{"id":"a76bdf59-8926-47b9-af3e-fd4aa57a6be9","arxiv_id":"2506.08667","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quasilinear equations mixing anisotropic and fractional p-Laplace operators satisfy a Pohozaev identity with coefficients n-p and n-sp, giving nonexistence for power nonlinearities.","lead":"The paper derives integral identities (Pohozaev identities) for a broad class of nonlinear elliptic equations mixing anisotropic and fractional p-Laplace operators in R^n, along with system versions. The identities force nonexistence of many entire solutions, giving a standard tool for quasilinear PDEs without requiring strong regularity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 2.5 uses test functions outside the weak form and a boundary-term vanishing that fails for the W^{1,p}∩W^{s,p} branch of X_{0,1}; identity not established for full stated class.","rationale":"The reader's verdict already flags the two technical gaps I independently identify. I regard the boundary-term issue as the most load-bearing because it touches the definition of X_{0,1}: the theorem explicitly allows W^{1,p}∩W^{s,p} solutions, yet the proof of the J1 estimate (3.19) needs Hölder regularity C^{0,l} with l>s, which that branch does not provide. The failure is not merely a missing detail: there are functions in W^{1,p}∩W^{s,p} for which the boundary integral diverges, so the I=0 step cannot be justified from the stated hypotheses. The inadmissible test-function issue is equally real and affects all cases, but it is in principle repairable by an approximation argument; the boundary estimate may require altering the solution space or proving extra regularity of weak solutions. Neither gap shows the final identity is false, but both mean the central claim is not established as stated. Since the reader's conditional verdict already calls for exactly this kind of repair, I do not change the verdict. The concrete test isolates the boundary-term step; if it is resolved (e.g., by proving weak solutions have the needed regularity), the remaining gap is the density argument for the difference-quotient test functions.","tokens_in":15903,"tokens_out":24610,"duration_ms":268358,"concrete_test":"Take n=3, p=2, s=1/2 and a compactly supported radial u with u(x)=|x|^{-a} near 0, 0<a<1/2. Then u∈W^{1,2}∩W^{1/2,2} (the W^{1,p}∩W^{s,p} branch of X_{0,1}) but u∉C^{0,l} for any l>1/2. Evaluate the boundary term I in (3.19) with Ψ_λ(x)=x and y=0 as μ→0+; scaling gives |I| ~ μ^{-(sp+ap)} = μ^{-(1+2a)} → ∞. This calculation determines whether the assertion I=0 in the proof is valid under the stated solution-space assumptions, or whether Theorem 2.5 requires either an additional regularity argument for weak solutions or a restriction of X_{0,1} to the C^{1,1} branch.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 2.5) is not proved for the full space X_{α,β} in (2.1). Two load-bearing gaps appear. (i) Definition 2.2 defines weak solutions only against C_c^1 test functions, but the proof of Theorem 2.5 tests (2.2) with φ=ψ_λ Σ_j x_j D_j u in (3.3) and with φ=Σ_j D_j u in the estimate of I_{3,2}+J_{2,2}; D_j is a finite difference quotient, so these φ are not C_c^1 and, for u merely in W^{1,p} or W^{s,p}, need not have a weak gradient, making the anisotropic term in (2.2) undefined. No density or approximation argument is supplied. (ii) In (3.19), the boundary term I is declared zero by using 'u∈C^{0,l} for some l>s'. But X_{0,1} contains the branch W^{1,p}∩W^{s,p}, which does not imply C^{0,l} for any l>s. Functions in this branch can have point singularities (e.g., |x|^{-a} with suitable a) for which the boundary integral diverges, so the proof's I=0 step fails. Unless X_{0,1} is restricted to the Hölder/C^{1,1} branch or an alternative estimate is provided, Theorem 2.5 is unproved for the stated class.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16260,"tokens_out":9033,"duration_ms":95269,"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":[{"comment":"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.","section":"Section 3, Eq. (3.3) and estimate of I_{3,2}+J_{2,2}"},{"comment":"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.","section":"Section 3, Eq. (3.19)"},{"comment":"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.","section":"Theorem 2.6 and Section 4"}],"minor_comments":[{"comment":"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.","section":"Section 3, Eq. (3.23)"},{"comment":"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.","section":"Section 3, Eq. (3.14)"},{"comment":"The paper contains several typographical errors, including 'hypotheis' in Remark 4.3 and inconsistent spacing in displayed formulas; a careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The author list and affiliations are appropriate, and the paper fits the scope of math.AP. The two main gaps are genuine and load-bearing; I do not think the paper should be rejected outright because the stated identities are