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Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation

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Pith's one-line read For the mixed local-fractional Laplacian, positive radial solutions exist just below the classical critical exponent, so the usual whole-space/domain duality breaks down.

desk verdict The new sign computation and the failure-duality claim are worth a referee, but the proof breaks for n ≤ 6 because the Talenti bubble is not in the working space X. read the letter →

arxiv 2507.12258 v1 pith:WDZP6PSU submitted 2025-07-16 math.AP

Begoña Barrios , Leandro M. Del Pezzo , Alexander Quaas This is my paper
classification math.AP MSC 35J6035R1135B33
keywords mixedlocal-nonlocaloperatorsfractionalLaplacianLane-EmdenequationcriticalexponentLyapunov-SchmidtreductionTalentibubbleradialsolutionsSobolev
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

The paper studies the Lane–Emden equation $-a\Delta u + b(-\Delta)^s u = u^p$ in $\mathbb{R}^n$ with $a,b>0$ and $s\in(0,1)$, and asks where the whole-space critical exponent lies. It establishes that for every small $\varepsilon>0$, with $p_1=(n+2)/(n-2)$ and $p=p_1-\varepsilon$, the equation has a positive radial solution that is a small perturbation of the Talenti bubble $U(x)=(1+|x|^2)^{-(n-2)/2}$. Because the same mixed operator on a bounded star-shaped domain still has critical exponent $p_1$, the whole-space exponent is strictly smaller than $p_1$. This contradicts the classical duality $p^*(\Omega,F)=p^*(F)$ that holds for $-\Delta$ and $(-\Delta)^s$ separately, and the paper presents it as the first example of an elliptic operator for which that duality fails.

What carries the argument

The argument is a Lyapunov–Schmidt reduction around the Talenti bubble $U=(1+|x|^2)^{-(n-2)/2}$. The parameter $\delta$ makes the fractional term small after rescaling, giving the perturbation $-\Delta u+\delta^{2(1-s)}(-\Delta)^s u$. The linearized operator at $U$ has a one-dimensional radial kernel spanned by $\psi=x\cdot\nabla U+\frac{n-2}{2}U$; the reduction solves the equation orthogonally to $\psi$, then uses the relation $\delta^{2(1-s)}=\varepsilon\lambda$ to kill the remaining projection. The load-bearing signs are $\int_{\mathbb{R}^n}\log(U)U^{p_1}\psi\,dx>0$ and $\langle U,\psi\rangle_s<0$, the second obtained from the explicit hypergeometric formula for $(-\Delta)^s U$; these signs fix the parameter $\lambda$ in the final step.

What would settle it

One can settle the membership question directly: the tail integral $\int_1^\infty r^{-(2n/(n+2))(n-2)+n-1}\,dr$ converges only for $n>6$, so in dimensions $3,4,5,6$ the bubble $U$ is not in $L^{2n/(n+2)}(\mathbb{R}^n)$; a valid theorem for those dimensions would need to show how the asserted solution is represented without this membership.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for $n\ge3$ and any small $\varepsilon>0$ there is a $\delta=\delta(\varepsilon)$ such that the rescaled problem $-\Delta u+\delta^{2(1-s)}(-\Delta)^s u=n(n-2)u^{p_1-\varepsilon}$ in $\mathbb{R}^n$ has a radial solution $z_\varepsilon=U+\phi_\varepsilon$ with $\phi_\varepsilon\to0$ in $D^{1,2}(\mathbb{R}^n)$, where $U$ is the Talenti bubble. Rescaling back gives a positive solution of the original mixed Lane–Emden equation that concentrates at the origin. The authors interpret this as showing that the whole-space critical exponent $p^*(L_{a,b})$ is strictly below $p_1=p^*(\Omega,L_{a,b})$ for star-shaped bounded domains, so the standard equality between whole-space and bounded-domain critical exponents is false for this operator.

Load-bearing premise

The load-bearing premise is that the Talenti bubble $U$, decaying like $|x|^{-(n-2)}$, lies in the solution space $X=D^{1,2}(\mathbb{R}^n)\cap L^{2n/(n+2)}(\mathbb{R}^n)$ where the fixed-point equation is solved; for $3\le n\le6$ this membership fails, and without it the ansatz $U+\phi$ cannot represent a solution in that space.

Editorial extensions

If this is right

  • The whole-space critical exponent of $L_{a,b}$ satisfies $p^*(L_{a,b}) < p_1 = p^*(\Omega,L_{a,b})$ for star-shaped bounded domains, so the standard duality between whole-space and bounded-domain critical exponents fails for this mixed operator.
  • For every sufficiently small $\varepsilon>0$ there are positive radial solutions of $-a\Delta u + b(-\Delta)^s u = u^{p_1-\varepsilon}$ in $\mathbb{R}^n$, and they concentrate at the origin as $\varepsilon\to0$.
  • The sign of $\langle U,\psi\rangle_s$ controls the balance between $\varepsilon$ and $\delta$; with the opposite sign the reduction would not produce a full solution.
  • The authors conjecture that $p^*(L_{a,b})=p_s=(n+2s)/(n-2s)$, with nonexistence below $p_s$ expected from a Pohozaev-type identity and existence above $p_s$ left as a perturbation problem.

