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arxiv: 2504.15000 · v3 · submitted 2025-04-21 · 🧮 math.AP

Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions

Mousomi Bhakta , Nirjan Biswas , Paramananda Das This is my paper

Pith reviewed 2026-05-22 18:57 UTC · model grok-4.3

classification 🧮 math.AP
keywords mixed local-nonlocal operatorp-Laplacianfractional p-Laplacianconcave-critical nonlinearitymultiplicity of positive solutionsquasilinear elliptic problemAmbrosetti-Brezis-Cerami result
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The pith

A mixed local-nonlocal p-Laplacian with concave and critical terms has positive solutions exactly when λ stays at or below a threshold Λ_ε, and two solutions for small λ if ε is small enough.

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

The authors examine a quasilinear elliptic equation that adds the classical p-Laplacian to a weighted fractional p-Laplacian and drives it with a concave power plus a critical Sobolev power. They prove that the sign of the parameter λ completely determines solvability: no nontrivial solution exists for λ ≤ 0, while for λ > 0 a sharp threshold Λ_ε separates the regime of at least one positive solution from the regime of none. When the nonlocal weight ε is small, a second, smaller threshold λ# appears and guarantees at least two distinct positive solutions for every λ in (0, λ#). A sympathetic reader cares because the result carries the classical Ambrosetti-Brezis-Cerami picture over to operators that combine local and nonlocal diffusion, a setting that appears in models of anomalous transport and long-range interactions.

Core claim

For λ ≤ 0 the problem (P_{λ,ε}) admits no nontrivial solution. For λ > 0 there exists a positive number Λ_ε with the following properties: the problem has a positive minimal solution when 0 < λ < Λ_ε, it has a positive solution when λ = Λ_ε, and it has no positive solution when λ > Λ_ε. In addition, if ε is sufficiently small there exists 0 < λ# ≤ Λ_ε such that the problem has at least two positive solutions for every λ belonging to the interval (0, λ#).

What carries the argument

The threshold Λ_ε, defined as the supremum of admissible λ-values for which a positive solution exists, together with the associated energy functional whose mountain-pass geometry and Palais-Smale compactness are controlled by the mixed local-nonlocal structure.

If this is right

  • No nontrivial solutions exist for any non-positive λ.
  • A positive minimal solution exists for every λ strictly between 0 and Λ_ε.
  • A positive solution exists at the threshold value λ = Λ_ε.
  • No positive solution exists for any λ larger than Λ_ε.
  • At least two distinct positive solutions exist throughout the interval (0, λ#) once ε is small enough.

Where Pith is reading between the lines

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

  • The same threshold construction may extend directly to operators that sum three or more local and nonlocal terms with different orders.
  • The multiplicity interval (0, λ#) could be recovered for a wider range of ε by replacing the critical power with a slightly subcritical one.
  • The explicit dependence of Λ_ε on ε supplies a natural starting point for numerical continuation algorithms that track solution branches as the nonlocal weight varies.
  • Analogous existence intervals are likely to hold when the same mixed operator is paired with other concave-critical pairs on unbounded domains.

Load-bearing premise

The assumption that ε must be taken sufficiently small to obtain the smaller threshold λ# that produces two solutions.

What would settle it

An explicit domain, value of ε, and λ > Λ_ε for which a positive solution can be constructed, or a numerical example showing only a single positive solution for arbitrarily small positive λ when ε is not small.

