Normalized Solutions on large smooth domains to the Schr\"{o}dinger equation with potential and general nonlinearity: Mass super-critical case

Xiaolu Lin; Zongyan Lv

REVIEW 3 major objections 5 minor 1 cited by

Normalized Solutions on large smooth domains to the Schr\"{o}dinger equation with potential and general nonlinearity: Mass super-critical case

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Pith's one-line read This paper proves that the mass-constrained nonlinear Schrödinger equation with a nonconstant potential admits positive normalized mountain-pass solutions on every sufficiently large star-shaped domain, and, for small masses, on all of…

desk verdict Plausible within-field extension of normalized NLS results to large domains with a potential, but Lemma 3.1's scaling is wrong for N≠3 and needs fixing before the main theorem holds. read the letter →

arxiv 2501.04893 v1 pith:BV2HUYK5 submitted 2025-01-06 math.AP

Xiaolu Lin , Zongyan Lv This is my paper · ORCID

classification math.AP MSC 35J6035J2035R25

keywords NormalizedsolutionsMasssupercriticalNonconstantpotentialLargesmoothdomainsStar-shapeddomainMountainpasssolutionConcentrationcompactnessNonlinearSchrödingerequation

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 existence of solutions with prescribed $L^2$ mass for the nonlinear Schrödinger equation $-\Delta u+V(x)u+\lambda u=g(u)$ when the nonlinearity is mass-supercritical and $V$ is nonconstant. Because the potential breaks the Pohozaev-manifold reduction used in the autonomous case, the authors work directly on the mass sphere in large star-shaped domains, using the monotonicity trick and a Pohozaev-type identity adapted to the potential. They prove that for every $c>0$ there is a radius $r_c$ such that the problem on every rescaled domain $\Omega_r$ with $r>r_c$ has a positive mountain-pass solution with positive energy. Under a quantitative smallness condition on the positive part of $x\cdot\nabla V$, they then take $r\to\infty$ and obtain a positive normalized solution on the whole space for small masses, with positive Lagrange multiplier. If correct, this supplies the missing mass-supercritical, nonconstant-potential case for large bounded domains and its whole-space limit.

What carries the argument

The carrying structure is the family of truncated energy functionals $E_{r,s}(u)=\frac12\int_{\Omega_r}|\nabla u|^2+\frac12\int_{\Omega_r} V u^2-s\int_{\Omega_r}G(u)$ on the mass sphere $S_{r,c}$, together with the monotonicity trick that produces bounded Palais-Smale sequences for almost every $s$. A Pohozaev-type identity, combined with the star-shaped geometry ($x\cdot n\ge 0$ on the boundary), converts the mountain-pass level into a uniform $H^1$ bound independent of $r$ and $s$. As $r\to\infty$, a concentration-compactness decomposition separates the whole-space solution $u_0$ from finitely many translated bubbles $w_k(\cdot-z_r^k)$ centered at points escaping to infinity; condition (V1), the exponential lower bound on $x\cdot\nabla V$ along rays, is precisely what rules those bubbles out, yielding strong convergence.

What would settle it

Fix $N=3$, take $V(x)=-\varepsilon(1+|x|^2)^{-1}$ and $g(u)=|u|^{p-2}u$ with $4<p<6$, and solve the constrained problem on $B_r$ for $c=1$ by a numerical mountain-pass algorithm for a sequence $r\to\infty$. The theorem predicts a positive solution with positive energy and $\lambda_r$ eventually positive for small $c$; finding a large $r$ with no such critical point, or a branch with $\lambda_r\to -\infty$ while $\|u_r\|_{\infty}$ stays bounded, would refute it.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is Theorem 1.1 and Theorem 1.2: for every $c>0$ there exists $r_c>0$ such that problem (1.1)--(1.2) on $\Omega_r$ possesses a positive mountain-pass type solution $(\lambda_{r,c},u_{r,c})$ with $E_V(u_{r,c})>0$, and the family obeys a uniform $L^\infty$ bound as $r\to\infty$. If additionally $\|\tilde V_+\|_{N/2}<2S$ with $\tilde V(x)=x\cdot\nabla V(x)$, then for $0<c<\tilde c$ the frequencies satisfy $\liminf_{r\to\infty}\lambda_{r,c}>0$, and passing to the limit gives a positive solution $(\lambda_c,u_c)$ on $\mathbb{R}^N$ with $\lambda_c>0$, $E_V(u_c)>0$, and $E_V(u_c)\to\infty$ as $c\to\infty$. The proof establishes this by showing that no bubbles can escape to infinity during the domain expansion, so the $r\to\infty$ limit is strong rather than a superposition of bubbles.

