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arxiv: 2605.19946 · v1 · pith:3SOMF6UPnew · submitted 2026-05-19 · 🧮 math.AP

Normalized groundstates for mixed (p,2)-Laplacian equations in mathbb R² with exponential critical growth

Jiankang Xia , Chao Zhong This is my paper

Pith reviewed 2026-05-20 03:33 UTC · model grok-4.3

classification 🧮 math.AP
keywords normalized groundstatesmixed p-Laplacianexponential critical growthPohozaev manifoldconstrained minimizationMoser iterationLagrange multiplier
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The pith

Normalized groundstates exist for any mass m>0 in mixed (p,2)-Laplacian equations with exponential critical growth, independent of the sign of λ.

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

The paper establishes the existence of normalized groundstates for the mixed (p,2)-Laplacian equation in R^2 subject to a fixed L^2 mass m for any m>0. The mixed operator lacks the regularity needed for standard identities and the exponential critical growth blocks compactness recovery. A refined Moser iteration is developed to obtain the Pohozaev identity from the weak assumption of local C^{1,α} regularity alone. Constrained minimization on the Pohozaev manifold inside a closed L^2 ball is then combined with a minimax argument to produce the groundstates. This construction succeeds without regard to whether the Lagrange multiplier λ is positive or negative.

Core claim

By introducing a refined Moser iteration technique adapted to exponential critical growth, the Pohozaev identity is established for weak solutions under the mere assumption of C_loc^{1,α}-regularity. Combining constrained minimization on the Pohozaev manifold within a closed L^2-ball with a minimax characterization then yields the existence of normalized groundstates for any prescribed mass m>0, and this holds independently of the sign of the Lagrange multiplier λ.

What carries the argument

Refined Moser iteration adapted to exponential critical growth that yields the Pohozaev identity from C_loc^{1,α} regularity, enabling constrained minimization on the Pohozaev manifold inside a closed L^2 ball.

If this is right

  • Normalized groundstates exist for every positive mass m.
  • The existence result holds whether the Lagrange multiplier λ is positive or negative.
  • The Pohozaev identity holds for weak solutions under only local C^{1,α} regularity.
  • Compactness is restored for the mixed operator despite the exponential critical growth.

Where Pith is reading between the lines

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

  • The same iteration-plus-manifold strategy may apply to other mixed or nonlocal operators with exponential nonlinearities.
  • The sign-independence of λ opens the door to studying orbital stability of these groundstates.
  • Numerical approximation of the minimizers could verify the predicted mass dependence for concrete choices of f and p.

Load-bearing premise

Weak solutions possess local C^{1,α} regularity so the refined Moser iteration can produce the Pohozaev identity.

What would settle it

A concrete weak solution that is not locally C^{1,α} and violates the Pohozaev identity, or a specific mass value m for which no critical point exists on the constrained Pohozaev manifold.

read the original abstract

We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*} \begin{cases} -\Delta_p u-\Delta u+\lambda u=f(u) & \text{in } \mathbb{R}^2, \displaystyle \int_{\mathbb{R}^2}|u|^2\,\mathrm{d}x=m, u\in H^1(\mathbb{R}^2)\cap D^{1,p}(\mathbb{R}^2), \end{cases} \end{align*} where $\Delta_p$ denotes the $p$-Laplacian with $1<p<2$, $\lambda\in\mathbb{R}$ represents a Lagrange multiplier and the nonlinerity $f$ exhibits exponential critical growth. Compared to the single-Laplacian case, the lack of regularity here precludes the Pohozaev identity, and the exponential critical growth severely compromises the restoration of compactness. To address these issues, we introduce a refined Moser iteration technique adapted to exponential critical growth, which establishes the Pohozaev identity for weak solutions under the mere assumption of $C_{\mathrm{loc}}^{1,\alpha}$-regularity. By combining constrained minimization on the Pohozaev manifold within a closed $L^2$-ball with a minimax characterization, we prove the existence of normalized groundstates for any prescribed mass $m>0$. Notably, our approach works independently of the sign of the Lagrange multiplier $\lambda$, thereby surmounting the fundamental barrier in recovering compactness for mixed Laplacian problems.

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 proves existence of normalized groundstates for the mixed (p,2)-Laplacian equation -Δ_p u - Δu + λu = f(u) in R^2 with ∫|u|^2 = m and exponential critical growth on f, for any m>0 and independently of the sign of λ. The strategy relies on a refined Moser iteration to recover the Pohozaev identity for weak solutions possessing only C_loc^{1,α} regularity, followed by constrained minimization on the Pohozaev manifold inside a closed L^2-ball combined with a minimax characterization.

