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Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity

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Pith's one-line read This paper proves two positive normalized solutions for a focusing–defocusing NLS equation on large bounded domains of arbitrary shape, and a ground state plus a high-energy solution in the Sobolev-critical focusing case on star-shaped…

desk verdict A genuine open-problem paper where the main claims are likely true but the mountain-pass compactness argument in Theorem 2.3 has a real gap that needs repair before the positive-energy solution is proven. read the letter →

arxiv 2411.17951 v1 pith:UTCRA6QN submitted 2024-11-27 math.AP

He Zhang , Haibo Chen , Shuai Yao , Juntao Sun This is my paper · ORCID
classification math.AP MSC 35J2035J6035Q55
keywords NLSequationsnormalizedsolutionsprescribedmassboundeddomainsL2-supercriticalnonlinearitySobolevcriticalexponentmonotonicitytrickmultiplicity
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

The paper studies standing waves of the nonlinear Schrödinger equation with an external potential and an inhomogeneous combined nonlinearity $|u|^{q-2}u+\beta|u|^{p-2}u$ on a bounded domain, with the mass $\int|u|^2$ prescribed. It claims that when $2+\frac4N

What carries the argument

The argument is carried by three mechanisms: the monotonicity trick of [8], which produces bounded Palais–Smale sequences at almost every level of a parameter $s\in[1/2,1]$; a rescaling $v_t(x)=t^{N/2}v_1(tx)$ of the first Dirichlet eigenfunction that converts the bounded domain into a large one and makes the energy functional track a one-variable function $h(t)$ whose sign change determines the mass threshold $a_V$; and Liouville/Pohozaev nonexistence on half-spaces and $\mathbb R^N$ used to rule out concentration along the boundary or at infinity. In the Sobolev-critical case, the Aubin–Talenti bubbles $U_\varepsilon$ and the ground state from the local minimization are used to estimate the mountain pass level below the first concentration threshold.

What would settle it

Look for a nonzero solution of $-\Delta w+\lambda w=|w|^{q-2}w-|w|^{p-2}w$ in the half-space $\{x_1>0\}$ with zero boundary condition, for some $N\ge3$, $q^*<q<p\le2^*$ and $\lambda\le0$; a numerical or variational construction of such a solution would invalidate the compactness step. Conversely, proving nonexistence for the sign-changing nonlinearity by the moving-plane or Pohozaev methods would close the gap.

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Extended reading notes

Core claim

The paper's central claim, stated as Theorem 1.2, is that for $N\ge3$, $q^*<q<p\le2^*$, $\beta=-1$, and a bounded smooth domain $\Omega$, under the assumption $(V0)$ on the potential, for every $a>a_V$ and every sufficiently large $r$, the constrained problem (4) has a positive mountain pass solution $u_r$ with $I_r(u_r)>0$ and a positive global minimizer $u_r$ with $I_r(u_r)<0$. Under an additional smallness condition involving $\tilde V(x)=\nabla V(x)\cdot x$, the corresponding Lagrange multipliers satisfy $\liminf_{r\to\infty}\lambda_r>0$. The paper further claims, in Theorems 1.4 and 1.5, that for $2<q<q^*<p=2^*$ and $\beta=1$, small mass on large star-shaped domains yields a local minimizer that is a ground state and, under boundedness of $\tilde V$, a second high-energy solution.

Load-bearing premise

The proof of compactness for the mountain pass sequence requires that the limiting equation $-\Delta w+\lambda w=|w|^{q-2}w-|w|^{p-2}w$ on a half-space (and on $\mathbb R^N$) has no nontrivial solution, and this is justified by a cited Liouville theorem that was proved only for pure power nonlinearities; a separate smallness condition on $\nabla V$ is also used without being assumed in the statements.

Editorial extensions

If this is right

  • For the focusing-defocusing range, large bounded domains of arbitrary shape carry two normalized positive solutions when the mass is large; this removes the star-shaped condition from earlier existence theorems.
  • The negative-energy solution is a global minimizer of the energy on the mass sphere, so a mass-supercritical combined nonlinearity can still admit a global minimizer on bounded domains.
  • In the Sobolev-critical focusing case, small mass on large star-shaped domains gives a ground state and a high-energy solution; the paper states this is new even when $V\equiv0$ in the Brézis-Nirenberg context.
  • Under the extra potential-gradient smallness, the Lagrange multipliers of the high-energy and minimizer solutions stay bounded away from zero as the domain expands, so the frequencies do not drift to zero or infinity.