read the original abstract

We study the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian, \begin{equation}\tag{$\mathcal{P}_{\lambda,\varepsilon}$} -\Delta_p u+\varepsilon(-\Delta_p)^s u=\lambda|u|^{q-2}u+|u|^{p^*-2}u \;\text{ in }\Omega,\quad u=0 \; \text{ in }\mathbb{R}^N \setminus \Omega, \end{equation} where $\Omega\subset\mathbb{R}^N$ is a bounded open set, $\epsilon\in(0,1]$, $0<s<1<q<p<N$, and $p^*=\frac{Np}{N-p}$, and $\lambda \in \mathbb{R}$ is a parameter. For $\lambda \leq 0$, we show that ($\mathcal{P}_{\lambda,\varepsilon}$) has no nontrivial solution. For $\lambda>0$, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of $\Lambda_\varepsilon$ such that ($\mathcal{P}_{\lambda,\varepsilon}$) has a positive minimal solution for $0<\lambda<\Lambda_\varepsilon$, a positive solution for $\lambda=\Lambda_\varepsilon$ and no positive solution for $\lambda>\Lambda_\varepsilon$. We also prove the existence of $0<\lambda^\#\leq\Lambda_\varepsilon$ such that ($\mathcal{P}_{\lambda,\varepsilon}$) has at least two positive solutions for $\lambda\in(0,\lambda^\#)$ provided $\varepsilon$ small enough. This extends the recent result of Biagi and Vecchi (Nonlinear Anal. 256 (2025),113795), Amundsen, et al. (Commun. Pure Appl. Anal., 22(10):3139-3164, 2023) from $p=2$ to the general $1<p<N$. Additionally, it extends the classical result of Azorero and Peral (Indiana Univ. Math. J., 43(3):947-957, 1994) to the mixed local-nonlocal quasilinear problems. Moreover, our results complements the multiplicity results for nonnegative solutions in da Silva, et al. (J. Differential Equations, 408:494-536, 2024).

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

2 major / 2 minor

Summary. The manuscript studies positive solutions to the mixed local-nonlocal quasilinear problem (P_{λ,ε}) driven by -Δ_p u + ε(-Δ_p)^s u with concave-critical nonlinearity λ|u|^{q-2}u + |u|^{p^*-2}u. For λ ≤ 0 there are no nontrivial solutions. For λ > 0 the authors establish an Ambrosetti-Brezis-Cerami-type threshold Λ_ε > 0 such that a minimal positive solution exists for 0 < λ < Λ_ε, a positive solution exists at λ = Λ_ε, and no positive solution exists for λ > Λ_ε. They further claim that for all sufficiently small ε > 0 there exists 0 < λ# ≤ Λ_ε such that at least two positive solutions exist for every λ ∈ (0, λ#). The results extend the local p-Laplacian case of Azorero-Peral and the p = 2 mixed case of Biagi-Vecchi to general 1 < p < N.

Significance. If the proofs are complete, the work supplies a nontrivial extension of multiplicity results for critical-exponent problems to mixed local-nonlocal operators. The adaptation of mountain-pass geometry, Nehari-manifold analysis, and sub-supersolution methods to the mixed setting is of interest to researchers working on nonlocal perturbations of quasilinear equations.

major comments (2)
  1. [Theorem 1.2 (multiplicity part) and the estimates in §4–§5] The multiplicity statement (existence of λ# > 0 with two solutions for λ ∈ (0, λ#)) is asserted only for ε sufficiently small, yet the manuscript provides no explicit quantitative bound on this smallness in terms of q, s, N, p, or |Ω|. Without such control it is impossible to verify whether the mountain-pass geometry or the separation of the two critical levels survives for any fixed ε > 0, which directly affects the validity of the two-solution interval.
  2. [§3 (variational setting) and §5 (multiplicity argument)] The proof that the mixed operator remains a compact perturbation of the local p-Laplacian for small ε relies on fractional Sobolev embeddings and interaction terms ε∫|u|^p; the constants arising from these embeddings are not tracked explicitly enough to confirm that the required inequality for the Nehari manifold or the sub-supersolution comparison holds uniformly down to a positive λ#.
minor comments (2)
  1. [Introduction] The abstract and introduction cite the extensions to Azorero-Peral and Biagi-Vecchi but do not include a concise table or paragraph contrasting the new technical obstacles (mixed operator, general p) with the earlier proofs.
  2. [§2 (preliminaries)] Notation for the fractional p-Laplacian and the critical exponent p^* is introduced without a preliminary reminder of the precise definition of the Gagliardo seminorm used throughout the estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and insightful comments on our manuscript. The observations regarding the multiplicity result and the tracking of constants are well-taken. We address each point below and indicate the revisions we are prepared to make.

read point-by-point responses

  1. Referee: [Theorem 1.2 (multiplicity part) and the estimates in §4–§5] The multiplicity statement (existence of λ# > 0 with two solutions for λ ∈ (0, λ#)) is asserted only for ε sufficiently small, yet the manuscript provides no explicit quantitative bound on this smallness in terms of q, s, N, p, or |Ω|. Without such control it is impossible to verify whether the mountain-pass geometry or the separation of the two critical levels survives for any fixed ε > 0, which directly affects the validity of the two-solution interval.

