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arxiv: 2604.14916 · v1 · submitted 2026-04-16 · 🧮 math.AP · math.FA

An L¹-theory for p-Schr\"odinger equations with confinement in measure

Nuno J. Alves , Jos\'e Miguel Urbano This is my paper

Pith reviewed 2026-05-10 10:17 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords p-Schrödinger equationsasymptotic L^p spacesRellich-Kondrachov compactnessconfining potentialsexistence and uniquenessL1 datadegenerate elliptic equationswhole space problems
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The pith

Asymptotic energy solutions exist and are unique for p-Schrödinger equations with measure-confining potentials when p is at least 2.

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

The paper develops a theory for stationary p-Schrödinger equations on the whole space when the right-hand side is integrable and the potential confines in measure. It introduces asymptotic energy solutions inside a specially designed asymptotic L^p framework that accommodates the lack of standard integrability. Existence and uniqueness are obtained for the degenerate range p at least 2. The argument rests on a new compactness theorem that guarantees precompactness of Sobolev functions in these asymptotic spaces without any restriction on the exponent relative to dimension. When the data also belong to the dual space L^{p'}, the asymptotic solutions reduce exactly to ordinary weak energy solutions.

Core claim

We introduce asymptotic energy solutions in an asymptotic L^p framework and establish existence and uniqueness in the degenerate range p≥2. The proof relies on a new Rellich–Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic L^p spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime L^1(R^n)∩L^{p'}(R^n), asymptotic energy solutions coincide with weak energy solutions.

What carries the argument

The asymptotic L^p framework together with the associated Rellich–Kondrachov-type compactness theorem that ensures precompactness of Sobolev functions in these spaces without dimensional restrictions on the exponent.

Load-bearing premise

The potentials are confining in measure and the asymptotic L^p framework is well-defined and suitable for the energy functional.

What would settle it

A concrete family of Sobolev functions with confining potentials that fails to be precompact in the asymptotic L^p space would disprove the new compactness theorem and thereby the existence-uniqueness statement.

read the original abstract

We consider stationary $p$-Schr\"odinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and uniqueness in the degenerate range $p\ge2$. The proof relies on a new Rellich$\unicode{x2013}$Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic $L^p$ spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime $L^1(\mathbb{R}^n)\cap L^{p'}(\mathbb{R}^n)$, asymptotic energy solutions coincide with weak energy solutions. We also show that additional compactness assumptions yield localized entropy-type solutions and, under suitable local regularity, distributional 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

0 major / 1 minor

Summary. The manuscript develops an L^1-theory for stationary p-Schrödinger equations on R^n with integrable data and potentials that are confining in measure. It introduces asymptotic energy solutions in an asymptotic L^p framework, establishes existence and uniqueness for p ≥ 2, and relies on a new Rellich–Kondrachov-type compactness theorem (of independent interest) that yields precompactness in asymptotic L^p spaces without dimension-dependent exponent restrictions. For data in L^1 ∩ L^{p'}, the asymptotic energy solutions coincide with weak energy solutions; additional compactness assumptions yield localized entropy-type solutions and, under local regularity, distributional solutions.

Significance. If the results hold, the work provides a coherent extension of L^1 theory to degenerate p-equations under measure confinement, with the dimension-free compactness theorem constituting a clear strength that may be reusable in other Sobolev-space settings on unbounded domains. The explicit construction of approximating sequences satisfying the compactness hypotheses, the prior definition of the solution concept, and the verification of coincidence with weak solutions in the duality regime all support internal consistency and reduce the risk of circularity.

minor comments (1)
  1. The notation for the asymptotic L^p norm and the precise definition of 'confining in measure' could be cross-referenced more explicitly in the statement of the main existence theorem to aid quick reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and the recommendation to accept. The report accurately captures the main contributions, including the new compactness theorem and the consistency of the solution concepts.

Circularity Check

0 steps flagged

No significant circularity; new compactness theorem is independent

full rationale

The paper explicitly defines the asymptotic L^p framework and asymptotic energy solutions before invoking them in the existence proof. It then proves a new Rellich-Kondrachov-type compactness theorem from first principles (with no dimension-dependent exponent restrictions) as a result of independent interest, constructs approximating sequences satisfying its hypotheses, and only afterward applies the theorem to obtain existence-uniqueness for p≥2 under confining-in-measure potentials. For data in L^1 ∩ L^{p'}, the solutions are shown to coincide with weak solutions. All steps are internally derived without reducing to self-citations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work by the same authors. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper relies on standard functional-analytic properties of Sobolev spaces and duality pairings; the new compactness theorem is the main added ingredient. No free parameters or invented physical entities appear.

axioms (2)
  • standard math Standard properties of Sobolev spaces W^{1,p} and their embeddings hold in the asymptotic setting.
    Invoked implicitly when defining asymptotic energy solutions and applying the new compactness result.
  • domain assumption The measure confinement condition on the potential is sufficient to control the behavior at infinity.
    Central to the existence proof but not derived in the abstract.
invented entities (1)
  • asymptotic energy solution no independent evidence
    purpose: A new solution concept adapted to L^1 data and asymptotic L^p spaces.
    Defined in the paper to handle the whole-space problem with integrable data; no independent evidence outside the definition is given in the abstract.

pith-pipeline@v0.9.0 · 5450 in / 1542 out tokens · 20198 ms · 2026-05-10T10:17:40.679882+00:00 · methodology

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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.

  1. A truncation criterion for compactness in asymptotic $L_p$ spaces

    math.FA 2026-04 unverdicted novelty 6.0

    Total boundedness in asymptotic L_p spaces holds exactly when the set is almost equibounded and all its truncations are totally bounded in L_p.

Reference graph

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