{"id":"dbd0f9c9-0ab2-4d7e-8ca4-d45b3e6bc093","arxiv_id":"2412.00755","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence and regularity are established for semilinear mixed local-nonlocal problems with variable singular exponent and measure-valued data, including the case of two simultaneous measure sources.","lead":"This paper proves existence and regularity of weak solutions for a class of mixed local-nonlocal elliptic equations where the singular exponent varies in space and the data are allowed to be measures. It extends prior work by allowing both the singular and the perturbed source terms to be measures simultaneously, even for constant singular exponent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the main theorems are supported by the a priori estimates, with only a repairable notational gap in the proof of (3.11).","rationale":"The reader's weakest_assumption identifies condition (P_epsilon,delta*) as the most fragile premise. I agree that this condition is load-bearing: without it, the boundary-layer estimates in Lemma 5.2 fail and the existence theorem is not obtained. However, this is an explicit hypothesis rather than an internal inconsistency; the paper clearly states and uses it. The reader also notes the min_Omega delta(x) > 0 issue in the proof of (3.11). This is a real but repairable gap: the proof can be fixed by taking the minimum over a compact set containing the support of the test function, where delta is continuous and positive. I found no additional errors that would change the verdict. The limiting arguments for the singular measure term rely on standard weak/dominated convergence, and the a priori estimates are technically sound. The theorems are new and the proof strategy is coherent. Therefore, the appropriate verdict remains CONDITIONAL, with the minor revisions suggested by the reader (repairing (3.11) and clarifying the admissibility of the test function in Lemma 5.3(b)).","tokens_in":31310,"tokens_out":30561,"duration_ms":259061,"concrete_test":"Re-derive the dominated convergence in (3.11) with r := min_{supp phi} delta(x) in place of min_Omega delta(x), and verify the bound |log t|/t^r <= C on [C(omega), infinity), where C(omega) is the uniform lower bound of u_n on supp(phi). If this estimate holds, the gap in (3.11) is closed and the proof of Theorem 2.12 is complete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the central existence and regularity arguments in good faith. The main claims in Theorems 2.12, 2.13, and 2.16 are supported by the uniform a priori estimates in Lemmas 5.2 and 5.3, and the limiting procedures in Section 3 are standard. The condition (P_epsilon,delta*) is used essentially to control the singular term near the boundary; this is an explicit assumption, not a hidden flaw. The step flagged by the reader as the most fragile, the dominated convergence in (3.11), does contain a repairable gap: the proof defines r := min_Omega delta(x) > 0, but delta need not have a positive minimum on the open domain Omega. However, the estimate is only needed on the compact set omega = supp(phi), where r_omega := min_omega delta(x) > 0 is guaranteed by continuity and positivity of delta. Replacing r by r_omega makes the bound |log(t)|/t^r bounded on [C(omega), infinity), and the rest of the argument proceeds unchanged. No step in the proof appears to invalidate the stated theorems. The reliance on self-cited lemmas is acceptable because they are independent published results, and the minor technical issues identified by the reader do not affect the main conclusions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies existence and regularity for the mixed local-nonlocal semilinear problem (1.1), in which the singular exponent δ(x) is variable and the data include measures: ν is a non-singular measure in M^p_0(Ω) and μ is a bounded Radon measure in the main variable-exponent theorem, while in the constant-exponent theorem both ν and μ are measures. The main existence results are Theorem 2.12 (variable δ under condition (P_{ε,δ*})), Theorem 2.13 (ν∈L^1), and Theorem 2.16 (constant δ with μ∈L^{N(δ+1)/(N+2δ)}), yielding weak solutions in the expected classes W^{1,p}_{loc}(Ω)∩L^1(Ω), with energy-type regularity for truncations when δ*>1. The proofs use a fixed-point approximation scheme, uniform a priori estimates for approximating solutions (Section 5), and passage to the limit via monotonicity, Marcinkiewicz bounds, and compactness. The paper also states regularity conclusions in Theorems 2.18–2.20 for integrable data. The central novelty claimed is the treatment of variable singular exponents with measure data, and the fact that both source terms can simultaneously be measures even for constant δ.","tokens_in":31525,"tokens_out":13128,"duration_ms":124258,"significance":"If the results are correct, the paper closes a