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REVIEW 4 major objections 3 minor 42 references

Ground state solutions of a class of (2,q)-Laplacian Schr\"odinger equations with inhomogeneous nonlinearity

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a class of (2,q)-Laplacian Schrödinger equations with singular inhomogeneous nonlinearity, prescribed-mass ground states exist exactly on one side of a sharp mass threshold in the subcritical range.

desk verdict Systematic and mostly solid treatment of normalized (2,q)-Laplacian ground states; the supercritical existence proof has a genuine gap that is probably repairable. read the letter →

arxiv 2505.20758 v2 pith:GJEFZYOJ submitted 2025-05-27 math.AP

Ying Huang , Tingjian Luo , Youde Wang This is my paper · ORCID
classification math.AP MSC 35J5035Q4135Q5537K45
keywords normalizedsolutionsgroundstates(2q)-LaplacianinhomogeneousnonlinearityGagliardo-NirenberginequalityPohozaevidentitymasscriticalexponentconstrainedvariationalmethods
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 and non-existence of L2-normalized ground states for the (2,q)-Laplacian Schrödinger equation with a singular inhomogeneous nonlinearity |u|^{p-2}u/|x|^b, under a prescribed mass constraint. In the mass-subcritical range it proves a sharp threshold: depending on p relative to 2(2-b)/N+2, minimizers of the energy on the L2-sphere exist for all masses, only for masses above a critical value c*_1, or exactly for masses at or above that critical value. In the mass-critical case p = p*_q := 2(q-b)/N+q, no minimizers exist for any mass. In the mass-supercritical case, the paper turns to a local minimization problem on the Pohozaev constraint and proves existence of minimizers that solve the equation, with precise asymptotic behavior of the energy and Lagrange multiplier, and also proves the existence of infinitely many normalized bound states. The whole analysis rests on a sharp inhomogeneous Lq-Gagliardo-Nirenberg inequality with an explicit optimizer.

What carries the argument

The key machinery is a sharp inhomogeneous Lq-Gagliardo-Nirenberg inequality: for 2 < p < q(N-b)/(N-q)_+, it states that ∫ |u|^p/|x|^b dx ≤ K_{N,p,q} ||∇u||$_q^{{σ_{p,q}}$} ||u||$_2^{{p-σ_{p,q}}$}, with sharp constant K_{N,p,q} = p/||Q_{p,q}||$_2^{{p-2}}$ attained by a ground state Q_{p,q} of σ_{p,q} Δ_q Q + (p-σ_{p,q})Q - |x|^{-b}|Q|^{p-2}Q = 0. This inequality controls the singular nonlinear term, produces the lower bounds that make m(c) finite, and identifies the mass-critical exponent p*_q = 2(q-b)/N+q. The second load-bearing mechanism is the Pohozaev identity P(u) = 0, with P(u) = ||∇u||$_2^{2}$ + (N(q-2)+2q)/(2q)||∇u||_q^q - (N(p-2)+2b)/(2p)∫|u|^p/|x|^b dx, which every weak solution must satisfy and which defines the constraint V(c) for the mass-supercritical local minimization γ(c). The proof that the constrained minimizer is a true critical point on S(c) is carried by the natural-constraint argument, which uses the strict positivity of p - p*_q.

What would settle it

Choose parameters satisfying 2(2-b)/N+2 < p < p*_q and solve the constrained minimization problem numerically for masses slightly below and above the computed c*_1; finding a minimizer for some c < c*_1, or failing to find one for some c ≥ c*_1, would disprove the sharp-threshold claim. Alternatively, exhibit a minimizer of γ(c) at which the transverse derivative of the Pohozaev constraint vanishes, which would show that the natural-constraint mechanism fails and that such a minimizer need not solve equation (1.1).