Significance. If the technical details hold, the result is significant for variational methods applied to quasilinear problems with mixed operators and critical exponential nonlinearities. It extends single-Laplacian results by handling the reduced regularity of the mixed operator and by removing the usual sign restriction on λ that obstructs compactness recovery. The constrained minimization on the Pohozaev manifold within an L^2-ball is a clean construction that avoids direct dependence on λ.

major comments (1)
  1. [Abstract and Moser iteration section] Abstract (paragraph on technical issues) and the section presenting the refined Moser iteration: the argument that this iteration yields the Pohozaev identity for weak solutions under mere C_loc^{1,α} regularity must separately control the degenerate p-term (|∇u|^p when |∇u| is small, since 1<p<2) versus the 2-term, together with the exponential remainder f(u)(x·∇u). Explicit estimates showing that the integrated identity passes to the limit in approximating domains are needed; without them the Pohozaev manifold construction for the minimizers obtained from the constrained problem is not justified.
minor comments (2)
  1. [Introduction] The notation for the mixed operator and the space H^1(R^2) ∩ D^{1,p}(R^2) is clear, but the precise definition of the exponential critical growth condition on f should be stated explicitly in the introduction rather than only referenced.
  2. [Main theorem] In the statement of the main theorem, the dependence of the groundstate on m should be made explicit (e.g., u_m) to avoid ambiguity when discussing the mass constraint.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We appreciate the referee's careful analysis and constructive feedback on our paper. The major comment correctly identifies the need for more detailed estimates in justifying the Pohozaev identity via the refined Moser iteration, particularly regarding the degenerate behavior of the p-Laplacian term. We will revise the manuscript to include these explicit estimates, ensuring the construction of the Pohozaev manifold is fully justified. No other major issues were raised.

read point-by-point responses

  1. Referee: [Abstract and Moser iteration section] Abstract (paragraph on technical issues) and the section presenting the refined Moser iteration: the argument that this iteration yields the Pohozaev identity for weak solutions under mere C_loc^{1,α} regularity must separately control the degenerate p-term (|∇u|^p when |∇u| is small, since 1<p<2) versus the 2-term, together with the exponential remainder f(u)(x·∇u). Explicit estimates showing that the integrated identity passes to the limit in approximating domains are needed; without them the Pohozaev manifold construction for the minimizers obtained from the constrained problem is not justified.

    Authors: We thank the referee for this precise observation. Upon re-examination, we acknowledge that while the refined Moser iteration in Section 3 is adapted to the mixed operator and exponential growth, the passage to the limit for the Pohozaev identity in approximating domains requires more explicit control on the terms mentioned. In the revised version, we will insert a new subsection (e.g., Subsection 3.4) that separately estimates the contribution of the p-Laplacian term in regions where |∇u| is small by using the integrability from D^{1,p}(R^2) and a cutoff function argument. For the 2-Laplacian term, standard integration by parts applies directly. The exponential remainder f(u)(x·∇u) will be bounded using the Trudinger-Moser inequality and the L^2-mass constraint to show uniform integrability. We will then demonstrate that the boundary terms on ∂B_R vanish as R→∞ and the identity holds for the weak solution. This revision will be made to justify the subsequent minimization on the Pohozaev manifold. revision: yes

Circularity Check

0 steps flagged

No significant circularity: direct variational construction with independent Moser iteration lemma

full rationale

The derivation proceeds by first proving a refined Moser iteration that yields the Pohozaev identity for any weak solution possessing only C_loc^{1,α} regularity; this identity is then applied to define the Pohozaev manifold on which constrained minimization (inside a closed L^2-ball) is performed, together with a minimax characterization. Both the iteration and the subsequent minimization are presented as explicit constructions from standard variational tools adapted to the mixed operator and exponential nonlinearity, without any reduction of the target existence statement to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The argument therefore remains self-contained against external benchmarks and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard Sobolev embeddings, Trudinger-Moser inequality, and variational calculus in H^1 ∩ D^{1,p}; no new free parameters or invented entities are introduced. The key new step is the adapted Moser iteration whose validity is asserted under C^{1,α}_loc regularity.

axioms (2)
  • standard math Standard Sobolev and Trudinger-Moser embeddings hold for the mixed space H^1(R^2) ∩ D^{1,p}(R^2)
    Invoked to control the exponential nonlinearity and compactness restoration.
  • domain assumption Weak solutions possess C_loc^{1,α} regularity sufficient for integration by parts in the Pohozaev identity
    Explicitly stated as the minimal assumption under which the refined Moser iteration works.

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