Reading between the lines

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

  • If the half-space nonexistence step can be justified for the sign-changing nonlinearity (or replaced by another compactness argument), the same two-solution pattern should hold for a whole family of inhomogeneous nonlinearities whose energy is bounded below on the mass sphere; the paper's geometric two-well structure is the load-bearing part.
  • The dichotomy $a>a_V$ versus $a<\tilde a_V$ is likely generic: when the highest-degree term is defocusing, large masses help create the two wells, whereas when it is focusing and critical, only small masses avoid concentration.
  • The existence of a global minimizer in a mass-supercritical regime looks like a bounded-domain effect: the volume term in the Gagliardo–Nirenberg inequality, together with the defocusing higher-power term, prevents the energy from escaping to $-\infty$, something impossible on $\mathbb R^N$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies normalized solutions of the NLS equation with a bounded potential V and the inhomogeneous nonlinearity |u|^{q-2}u + β|u|^{p-2}u on bounded domains Ω_r = rΩ. In the case q* < q < p ≤ 2* with β = -1, the authors claim, under an explicit smallness condition on V and for large mass a > a_V and large domains r, the existence of both a positive mountain-pass solution with positive energy and a positive global minimizer with negative energy, without assuming Ω is star-shaped. In the case 2 < q < q* < p = 2* with β = 1, they claim a local minimizer (ground state under extra assumptions) and a high-energy solution for small mass on star-shaped domains. The proofs use the monotonicity trick of Borthwick-Chang-Jeanjean-Soave, minimization on the sphere, and compactness analysis via blow-up and Liouville-type theorems.

Significance. If the results are correct, the paper makes a substantial contribution: it removes the star-shapedness assumption in the L2-supercritical combined-nonlinearity problem, which is an open problem raised by Bartsch, Qi and Zou, and it gives both positive- and negative-energy normalized solutions with explicit mass thresholds. The extension to the Sobolev-critical Brézis-Nirenberg regime for normalized ground states is also new. The strengths of the paper include explicit, parameter-free thresholds a_V and ~a_V, the use of an imported monotonicity trick rather than fitted parameters, and a clear presentation of the two different parameter regimes. However, the proof of the mountain-pass compactness in Theorem 2.3 contains a serious gap, and a key nonexistence assertion in Lemma 2.9 is not supported by the cited reference; these issues affect the central positive-energy existence claim of Theorem 1.2(i).

major comments (3)
  1. [Theorem 2.3, paragraph following (16)] The compactness argument for the bounded Palais-Smale sequence is not valid as written. The displayed identity mixing integrals over Ω_r and R^N, with the extra terms (q-2)/(2q)∫_{R^N} V u_{r,s}^2 dx - (p-q)/(pq)∫_{R^N} |u_{r,s}|^p dx, does not follow from (14)-(16). The subsequent inequality requires the condition q||~V_+||_{N/2} + N(q-2)||V_+||_{N/2} < S(2N-(N-2)q), which is not assumed in the statement of Theorem 2.3 and appears only in the separate 'Moreover' clause of Theorem 1.2(i). Since Theorem 2.5 and hence the positive-energy solution in Theorem 1.2(i) depend on this compactness, the existence proof is incomplete as written; a genuine re-derivation of the strong convergence is needed.
  2. [Lemma 2.9, equation (25)] The nonexistence assertion for the half-space limiting equation is not justified. Equation (25) is written as -Δu + λ∞u = |u|^{q-2}u - |u|^{q-2}u, which is evidently a typo; the intended equation is -Δu + λ∞u = |u|^{q-2}u - |u|^{p-2}u. The text attributes the nonexistence of nontrivial solutions to Esteban-Lions [14], but that reference treats pure power nonlinearities, not the combined sign-changing term with q ≠ p. Thus the conclusion lim inf_{r→∞} λr > 0 in Lemma 2.9, which is used in the 'Moreover' clause of Theorem 1.2(i), needs a different argument or a reference that actually covers this equation.
  3. [Theorem 3.1(ii), Lagrange multiplier passage] The proof of strong convergence of the minimizing sequence invokes the identity I'_r(u_r) - λ_r u_r = 0 before u_r has been shown to be a critical point. At that stage u_r is only the weak limit of {u_n}, and the claim that the limit of the Lagrange multipliers λ_n corresponds to u_r is circular. The compactness can be obtained by testing the Palais-Smale condition I'_r(u_n) - λ_n u_n → 0 against u_n - u_r, which avoids assuming u_r is critical in advance; the proof should be rewritten accordingly.
minor comments (5)
  1. [Lemma 3.2(i), displayed estimate for ~I_{1/t,s}(v_t)] The last term '- s/p ∫_{Ω} |v_t|^p dx' duplicates the critical term '- s/2* ∫_{Ω} |v_t|^{2*} dx' because p = 2* in this section; this appears to be a typographical error.
  2. [Lemma 2.9] The reference to 'Theorem 2.4' at the start of the proof should be 'Lemma 2.4'.
  3. [Lemma 2.6] The phrase 'Liouville's theorem [14]' is used for the nonexistence of positive solutions of -Δv = |v|^{q-2}v on a half-space or R^N; this is a valid use for a pure power equation, but the precise statement and range of q from [14] should be quoted.
  4. [Throughout] There are multiple typos: 'Pohozave identity' (Lemma 2.9), 'direction calculation' (Lemma 2.9), 'moutain pass' (Theorem 3.5), and 'Arze-Ascoli' (Lemma 2.6). These should be corrected.
  5. [Lemma 2.1(i)] The function u0,s is defined as v_{1/r*} and hence belongs to S_{r*,a}; for the conclusion stated for all r > r_a, the zero-extension of u0,s to Ω_r should be made explicit, since otherwise the mountain-pass endpoints are not literally in S_{r,a}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proofs are external-theorem-driven existence arguments with explicit thresholds, not fitted predictions.