Load-bearing premise

The load-bearing assumption is that the radial derivative $x\cdot\nabla V(x)$ stays positive and decays slower than every exponential as $|x|\to\infty$; if a potential satisfies the other hypotheses but violates this exponential growth condition, the proof's mechanism for excluding bubbles escaping to infinity no longer works, and the whole-space conclusion may fail.

Editorial extensions

If this is right

For any prescribed mass $c>0$, every sufficiently large star-shaped domain carries a positive normalized solution of mountain-pass type with positive energy, even though the energy is unbounded below on the mass sphere.
The uniform $L^\infty$ bound as $r\to\infty$ means the family of domain solutions does not develop spikes at the boundary; the whole-space solution is approached in a controlled way.
Under the smallness condition $\|\tilde V_+\|_{N/2}<2S$, small masses yield whole-space solutions with positive frequency $\lambda_c>0$, so they behave like bound states rather than zero-frequency limits.
The solution branch persists for all large masses in the whole space, with energy $E_V(u_c)\to\infty$ as $c\to\infty$.
The result covers general mass-supercritical nonlinearities satisfying two-sided power bounds, not just pure power nonlinearities.

Reading between the lines

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

If condition (V1) is weakened to polynomial decay of $x\cdot\nabla V$, the exponential-weight argument that kills the bubbles fails at a specific surface-integral estimate; a plausible consequence is that the $r\to\infty$ limit consists of a whole-space solution plus one or more bubbles at infinity, so the domain-size limit would be a superposition rather than a single solution.
The quantitative threshold $\|\tilde V_+\|_{N/2}<2S$ suggests a concrete test: compute the largest $c$ for which the whole-space solution's frequency stays positive and compare it with the constant $\tilde c$ that would follow from optimizing the Gagliardo-Nirenberg constants in the proof.
The same machinery should extend to coupled systems or potentials with singularities, as long as the mass-supercritical two-sided growth bounds and the exponential radial-growth condition hold; that extension is not stated in the paper.
One can try to construct an explicit potential $V(x)=-\varepsilon(1+|x|^2)^{-1}$ and a pure power $g(u)=|u|^{p-2}u$ with $2+4/N<p<2^*$, and solve the constrained problem numerically on large balls; matching the predicted mountain-pass level and positive $\lambda_r$ would test the quantitative constants, while observing bubble concentration would suggest the exponential condition is needed.

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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

The paper is a plausible within-field extension: it takes the standard normalized NLS machinery (monotonicity trick, Pohozaev identity, blow-up analysis, concentration compactness) and pushes it to large star-shaped domains with a general potential and mass-supercritical nonlinearity, then passes to the whole space as r→∞. That's a legitimate complement to Bartsch-Qi-Zou and Ding-Zhong, and the positioning looks honest.

The main problem is in Lemma 3.1. The trial function is v_t(x)=t^{3/2}v_1(tx), which only lies on the mass sphere if N=3. The exponent should be N/2. This error propagates into the estimates: h(t) and (3.2) use t^{3(d'-2)/2} instead of t^{N(d'-2)/2}, and (3.5) has t^{3(d-2)/4} instead of t^{N(d-2)/4}. Since the mountain-pass geometry in Lemma 3.1(iii) depends on this scaling, Theorem 1.1(i) is not established for N≠3 as written. The fix is straightforward if '3' is a typo for 'N' — and the presence of N elsewhere in the formulas suggests it is — but it has to be done.

There are also smaller mechanical issues: Theorem 1.2 states "lim_{c→∞} E_V(u_c)=∞" where it should be c→0+; the abstract promises a bifurcation result that never appears; and Lemma 3.5 contains a self-reference ('by an argument similar to that in Lemma 3.5') that should point to a different lemma.

On the substance, the strategy is standard and the estimates are mostly routine. Condition (V1), the exponential lower bound on the radial derivative of V, is doing a lot of work in ruling out bubbles moving off to infinity. It is strong and somewhat artificial, but that's a sufficient-condition issue, not a fatal one.