Reading between the lines

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

  • Inference: the proof's space $X$ is restrictive; the bubble $U$ fails to lie in $L^{2n/(n+2)}(\mathbb{R}^n)$ for $3\le n\le6$, so a low-dimensional theorem would need a modified ansatz or solution space.
  • Inference: the same parameter-balancing argument could be run with the ratio $a/b$ as the control parameter, giving a family of mixed operators whose critical-exponent gap depends continuously on the ratio.
  • Inference: the conjecture $p^*(L_{a,b})=p_s$ could be tested numerically by solving the radial equation across a grid of $p$ values and checking where positive solutions cease to exist.
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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

2 major / 4 minor

Summary. The paper studies the mixed Lane-Emden equation -Δu + δ^{2(1-s)}(-Δ)^s u = n(n-2)u^{p1-ε} in R^n, with p1 = (n+2)/(n-2). After a scaling, the authors propose a Lyapunov-Schmidt reduction around the Talenti bubble U in the space X = D^{1,2}(R^n) ∩ L^{2n/(n+2)}(R^n). The main theorem (Theorem 1.1) asserts that for every n ≥ 3 and small ε there exists δ(ε) such that the equation has a radial solution z = U + φ with φ → 0 in D^{1,2}; from this they conclude that the whole-space critical exponent for the mixed operator is strictly below p1, so the usual bounded-domain/whole-space duality fails. The proof reduces the problem to a fixed point for a map T_{ε,δ} on the radial subspace orthogonal to the kernel direction ψ, and the final choice of δ uses the signs of A = ∫ log(U) U^{p1} ψ dx and B = ⟨U, ψ⟩_s.

Significance. If the theorem were true for all n ≥ 3, it would disprove the natural conjecture p*(L_{a,b}) = p1 for mixed local/nonlocal operators and would provide the first example of a failure of the classical critical-exponent duality. The paper contains some genuine ingredients: Lemma 2.8 gives an explicit hypergeometric sign computation, the reduction follows a standard perturbation strategy, and the final parameter choice λ0 = -A/B involves no fitted parameters. The proof is not circular: the signs of A and B are cited from independent sources. However, the functional setting excludes the Talenti bubble in low dimensions, so the stated result is not established for n = 3, 4, 5, 6, and the proof as written cannot support the theorem in its present form.

major comments (2)
  1. [§2, Eq. (5); §3, Lemmas 2.4 and 3.4] The working space X = D^{1,2}(R^n) ∩ L^{2n/(n+2)}(R^n) does not contain the Talenti bubble U when 3 ≤ n ≤ 6. Since U(x) ∼ |x|^{-(n-2)} at infinity, U ∈ L^{2n/(n+2)} iff the integral ∫_1^∞ r^{n-1} r^{-2n(n-2)/(n+2)} dr converges, which holds exactly when n > 6. The fixed point is sought with φ ∈ K ⊂ X, so the ansatz z = U + φ would give z ∈ X and hence U = z - φ ∈ X, a contradiction for n = 3, 4, 5, 6. Consequently the object w_{ε,δ} = I_δ(g_{ε,δ}(U+φ)) in Lemma 2.4 is not defined for these dimensions, because g_{ε,δ}(U+φ) contains the term δ^{2(1-s)}U, which is not in the domain L^{2n/(n+2)} of I_δ. Lemma 3.4, which invokes Lemma 2.4, cannot be applied. The argument can at most begin for n ≥ 7; Theorem 1.1 as stated is unsupported.
  2. [§2, Lemma 2.1; §3, Lemmas 2.4 and 2.5] The uniform-in-δ estimates used in Lemma 2.4 and Lemma 2.5 are not justified. Lemma 2.1 states a bound with a constant C(n,δ) > 0, and the symbol of the operator in (7) is |ξ|^2 + δ^{2(1-s)}|ξ|^{2s} + δ^{2(1-s)}, whose value at ξ = 0 is δ^{2(1-s)}. The inverse operator I_δ can therefore have norm growing like δ^{-2(1-s)} as δ → 0, and a constant independent of δ does not follow from the cited estimate. The arguments then use estimates such as ∥z_δ∥ ≤ C δ^{2(1-s)} and ∥w - U∥ ≤ C(ε + δ^{2(1-s)}) with C independent of δ; without a proof of uniformity, the smallness conclusions for the δ-terms in Lemma 3.4 do not follow.
minor comments (4)
  1. [Theorem 1.1] The statement says 'Let n ≥ 3 and ε > 0. There exists δ(ε)', but the proof requires ε < 4/(n-2) (so that p1 - ε > 1), α > 1/(p1 - ε), and ε, δ sufficiently small (Lemmas 3.1-3.3 and 3.4). If smallness of ε is intended, it should be stated explicitly in the theorem.
  2. [Lemma 3.4] In the displayed expansion of ∫(f0(U)+f0'(U)φ_{ε,δ} - f_ε(U+φ_{ε,δ}))ψ dx, the last term ∫(f0'(U)-f_ε'(U))ψ dx is missing the factor φ_{ε,δ}; compare with the preceding line.
  3. [§3] The notation is inconsistent: Lemma 2.4 writes w_{ε,δ}, while Lemma 3.4 writes W_{ε,δ} for the same object, and the map T_{ε,δ} is written T_{δ,ε} in Lemmas 3.1-3.2.
  4. [§3, definition of T_{ε,δ}] The sentence 'by (11) is clear that (-Δ)^s U ∈ X so T_{ε,δ} is well defined' could mislead: (11) only gives the L^{2n/(n+2)} membership, and the relevant cancellation of the artificial δ^{2(1-s)}U term in the fixed-point formulation should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lyapunov-Schmidt construction is self-contained, and the key constants A and B are computed in the paper or cited from independent sources.

