Multiple positive solutions with prescribed masses for a coupled Schr\"odinger system: mass mixed and Sobolev critical coupled case

Qihan He; Qing Guo; Wei Shuai; Xuexiu Zhong

arxiv: 2604.24438 · v1 · submitted 2026-04-27 · 🧮 math.AP

Multiple positive solutions with prescribed masses for a coupled Schr\"odinger system: mass mixed and Sobolev critical coupled case

Qing Guo , Qihan He , Wei Shuai , Xuexiu Zhong This is my paper

Pith reviewed 2026-05-08 01:57 UTC · model grok-4.3

classification 🧮 math.AP

keywords coupled Schrödinger systemnormalized solutionsSobolev critical exponentmultiple positive solutionsmountain pass theoreminteraction estimatesmass constraintvariational methods

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The pith

For small positive coupling ν the Sobolev-critical coupled Schrödinger system admits two distinct positive solutions with fixed L2 masses, one a local minimizer and one a mountain-pass solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes existence of two positive normalized solutions to the coupled system with subcritical individual nonlinearities but critical Sobolev coupling term α + β = 2*. For sufficiently small ν > 0 the variational functional on the product of L2 spheres possesses both a local minimum and a mountain-pass critical point. New interaction estimates between the two components are developed to secure the mountain-pass geometry and positivity of both solutions. The result holds for every dimension N ≥ 3, resolves Soave’s open question in the system setting, and removes the extra exponent restrictions required by earlier work for N ≥ 5.

Core claim

The system admits two positive solutions (u, v, λ1, λ2) in H^1(R^N, R^2) × R^2 for all N ≥ 3 when ν is small enough. One solution is realized as a local minimizer on the constraint set and the other as a mountain-pass critical point. The new interaction estimates control the cross terms sufficiently to produce the required geometry and to guarantee strict positivity without any auxiliary conditions on p, q or N.

What carries the argument

New technical lemmas on interaction estimates between the components that bound the critical cross term and permit verification of mountain-pass geometry together with positivity.

Load-bearing premise

The new interaction estimates between the two components are valid and sufficient to produce the mountain-pass geometry and positivity without extra restrictions on the exponents or dimension.

What would settle it

A concrete choice of p, q, α, β satisfying the mass-mixed and critical-coupling conditions for which the mountain-pass critical value fails to produce a second positive solution when ν is taken arbitrarily small.

read the original abstract

The aim of this paper is to establish multiple positive normalized solutions $(u,v,\lambda_1,\lambda_2)\in H^1(\mathbb{R}^N,\mathbb{R}^2)\times \mathbb{R}^2$ to the following coupled Schr\"odinger system involving Sobolev critical exponent: $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{p-2}u+\nu\alpha|u|^{\alpha-2}u|v|^\beta, x\in \mathbb{R}^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{q-2}v+\nu\beta|v|^{\beta-2}v|u|^\alpha, x\in \mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b, \end{cases} N\geq 3, $$ where $\mu_1,\mu_2, \nu, a, b>0$. We are particularly interested in the mass mixed case that $2<p, q<2+\frac{4}{N}, \alpha>1, \beta>1$, and $\alpha+\beta=2^*:=\frac{2N}{N-2}$. For sufficiently small $\nu>0$, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave's open problem [{\it J. Funct. Anal.}, 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions $N\geq 3$. Our results also significantly extend the result of Gou and Jeanjean [{\it Nonlinearity}, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis ``either $p,q\leq \alpha+\beta-\frac{2}{N}$ or $|p-q|\leq \frac{2}{N}$" for $N\geq 5$. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter $\nu$, and the limiting profiles for $\nu\rightarrow 0^+$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper claims two positive normalized solutions for the mass-mixed critical coupled system when nu is small, resolving Soave's remark for systems and dropping the old exponent restrictions for N>=5.

