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arxiv: 2504.13434 · v2 · submitted 2025-04-18 · 🧮 math.AP

Global boundedness for Generalized Schr\"odinger-Type Double Phase Problems in mathbb{R}^N and Applications to Supercritical Double Phase Problems

Hoang Hai Ha , Ky Ho , Bui The Quan , Inbo Sim This is my paper

Pith reviewed 2026-05-22 19:39 UTC · model grok-4.3

classification 🧮 math.AP
keywords global boundednessdouble phase problemsvariable exponentsDe Giorgi iterationSchrödinger-type equationssupercritical growthlocalization methodweak solutions
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The pith

Weak solutions to generalized Schrödinger-type double phase problems in R^N are globally bounded under new critical growth conditions.

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

The paper proves two global boundedness results for weak solutions of generalized Schrödinger-type double phase equations with variable exponents over the whole space R^N. In the subcritical growth regime it applies De Giorgi iteration after a localization procedure to obtain uniform bounds and a decay estimate at infinity. For the critical growth regime the same iteration is run with a localization tailored to the critical case, again yielding global bounds. These bounds are then used to establish existence of weak solutions for the corresponding supercritical problems. The statements remain new even when all exponents are constant.

Core claim

Under the new critical growth conditions introduced in the cited works, any weak solution of the generalized Schrödinger-type double phase problem with variable exponents satisfies a uniform L^∞ bound on all of R^N. The proof proceeds by De Giorgi iteration: for subcritical growth a standard localization controls the variable-exponent terms at infinity and produces both the bound and a decay property; for critical growth a suitably adapted localization closes the iteration directly. The resulting global boundedness immediately yields existence of weak solutions for supercritical double-phase problems.

What carries the argument

De Giorgi iteration paired with a localization method in R^N that is adapted to the growth regime and the variable exponents.

If this is right

  • Weak solutions decay to zero at infinity when the growth is subcritical.
  • Existence of weak solutions follows for the associated supercritical double-phase problems.
  • The same boundedness holds when all exponents reduce to constants, giving new results in the whole-space constant-exponent setting.
  • The localization technique works uniformly without further restrictions on the variable exponents beyond the stated growth conditions.

Where Pith is reading between the lines

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

  • The localization method could be checked on radial test problems with explicit variable exponents to see how sharp the decay rate is.
  • Similar iteration-plus-localization arguments might apply to double-phase problems with nonlocal operators or on unbounded domains with different geometry.
  • The existence result for supercritical cases opens the possibility of studying multiplicity or symmetry breaking once boundedness is assured.

Load-bearing premise

The new critical growth conditions are compatible with the localization procedure so that the iteration can control the variable-exponent terms uniformly at infinity.

What would settle it

An explicit weak solution that remains unbounded on some sequence of points going to infinity in R^N, while satisfying the equation and the growth conditions only marginally violated, would disprove the global boundedness claim.

read the original abstract

We establish two global boundedness results for weak solutions to generalized Schr\"{o}dinger-type double phase problems with variable exponents in $\mathbb{R}^N$ under new critical growth conditions optimally introduced in [26, 32]. More precisely, for the case of subcritical growth, we employ the De Giorgi iteration with a suitable localization method in $\mathbb{R}^N$ to obtain a-priori bounds. As a byproduct, we derive the decay property of weak solutions. For the case of critical growth, using the De Giorgi iteration with a localization adapted to the critical growth, we prove the global boundedness. As an interesting application of these results, the existence of weak solutions for supercritical double phase problems is shown. These results are new even for problems with constant exponents in $\mathbb{R}^N$.

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 establishes two global boundedness results for weak solutions to generalized Schrödinger-type double phase problems with variable exponents in R^N. For subcritical growth, De Giorgi iteration combined with a localization method in R^N yields a priori L^∞ bounds together with a decay property. For critical growth, the iteration is adapted to the critical regime to obtain global boundedness. These estimates are applied to prove existence of weak solutions for the corresponding supercritical problems. The results are asserted to be new even in the constant-exponent case.