full rationale

This is a pure existence paper with no fitted parameters, data, or empirical predictions. The thresholds a_V and ~a_V are explicit constants defined from Sobolev and Gagliardo-Nirenberg constants, the principal eigenvalue, the volume of Omega, and norms of V; they are not calibrated to the solutions being constructed. The mountain-pass machinery is imported from the independent source [8,9] (Borthwick-Chang-Jeanjean-Soave), and the key compactness arguments cite Esteban-Lions [14] and follow the approach of Bartsch-Qi-Zou [4], neither of which has author overlap with the present paper. The only self-citations ([30,32,33]) appear in the introduction as background and are not load-bearing in the proofs. There is a genuine proof gap in Section 2.1, Theorem 2.3: the displayed identity obtained by 'combining (14)-(16)' adds potential and p-power integrals that do not follow from those equations, and the subsequent negative bound requires the extra smallness condition on q||V_+||_{N/2} + N(q-2)||V_+||_{N/2} that is not assumed in the theorem statement. However, this is a correctness or rigor defect, not circularity: it does not make any claimed output equal by construction to an input. No calculation in the paper fits a parameter to a subset of data and then reports a related quantity as a prediction, and no load-bearing uniqueness claim or ansatz is imported from the authors' own prior work. The circularity score is therefore 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented physical entities. Its central claims rest on standard variational machinery plus several domain assumptions on the potential and on the validity of the cited half-space nonexistence result. The most fragile assumption is the Esteban-Lions citation, which appears misapplied to a sign-changing nonlinearity.

assumptions (6)
  • standard math The principal eigenvalue theta of -Delta with Dirichlet condition on Omega has a positive eigenfunction v1 used to build explicit test functions.
    Invoked in Lemma 2.1 and Theorem 3.1 to construct v_t and v_r with controlled energy.
  • standard math Aubin-Talenti constant S and Gagliardo-Nirenberg constants C_s exist and satisfy the stated inequalities.
    Used throughout to bound the energy functional and to estimate critical and subcritical terms.
  • standard math The monotonicity trick theorem (Theorem 2.2) from Borthwick-Chang-Jeanjean-Soave produces bounded Palais-Smale sequences on the constraint manifold.
    Applied in Theorems 2.3 and 3.5 to obtain solutions of the modified problems for almost every parameter s.
  • standard math Pohozaev identities hold for solutions on bounded star-shaped domains, on R^N, and on half-spaces with the relevant boundary terms.
    Used in Lemma 2.9 and in the proof of Theorem 1.4(ii) to relate the Lagrange multiplier, the gradient norm, and the power integrals.
  • domain assumption The Esteban-Lions nonexistence result [14] applies to the limiting half-space equation -Delta w + lambda w = w^{q-1} - w^{p-1}.
    The paper cites [14] for the claim that equation (25) has no nontrivial solution on a half-space, but [14] addresses pure power nonlinearities; the present sign-changing nonlinearity is not covered and the claim is not verified. This is a load-bearing assumption in Lemma 2.9.
  • domain assumption The potential V satisfies the smallness conditions (V0) and, where needed, (22) with the additional regularity V in C^1 and boundedness of ~V = grad V . x.
    These hypotheses control the potential terms in the energy and in the Pohozaev identities. Theorem 1.4(ii) omits the boundedness of ~V while the proof uses it.

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Pith. "Pith review of Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity." pith.science (2026). https://pith.science/paper/UTCRA6QN

@misc{pith2026241117951,
  author       = {Pith},
  title        = {Pith review of: Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTCRA6QN}},
  note         = {Machine review of arXiv:2411.17951}
}
abstract

We investigate normalized solutions for a class of nonlinear Schr\"{o}dinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+\beta |u|^{p-2}u$ on a bounded domain $\Omega$. Firstly, when $2+\frac{4}{N}<q<p\leq2^*:=\frac{2N}{N-2}$ and $\beta=-1$, under an explicit smallness assumption on $V$, we prove the existence of a global minimum solution and a high-energy solution if the mass is large enough. For this case we do not require that $\Omega$ is star-shaped, which partly solves an open problem by Bartsch et al. [Math. Ann. 390 (2024) 4813--4859]. Moreover, we find that the global minimizer also exists although the nonlinearity is $L^2$-supercritical. Secondly, when $2<q<2+\frac{4}{N}<p=2^*$ and $\beta=1$, under the smallness and some extra assumptions on $V$, we prove the existence of a ground state and a high-energy solution if $\Omega$ is star-shaped and the mass is small enough. It seems to be new in the study of normalized ground state in the context of the Br\'{e}zis-Nirenberg problem, even for the autonomous case of $V(x)\equiv0$.

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Forward citations

Cited by 1 Pith paper

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

  1. 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.

Reference graph

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