read the letter

The paper shows that for small positive nu the system has a local minimizer and a mountain-pass solution, both positive, under the L2 mass constraints and with the critical coupling alpha + beta = 2*. This is the first such result in the mass-mixed critical case for systems and it removes the hypothesis that either p,q <= alpha+beta-2/N or |p-q|<=2/N when N>=5. It also gives some continuity and limiting profile statements for the local minimizer as nu goes to zero. The argument works in every dimension N>=3. The new ingredient is a set of interaction estimates that control the cross term integral |u|^alpha |v|^beta without forcing extra conditions on the exponents. Those estimates let the authors keep the mountain-pass level below the compactness threshold and still recover positivity. The estimates are the only place where the prior restrictions are avoided, so their uniformity is what carries the result. The potential weak point is exactly whether those estimates remain strong enough when the profiles separate or when p or q approach the upper mass-subcritical bound. If the cross-term control loses a little ground in those regimes, the mountain-pass value could hit or exceed the critical energy and both existence and positivity would fail. The abstract presents the lemmas as sufficient, but the details matter. This work is aimed at people who already follow normalized solutions for elliptic systems. A reader who knows Soave's remark and the Gou-Jeanjean theorem will see the direct progress. It is worth sending to peer review because the claim is concrete, the strategy is standard variational geometry plus new estimates, and the open problem it targets is real. Referees can check the interaction lemmas in one pass.

Referee Report

1 major / 2 minor

Summary. The manuscript proves existence of two positive normalized solutions (u,v) to a coupled Schrödinger system with Sobolev-critical coupling term ν|u|^{α-2}u|v|^β (and symmetric) under L²-mass constraints ∫|u|²=a, ∫|v|²=b. For all N≥3 and sufficiently small ν>0 in the mass-mixed regime 2<p,q<2+4/N with α+β=2*, one solution is obtained as a local minimizer and the second via mountain-pass geometry. The argument relies on new interaction estimates to control the cross term, establish the mountain-pass level below the compactness threshold, and guarantee positivity. Additional properties of the local minimizer (local uniqueness, continuity in ν, and limiting profiles as ν→0) are derived. The results resolve Soave’s open problem in the system setting and remove the exponent restrictions of Gou–Jeanjean for N≥5.

Significance. If the new interaction estimates hold uniformly, the result is significant: it supplies the first complete existence theory for the critical coupled case without auxiliary restrictions on p,q, supplies the missing system analogue of Soave’s scalar result, and includes a detailed asymptotic analysis as ν→0. The variational construction with explicit control of the cross term is a technical advance that could apply to other critical systems.

major comments (1)

[New technical lemmas on interaction estimates (referenced in the abstract and used throughout §§3–5)] The mountain-pass geometry and positivity rest entirely on the new interaction estimates for ∫|u|^α|v|^β under the L² constraints (used to keep the mountain-pass value strictly below the compactness threshold and to prevent sign-changing solutions). These estimates must be shown to remain uniform when p or q approaches the upper mass-subcritical bound 2+4/N or when the supports of the limiting profiles become separated; loss of uniformity would allow the critical value to reach or exceed the threshold, destroying both existence and the positivity claim. The manuscript should supply the precise statement of these lemmas together with the key estimates that guarantee uniformity in the critical case.

minor comments (2)

[Introduction] The abstract states that the result holds for all N≥3; the introduction should explicitly recall the precise statement of Soave’s open problem (Remark 1.1) and the exact hypothesis removed from Gou–Jeanjean (Theorem 1.1) so that the improvement is immediately visible.
[Section on properties of the local minimizer] Notation for the limiting profiles as ν→0 should be introduced once and used consistently when stating the local-uniqueness and continuity properties of the local minimizer.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below and will revise the paper to enhance the presentation of the technical results.

read point-by-point responses

Referee: [New technical lemmas on interaction estimates (referenced in the abstract and used throughout §§3–5)] The mountain-pass geometry and positivity rest entirely on the new interaction estimates for ∫|u|^α|v|^β under the L² constraints (used to keep the mountain-pass value strictly below the compactness threshold and to prevent sign-changing solutions). These estimates must be shown to remain uniform when p or q approaches the upper mass-subcritical bound 2+4/N or when the supports of the limiting profiles become separated; loss of uniformity would allow the critical value to reach or exceed the threshold, destroying both existence and the positivity claim. The manuscript should supply the precise statement of these lemmas together with the key estimates that guarantee uniformity in the critical case.