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Extended reading notes

Core claim

The central claim is that, in the subcritical regimes, the value of the prescribed mass c relative to a critical value c*_1 := inf{c>0 : m(c)<0}, where m(c) is the infimum of the energy on the L2-sphere S(c), completely determines whether a ground state exists. Concretely, if 2(2-b)/N+2 < p < p*_q, then 0 < c*_1 < +∞ and m(c) admits a minimizer if and only if c ≥ c*_1, with the borderline minimizer present; if p = 2(2-b)/N+2, minimizers exist if and only if c > c*_1; and if 2 < p < 2(2-b)/N+2, minimizers exist for every c > 0. In the mass-critical case p = p*_q, the paper proves that m(c) = 0 for 0 < c ≤ c*_2, m(c) = -∞ for c > c*_2, and that the functional has no critical points for c ≤ c*_2, so no normalized ground state exists for any c > 0. In the mass-supercritical case p > p*_q, the paper proves that for N = 1, 2 with any p > p*_q, or for N ≥ 3 with q < 2($N^{2}$-2b)/($N^{2}$-4) and p < 2(N-b)/(N-2), the Pohozaev-constrained minimization problem γ(c) admits a minimizer, which is a genuine solution with negative Lagrange multiplier, satisfying I(u_c) → +∞ and λ_c → -∞ as c → 0+, while I(u_c) → 0 as c → ∞.

Load-bearing premise

The load-bearing premise is that the constrained set defined by the Pohozaev identity is smooth enough for the Lagrange multiplier rule to apply at a minimizer, and that every weak limit of a minimizing sequence genuinely solves the original equation rather than merely the constraint.

Editorial extensions

If this is right

  • In the subcritical window 2(2-b)/N+2 < p < p*_q, the mass boundary is exact: ground states exist at and above c*_1 and fail below it, making c*_1 a genuine phase transition for the constrained energy.
  • At the mass-critical exponent p = p*_q, the least energy is zero for small masses and unbounded below for large masses, with no critical points at all on S(c) for c ≤ c*_2, so normalized ground states do not exist in this regime.
  • In the mass-supercritical regime covered by Theorem 1.4, the Pohozaev-constrained minimizer is a true solution with negative Lagrange multiplier and has the asymptotic signatures I(u_c) → ∞ as c → 0+ and I(u_c) → 0 as c → ∞.
  • The same method produces infinitely many normalized bound states with λ_n < 0 and I(u_n) → ∞, so the constrained problem admits not only a ground state but also excited states of arbitrarily high energy.
  • For subcritical masses away from the threshold, minimizing sequences are precompact, which gives orbital stability of the corresponding standing waves for the associated time-dependent problem.

Reading between the lines

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

  • The sharp Gagliardo-Nirenberg inequality with optimizer Q_{p,q} suggests that the same threshold phenomenon should hold for the Lq-normalized version of the problem mentioned in the paper, since the authors note the arguments extend with little change; a direct adaptation would yield an Lq-analogue of the critical mass c*_1.
  • The paper leaves implicit that the mass-critical exponent p*_q = 2(q-b)/N+q shifts the classical threshold by exactly the inhomogeneity term 2b/N; extrapolating, other double-phase models with a |x|^{-b} weight should show a similar shift in their critical exponents.
  • If the transversality of the Pohozaev constraint could be verified, the hypotheses of Theorem 1.4 could likely be relaxed to the full range p < q(N-b)/(N-q)_+ for small masses, matching the paper's stated conjecture; a numerical search for minimizers of γ(c) at small c in the currently excluded range would directly test that possibility.
  • The non-attainment at mass-critical p means the least-energy level is realized only as a limit of concentrating or diffusing sequences, which suggests that nearby subcritical problems may exhibit instability or symmetry-breaking behavior as p approaches p*_q.
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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

4 major / 3 minor

Summary. The paper studies L2-normalized ground states for the inhomogeneous (2,q)-Laplacian Schrödinger equation −Δu − Δ_q u = λu + |u|^{p−2}u/|x|^b on R^N, under the constraint ∫|u|^2 = c. The main results are: (i) in the mass-subcritical range, a sharp threshold c_1^* separating existence from non-existence of minimizers of the global minimization problem m(c); (ii) in the mass-critical case, non-existence of critical points for c ≤ c_2^* and non-attainment of m(c); (iii) in the mass-supercritical case, existence of minimizers for the Pohozaev-constrained problem γ(c), together with a negative Lagrange multiplier and asymptotic behavior as c → 0 and c → ∞; and (iv) a multiplicity result for bound state solutions. The proofs rely on a sharp inhomogeneous Lq-Gagliardo–Nirenberg inequality, concentration-compactness arguments, and a natural-constraint (Pohozaev manifold) approach.

