^1(\Omega), \quad \int_{\Omega} u^2dx=\rho, \end{cases} \end{equation*} where $N \geqslant 6$, $2^*=2 N /(N-2)$ is the critical Sobolev exponent, $\rho>0$ is a given small constant and $\lambda_\rho>0$ acts as an Euler-Lagrange multiplier. For any $k\in \mathbb{R}^+$, we construct a $k$-spike solutions in some suitable bounded domain $\Omega$. Our results extend those in \cite{BHG3,DGY,SZ}, where the authors obtained one or two positive solutions corresponding to the (local) minimizer or mountain pass type critical point for the energy functional of above equation. Furthermore, using blow-up analysis and local Pohozaev identities arguments, we prove that the $k$-spike solutions are locally unique. Compared to the standard Br\'{e}zis-Nirenberg problem without the mass constraint, an additional difficulty arises in estimating the error caused by the differences in the Euler-Lagrange multipliers corresponding to different solutions. We overcome this difficulty by introducing novel observations and estimates related to the kernel of the linearized operators.

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