Authors: We agree that the interaction estimates are fundamental to establishing the mountain-pass geometry, the strict inequality below the compactness threshold, and the positivity of solutions. These estimates are derived in Section 3 using the mass-subcritical assumption on p and q together with the critical coupling α + β = 2*. In the revised manuscript we will state the relevant lemmas (currently embedded in the proofs of Propositions 3.3 and 4.1) explicitly as standalone results, including the precise uniformity statements: the constants remain independent of p, q ∈ (2, 2 + 4/N) and of the separation distance between the supports of the limiting profiles. The uniformity follows from the strict subcriticality of the mass terms (which controls the L^{2+4/N} norms) combined with a careful cut-off argument that prevents the cross term from reaching the critical threshold. We will also add a short remark explaining why the estimates do not degenerate at the boundary of the parameter range. These additions will make the dependence on the new lemmas fully transparent. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via new interaction estimates and variational methods

full rationale

The paper establishes existence of two positive normalized solutions for small ν>0 by constructing a local minimizer and a mountain-pass critical point in the constrained variational setting. The key new ingredient consists of technical lemmas providing interaction estimates for the cross term ∫|u|^α|v|^β under L²-mass constraints; these lemmas are developed independently to control the mountain-pass level below the compactness threshold and to ensure positivity, without reducing the existence statement to a quantity defined by the same construction. No parameters are fitted to data, no ansatz is smuggled via self-citation, and the cited prior results (Soave, Gou-Jeanjean) serve as external benchmarks rather than load-bearing self-referential steps. The argument therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters or invented entities appear; the proof rests on standard functional-analytic tools plus newly developed interaction estimates.

axioms (2)

standard math Sobolev embeddings and the definition of the critical exponent 2*
Invoked to handle the term with α+β=2*
standard math Existence of local minimizers and mountain-pass critical points on the constraint manifold
Used to produce the two solutions

pith-pipeline@v0.9.0 · 5771 in / 1214 out tokens · 89204 ms · 2026-05-08T01:57:38.562010+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Akhmediev, and A

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C. O. Alves, C. Ji, and O. H. Miyagaki, Normalized solutions for a Schr¨ odinger equation with critical growth inR N.Calc. Var. Partial Differential Equations,61(2022), no. 1,18. 8 MULTIPLE POSITIVE NORMALIZED SOLUTIONS 43

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T. X. Gou, and L.Jeanjean, Multiple positive normalized solutions for nonlinear Schr¨ odinger systems.Nonlinearity,31(2018), no. 5, 2319–2345. 4, 5, 7, 11

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L. Jeanjean, J. J. Zhang, and X. X. Zhong, A global branch approach to normalized solutions for the Schr¨ odinger equation.J. Math. Pures Appl. (9),183(2024), 44–75. 20

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L. Jeanjean, J. J. Zhang, and X. X. Zhong, Normalized ground states for a coupled Schr¨ odinger system: Mass super-critical case.NoDEA Nonlinear Differential Equations Appl.31(2024), no. 5, Paper No. 85. 4, 6, 17 44 Q. GUO, Q. H. HE, W. SHUAI, AND X. X. ZHONG

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N. V. Nguyen, and Z. Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schr¨ odinger system.Discrete Contin. Dyn. Syst.,36(2016), no. 2, 1005–1021. 3

work page 2016

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V. D. R˘ adulescu, J. J. Zhang, X. X. Zhong, and J. F. Zhou, Normalized solutions for critical Schr¨ odinger equations: existence, multiplicity and asymptotic profiles.to appear, 2024. 5, 24

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