Overall, this is solid within-field progress, likely correct after the scaling typo is fixed. It doesn't open new territory, but it fills a real gap. I'd send it to a competent referee — the scaling error is exactly the kind of thing a referee should catch — but I wouldn't accept it until the Lemma 3.1 issue is resolved. If the authors confirm the N/2 fix, the paper deserves publication.

Referee Report

3 major / 5 minor

Summary. The paper studies normalized solutions (fixed L^2 mass) of the nonlinear Schrödinger equation -Δu+V(x)u+λu=g(u) on large star-shaped bounded domains Ω_r and on R^N, under mass-supercritical growth assumptions on the general nonlinearity g. The authors use the monotonicity trick and a mountain-pass geometry on the mass constraint manifold to prove, for every c>0, existence of a positive mountain-pass type solution on Ω_r for r sufficiently large; under an additional small-mass condition and a strong exponential decay/growth condition (V1), they obtain a positive solution on R^N with λ>0. The paper is framed as a complement to a recent work of Bartsch-Qi-Zou.

Significance. If the results are fully established, they constitute a useful extension of the normalized-solution theory to nonconstant potentials and general mass-supercritical nonlinearities on large domains, including a passage to the whole space with positive Lagrange multiplier. The proof strategy is standard and the paper contains no fitted free parameters; the claims are concrete and falsifiable. However, the current proof contains a dimensional-scaling error that, as written, restricts the main existence result to N=3, so the significance for the stated N≥3 depends on a repair of that argument.

major comments (3)

[Section 3, Lemma 3.1] The trial function v_t(x)=t^{3/2}v_1(tx) is mass-preserving only in dimension N=3. Indeed, ∫_{Ω_{1/t}}|v_t|^2 dx = t^{3-N}c, so v_t∉S_{1,c} for N≠3. Because the estimates (3.2)–(3.7), the choice of endpoints u_0 and u_1, and the path γ_0 in part (iii) all use the same 3/2 scaling and the exponent 3(d'-2)/2 in the definition of h(t), the mountain-pass geometry of Lemma 3.1 is established only for N=3. Since Theorem 3.2 and Lemma 3.5(i) rely on this geometry, Theorem 1.1(i) is not proved for the stated N≥3. The proof can likely be repaired by replacing 3 with N throughout, but as written the argument covers only the three-dimensional case.
[Theorem 1.2] The final assertion 'lim_{c→∞} E_V(u_c)=∞' is inconsistent with the quantifier 'for any 0<c<\tilde c' preceding it, and Section 4 proves only the passage r→∞ for fixed c; no proof of an energy limit in c appears anywhere. If the intended limit is c→0+, it still requires an argument; if not, the assertion should be removed or corrected.
[Section 4, Lemma 4.1, Step 2] The statement 'It follows from (3.15) that m0>0' is not justified: (3.15) concerns lim inf of max_{Ω_r} u_{r,c}, whereas z^1_r is a concentration center of the remainder ν^1_r=u_r-u_0, and there is no evident relation between the two. The half-space exclusion may be obtainable directly for m0=0 as well, but the manuscript needs to supply the missing reasoning or clarify the role of (3.15).

minor comments (5)

[Abstract] The abstract advertises 'multiplicity' and 'bifurcation property', but no theorem in the manuscript concerns multiplicity or bifurcation; the abstract should be aligned with the actual results.
[Lemma 3.5] In the proof of (ii), Case 2, the phrase 'by an argument similar to that in Lemma 3.5' is a self-reference; presumably another lemma (e.g., Lemma 3.4) is meant, and the citation should be corrected.
[Section 4, before (4.5)] The phrase 'for any ϕ∈C_c^∞(Ω_r)' should read 'for any ϕ∈C_c^∞(Σ)' to make the limiting argument meaningful.
[Lemma 3.4] The citation '[20, Theorem 9.11]' for L^p estimates should be the Gilbarg-Trudinger reference [21], since [20] is a different paper.
[Lemma 3.1(iii)] The sentence 'Since E_{r,s}(u1)≤0 for any γ∈Γ_{r,c}, we have ...' is garbled; it should be phrased as 'for every γ∈Γ_{r,c}, the path must cross the level set...' or similar.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence theorems are derived from (V0)-(V1), (G1)-(G3) via standard external variational and elliptic results, with no fitted parameter or load-bearing self-citation.