genuine gap in the literature: mixed local-nonlocal singular problems with variable exponent had not been treated with measure data, and the simultaneous measure case for the perturbed problem appears new even for constant δ. The proofs are substantial and mostly transparent: the a priori estimates in Lemmas 5.2, 5.3, and 5.6–5.8 are detailed, the approximation scheme in Section 5.1 follows standard fixed-point and comparison arguments, and the limiting procedures in Section 3 are standard once the estimates are in hand. A particular strength is that the main regularity conclusions are obtained through explicit comparison estimates between the perturbed solution and the purely singular and purely non-singular auxiliary solutions. The paper also clearly states the boundary-layer condition (P_{ε,δ*}) that makes the variable-exponent estimates work; this is an explicit hypothesis rather than a hidden assumption.","major_comments":[{"comment":"The passage to the limit in the term containing ∇δ defines r := min_Ω δ(x)>0 and then uses boundedness of log(x)/x^r on [C,∞). Since Ω is open and δ is only continuous, the minimum over Ω need not be attained, so the argument as written is not fully justified. However, the estimate is needed only on ω=supp φ, where r_ω:=min_ω δ(x)>0 is guaranteed by continuity and positivity of δ, and |∇δ|∈L∞(ω) follows from the local Lipschitz hypothesis. Replacing r by r_ω and applying dominated convergence on ω repairs the gap. This is a local but load-bearing point in the proof of Theorem 2.12, so it should be corrected explicitly.","section":"§3.1, Eq. (3.11)"}],"minor_comments":[{"comment":"The notation T_k^{(δ*+1)/2}(u) is used before being defined; it should be stated explicitly that this means (T_k(u))^{(δ*+1)/2}, not T_k evaluated at a power of u.","section":"Section 2, Lemma 5.2(b), Theorems 2.12–2.13"},{"comment":"There are repeated spelling errors: 'purturbed' should be 'perturbed' in the discussion of equations (1.6), (1.8), and (1.11).","section":"Section 1"},{"comment":"In the same step, the sentence 'using this together with the fact lim_n ∫_Ω |G_n| = ∫_Ω |G|' should explicitly mention that strong convergence in L^{p'} on a bounded domain gives convergence in L^1, so that a generalized dominated convergence theorem applies to the terms |G_n| times a uniformly bounded factor.","section":"Section 3.1, Eq. (3.11)"},{"comment":"The proof of Theorem 2.16 is very terse, saying only that the limit is passed 'along the lines of the proof of Theorem 2.12.' Since the exponent q and the test functions in Lemma 5.3 differ from those in Lemma 5.2, a short indication of how the nonlocal term and the singular measure term are handled in this case would improve verifiability.","section":"Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"The only substantive issue I found is the min_Ω δ(x) step in the proof of (3.11); it is repairable by restricting to the support of the test function, but it is part of the central existence proof and should be fixed before publication. The reliance on the authors' prior results [31, Lemma 4.6], [31, Theorem 3.10], and [35, Lemmas 3.1–3.2] is explicit and appears appropriate. With that correction, I would support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a careful read. The genuinely new thing is existence and regularity for (1.1) when the singular exponent varies and the data are measures, and in the constant-exponent case when both sources are measures. That fills a real gap: variable exponent was previously treated only with integrable data, and constant exponent with one measure. The main theorems 2.12, 2.13, and 2.16 say what they claim.\n\nThe technical core is the a priori estimates in Section 5. I checked the Marcinkiewicz bounds and truncation estimates; they are coherent, and the limiting arguments in Section 3 follow the standard approximation route. The reliance on [31] and [35] for positivity, the strong maximum principle, and existence of bounded approximating solutions is acceptable, since those are published parameter-free results. The condition (P_epsilon,delta*) is explicit and used where it matters, at the boundary layer. No hidden assumption.\n\nSoft spots are minor. In the proof of (3.11) the authors define r = min_Omega delta(x) > 0, but delta is only continuous positive on an open domain; the minimum need not be attained. The estimate is only needed on supp phi, where the minimum is attained, so replacing r by r_omega fixes it. The dominated convergence argument then needs |log t|/t^r bounded on [C(omega), infinity), which is true. The proof should say this. Lemma 5.3(b) also uses a test function (u_n+epsilon)^delta - epsilon^delta; as written it is a Lipschitz function of u_n, so admissible, but the justification would be better stated. These are not load-bearing.