Normalized solutions on large smooth domains to the Schr\"{o}dinger equations with potential and combined nonlinearities: The Sobolev critical case

Xiaolu Lin; Yanjun Liu; Zongyan Lv

REVIEW 3 major objections 5 minor 42 references

Normalized solutions on large smooth domains to the Schr\"{o}dinger equations with potential and combined nonlinearities: The Sobolev critical case

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper establishes that a nonlinear Schrödinger equation with an external potential and both subcritical and Sobolev-critical power nonlinearities admits solutions with a prescribed L² mass on large star-shaped domains, and that these…

desk verdict Genuinely novel critical-case program, but the compactness proof leans on a false Liouville step and a sign error, so the main existence claims do not hold as written. read the letter →

arxiv 2412.03296 v2 pith:3U6T55J4 submitted 2024-12-04 math.AP

Xiaolu Lin , Yanjun Liu , Zongyan Lv This is my paper · ORCID

classification math.AP MSC 35J6035J2035R25

keywords NormalizedsolutionsprescribedmassSobolevcriticalexponentcombinednonlinearitiespotentiallargesmoothdomainsmountainpasscompactness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that a nonlinear Schrödinger equation with an external potential and both subcritical and Sobolev-critical power nonlinearities admits solutions with a prescribed L² mass on large star-shaped domains, and that these solutions persist as the domain expands to all of R^N. Because the potential makes the usual Pohozaev-manifold method unavailable, the authors work directly on the mass sphere and control the energy by an explicit mass threshold: for small mass the energy stays below the critical bubble level, which restores compactness. For β>0 they find a positive local-minimum solution with negative energy; for β>0 and for β≤0 they find positive mountain-pass solutions with positive energy. The solution families are uniformly bounded in L∞ and have Lagrange multiplier λ bounded away from zero as the radius tends to infinity, which is the step that yields whole-space solutions.

What carries the argument

The central object is the mass-threshold function h(t) (and its descendants f, g) that bounds the constrained energy from below in terms of ‖∇u‖². The threshold α_V is chosen so that h has a positive global maximum with two roots R1<R2, which localizes the minimization set B_{r,α} and keeps its energy negative. In the mountain-pass arguments, the same kind of bound shows the min-max level m_{r,s}(α) lies between a positive value and the critical bubble threshold (1/N)$s^{{(2-N)/2}}$$S^{{N/2}}$, so a Palais–Smale sequence from the monotonicity trick cannot lose a bubble and converges. Lemma 3.3 then supplies uniform L∞ bounds via a blow-up argument that invokes Liouville-type non-existence on R^N or a half-space. The bubbles themselves are the explicit family U_ε(x) = (ε/(ε²+|x|²))^{(N-2)/2}.

What would settle it

Evaluate Lemma 3.3's Liouville step: for the blow-up limit Σ=R^N, the lemma claims the only nonnegative solution of -Δω = $ω^{{2*-1}}$ is ω≡0, but substituting the explicit bubble U_ε (a positive solution with ω(0)=1 after rescaling) into equation (3.6) with V≡0 gives a valid nonzero limit, so the lemma's conclusion cannot be true; hence the uniform L∞ bound, and with it the whole-space passage, is not established by this proof.

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Extended reading notes

Core claim

The paper's central claim is that equation (1.1), with q=2*, has normalized solutions of mass α on the scaled star-shaped domain Ω_r whenever α lies below a threshold built from the Sobolev constant S, the Gagliardo-Nirenberg constant, the mass-subcritical exponent p, and the $L^{{N/2}}$ norm of the negative part of V. For β>0, the energy functional restricted to the mass sphere has a local minimum inside the sublevel ball B_{r,α} = {‖∇u‖²≤T_α²}, giving a positive solution with negative energy; crossing from one side of the mountain yields a second, positive-energy solution. For β≤0 the same crossing argument works with the critical term alone driving the mountain pass. What makes the critical exponent tractable is the observation that any loss of compactness costs at least $S^{{N/2}}$/N in energy; the thresholds α_V, ᾱ_V, α1, α2 are chosen so that all energy levels considered lie strictly below that cost, so Palais–Smale sequences converge strongly. Expanding r→∞ then gives normalized solutions on all of R^N.

Load-bearing premise

The uniform bounds that make the r→∞ limit work assume that the rescaled blow-up limit has no positive solution in R^N (or in a half-space) and that the potential term vanishes at the concentration scale; if the potential does not vanish at the blow-up point, that assumption fails and the boundedness estimates can collapse.

