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Pohozaev-type identities for classes of quasilinear elliptic local and nonlocal equations and systems, with applications

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Pith's one-line read The paper proves an exact Pohozaev-type identity for weak solutions of quasilinear equations mixing anisotropic and fractional p-Laplace operators, and derives nonexistence results from it.

desk verdict Useful new identity for mixed local/nonlocal p-Laplace equations, but the proof has two technical gaps that leave Theorem 2.5 unproved for part of the stated solution space. read the letter →

arxiv 2506.08667 v1 pith:TUO6MJ5C submitted 2025-06-10 math.AP

Gurdev Chand Anthal , Prashanta Garain This is my paper
classification math.AP MSC 35R1135J9235A0135J62
keywords Pohozaevidentityanisotropicp-LaplaceequationfractionalmixedlocalandnonlocalsystemofquasilinearequationsnonexistenceFinsler-Minkowskinormweaksolutions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that every weak solution in the stated energy space of the equation \(-\$\alpha$ H_p u+\$\beta$(-\$\Delta$)_p^s u=f(u)\) in \(\mathbb R^n\) satisfies an exact balance identity, \(\frac{\$\alpha$(n-p)}{p}\|H(\nabla u)\|^p_{L^p}+\frac{\$\beta$(n-sp)}{p}[u]^p_{s,p}=n\int_{\mathbb R^n}F(u)\,dx\), where \(H\) is any Finsler-Minkowski norm and \(F'=f\). The identity is proved for the purely anisotropic case \((\$\alpha$,\$\beta$)=(1,0)\), the purely fractional case \((0,1)\), and the mixed local-nonlocal operator \((1,\gamma)\); the mixed case is claimed to be new even for \(p=2\). The same balance is extended to systems. A sympathetic reader cares because Pohozaev identities are the standard route to nonexistence and decay statements, and the paper applies them to rule out many power-type solutions.

What carries the argument

The carrying object is the Finsler-Minkowski norm \(H\), a strictly convex, positively homogeneous function whose Euler identity \(x\cdot\nabla H(x)=H(x)\) turns the local operator's integrals into \(\|H(\nabla u)\|^p_{L^p}\). The fractional side is carried by the Gagliardo seminorm \([u]^p_{s,p}\) and the kernel \(J_p(r)=|r|^{p-2}r\). The proof's engine is a pair of test functions built from difference quotients: \(\psi_\$\lambda$\sum_{j=1}^n x_j D_j u\) and \(\sum_{j=1}^n D_j u\), where \(D_j\) is the coordinate difference quotient and \(\psi_\$\lambda$\) is a smooth cutoff that approaches one as \(\$\lambda$\to0\). Convexity makes the error terms \(I_{3,2}+J_{2,2}\) nonnegative; dominated convergence and the growth condition on \(f\) force them to vanish in the limit \(h\to0\). Integration by parts against the vector field \(\Psi_\$\lambda$=\psi_\$\lambda$ x\) produces the boundary terms, and an estimate in the style of [2] makes the remaining boundary integral vanish under the Hölder regularity required in \(X_{\$\alpha$,\$\beta$}\). Sending \(\$\lambda$\to0\) leaves exactly the stated identity.