\n\nOne caveat: the regularity results are stated for solutions obtained in Theorem 2.13 and use comparison with resolved problems. I did not see an issue there. The paper is not polished in places, but the mathematics is honest and the claims are supported.\n\nFor whom? Researchers in singular elliptic equations and mixed local-nonlocal problems. It is a serious contribution and deserves peer review. I would engage with it, and cite it if I worked in the area.","headline":"New existence and regularity for mixed local-nonlocal singular problems with variable exponent and measure data; the proof is sound apart from two small repairable gaps.","tokens_in":32077,"tokens_out":1403,"would_cite":true,"duration_ms":14316,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35M10","35M12","35J75","35R06","35R11","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The mixed local-nonlocal singular equation has weak solutions even when the singular exponent varies and both source terms are measures.","keywords":["mixed local-nonlocal operator","singular nonlinearity","variable singular exponent","measure data","weak solution","existence","regularity","capacity"],"falsifier":"For the unit ball with constant exponent $\\delta=1$, take $\\nu=dx$ (Lebesgue measure) and $\\mu=c\\,\\delta_0$ (Dirac mass at the centre, $c>0$). Theorem 2.16 predicts a weak solution in $W^{1,2}_0(\\Omega)$. Using the Green's function of $-\\Delta+(-\\Delta)^s$, which has the same $|x-y|^{2-N}$ singularity as the Laplacian, compute the Dirichlet energy of the candidate limit and the value of $\\int \\varphi/u\\,d\\mu$; if no finite-energy limit exists or that integral diverges for every admissible $\\varphi$, the simultaneous-measure existence claim is wrong.","tokens_in":31069,"feed_emoji":"","tokens_out":17297,"duration_ms":151015,"temperature":0.7,"pith_summary":"This paper establishes that the mixed local-nonlocal semilinear problem with a singular nonlinearity whose exponent varies from point to point has weak solutions even when the data are measures. The central result, Theorem 2.12, proves existence of a solution in $W^{1,p}_{\\mathrm{loc}}(\\Omega)\\cap L^1(\\Omega)$ for every $1<p<N/(N-1)$, provided the singular measure $\\nu$ is non-singular and belongs to a $p$-capacity class, and the singular exponent $\\delta(x)$ is bounded above in a boundary layer. A companion theorem shows that when $\\delta$ is constant, both the singular source and the perturbing source may simultaneously be bounded Radon measures, a configuration the authors note is new even for constant exponents. The paper also supplies explicit integrability regularity for the solutions in terms of the integrability of the data, including $L^\\infty$ bounds when both sources are integrable with exponent larger than $N/2$. This closes a known gap: variable singular exponents were previously handled only for integrable data, and mixed local-nonlocal problems with simultaneous measure sources were open.","feed_headline":"Mixed singular PDEs solved when both sources are measures","feed_subtitle":"Earlier results needed integrable data; this one handles variable exponents and two measures at once.","key_machinery":"The argument is carried by an approximation scheme combined with a priori estimates that are stable under passage to the limit. For each $n$, the singular data are truncated and regularized, producing problems with data $T_n(f)+h_n$ and $g_n$, whose unique weak solutions $u_n\\in W^{1,2}_0(\\Omega)$ are obtained by Schauder's fixed point theorem (Lemma 5.1). The measure $\\nu$ is decomposed via capacity theory as $\\nu = f - \\operatorname{div} G$ with $f\\in L^1$ and $G\\in (L^{p'})^N$, following [13]; this decomposition lets the singular term be distributed across a density and a divergence, and the approximation inherits this structure. The core estimates, Lemma 5.2 and Lemma 5.3, use the truncation operators $T_k$ and $G_k$ and the boundary-layer condition $(P_{\\varepsilon,\\delta_*})$ to prove uniform boundedness: Marcinkiewicz-type bounds on $\\nabla u_n$ and $\\nabla G_k(u_n)$ in $M^{N/(N-1)}(\\Omega)$, local $W^{1,2}$ bounds on $T_k(u_n)$, and global $W^{1,2}$ bounds on $T_k(u_n)^{(\\delta_*+1)/2}$. These bounds give compactness and, after passing to the limit in the weak formulation with carefully chosen test functions, produce the weak solution. The regularity theorems are built on comparison with solutions of the singular-only and perturbation-only problems, using the pointwise estimate $u_n \\le v_n + w_n$ and Moser-type iteration arguments.","core_discovery":"On its own terms, the paper proves that for a bounded Lipschitz domain $\\Omega$ and the operator $M=-\\operatorname{div}(A\\nabla)+B$ with uniformly elliptic $A$ and a symmetric kernel $K$ satisfying the fractional bounds (1.3), the problem $Mu = \\nu/u^{\\delta(x)} + \\mu$ with $u=0$ in $\\mathbb{R}^N\\setminus\\Omega$ and $u>0$ admits a weak solution under the following hypotheses: $\\delta:\\Omega\\to(0,\\infty)$ is continuous (locally Lipschitz in Theorem 2.12) and satisfies the boundary-layer condition $(P_{\\varepsilon,\\delta_*})$; $\\nu$ is a non-negative bounded Radon measure, non-singular with respect to Lebesgue measure and belonging to $M^p_0(\\Omega)$ for some $1<p<N/(N-1)$; $\\mu$ is a non-negative bounded Radon measure. The solution lies in $W^{1,p}_{\\mathrm{loc}}(\\Omega)\\cap L^1(\\Omega)$, with $u\\in W^{1,p}_0(\\Omega)$ if $\\delta_*=1$ and with the truncations $T_k(u)\\in W^{1,2}_{\\mathrm{loc}}(\\Omega)$ and $T_k(u)^{(\\delta_*+1)/2}\\in W^{1,2}_0(\\Omega)$ if $\\delta_*>1$. When $\\delta$ is constant, Theorem 2.16 relaxes the assumptions: $\\mu$ may be in a Lebesgue space and $\\nu$ may be a measure in a capacity class, and again a weak solution exists with regularity depending on whether $\\delta\\le 1$ or $\\delta>1$. The regularity results, Theorems 2.18–2.20, assert that when the data are integrable, the solution inherits explicit Lebesgue integrability: for instance, $r,m>N/2$ forces $L^\\infty$, and otherwise the solution lies in spaces such as $L^{m^{**}}$ or $L^{Nr(\\delta+1)/(N-2r)}$. The simultaneous-measure phenomenon, in which both the singular source and the perturbing source are measures, is new even in the constant exponent case.","pith_inferences":["The non-singularity assumption on $\\nu$ may be inessential: the capacity decomposition already handles the singular part, so a different limiting argument might extend Theorem 2.16 to purely singular $\\nu$, making the simultaneous-measure result hold for all bounded Radon measures.","The boundary-layer condition $(P_{\\varepsilon,\\delta_*})$ is used only through uniform lower bounds away from the boundary and control of $\\delta$ near $\\partial\\Omega$; this suggests the theorem could generalize to non-Lipschitz domains if the boundary layer were replaced by a weight, though the Marcinkiewicz estimates would need to be reworked.","Because the mixed operator combines a local second-order term and a nonlocal fractional term, the regularity thresholds in Theorems 2.18–2.20 are set by the local operator ($N/2$); a natural test is whether the fractional order $s$ affects intermediate regularity, for instance by checking if the $L^\\infty$ condition can be relaxed when $s$ is close to $1$.","The comparison estimate $u_n\\le v_n+w_n$ in Lemma 5.4 is a linearization device; the same device could yield gradient estimates for the singular term alone, potentially leading to Hölder regularity under stronger integrability of the data."],"forward_implications":["For any bounded Lipschitz domain and any continuous $\\delta$ satisfying $(P_{\\varepsilon,\\delta_*})$ with $\\delta_*=1$, Theorem 2.12 gives a solution in $W^{1,p}_0(\\Omega)$ for every $1<p<N/(N-1)$, so the boundary condition is attained in the usual Sobolev sense.","If $\\delta_*>1$, the solution still belongs to $W^{1,p}_{\\mathrm{loc}}(\\Omega)\\cap L^1(\\Omega)$, with the nonlinear truncation $T_k(u)^{(\\delta_*+1)/2}$ in $W^{1,2}_0(\\Omega)$; this quantifies how much regularity survives when the singular exponent is large near the boundary.","In the constant-exponent case, both $\\nu$ and $\\mu$ can be measures at the same time (with $\\nu$ non-singular and in $M^q_0$), which extends the mixed local-nonlocal theory beyond prior results where at least one source was integrable.","When $\\nu\\in L^r$ and $\\mu\\in L^m$, the solution inherits explicit Lebesgue regularity: $r,m>N/2$ gives $L^\\infty$, and otherwise the solution lies in the stated spaces $L^{m^{**}}$, $L^{Nr(\\delta+1)/(N-2r)}$, or their minima, so the integrability threshold $N/2$ of the local Laplacian is preserved in the mixed setting.","Remark 2.17 notes that the same existence statements hold for the purely local operator $A$ alone, so the results are new for local variable-exponent singular equations with measure data as well."],"supporting_citations":[{"why":"supplies the approximation and existence approach for semilinear singular equations with integrable data, which the paper adapts to measures.","marker":"[14]"},{"why":"provides the measure-data setting and the Lazer–McKenna-type problem whose methods are extended to the mixed local-nonlocal operator.","marker":"[45]"},{"why":"gives the singular-equation-with-measure-source framework used for the perturbed problem and the limit passage.","marker":"[42]"},{"why":"yields the capacity-based decomposition ν=f−div G and the approximation lemma that let the singular measure be distributed.","marker":"[13]"},{"why":"introduces the variable singular exponent semilinear problem and the boundary-layer condition that the paper adapts as (P_{ε,δ*}).","marker":"[19]"},{"why":"establishes existence and regularity