full rationale

The derivation chain is self-contained relative to its stated assumptions. Lemma 3.1 constructs mountain-pass geometry from explicit trial functions and the Gagliardo-Nirenberg inequality; Theorem 3.2 applies the external Monotonicity Trick (Theorem 2.2, cited to Borthwick-Chang-Jeanjean-Soave and Chang-Jeanjean-Soave) to obtain a bounded Palais-Smale sequence; Lemma 3.3 and Lemma 3.4 provide uniform gradient and sup-norm bounds from the Pohozaev identity and standard elliptic estimates; Lemma 3.5 passes s to 1 to get Theorem 1.1; Lemma 4.1 and the proof of Theorem 1.2 pass r to infinity and use (V1) to rule out bubble concentration. No parameter is fitted to the target quantity, and no theorem used as input asserts the desired existence. Citations to Bartsch-Qi-Zou [7] and Bartsch-Molle-Rizzi-Verzini [8] are context and comparison, not prior work of the present authors, and they are not used to justify the main conclusions. The only self-reference in the text is the typographical slip in Lemma 3.5, which says 'by an argument similar to that in Lemma 3.5'; this is a cross-reference error rather than a circular derivation. The t^{3/2} scaling in Lemma 3.1, if taken literally for N not equal to 3, is a dimensional or correctness gap, but it is not an instance of an output being equivalent to an input by construction; such an error would not raise the circularity score.

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

The paper introduces no physical entities or fitted constants. All active hypotheses are listed above; the proof uses standard variational and elliptic results. No data-fitting is present, so the circularity burden is low.

assumptions (7)

domain assumption (V0): V ∈ C(R^N) ∩ L^{N/2}(R^N) is bounded and ||V_-||_{N/2} < S.
Controls the potential term in the energy and the Pohozaev estimates; introduced in the introduction.
domain assumption (G1)-(G3): g is continuous and odd, and there exist 2+4/N < α ≤ β < 2* with 0 < αG(s) ≤ g(s)s ≤ βG(s), and g(s)/s^{β-1} → B > 0 as s→∞.
Mass-supercritical growth needed for the mountain pass geometry and for blow-up limits.
domain assumption (V1): V ∈ C^1, V(x)→0 at infinity, and lim inf_{|x|→∞} inf_{y∈B(x,ρ|x|)} (x·∇V(y)) e^{τ|x|} > 0 for every τ>0.
Excludes bubbles at infinity in the R^N limit; used in Lemma 4.1 and Theorem 1.2.
domain assumption Ω is bounded, smooth, and star-shaped with respect to 0.
Used to drop the boundary term in the Pohozaev identity in Lemma 3.3.
standard math Monotonicity trick (Theorem 2.2) as stated by Borthwick-Chang-Jeanjean-Soave.
Provides bounded PS sequences for almost every parameter s.
standard math Liouville and classification results: nonnegative solutions of -Δω = B|ω|^{β-2}ω on R^N or half-spaces are trivial for β < 2* (Chen-Li; Esteban-Lions).
Used in Lemma 3.4 blow-up analysis to rule out concentration at maxima.
standard math Elliptic regularity and L^p estimates from Gilbarg-Trudinger, Gagliardo-Nirenberg inequality, and Sobolev embeddings.
Used throughout Sections 3-4 for compactness and decay estimates.

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Pith. "Pith review of Normalized Solutions on large smooth domains to the Schr\"{o}dinger equation with potential and general nonlinearity: Mass super-critical case." pith.science (2026). https://pith.science/paper/BV2HUYK5

@misc{pith2026250104893,
  author       = {Pith},
  title        = {Pith review of: Normalized Solutions on large smooth domains to the Schr\"odinger equation with potential and general nonlinearity: Mass super-critical case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BV2HUYK5}},
  note         = {Machine review of arXiv:2501.04893}
}

abstract

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schr\"{o}dinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -\Delta u+V(x)u+\lambda u=g(u),\\ \|u\|_2^2=\int|u|^2\mathrm{d}x=c, \end{cases} \] both on large bounded smooth star-shaped domain $\Omega\subset\mathbb{R}^N$ and on $\mathbb{R}^N$, where $V(x)$ is the potential and the nonlinearity $g(\cdot)$ considered here are very general and of mass super-critical. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential $V(x)$. In addition, our study can be considered as a 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).

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Normalized solutions for fractional Choquard equation with critical growth on bounded domain
math.AP 2025-09 conditional novelty 6.0 of 10

For the critical fractional Choquard equation with a perturbation on a bounded star-shaped domain, at least two positive normalized solutions exist when the prescribed mass lies below an explicit threshold.

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Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

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