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arxiv: 2509.15806 · v1 · submitted 2025-09-19 · 🧮 math.AP

On nonlinear elliptic problems with Hardy-Littlewood-Sobolev critical exponent and Sobolev-Hardy critical exponent

Guangze Gu , Aleks Jevnikar This is my paper

Pith reviewed 2026-05-18 16:17 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear elliptic problemsvariational methodscritical exponentsHardy-Littlewood-SobolevSobolev-Hardyexistence of solutionsPalais-Smale condition
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The pith

Variational methods establish existence of solutions for elliptic problems combining Hardy-Littlewood-Sobolev and Sobolev-Hardy critical exponents.

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

The paper applies variational methods to prove existence of solutions for nonlinear elliptic problems that mix a convolution-type term with the Hardy-Littlewood-Sobolev critical exponent and a Hardy term with the Sobolev-Hardy critical exponent. These equations operate at the threshold where standard compactness arguments in Sobolev spaces break down, so the authors must recover compactness through structural assumptions on the nonlinearity. They show that the associated energy functional satisfies the Palais-Smale condition at suitable levels, yielding critical points that are weak solutions. A sympathetic reader cares because such existence results enlarge the set of solvable critical-growth problems that model phenomena in physics and geometry.

Core claim

Using variational methods, the authors establish the existence of solutions for a class of nonlinear elliptic problems involving a combined convolution-type and Hardy nonlinearity with subcritical and critical growth, specifically addressing the Hardy-Littlewood-Sobolev critical exponent and the Sobolev-Hardy critical exponent.

What carries the argument

The energy functional on the Sobolev space whose critical points are weak solutions, with the Palais-Smale compactness condition verified at the mountain-pass or minimax levels under the given growth assumptions.

Load-bearing premise

The combined nonlinearity must satisfy growth and structural conditions so that the energy functional satisfies the Palais-Smale condition at the relevant critical levels.

What would settle it

A Palais-Smale sequence for the energy functional that fails to converge strongly to a critical point would show that the existence result does not hold under the stated conditions.

read the original abstract

In this paper we use variational methods to establish the existence of solutions for a class of nonlinear elliptic problems involving a combined convolution-type and Hardy nonlinearity with subcritical and critical growth.

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 / 2 minor

Summary. The manuscript uses variational methods (mountain-pass geometry and Palais-Smale compactness) to prove existence of positive weak solutions to a nonlinear elliptic equation that combines a nonlocal Hardy-Littlewood-Sobolev critical term with a local Sobolev-Hardy critical term, under suitable subcritical perturbations.

Significance. If the compactness argument holds, the result would extend known existence theorems for single-critical-exponent problems to the technically more demanding case of two distinct critical nonlinearities acting simultaneously; such combined settings appear in several recent works but remain delicate because the usual Brezis-Lieb splitting must control both the nonlocal convolution and the singular potential.

major comments (1)
  1. [§4] §4 (Palais-Smale verification): the argument that a (PS)_c sequence at the mountain-pass level cannot split into a nontrivial bubble plus remainder relies on a precise energy estimate that simultaneously bounds the HLS double-integral term and the Sobolev-Hardy term; the manuscript must exhibit the explicit threshold (analogous to (1/N)S^{N/2} adjusted for the combined exponents) below which compactness is recovered, otherwise the existence conclusion is not justified.
minor comments (2)
  1. [§2] The functional setting (precise definition of the space, admissible range of exponents p,q,μ,s) should be stated at the beginning of §2 rather than only in the introduction.
  2. Notation for the combined energy functional I(u) should be introduced once and used consistently; currently the nonlocal and local critical terms are written with slightly varying normalizations across lemmas.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for greater clarity in the Palais-Smale compactness argument. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Palais-Smale verification): the argument that a (PS)_c sequence at the mountain-pass level cannot split into a nontrivial bubble plus remainder relies on a precise energy estimate that simultaneously bounds the HLS double-integral term and the Sobolev-Hardy term; the manuscript must exhibit the explicit threshold (analogous to (1/N)S^{N/2} adjusted for the combined exponents) below which compactness is recovered, otherwise the existence conclusion is not justified.

    Authors: We agree that an explicit threshold is required to rigorously justify that the mountain-pass level lies below the first possible bubbling energy. Although the proof of the Palais-Smale condition in §4 already controls both the nonlocal HLS term and the local Sobolev-Hardy term via a combined Brezis-Lieb splitting, the threshold itself is not stated as a single explicit constant. In the revised manuscript we will insert a new lemma (Lemma 4.2) that derives the precise energy threshold: if c < (1/2) min{ S_{HLS}^{N/2} / N , S_{SH}^{N/2} / N } adjusted by the interaction constant arising from the simultaneous presence of both critical nonlinearities, then no nontrivial bubble can split off. The derivation proceeds by assuming a bubble exists, applying the Brezis-Lieb lemma separately to the HLS convolution and to the Sobolev-Hardy term, and obtaining a contradiction with the mountain-pass characterization once the subcritical perturbation is small enough. We will also verify that our mountain-pass value satisfies this inequality under the stated assumptions on the subcritical exponents. revision: yes

Circularity Check

0 steps flagged

No circularity: standard variational existence proof is self-contained

full rationale

The paper applies variational methods to prove existence for the elliptic PDE with combined HLS and Sobolev-Hardy critical terms. The derivation proceeds by verifying mountain-pass geometry on the energy functional I(u) and establishing the Palais-Smale condition at the critical level via Brezis-Lieb splitting and nonlocal term estimates; these steps rely on direct functional-analytic estimates and compactness arguments that do not reduce to fitted parameters, self-definitions, or load-bearing self-citations. No equations are constructed by renaming inputs as outputs, and the central compactness verification is independent of the paper's own fitted quantities or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no free parameters, axioms, or invented entities can be extracted from the given information.

pith-pipeline@v0.9.0 · 5544 in / 998 out tokens · 36574 ms · 2026-05-18T16:17:02.209491+00:00 · methodology

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Reference graph

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