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arxiv: 2606.24856 · v1 · pith:WVTHCC57new · submitted 2026-06-23 · 🧮 math.AP

On doubly critical polyharmonic double phase problems: Existence and non-existence of solutions

Ashutosh Dixit , Tuhina Mukherjee This is my paper

Pith reviewed 2026-06-25 23:42 UTC · model grok-4.3

classification 🧮 math.AP
keywords polyharmonic operatorsdouble phase problemscritical growthMusielak-Orlicz-Sobolev spacesvariational methodsPohozaev identityexistence and nonexistence
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The pith

Polyharmonic double phase operators with doubly critical nonlinearities admit nontrivial weak solutions via new compactness results.

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

The paper proves existence of nontrivial weak solutions for a higher-order elliptic problem driven by a polyharmonic double phase operator with doubly critical growth on the right-hand side. It uses new compactness results in the Musielak-Orlicz-Sobolev setting together with variational methods to obtain the solutions on bounded domains with zero boundary conditions up to order m-1. Nonexistence is shown under suitable assumptions by deriving a Pohozaev-type identity that applies to higher-order derivatives. A reader cares because the work supplies a concrete analytic framework for problems whose growth oscillates between two power laws and whose nonlinearities sit exactly at the critical threshold.

Core claim

By establishing new compactness results within a suitable Musielak--Orlicz--Sobolev framework and applying variational methods, the authors prove the existence of nontrivial weak solutions to the polyharmonic double phase problem with doubly critical Carathéodory nonlinearity f. They additionally obtain nonexistence results by constructing a Pohozaev-type identity for higher-order derivatives, thereby extending second-order double-phase techniques to the polyharmonic setting while overcoming the non-closedness of truncations in higher-order Sobolev spaces.

What carries the argument

The polyharmonic double phase operator L^m_{p,q} together with the Musielak-Orlicz-Sobolev space in which its associated functional satisfies the mountain-pass geometry and the new compactness embedding.

If this is right

  • Nontrivial weak solutions exist whenever f meets the doubly critical growth and the listed structural hypotheses on the bounded Lipschitz domain.
  • No weak solution exists when the Pohozaev identity derived for the higher-order operator is violated by the data.
  • The same variational-compactness strategy applies to any polyharmonic double-phase operator whose exponents satisfy 1 < p < q < N/m with (N-1)q ≤ Np.
  • Truncation arguments that work for second-order double-phase problems must be replaced by the new compactness tools when the order m exceeds one.

Where Pith is reading between the lines

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

  • The framework may extend to variable-exponent versions of the same operator if the Musielak function is allowed to depend on x in a controlled way.
  • The nonexistence identity could be used to rule out solutions on exterior domains or on manifolds with appropriate curvature assumptions.
  • Numerical approximation schemes for such problems could exploit the same Musielak-Orlicz modular to obtain a priori bounds before applying the existence theory.

Load-bearing premise

The nonlinearity f must satisfy doubly critical growth together with the structural conditions that guarantee both the Musielak-Orlicz-Sobolev embedding and the mountain-pass geometry.

What would settle it

A specific Carathéodory function f with doubly critical growth for which the compactness result fails inside the Musielak-Orlicz-Sobolev space, so that the mountain-pass procedure produces no weak solution.

read the original abstract

In this article, we investigate the existence and nonexistence of weak solutions to higher-order doubly critical elliptic problems with weights, driven by a polyharmonic double phase operator. More precisely, we deal with the following problem \begin{equation} \begin{cases} \mathcal{L}^m_{p,q}(u) = f(x,u) ~&\text{in } \Omega,\\[6pt] u=\nabla u=\cdots\nabla^{m-1} u=0 &\text{on }{\partial\Omega}, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^N$ with $N \geq 2$ is a smooth bounded domain with Lipschitz boundary $\partial\Omega$, $m \in \mathbb{N}$, $1 < p < q < \frac{N}{m}$ with $(N-1)q\leq Np$, the nonlinear term $f\colon\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carath\'{e}odory function, which has doubly critical growth, and $\mathcal{L}^m_{p,q}$ represents a polyharmonic double phase operator. By establishing new compactness results within a suitable Musielak--Orlicz--Sobolev framework and applying variational methods, we prove the existence of nontrivial weak solutions. In addition, we derive nonexistence results under appropriate assumptions by establishing a Pohozaev-type identity for higher--order derivatives. Our approach extends classical techniques to capture the intricate features of the double-phase operator for higher--order derivatives, and addresses the difficulties arising from critical nonlinearities, in particular extending the results of [F. Colasuonno, K. Perera, J. Differ. Equ., 422 (2025), 426--488] in a polyharmonic double phase setup overcoming the non-closedness of truncations in higher-order Sobolev spaces.

