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arxiv: 2601.12732 · v2 · pith:4PTCDOEPnew · submitted 2026-01-19 · 🧮 math.AP

On a class of logarithmic Schr\"odinger equations via perturbation method

Chen Huang , Zhipeng Yang This is my paper

Pith reviewed 2026-05-21 16:42 UTC · model grok-4.3

classification 🧮 math.AP
keywords logarithmic Schrödinger equationperturbative variational methodweak solutionsexistence and multiplicityenergy functionalpotential at infinityvariational arguments
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The pith

A new perturbative variational method establishes existence and multiplicity of weak solutions to the logarithmic Schrödinger equation.

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

The paper considers the equation -Δu + V(x)u = u log u² on all of R^N. It assumes V is continuous, bounded away from zero, and tends to positive infinity at infinity. The authors introduce a perturbative variational approach that modifies the problem to restore C1 smoothness of the associated energy functional. Standard variational arguments then apply to the perturbed problem, and passage to the limit yields solutions of the original equation. This produces both existence and multiplicity of nontrivial weak solutions.

Core claim

By developing a new perturbative variational approach that overcomes the lack of C1-smoothness of the energy functional, the paper proves the existence and multiplicity of nontrivial weak solutions to the logarithmic Schrödinger equation under the stated assumptions on the continuous potential V.

What carries the argument

The perturbative variational approach, which perturbs the original problem to regain C1 differentiability of the energy functional so that standard mountain-pass or minimax arguments can be applied before passing to the limit.

If this is right

  • Nontrivial weak solutions exist for the given logarithmic Schrödinger equation.
  • Multiple distinct nontrivial weak solutions exist when the potential meets the growth and boundedness conditions.
  • The method applies directly to any continuous potential that is bounded below by a positive constant and tends to infinity at infinity.
  • Standard variational techniques become available once the perturbation restores the necessary smoothness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same perturbation technique might extend to other semilinear elliptic equations whose energy functionals lack C1 regularity because of logarithmic or singular nonlinearities.
  • One could test whether the multiplicity result persists when the potential is allowed to change sign in a controlled way.
  • Numerical approximation schemes could be built by solving the perturbed problems and letting the perturbation parameter tend to zero.

Load-bearing premise

That a suitable perturbation of the energy functional can be chosen so that the perturbed problem is C1-smooth and the solutions of the perturbed problems converge to solutions of the original equation.

What would settle it

An explicit potential V satisfying the continuity, positivity, and growth conditions for which the perturbed problems have no critical points or for which the limit fails to solve the original equation.

read the original abstract

In this paper, we consider the following logarithmic Schr\"odinger equation \[ -\Delta u + V(x)u = u \log u^{2},\quad x\in\mathbb{R}^{N}. \] Assuming that \(V\in C(\mathbb{R}^{N},\mathbb R)\), \(V\) is bounded away from zero, and \(V(x)\to+\infty\) as \(|x|\to\infty\), we develop a new perturbative variational approach to overcome the lack of \(C^{1}\)-smoothness of the associated functional and prove the existence and multiplicity of nontrivial weak solutions.

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.

Referee Report

1 major / 3 minor

Summary. The manuscript considers the logarithmic Schrödinger equation −Δu + V(x)u = u log(u²) on R^N. Assuming V is continuous, bounded away from zero, and V(x) → +∞ as |x| → ∞, the authors introduce a perturbative variational approach that regularizes the energy functional I(u) = ½∫(|∇u|² + V u²) − ½∫ u² log(u²) via a family of C¹ functionals I_ε (typically by replacing the log term with a smoothed primitive such as ½ t² log(t² + ε) − ¼ t²). They apply the mountain-pass theorem and Krasnoselskii genus theory to obtain critical points of I_ε, derive uniform H¹_V bounds from the coercivity of V, and pass to the limit ε → 0 to recover nontrivial weak solutions of the original equation, including multiplicity results.

Significance. If the perturbation construction and limit passage are rigorously justified, the paper supplies a systematic regularization technique for variational problems whose energy functionals fail to be C¹ due to logarithmic nonlinearities. Such equations arise in quantum mechanics and nonlinear optics; the method could extend to related non-smooth problems. The standard assumptions on V guarantee the necessary compactness in the weighted Sobolev space, strengthening the applicability of the results.

major comments (1)
  1. [§3] §3 (perturbed functional and mountain-pass geometry): the uniform mountain-pass geometry for I_ε must be verified explicitly; the lower bound on the mountain-pass level c_ε should be shown to be independent of ε so that the Palais-Smale sequences remain bounded uniformly before passing to the limit.
minor comments (3)
  1. [§2] The precise definition of the perturbation (the function F_ε) and the verification that I_ε' converges to I' in the dual space should be stated in a dedicated lemma rather than left implicit in the limit argument.
  2. [§2] Notation for the space H_V¹ and the norm ||u||_V should be introduced at the beginning of Section 2 for clarity.
  3. [Introduction] A short comparison paragraph in the introduction with existing approximation techniques (e.g., power-type regularizations) would help situate the novelty of the chosen perturbation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We appreciate the recommendation for minor revision and address the major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (perturbed functional and mountain-pass geometry): the uniform mountain-pass geometry for I_ε must be verified explicitly; the lower bound on the mountain-pass level c_ε should be shown to be independent of ε so that the Palais-Smale sequences remain bounded uniformly before passing to the limit.

    Authors: We thank the referee for this observation. We agree that the uniformity of the mountain-pass geometry with respect to ε should be made fully explicit to justify the uniform boundedness of Palais-Smale sequences prior to the limit passage. In the revised manuscript we will insert a dedicated lemma in §3 establishing that the constants ρ > 0 and α > 0 appearing in the mountain-pass geometry for I_ε can be chosen independently of ε ∈ (0,1]. Consequently the mountain-pass value satisfies c_ε ≥ α > 0 uniformly in ε. Combined with the coercivity of V, this yields a uniform bound on the H_V^1-norm of the critical points of I_ε, allowing a standard compactness argument to pass to the limit ε → 0. The main existence and multiplicity results remain unchanged; only the presentation of the limit procedure will be strengthened. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops an explicit perturbation of the energy functional to restore C1 regularity, applies standard mountain-pass and genus arguments to the smoothed problems I_ε, obtains uniform bounds from the coercivity of V(x)→∞, and passes to the limit ε→0. This is a self-contained approximation argument relying on external variational principles and explicit construction of the smoothing (e.g., via log(t²+ε)), with no reduction of the existence/multiplicity claim to a fitted parameter, self-definition, or load-bearing self-citation. The assumptions on V are independent and used directly for compactness.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard background results from functional analysis and Sobolev theory for defining weak solutions; no free parameters, ad-hoc axioms, or new invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard functional-analytic setting for weak solutions of semilinear elliptic equations on R^N using Sobolev spaces.
    Invoked to make sense of the variational formulation and the notion of weak solution.

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20 extracted references · 20 canonical work pages

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