full rationale

The paper's central claim is the existence of a radial solution to the mixed Lane-Emden equation for p = p1 - epsilon. This is proved by a Lyapunov-Schmidt reduction in which the perturbation phi_epsilon is found as a fixed point of the operator T_{delta,epsilon} on a small ball in K. The reduction has no fitted parameters: the constants A = integral log(U) U^{p1} psi dx and B = <U, psi>_s are computed explicitly or cited from independent sources ([37], [28], [35]), and the parameter delta is chosen through the relation delta^{2(1-s)} = epsilon lambda_0 with lambda_0 = -A/B. This is a standard parameter selection to cancel the leading-order terms in the projection, not a prediction of a fitted quantity. The only self-citations in the proof, such as [21] for strict positivity in Lemma 2.2, are peripheral and not load-bearing for the existence argument. The possible issue that the Talenti bubble U may not lie in the space X for n <= 6 is a domain/well-posedness concern about the proof as written, not a circularity: it does not make the conclusion an input of the derivation. Hence the derivation is not circular.

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

All assumptions are listed. The most consequential is the implicit assumption U ∈ X, which fails for n ≤ 6. No free parameters are fitted; δ is determined by the Lyapunov-Schmidt reduction through λ0 = -A/B. No new entities are introduced.

assumptions (4)
  • standard math The Talenti bubble U solves -ΔU = n(n-2)U^{p1} and spans the radial kernel of the linearized operator.
    Used throughout the construction as the base profile; standard result cited as [45] and [16].
  • ad hoc to paper The bubble U belongs to the working space X = D^{1,2}(R^n) ∩ L^{2n/(n+2)}(R^n).
    Assumed implicitly in Lemmas 2.4 and 3.1 through the term δ^{2(1-s)}U in g_{ε,δ}(U+φ). This is false for n ≤ 6, where U decays too slowly to be in L^{2n/(n+2)}.
  • domain assumption The linearized operator L_0 is invertible on K and L_δ converges to L_0 in operator norm as δ→0.
    Invertibility is cited to [37]; convergence is proved in Lemma 2.5 using estimates cited to [44] and [28].
  • domain assumption The Pohozaev identity for the mixed operator yields p*(Ω,L)=p1 for star-shaped domains.
    Cited to [41, Remark 3.1], used in the introduction to frame the duality claim that the main theorem aims to refute.

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Pith. "Pith review of Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation." pith.science (2026). https://pith.science/paper/WDZP6PSU

@misc{pith2026250712258,
  author       = {Pith},
  title        = {Pith review of: Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDZP6PSU}},
  note         = {Machine review of arXiv:2507.12258}
}
abstract

In this paper, we investigate a mixed elliptic equation involving both local and nonlocal Laplacian operators, with a power-type nonlinearity. Specifically, we consider a Lane-Emden type equation of the form \[-\Delta u + (-\Delta)^s u = u^p,\quad\mbox{ in }\mathbb{R}^n.\] where the operator combines the classical Laplacian and the fractional Laplacian. We establish the existence of solutions for exponents slightly below the critical local Sobolev exponent, that is, for $p < \frac{n+2}{n-2}$, with $p$ close to $\frac{n+2}{n-2}$. Our results show that, due to the interaction between the local and nonlocal operators, this mixed Lane-Emden-Fowler equation does not admit a critical exponent in the traditional sense. The existence proof is carried out using a Lyapunov-Schmidt type reduction method and, as far as we know, provide the first example of an elliptic operator for which the duality between critical exponents fails.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''

    math.AP 2026-01 accept novelty 6.0 of 10

    In high dimensions n≥5, any small negative fractional correction yields a nontrivial critical solution; in low dimensions n=3,4, existence is guaranteed once the correction is large enough.

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