Significance. If the results are correct, the paper provides a systematic and largely sharp theory of normalized solutions for a double-phase operator with inhomogeneous nonlinearity, including a critical mass threshold that is new in this setting. The self-contained proof of the sharp inhomogeneous Gagliardo–Nirenberg inequality is a useful contribution, and the supercritical analysis via the Pohozaev constraint is a natural extension of recent techniques. However, several load-bearing steps in the proofs are not justified as written, so the significance is conditional on substantial repair.

major comments (4)
  1. [§2, Lemma 2.1 and Theorem 1.1] In Proposition 3.1 the key inequality (3.18) is invalid. Since t_0 > 1 and t_n > 1, one has t_0^{−N} + t_n^{−N} > 1; because m(c) < 0, the product m(c)(t_0^{−N}+t_n^{−N}) is strictly less than m(c). The second inequality in (3.18), which replaces this product by m(c), therefore has the wrong direction, and the claimed contradiction 'Since p > b and t_0 > 1, then (3.18) is impossible' does not follow. The pre-compactness of any minimizing sequence for m(c) and the subsequent existence statements in Theorem 1.2 are not established by the given argument.
  2. [§5.1, Proposition 5.1] The assertion 'Since u_n ⇀ u_0 in X, then u_0 is a weak solution of Eq. (1.1)' is unjustified. A weak limit of a minimizing sequence for I restricted to V(c) is not automatically a critical point of I|S(c), and no Palais–Smale condition, Ekeland principle, or natural-constraint argument is supplied at that point. This step is load-bearing: it is used to infer P(u_0) = 0, to identify u_0 as a minimizer of γ(∥u_0∥_2^2), and then with Lemma 5.5 to conclude ∥u_0∥_2^2 = c. Since Lemma 5.5 itself assumes the minimizer is a weak solution with a negative Lagrange multiplier, the argument is circular as written.
  3. [§5.1, Lemma 5.7] The proof applies the Lagrange multiplier rule to the two-constraint set V(c) = {u ∈ S(c) : P(u) = 0} but never verifies that the constraint map (u ↦ (∥u∥_2^2 − c, P(u))) is a submersion. In particular, the paper does not prove that P'(u) is not a scalar multiple of the L2-functional v ↦ 2∫uv, which is necessary for the existence of multipliers λ, μ in (5.28). Without this regularity check, the conclusion μ = 0, and hence the statement that a critical point of I|V(c) is a critical point of I|S(c), is not justified. This affects Theorem 1.4 and the ground-state interpretation of γ(c)-minimizers.
  4. [§5.1, Lemma 5.4] The proof of Lemma 5.4 does not treat the cases N = 1, 2 stated in part (1) of the lemma. The argument 'Since 2_b^* < q_b^*, ... if p ≤ 2_b^*, then λ_c < 0' relies on the quantity 2_b^* = 2(N−b)/(N−2), which is undefined for N = 1, 2. No separate argument is given for these dimensions, yet the negativity of λ_c for N = 1, 2 is used in Proposition 5.2 and Theorem 1.5.
minor comments (3)
  1. [§5.2, Lemma 5.8] The orthogonal space V_n^⊥ is defined in H_r^1(R^3), but the ambient space in this paper is X = H^1(R^N) ∩ D^{1,q}(R^N), and the dimension N need not be 3. This appears to be a typographical artifact from a previous source and should be corrected.
  2. [Throughout] There are several typos and grammatical slips, for example 'fucus', 'week solutions', 'nagetive', 'r igours proof', 'Fanally', and '22' in reference [41]. The paper would benefit from a careful proofreading.
  3. [§1, Theorem 1.2] The phrase 'describes sharply' in Remark 1.1 and the phrase 'the existence, non-existence, and multiplicity of L2-normalized solutions' in the introduction are slightly awkward; consider 'describes sharp results' and 'the existence, non-existence, and multiplicity of L2-normalized solutions'. These are presentation issues only.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the variational thresholds are derived from internal minimization problems, not fitted; the only self-citation is a minor, non-load-bearing reference for a standard monotonicity property.