    Authors: We agree that an explicit numerical bound on ε is not computed in the manuscript. The smallness condition arises because the perturbation terms ε∫|∇u|^p and the fractional interaction terms must remain strictly smaller than the leading local p-Laplacian contributions in the mountain-pass geometry and in the separation of the two critical values on the Nehari manifold. All embedding constants appearing in §§4–5 depend only on the fixed parameters q, s, N, p and |Ω| and are therefore independent of ε. Consequently, there exists ε0>0 (determined by these fixed constants) such that the required inequalities hold uniformly for all ε∈(0,ε0). While we elected not to display the lengthy but straightforward calculation of ε0 in order to keep the exposition readable, the existence of such an ε0 follows directly from the continuity of the estimates with respect to ε. We will add a short remark after the statement of Theorem 1.2 that explicitly indicates how ε0 is determined by the constants already present in the proofs of §§4–5. revision: partial

  2. Referee: [§3 (variational setting) and §5 (multiplicity argument)] The proof that the mixed operator remains a compact perturbation of the local p-Laplacian for small ε relies on fractional Sobolev embeddings and interaction terms ε∫|u|^p; the constants arising from these embeddings are not tracked explicitly enough to confirm that the required inequality for the Nehari manifold or the sub-supersolution comparison holds uniformly down to a positive λ#.

    Authors: The referee correctly notes that the constants are not written out in full detail. In §3 we use the compact embedding of W^{1,p}_0(Ω) into L^r(Ω) for r<p* together with the fractional Sobolev embedding for the term ε(-Δ_p)^s u. These embeddings produce constants C=C(N,p,s,|Ω|) that are independent of ε and of λ. In the Nehari-manifold analysis of §5 the quadratic and higher-order terms are estimated by absorbing the ε-terms into the local p-term provided ε is smaller than a positive threshold depending on those same constants. The sub-supersolution comparison likewise holds once the minimal solution is known to lie below the mountain-pass solution, which is guaranteed by the same ε-smallness. We will revise §5 to include one additional sentence that records the explicit dependence of the threshold on the embedding constants already introduced in §3, thereby making the uniform validity down to a positive λ# transparent. revision: partial

Circularity Check

0 steps flagged

No significant circularity; independent adaptation of variational methods to mixed operator

full rationale

The paper establishes the existence of thresholds Λ_ε and λ# (with multiplicity for small ε) for the mixed p-Laplacian plus fractional p-Laplacian problem by adapting standard Ambrosetti-Brezis-Cerami arguments, mountain-pass geometry, Nehari manifold analysis, and sub-supersolution comparison from the cited local case (Azorero-Peral) and p=2 mixed case (Biagi-Vecchi). These extensions rely on the compactness of the embedding and the perturbation structure for small ε, but the derivation does not reduce any claimed prediction or threshold to a fitted parameter, self-definition, or load-bearing self-citation. The smallness condition on ε is an explicit technical hypothesis required for the nonlocal term to preserve the geometry of the local problem; it is not smuggled in via prior self-work or used to rename a known result. All load-bearing steps are externally supported by the referenced independent literature and standard functional-analytic tools, making the central claims self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard functional-analytic axioms for the mixed operator; no free parameters or new postulated entities appear in the abstract.

axioms (2)
  • standard math Sobolev embeddings and compact embeddings hold for the space associated with the mixed local-nonlocal operator
    Required to define the critical exponent p* and to obtain compactness for the variational formulation.
  • domain assumption The mixed operator -Δ_p + ε(-Δ_p)^s generates a coercive functional on the appropriate Sobolev space
    Necessary for the mountain-pass and minimization arguments used to obtain solutions.

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