What would settle it

Take a weak solution of the mixed or fractional equation lying in the \($W^{{1,p}}$\cap $W^{{s,p}}$\) branch of \(X_{\$\alpha$,\$\beta$}\) but not in \($C^{{0,l}}$\) for any \(l>s\). For such a solution, compute the boundary integral \(I\) in (3.19): if \(I\neq0\), the identity acquires an extra surface term and Theorem 2.5 fails as stated. A simpler concrete check is to compute both sides of the identity for a known explicit solution, for example a ground state of the pure power equation at the borderline exponent, and verify equality; a mismatch in any case where the proof's test functions are not admissible would falsify the claimed universality.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 2.5: if \(0<s<1<p<\infty\), \(f\) is continuous with \(|f(t)|\le C|t|^{p-1}\), and \(u\in X_{\$\alpha$,\$\beta$}\) is a weak solution of (1.1), then the displayed identity holds. Here \(X_{\$\alpha$,\$\beta$}\) is \($W^{{1,p}}$(\mathbb R^n)\) for \((\$\alpha$,\$\beta$)=(1,0)\), \($W^{{1,p}}$\cap $W^{{s,p}}$\cap $C^{{0,l}}$\) with \(l>s\) for \((1,\gamma)\), and \(($W^{{s,p}}$\cap $C^{{1,1}}$)\cup($W^{{1,p}}$\cap $W^{{s,p}}$)\) for \((0,1)\). The companion Theorem 2.6 gives the two-component analogue: for a weak solution \((u,v)\) of the system, the same expression with \(u\) and \(v\) added equals \(n\int_{\mathbb R^n}g(u,v)\,dx\). The paper states that in the mixed case the identity is new even for \(p=2\).

Load-bearing premise

The load-bearing premise is that the non-\($C^{1}$\) difference-quotient test functions used in the proof are admissible for every weak solution in \(X_{\$\alpha$,\$\beta$}\), and that the Hölder-type regularity needed to kill the boundary term holds on the full range of the stated spaces, neither of which is established by a density or approximation argument.

Editorial extensions

If this is right

  • For \((\alpha,\beta)=(1,0)\), every nontrivial bounded weak solution of \(-H_p u=\lambda (t_+)^{q-1}-\mu(t_-)^{q-1}\) is ruled out unless \(q=p^*=np/(n-p)\).
  • For \((\alpha,\beta)=(0,1)\), the same nonexistence conclusion holds unless \(q=p^*_s=np/(n-sp)\).
  • For the mixed operator with \((\alpha,\beta)=(1,1)\), no nontrivial bounded weak solution exists when \(q\ge p^*\) or \(q\le p^*_s\), because the two coefficients in the identity have the same sign and force the energy to vanish.
  • The system version gives the same critical-exponent obstruction for \(g(u,v)=(\lambda|u|^q+\mu|v|^q)/q\), restricted to sign-definite solutions, and leaves the window \(p^*_s<q<p^*\) open in the mixed case.
  • When \(p=q\), these results become nonexistence statements for the eigenvalue-type problem \(-\alpha H_p u+\beta(-\Delta)_p^s u=\lambda u_+^{p-1}-\mu u_-^{p-1}\) and its system counterpart.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same difference-quotient scheme should produce Pohozaev identities for systems whose two components carry different local and fractional orders, or different exponents \(p_1,p_2\), after adjusting the weighting coefficients.
  • Beyond the paper, adding a Hardy-type potential \(|x|^{-a}u\) or a sign-changing weight to (1.1) should introduce an extra boundary or gradient term in the identity; such identities are the usual first step toward classification of singular solutions.
  • Beyond the paper, the proof's reliance on Hölder regularity suggests that the identity may actually hold on the full \(W^{1,p}\cap W^{s,p}\) branch of \(X_{0,1}\) once a regularization argument is supplied, making the split definition of the solution space unnecessary.
  • Beyond the paper, the open window \(p^*_s<q<p^*\) in the mixed case is where the nonexistence argument gives no conclusion and where possible critical or multiple-solution phenomena for mixed operators should be investigated.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper derives Pohozaev-type identities for weak solutions of quasilinear elliptic equations and systems involving the anisotropic p-Laplace operator, the fractional p-Laplace operator, and a mixed local-nonlocal combination. The main results (Theorems 2.5 and 2.6) state a scaling identity relating the H(∇u) L^p norm and the fractional seminorm to the integral of the nonlinearity. The proofs use difference quotients of the solution as test functions and pass to limits h→0 and λ→0. The paper also gives nonexistence applications for model nonlinearities and eigenvalue-type problems. The claimed novelty is the mixed-case identity, even for p=2.