Editorial extensions

If this is right

If the theorems hold, equation (1.1) on R^N has positive normalized solutions of both local-minimum and mountain-pass type for every sufficiently small mass α, obtained as limits of the domain solutions as r→∞.
The explicit thresholds (α_V, ᾱ_V, α1, α2) express the smallness condition on the mass in terms of V, p, N, and the Sobolev constant, giving a quantitative answer to the open problem raised in Bartsch et al.
The uniform L∞ bounds and liminf λ>0 mean these whole-space limits are genuine localized waves, not bubbling peaks escaping to infinity.
For β≤0 the mountain-pass result extends the known autonomous case to potentials, showing the focusing critical term can be stabilized by small mass even when the subcritical term is defocusing.
The method avoids the Pohozaev manifold entirely, so it can be applied to bounded domains and to potentials where the Pohozaev identity gives no constraint.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the blow-up analysis ignores the potential at the concentration scale, a natural testable extension is whether the uniform bounds persist for non-vanishing potentials, e.g., V(0)≠0; the scaling in Lemmas 4.2 and 5.2 suggests they may need a corrected potential term or a stronger smallness condition.
The threshold structure suggests a possible dichotomy: for masses above α_V, either no normalized solution exists or the energy reaches the bubble level and compactness is lost, so solutions would concentrate and the whole-space limit would fail; checking the autonomous case V≡0 against Soave's results could reveal whether the thresholds are sharp or merely sufficient.
The same energy-below-bubble-threshold strategy could transfer to normalized solutions for systems, to magnetic Schrödinger operators, or to problems on domains with different symmetries, wherever the Pohozaev constraint is unavailable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is the first attack on the Sobolev-critical normalized problem with a nonconstant potential on large domains, and that is a real gap in the literature. But the central compactness machinery has three load-bearing errors. I would not trust the main theorems as written.

What is new and good: the paper correctly identifies the Bartsch–Qi–Zou subcritical result and extends it to q = 2*, which is not a restatement. The local-minimum existence for β > 0 and small mass (Theorem 1.1(i)) is plausibly correct: the lower bound on E_r on B_{r,α}, the Ekeland argument, and the energy gap excluding a bubble are standard and the signs there check out. The overall variational setup is honest and the exposition is readable.

Where it breaks: Lemma 3.3 is the load-bearing uniform L^∞ estimate. The blow-up rescaling at a maximum leads to a limit solving -Δω = |ω|^{2*-2}ω on R^N. The proof invokes a Liouville theorem to get ω ≡ 0 and a contradiction with ω(0) = 1. No such theorem exists: the Aubin–Talenti bubble is a positive solution. So the contradiction fails, and every result that relies on uniform L^∞ (parts (ii) of 1.1, (i) of 1.2, (i) of 1.4, whole-space limits) is unsupported. The lower bound on λ_r in Theorem 3.2 does not repair this.

Lemma 5.2 has a sign error. For β ≤ 0 the term -β/p ∫|v_ε|^p is nonnegative, but the proof treats it as a negative contribution and concludes the mountain-pass level is below S^{N/2}/N. Also the potential term ∫ V v_ε^2 is O(1) (roughly α V(0)) for continuous V, not O(ε^{N-2}), so the upper bound in Lemma 5.2 (and Lemma 4.2) is not valid. For N = 3 the proof is not even completed. These are not cosmetic gaps; they are exactly the estimates needed to rule out bubbles in the mountain-pass concentration argument.

Net: the claimed theorems are not established. The β > 0 local minimum part may survive, and the question itself deserves attention. But the paper needs a different compactness argument, not copy-editing. I would still send it to a serious referee—this is a known-hard problem and the authors are aiming at the right target—but I would expect heavy revision or a substantially weakened set of results.

Referee Report

3 major / 5 minor

Summary. The manuscript studies prescribed-mass (normalized) solutions to the nonlinear Schrödinger equation with a potential and combined subcritical/critical power nonlinearities, on large smooth star-shaped bounded domains Ω_r and then in the whole space R^N. The main results (Theorems 1.1, 1.2, 1.4) assert existence of local-minimum and mountain-pass type solutions for small mass, uniform L^∞ bounds as r→∞, and strictly positive Lagrange multipliers in the limit, with different signs of β. The proof strategy uses energy estimates on the constraint manifold, the monotonicity trick of Jeanjean, and a blow-up argument to rule out concentration. The central compactness mechanism is Lemma 3.3, which claims a uniform L^∞ bound by rescaling at a maximum point and invoking a Liouville theorem for the critical equation in R^N.