plausible and the method is standard, but the proof currently does not establish the theorems for the full classes claimed. The authors should be asked to either restrict the solution spaces, notably X_{0,1}, or supply the missing approximation/regularity arguments. In my view this is a major-revision situation, not a reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper derives Pohozaev identities for mixed anisotropic and fractional p-Laplace equations and systems in R^n, and the mixed-case identity is new even for p=2. It also removes the C^1 requirement for the anisotropic case. But the proof as written does not establish the identity for the full stated solution space. The test functions used in the proof are not admissible under Definition 2.2, and one boundary term needs C^{0,l} regularity that part of X_{0,1} does not provide.\n\nWhat's genuinely new and good: the mixed operator case in R^n is a real extension of the bounded-domain results in [3,4,6] and the pure fractional result in [2]. The method — difference quotients plus the Pohozaev multiplier — is a direct synthesis of Ambrosio and Liu-Liu, and the computations are coherent. The identity has the correct variational scaling, and the nonexistence applications are standard consequences that follow cleanly once the identity is in hand.\n\nThe soft spots are technical but load-bearing. Definition 2.2 only admits C_c^1 test functions, yet the proof plugs in φ = ψ_λ Σ x_j D_j u and φ = Σ D_j u without a density argument. In (3.19), the boundary term I is set to zero using u ∈ C^{0,l}, l>s, but X_{0,1} explicitly includes the W^{1,p}∩W^{s,p} branch, which does not imply that regularity. So Theorem 2.5 is unproved for that whole branch of the pure fractional case. Both gaps look repairable — add a density lemma for difference-quotient test functions, and restrict X_{0,1} to the Hölder branch or supply an alternative estimate for I — but as it stands the central theorem overclaims its coverage.\n\nThe citation pattern is honest and the paper does not fit constants or hide circularity. I'd bring it to a reading group to poke at the regularity issue, but I would not cite the current version for the identity in W^{1,p}∩W^{s,p}. Once the gaps are fixed, I would.\n\nVerdict: it deserves a serious referee. Send it to review, with the expectation of revision.","headline":"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.","tokens_in":16729,"tokens_out":4581,"would_cite":false,"duration_ms":51321,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35J92","35A01","35J62"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Pohozaev identity","anisotropic p-Laplace equation","fractional p-Laplace equation","mixed local and nonlocal p-Laplace equation","system of quasilinear equations","nonexistence","Finsler-Minkowski norm","weak solutions"],"falsifier":"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}}$\\) for any \\(l>s\\). For such a solution, compute the boundary integral \\(I\\) in (3.19): if \\(I\\neq0\\), the identity acquires an extra surface term and Theorem 2.5 fails as stated. A simpler concrete check is to compute both sides of the identity for a known explicit solution, for example a ground state of the pure power equation at the borderline exponent, and verify equality; a mismatch in any case where the proof's test functions are not admissible would falsify the claimed universality.","tokens_in":15718,"feed_emoji":"⚖️","tokens_out":16396,"duration_ms":169570,"temperature":0.7,"pith_summary":"This paper proves that every weak solution in the stated energy space of the equation \\(-\\$\\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$)=(1,0)\\), the purely fractional case \\((0,1)\\), and the mixed local-nonlocal operator \\((1,\\gamma)\\); the mixed case is claimed to be new even for \\(p=2\\). The same balance is extended to systems. A sympathetic reader cares because Pohozaev identities are the standard route to nonexistence and decay statements, and the paper applies them to rule out many power-type solutions.","feed_headline":"Weak solutions of mixed p-Laplace equations obey exact energy balance","feed_subtitle":"The mixed case is new even for p=2; away from critical exponents, pure-power solutions are ruled out.","key_machinery":"The carrying object is the Finsler-Minkowski norm \\(H\\), a strictly convex, positively homogeneous function whose Euler identity \\(x\\cdot\\nabla H(x)=H(x)\\) turns the local operator's integrals into \\(\\|H(\\nabla u)\\|^p_{L^p}\\). The fractional side is carried by the Gagliardo seminorm \\([u]^p_{s,p}\\) and the kernel \\(J_p(r)=|r|^{p-2}r\\). The proof's engine is a pair of test functions built from difference quotients: \\(\\psi_\\$\\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$\\to0\\) leaves exactly the stated identity.","core_discovery":"On the paper's own terms, the central result is Theorem 2.5: if \\(0<s<1<p<\\infty\\), \\(f\\) is continuous with \\(|f(t)|\\le C|t|^{p-1}\\), and \\(u\\in X_{\\$\\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\\).