Significance. If established in full generality, the identities would constitute a useful extension of the known Pohozaev identities for the p-Laplacian and fractional p-Laplacian to the anisotropic and mixed settings, with the mixed case being genuinely new to the best of the authors' and my knowledge. The proof strategy is transparent, the scaling of the final identity is consistent with a variational picture, and the material contains no fitted parameters or circular normalizations. The nonexistence applications are standard in form but would follow directly from the identities. However, the proof as written has two substantial gaps that affect the central theorems, so the significance is conditional on repairing those gaps.

major comments (3)
  1. [Section 3, Eq. (3.3) and estimate of I_{3,2}+J_{2,2}] The proof of Theorem 2.5 tests the weak form (2.2) with φ = ψ_λ Σ_j x_j D_j^h u in (3.3) and, later, with φ = Σ_j D_j^h u in the estimate of I_{3,2}+J_{2,2}. Definition 2.2 only admits C_c^1 test functions, and for u ∈ W^{1,p} or u ∈ W^{s,p} the difference quotient D_j^h u need not have a weak gradient, so the anisotropic term in (2.2) is not defined for these φ. No density or approximation argument is supplied. The formal h→0 limit is therefore not justified for the full class X_{α,β}; this gap affects all three branches of the solution space, including the purely anisotropic case.
  2. [Section 3, Eq. (3.19)] The boundary term I in the J_1 estimate is declared zero using 'u ∈ C^{0,l} for some l > s'. This regularity is part of the definition of X_{1,γ} and of the first branch of X_{0,1}, but X_{0,1} also contains the branch W^{1,p} ∩ W^{s,p}, for which C^{0,l} regularity is not guaranteed for any l > s. Functions in this branch can have point singularities for which the boundary integral does not vanish, so the I = 0 step is not established for the stated class. The theorem needs either a restriction of X_{0,1} to the Hölder/C^{1,1} branch or an alternative estimate for I.
  3. [Theorem 2.6 and Section 4] Theorem 2.6 and the nonexistence applications in Propositions 4.1 and 4.2 inherit both gaps described above. The system proof repeats the same non-admissible test functions, and Proposition 4.2 applies Theorem 2.6 to solutions in X_{0,1} ∩ L^∞ ∩ L^q, which may lie in the W^{1,p} ∩ W^{s,p} branch of X_{0,1}. The applications are therefore conditional on closing the gaps in the proof of Theorem 2.5.
minor comments (3)
  1. [Section 3, Eq. (3.23)] The expression 'n+qs' contains the symbol q, which is never defined in the paper; it should presumably read 'n+sp' to match the exponent in the fractional seminorm.
  2. [Section 3, Eq. (3.14)] The chain of equalities '∫ D_j(H(∇u)^p) dx = ∫ D_j(f(u)u) dx = ∫ D_j F(u) dx = 0' is not justified as written: the first equality is not a consequence of (3.9). The subsequent estimate is correct only after also using the weak form with φ = D_j^h u and the identity ∫ D_j F(u) = 0; this should be explained explicitly to avoid confusion.
  3. [Throughout] The paper contains several typographical errors, including 'hypotheis' in Remark 4.3 and inconsistent spacing in displayed formulas; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Pohozaev identity is derived from the weak form by explicit difference-quotient multipliers; the noted regularity gaps are correctness risks, not self-referential reasoning.