Significance. If the localization argument succeeds in producing uniform constants, the work would supply useful global a priori estimates for double-phase problems on unbounded domains, extending local regularity theory and enabling existence results in the supercritical range. The explicit use of the critical growth conditions from the cited references [26,32] and the decay byproduct are positive features when the uniformity can be verified.

major comments (2)
  1. [Localization procedure (Sections 3–4)] The central localization argument (Sections 3–4) applies De Giorgi iteration to obtain global bounds from local estimates. The Caccioppoli-type inequalities and modular estimates contain constants that depend on the local values of the variable exponents p(x) and q(x). Without an explicit hypothesis that p(x) and q(x) possess a limit at infinity or have uniformly controlled oscillation, these constants become position-dependent; the passage from local to global bounds then requires an additional uniformity argument that is not visible in the abstract and may be missing from the growth conditions imported from [26,32]. Please identify the precise location where this uniformity is established or add the necessary decay/limit assumption on the exponents.
  2. [Growth conditions and main theorems] Table or theorem statements that import the critical growth conditions from [26,32] should include a short verification that these conditions remain compatible with the whole-space cut-off functions and do not introduce position-dependent factors that would invalidate the iteration constants at infinity.
minor comments (2)
  1. [Abstract] The abstract refers to “new critical growth conditions optimally introduced in [26, 32]” without a one-sentence description; adding a brief indication of the form of these conditions would improve readability.
  2. [Introduction] Notation for the variable exponents and the precise range of p(x), q(x) should be stated explicitly in the introduction before the statement of the main results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The observations regarding uniformity of constants in the localization argument and compatibility of growth conditions have prompted us to clarify and strengthen the presentation. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Localization procedure (Sections 3–4)] The central localization argument (Sections 3–4) applies De Giorgi iteration to obtain global bounds from local estimates. The Caccioppoli-type inequalities and modular estimates contain constants that depend on the local values of the variable exponents p(x) and q(x). Without an explicit hypothesis that p(x) and q(x) possess a limit at infinity or have uniformly controlled oscillation, these constants become position-dependent; the passage from local to global bounds then requires an additional uniformity argument that is not visible in the abstract and may be missing from the growth conditions imported from [26,32]. Please identify the precise location where this uniformity is established or add the necessary decay/limit assumption on the exponents.

    Authors: We appreciate the referee's identification of this subtlety. The manuscript assumes p(x) and q(x) are continuous and globally bounded (as is standard for variable-exponent double-phase problems), and the growth conditions from [26,32] are stated for the whole-space setting. Nevertheless, to render the uniformity of iteration constants fully explicit, we have added in the revised Section 2 an assumption that p(x) and q(x) admit limits as |x|→∞. This guarantees that local constants stabilize at infinity and can be bounded uniformly by their global supremum. With this hypothesis in place, the localization procedure in Sections 3–4 proceeds with position-independent constants, yielding the claimed global bounds and decay. A clarifying remark has also been inserted after the statement of the main theorems. revision: yes

  2. Referee: [Growth conditions and main theorems] Table or theorem statements that import the critical growth conditions from [26,32] should include a short verification that these conditions remain compatible with the whole-space cut-off functions and do not introduce position-dependent factors that would invalidate the iteration constants at infinity.

    Authors: We agree that an explicit verification improves readability. In the revised manuscript we have augmented the statements of Theorems 1.1 and 1.2 with a short paragraph verifying compatibility. Because the cut-off functions employed in the localization have compact support and the exponents are globally bounded and continuous, the modular estimates absorb any local variation without generating unbounded position-dependent factors. Consequently, the constants in the De Giorgi iteration remain uniform throughout ℝ^N, including at infinity. This verification is fully consistent with the hypotheses already present in [26,32] and does not alter the validity of the results. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for growth conditions; De Giorgi localization derivation remains independent

full rationale

The paper derives global boundedness via De Giorgi iteration plus localization in R^N, adapting standard techniques to variable-exponent double-phase Schrödinger problems. The central steps rely on modular inequalities, Caccioppoli estimates, and iteration that are constructed directly from the weak formulation and the assumed growth conditions imported from [26,32]. No equation or claim reduces by construction to a fitted parameter, self-definition, or prior result by the same authors that would make the boundedness tautological. The self-citation is limited to the statement of critical growth and is not invoked as a uniqueness theorem or to close the derivation loop. The argument is therefore self-contained against external benchmarks once the growth hypotheses are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on the growth conditions from the cited references and on the structural assumptions that make the double-phase operator amenable to De Giorgi iteration in R^N.

axioms (1)
  • domain assumption The double-phase operator satisfies standard ellipticity and growth conditions with variable exponents p(x) and q(x) that allow the De Giorgi iteration to close.
    Invoked implicitly when the authors state that the new critical growth conditions permit the localization method.

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