for quasilinear nonlocal equations with variable singular exponent, the prior art that this paper extends to mixed operators and measures.","marker":"[34]"},{"why":"is the most recent mixed local-nonlocal singular result with integrable f and measure g; Theorem 2.13 explicitly extends its Theorem 1.1.","marker":"[7]"},{"why":"provides mixed local-nonlocal Sobolev inequalities and quasilinear singular elliptic results used for uniform positivity and the regularity bootstrap.","marker":"[35]"},{"why":"treats mixed local and nonlocal equations with measure data in the absence of singularities, supplying the operator framework for measure data.","marker":"[16]"},{"why":"offers the regularity theory and comparison principles for mixed p-Laplace and singular problems that underpin Lemma 5.1 and the lower bounds.","marker":"[31]"}],"fun_headline_variants":["Variable singular exponents & dual measures in PDEs solved","New results for mixed PDEs with two measure sources","Both sources as measures now allowed in singular PDEs","PDEs with variable singularity and measure data resolved","Measure-valued data for both terms in mixed equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the singular exponent $\\delta(x)$ is bounded above by some $\\delta_*\\ge 1$ in a thin boundary layer of the domain; without this bound, the a priori estimates that control the singular term near the boundary and allow passage to the limit are not established.","fun_headline_variants_meta":{"raw":{"variants":["Variable singular exponents & dual measures in PDEs solved","New results for mixed PDEs with two measure sources","Both sources as measures now allowed in singular PDEs","PDEs with variable singularity and measure data resolved","Measure-valued data for both terms in mixed equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2770,"prompt_tokens":1102,"completion_tokens":1668,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":1593}},"tokens_in":718,"tokens_out":1668,"duration_ms":10988,"temperature":1.0,"reasoning_tokens":1593,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:04:07.149575+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the unit ball with constant exponent $\\delta=1$, take $\\nu=dx$ (Lebesgue measure) and $\\mu=c\\,\\delta_0$ (Dirac mass at the centre, $c>0$). Theorem 2.16 predicts a weak solution in $W^{1,2}_0(\\Omega)$. Using the Green's function of $-\\Delta+(-\\Delta)^s$, which has the same $|x-y|^{2-N}$ singularity as the Laplacian, compute the Dirichlet energy of the candidate limit and the value of $\\int \\varphi/u\\,d\\mu$; if no finite-energy limit exists or that integral diverges for every admissible $\\varphi$, the simultaneous-measure existence claim is wrong.","supporting_citations":[{"cited_title":"Semilinear elliptic e quations with singular nonlinear- ities","cited_arxiv_id":null,"evidence_quote":"supplies the approximation and existence approach for semilinear singular equations with integrable data, which the paper adapts to measures."},{"cited_title":"A Lazer-McKenna ty pe problem with measures","cited_arxiv_id":null,"evidence_quote":"provides the measure-data setting and the Lazer–McKenna-type problem whose methods are extended to the mixed local-nonlocal operator."},{"cited_title":"On singu lar elliptic equations with measure sources","cited_arxiv_id":null,"evidence_quote":"gives the singular-equation-with-measure-source framework used for the perturbed problem and the limit passage."},{"cited_title":"Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure da ta","cited_arxiv_id":null,"evidence_quote":"yields the capacity-based decomposition ν=f−div G and the approximation lemma that let the singular measure be distributed."},{"cited_title":"Quasilinear no nlocal elliptic problems with variable singular exponent","cited_arxiv_id":null,"evidence_quote":"establishes existence and regularity for quasilinear nonlocal equations with variable singular exponent, the prior art that this paper extends to mixed operators and measures."},{"cited_title":"Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular ellipti c problems","cited_arxiv_id":null,"evidence_quote":"provides mixed local-nonlocal Sobolev inequalities and quasilinear singular elliptic results used for uniform positivity and the regularity bootstrap."},{"cited_title":"Mixed local and nonlocal e quations with measure data","cited_arxiv_id":null,"evidence_quote":"treats mixed local and nonlocal equations with measure data in the absence of singularities, supplying the operator framework for measure data."},{"cited_title":"On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular proble ms","cited_arxiv_id":null,"evidence_quote":"offers the regularity theory and comparison principles for mixed p-Laplace and singular problems that underpin Lemma 5.1 and the lower bounds."}],"review_version":1}