full rationale

The derivation chain of Theorem 2.5 is self-contained and non-circular. The identity α(n−p)/p ‖H(∇u)‖^p_{L^p} + β(n−sp)/p [u]^p_{s,p} = n∫F(u) dx is obtained by testing the weak form (2.2) with the explicit multiplier φ = ψ_λ Σ_j x_j D_j u, then computing each resulting term (I1, I2, I3,1, J1, J2,1, L) in the h→0 and λ→0 limits, and showing the remainder I3,2 + J2,2 vanishes by convexity (3.12)–(3.13) together with the weak form tested against Σ_j D_j u. Nothing in this chain is defined in terms of the target identity: F is the ordinary antiderivative of f and the norms are the standard ones. There are no fitted parameters and no prediction forced by construction. The nonexistence applications (Propositions 4.1–4.2) compare two independent consequences of the equation — the u-test identity (4.1) and the Pohozaev identity (4.2) — so the vanishing conclusion is not an input of the theorem. Regarding citations: the only self-citation is [3] (Anthal–Giacomoni–Sreenadh), which appears solely as literature context for mixed local–nonlocal Pohozaev identities in bounded domains and plays no role in the proofs. The genuinely load-bearing citation, for the boundary term I = 0 in (3.19), is [2] (Ambrosio, BLMS 2024) — an external, independent proof invoked with its hypotheses explicitly named (u ∈ C^{0,l}, l > s); the methodology citations [20] are likewise external. Per the rubric, external citations with stated assumptions are independent support and do not constitute circularity. The reader's concerns — that Definition 2.2 admits only C_c^1 test functions while the proof uses difference-quotient multipliers, and that the W^{1,p}∩W^{s,p} branch of X_{0,1} need not satisfy the C^{0,l} hypothesis needed for I = 0 — are genuine correctness/regularity risks (the theorem may be over-stated for that branch), but they are not circularity: the identity does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard functional analysis, a regularity assumption on solutions that is not fully stated in the main theorem, and no fitted parameters or invented entities. The main burden is the unstated density of test functions and the Holder regularity needed for the boundary term.

assumptions (4)
  • domain assumption Finsler-Minkowski norm properties (H1)-(H3) for H, including Euler relation x·nabla H(x)=H(x) and the Lipschitz bound |nabla H(x)| <= C.
    Used throughout Lemma 2.1 and in the I1/I2 estimates; standard for anisotropic p-Laplace operators.
  • ad hoc to paper Weak solutions in X_{alpha,beta} have enough regularity for the boundary term in the J1 estimate, namely C^{0,l} for some l>s, or C^{1,1} in the pure fractional branch.
    The J1 boundary term I=0 in (3.19) is justified only with u in C^{0,l}, l>s, but Theorem 2.5 is stated for W^{1,p} cap W^{s,p} solutions in the (0,1) case without that Holder condition.
  • standard math Translation invariance of Lebesgue integrals gives int D_j K = 0 for K in L^1, and the divergence theorem applies on R^n \ B_mu(y).
    Used in (3.9), (3.11), (3.14), (3.15) and (3.19); standard but unproved.
  • standard math Convexity of t maps to |t|^p and xi maps to H(xi)^p.
    Used in (3.12) and (3.13) to identify nonnegative error terms that vanish in the limit.

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Cite this review

Pith. "Pith review of Pohozaev-type identities for classes of quasilinear elliptic local and nonlocal equations and systems, with applications." pith.science (2026). https://pith.science/paper/TUO6MJ5C

@misc{pith2026250608667,
  author       = {Pith},
  title        = {Pith review of: Pohozaev-type identities for classes of quasilinear elliptic local and nonlocal equations and systems, with applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUO6MJ5C}},
  note         = {Machine review of arXiv:2506.08667}
}
abstract

In this article, we establish Pohozaev-type identities for a class of quasilinear elliptic equations and systems involving both local and nonlocal $p$-Laplace operators. Specifically, we obtain these identities in $\mathbb{R}^n$ for the purely anisotropic $p$-Laplace equations, the purely fractional $p$-Laplace equations, as well as for equations that incorporate both anisotropic and fractional $p$-Laplace features. We also extend these results to the corresponding systems. To the best of our knowledge, the identities we derive in the mixed case are new even when $p=2$. Finally, we illustrate some of the applications of our main results.

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Forward citations

Cited by 1 Pith paper

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