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the boundary-estimate argument used to show that the boundary integral \\(I\\) in (3.19) vanishes for the fractional part.","marker":"[2]"},{"why":"Provides the direct difference-quotient proof method for Pohozaev identities in \\(\\mathbb R^n\\) that the paper explicitly follows.","marker":"[20]"},{"why":"Establishes the anisotropic Pohozaev identity for \\(C^1\\) solutions in \\(\\mathbb R^n\\), the regularity level this paper aims to weaken.","marker":"[23]"},{"why":"The classical Pohozaev identity that defines the object being generalized.","marker":"[24]"},{"why":"General variational identity for quasilinear operators that this result extends to weak solutions and mixed operators.","marker":"[25]"},{"why":"Earlier anisotropic Pohozaev identity in bounded domains that is here extended to the whole space and to weak solutions.","marker":"[28]"},{"why":"Provides the weak regularity scale \\(W^{1,p}\\) for p-Laplace Pohozaev identities that the anisotropic case generalizes.","marker":"[16]"},{"why":"Source of the Finsler-norm identities, including Euler homogeneity, used in Lemma 2.1.","marker":"[15]"},{"why":"Source of the analytic facts about the norm used in Lemma 2.1, notably boundedness of the gradient of \\(H\\).","marker":"[17]"}],"fun_headline_variants":["Mixed p-Laplace equations get new Pohozaev identities","Pohozaev identities for anisotropic fractional systems","New energy identities for mixed p-Laplace even for p=2","Pohozaev-type laws for local and nonlocal p-Laplace systems","Identities for mixed p-Laplace equations new at p=2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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$}\\), and that the Hölder-type regularity needed to kill the boundary term holds on the full range of the stated spaces, neither of which is established by a density or approximation argument.","fun_headline_variants_meta":{"raw":{"variants":["Mixed p-Laplace equations get new Pohozaev identities","Pohozaev identities for anisotropic fractional systems","New energy identities for mixed p-Laplace even for p=2","Pohozaev-type laws for local and nonlocal p-Laplace systems","Identities for mixed p-Laplace equations new at p=2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000416,"raw_usage":{"total_tokens":2118,"prompt_tokens":890,"completion_tokens":1228,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":1140}},"tokens_in":506,"tokens_out":1228,"duration_ms":9747,"temperature":1.0,"reasoning_tokens":1140,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:07:38.729747+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a weak solution of the mixed or fractional equation lying in the \\($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.","supporting_citations":[{"cited_title":"On the Pohozaev identity for the fractionalp-Laplacian operator inR N","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary-estimate argument used to show that the boundary integral \\(I\\) in (3.19) vanishes for the fractional part."},{"cited_title":"On the eigenvalue problem for thep-Laplacian operator in RN .J","cited_arxiv_id":null,"evidence_quote":"Provides the direct difference-quotient proof method for Pohozaev identities in \\(\\mathbb R^n\\) that the paper explicitly follows."},{"cited_title":"Pohozaev identity for Finsler anisotropic problems","cited_arxiv_id":null,"evidence_quote":"Establishes the anisotropic Pohozaev identity for \\(C^1\\) solutions in \\(\\mathbb R^n\\), the regularity level this paper aims to weaken."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical Pohozaev identity that defines the object being generalized."},{"cited_title":"A general variational identity.Indiana Univ","cited_arxiv_id":null,"evidence_quote":"General variational identity for quasilinear operators that this result extends to weak solutions and mixed operators."},{"cited_title":"The Pohozaev identity for the anisotropicp-Laplacian and estimates of the torsion function.Rev","cited_arxiv_id":null,"evidence_quote":"Earlier anisotropic Pohozaev identity in bounded domains that is here extended to the whole space and to weak solutions."},{"cited_title":"Quasilinear elliptic equations involving critical Sobolev exponents.Nonlinear Anal., 13(8):879–902, 1989","cited_arxiv_id":null,"evidence_quote":"Provides the weak regularity scale \\(W^{1,p}\\) for p-Laplace Pohozaev identities that the anisotropic case generalizes."},{"cited_title":"An existence result for singular Finsler double phase problems.J","cited_arxiv_id":null,"evidence_quote":"Source of the Finsler-norm identities, including Euler homogeneity, used in Lemma 2.1."},{"cited_title":"Dover Publications, Inc., Mineola, NY, 2006","cited_arxiv_id":null,"evidence_quote":"Source of the analytic facts about the norm used in Lemma 2.1, notably boundedness of the gradient of \\(H\\)."}],"review_version":1}