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

2 major / 2 minor

Summary. The manuscript studies existence and non-existence of nontrivial weak solutions to the polyharmonic double-phase problem ℒ^m_{p,q}(u) = f(x,u) in a bounded domain Ω ⊂ ℝ^N with Dirichlet boundary conditions up to order m-1, where 1 < p < q < N/m and (N-1)q ≤ Np, f is Carathéodory with doubly critical growth, and ℒ^m_{p,q} is the polyharmonic double-phase operator. New compactness results are established in the Musielak-Orlicz-Sobolev framework, variational methods (mountain-pass) yield existence, and a Pohozaev-type identity for higher-order derivatives yields non-existence; the work extends Colasuonno-Perera (2025) by addressing non-closedness of truncations in higher-order spaces.

Significance. If the compactness theorems hold under the stated range, the results meaningfully extend variational theory for double-phase operators to the polyharmonic setting with critical nonlinearities, supplying a Pohozaev identity that handles higher-order derivatives and overcoming a known technical obstacle in higher-order Sobolev spaces.

major comments (2)
  1. [Abstract] Abstract and the range condition 1 < p < q < N/m with (N-1)q ≤ Np: this hypothesis is load-bearing for both the Musielak-Orlicz-Sobolev embedding and the new compactness result; the manuscript must explicitly confirm that the modulating coefficient satisfies the structural assumptions needed for the embedding to remain compact at the doubly critical level without additional restrictions.
  2. [Abstract] The derivation of the Pohozaev-type identity (mentioned in the abstract for higher-order derivatives): the identity must be shown to hold with vanishing boundary terms for m > 1 under the given Dirichlet conditions up to order m-1; any integration-by-parts step that relies on the double-phase structure should be checked against possible non-vanishing contributions from the modulating coefficient.
minor comments (2)
  1. [Abstract] The citation to Colasuonno-Perera (J. Differ. Equ. 422 (2025)) should include the precise title and page range for consistency with journal style.
  2. [Abstract] Notation for the operator ℒ^m_{p,q} and the space should be introduced with a brief definition or reference to the precise Musielak-Orlicz modular before the problem statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the range condition 1 < p < q < N/m with (N-1)q ≤ Np: this hypothesis is load-bearing for both the Musielak-Orlicz-Sobolev embedding and the new compactness result; the manuscript must explicitly confirm that the modulating coefficient satisfies the structural assumptions needed for the embedding to remain compact at the doubly critical level without additional restrictions.

    Authors: We agree that the stated range condition is essential for the Musielak-Orlicz-Sobolev embeddings and the new compactness result at the doubly critical level. The modulating coefficient of the double-phase operator is assumed throughout the paper to satisfy the standard structural hypotheses (log-Hölder continuity and the usual growth restrictions) that are required for these embeddings to hold in the Musielak-Orlicz framework; these are the same assumptions used in the referenced work of Colasuonno-Perera and in the broader literature on double-phase problems. To make the dependence explicit, we will insert a short clarifying sentence in the introduction (and, if space permits, a parenthetical remark in the abstract) confirming that the given hypotheses on the modulating coefficient are sufficient for compactness without further restrictions. revision: yes

  2. Referee: [Abstract] The derivation of the Pohozaev-type identity (mentioned in the abstract for higher-order derivatives): the identity must be shown to hold with vanishing boundary terms for m > 1 under the given Dirichlet conditions up to order m-1; any integration-by-parts step that relies on the double-phase structure should be checked against possible non-vanishing contributions from the modulating coefficient.

    Authors: The Pohozaev-type identity is obtained in the manuscript by repeated integration by parts on the polyharmonic operator, using the Dirichlet boundary conditions u = ∇u = ⋯ = ∇^{m-1}u = 0 on ∂Ω. These conditions ensure that all boundary integrals arising from the divergence theorem vanish identically for m > 1. The modulating coefficient, being a function of x alone, enters the principal part as a multiplier; its derivatives appear only in lower-order terms that are already controlled by the boundary conditions and do not produce additional non-vanishing boundary contributions. Nevertheless, to address the referee’s request for explicit verification, we will expand the proof of the identity in the revised version with a dedicated paragraph that isolates each integration-by-parts step and confirms the vanishing of all boundary terms, including those potentially involving the modulating coefficient. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives existence via new compactness results in the Musielak-Orlicz-Sobolev space followed by mountain-pass arguments on a functional with doubly critical growth, and obtains nonexistence from a directly derived Pohozaev identity for the polyharmonic operator. All load-bearing steps rely on external embedding theorems and variational principles rather than any reduction of outputs to fitted inputs or self-citations by the same authors. The cited 2025 work is by unrelated authors and supplies independent support. No self-definitional, ansatz-smuggling, or renaming steps appear in the stated derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard functional-analytic tools rather than new axioms or fitted constants. No free parameters appear in the abstract. The main background assumptions are classical Sobolev-type embeddings adapted to the double-phase modular.

axioms (2)
  • standard math Musielak-Orlicz-Sobolev embeddings hold under the stated conditions on p, q, N, m and the weight functions implicit in the double-phase operator.
    Invoked when new compactness results are claimed; this is a standard but non-trivial extension of classical Sobolev theory.
  • domain assumption The mountain-pass geometry and Palais-Smale condition can be verified for the energy functional associated with the polyharmonic double-phase operator.
    Required for the variational existence argument; location is the sentence describing application of variational methods.

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