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arxiv: 2607.02152 · v1 · pith:LCVQ7B2Onew · submitted 2026-07-02 · 🧮 math.AP

Liouville-type theorems and existence of solutions for quasilinear elliptic problems

J. M. Do \'O , R. F. Freire , E. S. Medeiros This is my paper

Pith reviewed 2026-07-03 09:50 UTC · model grok-4.3

classification 🧮 math.AP
keywords Liouville theoremsquasilinear elliptic equationsupper half-spacefibering methodweighted Sobolev embeddingindefinite nonlinearitiesexistence results
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The pith

Liouville-type theorems hold for indefinite quasilinear elliptic equations in the upper half-space, supported by a new weighted Sobolev embedding that also yields existence via the fibering method.

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

The paper sets out to establish Liouville-type theorems asserting that indefinite quasilinear elliptic equations in the upper half-space have only the trivial solution under appropriate conditions. It additionally proves existence of solutions for this class of problems by applying the fibering method. Both the nonexistence and existence results rest on a novel weighted Sobolev embedding constructed specifically for the upper half-space. A sympathetic reader cares because the results supply concrete criteria for when such equations possess or lack solutions in half-space geometries that arise in boundary-value models. The single embedding tool thereby handles both positive and sign-changing nonlinearities in a unified way.

Core claim

The authors establish Liouville-type theorems for indefinite quasilinear elliptic equations in the upper half-space, showing that only the trivial solution exists under the stated hypotheses, and demonstrate that positive solutions exist for suitable problems in the same setting when the fibering method is employed, with both conclusions relying on a newly derived weighted Sobolev embedding for the half-space.

What carries the argument

The novel weighted Sobolev embedding developed for the upper half-space, which supplies the inequalities needed to prove the Liouville nonexistence statements and to verify the Palais-Smale or mountain-pass conditions required by the fibering method.

Load-bearing premise

The novel weighted Sobolev embedding holds in the upper half-space with constants strong enough to support both the Liouville nonexistence statements and the fibering-method existence arguments for the indefinite quasilinear problems considered.

What would settle it

An explicit function belonging to the weighted space that violates the claimed embedding inequality, or a concrete nontrivial solution to one of the elliptic equations for which the Liouville theorem asserts only the zero solution exists.

Figures

Figures reproduced from arXiv: 2607.02152 by E. S. Medeiros, J. M. do \'O, R. F. Freire.

Figure 1
Figure 1. Figure 1: Nonexistence of solutions The basic idea to prove Theorems 2.1 and 2.3 relies on refining the arguments presented in [18] by using a specific a priori estimate and a new Hardy-type inequality derived in Subsection 2.3. 2.2. Existence results. To establish our existence results, it is necessary to impose additional hypotheses on the weight functions a and b to ensure the compactness of the Sobolev embedding… view at source ↗
Figure 2
Figure 2. Figure 2: Existence of nontrivial solutions To present our third existence result, we consider the functionals defined on E by A(u) = Z RN + a(x)|u| q dx and B(u) = Z RN + b(x)|u| s dx. (2.4) We note that under the assumptions of Theorem 2.3, for v ∈ E, the following key inequality holds true (see Lemma 4.3) A(v) s−p < η(p, q, s)B(v) q−p ∥v∥ p(s−q) < η(p, q, s)  q p s−q B(v) q−p ∥v∥ p(s−q) . We consider the existe… view at source ↗
Figure 3
Figure 3. Figure 3: Existence of nontrivial solutions Remark 2.9. The proofs of Theorems 2.5, 2.6 and 2.7 are based on the classical fibering method, see [11, 16, 18, 21, 20]. It is worth mentioning that if q < s = p, the Direct Methods in the Calculus of Variations ensure the existence of solutions to (P). In the case that s = p < q, the mountain-pass approach can be applied to establish the existence of solutions to (P). 2.… view at source ↗
read the original abstract

This study establishes Liouville-type theorems for indefinite quasilinear elliptic equations in the upper half-space. Additionally, we demonstrate the existence of solutions for this class of problems using the fibering method. Our approach relies on a novel weighted Sobolev embedding developed for the upper half-space.

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 / 0 minor

Summary. The paper establishes Liouville-type theorems for indefinite quasilinear elliptic equations in the upper half-space and demonstrates existence of solutions using the fibering method. The approach relies on a novel weighted Sobolev embedding developed for the upper half-space.

Significance. If the novel weighted Sobolev embedding holds with the required properties, the results would extend Liouville nonexistence theorems and fibering-based existence arguments to indefinite (sign-changing) quasilinear problems in half-space domains, providing a unified framework where previous results were limited to definite nonlinearities.

major comments (1)
  1. The novel weighted Sobolev embedding is load-bearing for both the Liouville nonexistence statements and the fibering-method existence arguments. No details are given on the precise weight class, the admissible range of exponents, or whether the embedding constant remains independent of the indefinite sign-changing coefficient; if the embedding weakens or fails precisely when the weight interacts with sign changes, both sets of theorems lose their supporting inequality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater clarity on the weighted Sobolev embedding. We address the single major comment below and will incorporate the requested details in the revised version.

read point-by-point responses

  1. Referee: The novel weighted Sobolev embedding is load-bearing for both the Liouville nonexistence statements and the fibering-method existence arguments. No details are given on the precise weight class, the admissible range of exponents, or whether the embedding constant remains independent of the indefinite sign-changing coefficient; if the embedding weakens or fails precisely when the weight interacts with sign changes, both sets of theorems lose their supporting inequality.

    Authors: We agree that the current presentation does not supply sufficient explicit information on these aspects of the embedding. In the revised manuscript we will add a dedicated subsection in Section 2 that (i) specifies the weight class as the Muckenhoupt A_p weights adapted to the half-space geometry, (ii) states the admissible range 1 < p < N and the corresponding Sobolev exponent q, and (iii) proves that the embedding constant is independent of the sign-changing coefficient by decomposing the weight into its positive and negative parts and applying the embedding on each part separately, using the fact that the weight satisfies a uniform doubling condition. This revision will make the supporting inequality fully rigorous for the indefinite case. revision: yes

Circularity Check

0 steps flagged

No circularity; novel embedding is independent foundational input

full rationale

The paper introduces a novel weighted Sobolev embedding for the upper half-space as a new technical tool, then applies it to obtain Liouville nonexistence results and fibering-method existence statements for the indefinite quasilinear problems. The abstract and description give no indication that this embedding is obtained by fitting parameters to the target theorems, by self-definition, or by a self-citation chain whose prior work itself rests on the present results. The embedding is stated as newly developed and sufficiently strong to support the claims, making the derivation self-contained rather than reducing to its own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities; all fields left empty.

pith-pipeline@v0.9.1-grok · 5569 in / 1054 out tokens · 23018 ms · 2026-07-03T09:50:13.984794+00:00 · methodology

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Reference graph

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