Significance. If correct, the paper would extend the recent work of Bartsch–Qi–Zou to the Sobolev critical case and would provide the first normalized-solution results with critical growth on large bounded domains with a nonconstant potential, including a passage to the whole space. The local-minimum existence for β>0 (Theorem 1.1(i)) appears to be a plausible and potentially useful contribution. However, the mountain-pass existence results and the uniform bounds that support the asymptotic and whole-space statements rest on several load-bearing estimates that contain mathematical errors. The false Liouville step and the sign error in the β≤0 estimate are not cosmetic; they invalidate the proofs of Theorems 1.2 and 1.4 as written. The significance of the paper would be high if the gaps could be repaired, but in its current form the central package is not established.

major comments (3)

[Section 3, Lemma 3.3] The uniform L^∞ bound is proved by a blow-up argument that invokes a Liouville theorem for the limit equation -Δω = |ω|^{2*-2}ω in Σ. When Σ = R^N, the text says 'from the Liouville theorem, there holds ω=0 in Σ,' contradicting ω(0)=1. This is false: the Aubin–Talenti bubble U_c(x) = c(1+|x|^2)^{-(N-2)/2}, with c=(N(N-2))^{(N-2)/4}, is a positive smooth solution of exactly that equation in R^N. Thus the claimed contradiction does not follow. This step is load-bearing for the uniform L^∞ bounds in Theorems 1.1(ii), 1.2(i), and 1.4(i), and indirectly for the passage to whole-space limits. The energy-based lower bound on λ_r in Theorem 3.2 does not repair this gap because the uniform bound is needed for the solution family itself.
[Section 5, Lemma 5.2] In the estimate for m_{r,s}(α), the term -β/p T_1^{N(p-2)/2} ∫|v_ε|^p dx is nonnegative when β≤0, since -β ≥ 0. However, in Cases 1 and 2 the proof treats this term as a negative contribution and concludes that the whole expression is < 0. This is a sign error. Consequently the strict upper bound m_{r,s}(α) < (1/N) s^{(2-N)/2} S^{N/2} is not established, and the exclusion of bubble concentration in Theorem 5.3 fails.
[Section 4, Lemma 4.2 and Section 5, Lemma 5.2] The potential term (1/2)∫_Ω V(x) v_ε^2 dx is not higher-order in ε at the bubble scale. Since v_ε = (√α/||u_ε||_2) U_ε and ||U_ε||_2^2 = C ε^2 (for N≥5), while ∫ V U_ε^2 dx ~ V(0) ||U_ε||_2^2 when V(0)≠0, this term tends to (α/2)V(0) as ε→0, which is O(1), not O(ε^{N-2}). In Lemma 4.2 this term is absorbed into O(ε^{N-2}) without justification, and in Lemma 5.2 the attempted cancellation via the β term has the wrong sign (see the previous comment). The upper-bound estimates for the mountain-pass levels in Theorems 1.2 and 1.4 are therefore not justified.

minor comments (5)

[Throughout] There are numerous typos and notation issues, e.g., 'Pincaré' should be 'Poincaré', 'Brezis-Lieb' should be 'Brézis-Lieb', and the Abstract contains 'Besides, Our study' with an improper capital letter.
[Section 2, Theorem 2.2] The statement of the monotonicity trick contains a garbled line '֒→֒→E′' that appears to be a formatting error; the intended embedding E into H should be stated correctly.
[Section 3, Lemma 3.3] The proof states 'lim inf_{r→∞} dist(x_r, ∂Ω_r)/τ_r > 0' and says it follows from a standard direct method ('So we omit it here'). Since this claim is used to identify the limit domain Σ, it should be either proved or given a precise reference.
[Section 5, Lemma 5.2] The proof refers to 'the proof of (iii)' but the Lemma has no item (iii); this appears to be a remnant from an earlier version and should be corrected.
[Section 4, Lemma 4.2] After defining the path γ(τ), the proof uses γ0(t) and max_{t≥0} J_{r,s}(v_t); the notation for the path and the variable t should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation is self-contained and no claim reduces to its own inputs; the main risk is a mathematical gap in Lemma 3.3, not circular reasoning.

full rationale

The paper's derivation chain does not reduce any claim to its own inputs. The existence results and uniform bounds are obtained from variational minimax arguments, energy estimates, Sobolev and Gagliardo-Nirenberg inequalities, and a blow-up/Liouville compactness argument. No parameter is fitted to the target conclusions, and no load-bearing premise is justified only by a self-citation: the cited prior works (Bartsch-Qi-Zou, Jeanjean-Le, Soave, etc.) are by other authors and are used as external tools or as inspiration. The skeptical concern that Lemma 3.3 invokes a Liouville theorem that is false for the critical equation in R^N, since the Aubin-Talenti bubble is a positive solution, is a substantive mathematical correctness issue in the compactness proof, but it is not circularity: the claimed theorem is not assumed in the hypotheses, and the contradiction derivation is simply invalid rather than definitionally forced. Accordingly, no circular step is identified, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted data or invented entities; the paper is a pure existence proof. The load-bearing background inputs are standard inequalities plus two questionable assumptions: the Liouville nonexistence in R^N and the neglect of O(1) potential terms at bubble scale.