full rationale

The paper's central claims are powered by internally derived variational quantities rather than by fitted inputs. The sharp inhomogeneous Gagliardo-Nirenberg inequality (Theorem 1.1) is proved self-containedly in Lemma 2.1 via a Weinstein-type functional, and the thresholds c*_1 and c*_2 are defined by the variational problems themselves (c*_1 := inf{c>0 | m(c)<0} and (4.3) using K*_{N,q}), not calibrated to the existence/non-existence answers. Theorem 1.2's dichotomy follows from Lemma 3.1, Proposition 3.1, Corollary 3.1, and Lemma 3.2; the mass-critical non-attainment follows from the Pohozaev identity and the sharp inequality. The one self-citation occurs in Lemma 3.1: 'the proof of the continuity and non-increasing of m(c) is somehow standard, here we only refer the readers to the reference such as [7, Theorem 1.2 (4)]'. This cites prior work by co-author Luo, but it is used only for a standard auxiliary monotonicity property, not for the paper's central existence claim, and no main result reduces through it. There is a genuine gap in the supercritical section: Proposition 5.1 states 'Since u_n ⇀ u_0 in X, then u_0 is a weak solution of Eq. (1.1)', which is not a valid implication and is the missing natural-constraint/Lagrange-multiplier step; this is an unproved bridge rather than a self-referential definition or a fitted-parameter renaming. The paper also candidly flags related open cases in Remark 1.4 ('a rigours proof is still open for us'). Thus no step is circular by construction, and the score reflects only the minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central results are derived through variational principles; the only non-standard inputs are the unproven existence of the auxiliary ground state Q*_q and the unverified regularity of the Pohozaev constraint. No numerical parameters are fitted.

assumptions (3)
  • ad hoc to paper Existence of a ground state solution Q*_q for Eq. (4.2), which defines the critical mass c*_2 in (4.3).
    Lemma 4.1 assumes Q*_q exists to define the critical mass c*_2 in (4.3); the paper does not prove this existence, though it likely follows from the same Weinstein-type minimization as Lemma 2.1.
  • ad hoc to paper The Pohozaev constraint V(c) is a regular manifold for the Lagrange multiplier rule.
    Proposition 5.1 and Lemma 5.7 rely on the existence of multipliers λ, μ in (5.28); the surjectivity of the constraint differential is not verified.
  • standard math Local compact embedding: X = H^1(R^N) ∩ D^{1,q}(R^N) embeds compactly into L^r_loc(R^N) for r < q(N-b)/(N-q)_+.
    Used in Lemma 7.1 to justify convergence of weighted integrals for weakly convergent sequences; follows from Rellich-Kondrachov for D^{1,q} and interpolation arguments.

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Pith. "Pith review of Ground state solutions of a class of (2,q)-Laplacian Schr\"odinger equations with inhomogeneous nonlinearity." pith.science (2026). https://pith.science/paper/GJEFZYOJ

@misc{pith2026250520758,
  author       = {Pith},
  title        = {Pith review of: Ground state solutions of a class of (2,q)-Laplacian Schr\"odinger equations with inhomogeneous nonlinearity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GJEFZYOJ}},
  note         = {Machine review of arXiv:2505.20758}
}
read the original abstract

In this paper, we systematically investigate the ground state solutions of a class of (2,q)-Laplacian Schr\"odinger equations with inhomogeneous nonlinearity. By analyzing global and local constrained variational problems, we establish the existence, non-existence, and asymptotic behavior of ground states, addressing the mass-subcritical,mass-critical, and mass-supercritical regimes. As a byproduct, we prove a multiplicity of bound states with prescribed mass. Some of our existence results are sharp. The proofs are based primarily on constrained variational techniques.

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