assumptions (7)

domain assumption Ω is a bounded smooth star-shaped domain containing 0; Ω_r = rΩ
Used throughout: Poincaré inequality with eigenvalue θ/r², boundary sign x·n≥0 in Pohozaev identities, and the rescaling construction of test functions.
domain assumption Potential condition (V0): V∈C(R^N)∩L^{N/2}(R^N) bounded, ‖V_-‖_{N/2}<S
Underlies the coercivity lower bounds (3.3), (4.5) and (5.5).
domain assumption For Theorems 1.2 and 1.4, V is C^1 and \tilde V(x)=∇V(x)·x is bounded
Needed in Pohozaev identity (4.13) to control the term ∫\tilde V u² uniformly.
standard math Gagliardo-Nirenberg inequality (Lemma 2.1) and Sobolev inequality with sharp constant S
Used to derive the h(t) and f(t) lower bounds for the energy on the sphere.
standard math Monotonicity trick (Theorem 2.2) yields bounded Palais-Smale sequences for almost every parameter s
Provides the bounded Palais-Smale sequences used in Sections 4 and 5.
standard math Pohozaev identity for the critical equation in bounded star-shaped domains, formula (4.13)
Gives the uniform H1 bounds in Lemmas 4.4 and 5.4; boundary term is nonnegative due to star-shape.
ad hoc to paper No nonzero nonnegative solution of -Δω=ω^{2*-1} in R^N with ω(0)=1 (invoked as a Liouville theorem in Lemma 3.3)
False: the standard bubble (1+|x|²)^{-(N-2)/2} is a positive solution with value 1 at the origin; this invalidates the L∞ bound proof.

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Cite this review

Pith. "Pith review of Normalized solutions on large smooth domains to the Schr\"{o}dinger equations with potential and combined nonlinearities: The Sobolev critical case." pith.science (2026). https://pith.science/paper/3U6T55J4

@misc{pith2026241203296,
  author       = {Pith},
  title        = {Pith review of: Normalized solutions on large smooth domains to the Schr\"odinger equations with potential and combined nonlinearities: The Sobolev critical case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3U6T55J4}},
  note         = {Machine review of arXiv:2412.03296}
}

read the original abstract

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrodinger equations with mixed nonlinearities. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential. Besides, Our study can be regarded as a Sobolev critical case complement of Bartsch-Qi-Zou (Math Ann 390, 4813-4859, 2024), which has addressed an open problem raised in Bartsch et al.(Commun Partial Differ Equ 46(9):1729--1756,2021).

Discussion (0). Continue with ORCID to comment.

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Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Akahori,T., Ibrahim, S., Ikoma,N., Kikuchi,H.; Uniqueness and nondeg eneracy of ground states to nonlinear scalar ﬁeld equations involving the Sobolev critical exponen t in their nonlinearities for high frequencies, Calc. Var. Partial Diﬀer. Equ., 58 (2019), 120

2019

[2] [2]

Alves, C.O., Ji, C.; Normalized solutions for the Schr¨ odinger equat ions with L2-subcritical growth and diﬀerent types of potentials, J. Geom. Anal., 32 (2022), Paper No. 165, 25pp

2022

[3] [3]

Alves, C.O., Thin, N.V.; On existence of multiple normalized solutions to a class of elliptic problems in whole RN via Lusternik-Schnirelmann category, SIAM J. Math. Anal., 55(2) (2023) 1264–1283

2023

[4] [4]

(French)

Aubin, T.: Probl` emes isop´ erim´ etriques et espaces de Sobolev. (French). J. Diﬀer. Geom. 11(4), 573– 598 (1976) 26 XIAOLU LIN, YANJUN LIU, AND ZONGYAN L V

work page 1976

[5] [5]

Science 269, 198–201 (1995)

Anderson, M., Ensher, J., Matthews, M., Wieman, C., Cornell, E.: Ob servation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995)

work page 1995

[6] [6]

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schr¨ odinger systems

Bartsch, T.; Li, H.W.; Zou, W.M. Existence and asymptotic behavior of normalized ground states for Sobolev critical Schr¨ odinger systems. Calc. Var. Partial Diﬀer. E qu., 62(1) Paper No. 9, 34 pp,(2023)

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Normalized solutions to Schr¨ odinger e quations with potential and inhomogeneous nonlinearities on large smooth domains

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