Existence and convergence of ground state solutions for Choquard-type systems on lattice graphs

Lidan Wang

REVIEW 3 major objections 5 minor 29 references

Existence and convergence of ground state solutions for Choquard-type systems on lattice graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that a Choquard-type p-Laplacian system on the integer lattice has ground state solutions for large coupling, and that these solutions converge onto the potential wells as the coupling grows.

desk verdict Solid extension of the graph variational program, but the proof of Theorem 1.1 has a genuine gap that needs a standard radial-graph argument. read the letter →

arxiv 2507.20464 v1 pith:BORGNQ5S submitted 2025-07-28 math.AP

Lidan Wang This is my paper · ORCID

classification math.AP MSC 35J5035R02

keywords latticegraphsp-LaplaciansystemChoquard-typenonlinearitygroundstatesolutionsNeharimanifoldpotentialwellsconcentrationofdiscreteHardy-Littlewood-Sobolevinequality

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 the discrete Choquard-type system with the $p$-Laplacian on $\mathbb{Z}^N$ has a ground state solution for every sufficiently large $\lambda$ and every $p\ge 2$, under assumptions (F1), (A1), and (A'2). It further proves that as $\lambda\to\infty$, any sequence of ground states converges strongly in $W^{1,p}\times W^{1,p}$ to a ground state of the limit problem posed on the potential wells $\Omega_a,\Omega_b$ with zero boundary data. The paper identifies this as the first treatment of a Choquard-type system on graphs, and the potential condition (A'2)—finite, nonempty sublevel sets—is weaker than the decay or summability conditions used in prior graph results, so compactness has to be recovered by a tail estimate rather than an embedding. The Nehari manifold provides the variational framework: one minimizes the energy on the set where the radial derivative vanishes, and the tail estimate converts Palais-Smale sequences into convergent ones.

What carries the argument

The load-bearing machinery is the Nehari manifold $N_\lambda=\{(u,v)\ne(0,0): \langle J'_\lambda(u,v),(u,v)\rangle=0\}$, which turns the search for ground states into a constrained minimization problem. The discrete Hardy-Littlewood-Sobolev inequality controls the nonlocal term $\int (R_\alpha*F(u,v))F(u,v)\,d\mu$ by the norm $\|(u,v)\|_\lambda^{2\gamma}$, and the tail estimate of Lemma 3.2 splits $\mathbb{Z}^N\setminus B_r$ into the sets where $a\ge M_1$ or $b\ge M_2$—controlled for large $\lambda$—and the finite sets where $a<M_1$ or $b<M_2$—controlled for large $r$. Together these give the Palais-Smale compactness condition at bounded energy levels, which converts a minimizing sequence into a true ground state and later identifies the limit profile with the Dirichlet problem on the wells.

What would settle it

Take a concrete pair of potentials satisfying (A1) and (A'2)—for example $a(x)=b(x)=0$ on a finite cube and $1$ outside—and solve the variational problem on large finite boxes with $p=2$, $N=3$, $\alpha=1$, $\gamma=3$. If the numerically computed Nehari minimum $m_\lambda$ does not converge to the ground-state level of the Dirichlet limit problem on the cube as $\lambda\to\infty$, or if a bounded Palais-Smale sequence at a level below the threshold keeps mass in every large ball instead of converging, the central claim would be refuted.

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Extended reading notes

Core claim

The central claim is that the parameter-dependent problem is variational and has minimal-energy solutions that do not escape to infinity. Theorem 1.1 gives, for $\lambda\ge\lambda_0$ and $p\ge 2$, a ground state $(u_\lambda,v_\lambda)$ at the Nehari level $m_\lambda$, meaning a nontrivial critical point of the energy $J_\lambda$ with minimal energy on the Nehari manifold. Theorem 1.2 says that along any sequence $\lambda_k\to\infty$, a subsequence of these ground states converges in $W^{1,p}(V)\times W^{1,p}(V)$ to a ground state of the Dirichlet problem (3) on $\Omega_a\times\Omega_b$, and the convergence is accompanied by convergence of the energy levels $m_\lambda\to m_\Omega$. In the paper's own framing, the nonlocal Choquard interaction does not destroy the semiclassical concentration picture familiar from local Schrödinger-type problems on graphs.

Load-bearing premise

The results rest on (A'2): the sets $\{x: a(x)\le M_1\}$ and $\{x: b(x)\le M_2\}$ must be finite and nonempty, because the proof's control of energy outside large balls and its compactness step both need those sets to be negligible at infinity.

Editorial extensions

If this is right

For large $\lambda$ the system has a nontrivial solution at the minimum energy level, so the nonlocal Choquard coupling does not destroy the variational ground-state structure on graphs.
As $\lambda\to\infty$, ground states concentrate on the potential wells: the limiting profile vanishes outside $\Omega_a\times\Omega_b$ and satisfies the Dirichlet problem (3), so the wells act as the effective domain in the strong-coupling limit.
The energy levels satisfy $m_\lambda\to m_\Omega$, giving a quantitative identity linking the parameter-dependent problem to the limit problem on the wells.
The compactness threshold needed for the Palais-Smale condition is uniform in $\lambda$ at each bounded energy level, so the existence result holds for all sufficiently large $\lambda$ rather than only in a single parameter regime.
The weaker hypothesis (A'2) replaces the decay or summability assumptions of earlier graph results, so the concentration phenomenon is shown for a broader class of potentials.

Reading between the lines

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

This suggests the same tail-splitting argument should transfer to more general locally finite graphs of polynomial growth, provided a discrete Hardy-Littlewood-Sobolev inequality and finite sublevel sets are available; the paper itself proves the result only on $\mathbb{Z}^N$.
Because the compactness threshold $c_0$ in Lemma 3.1 does not depend on $\lambda$, the proof likely yields uniform bounds on ground-state norms for all large $\lambda$, which could support quantitative rate-of-concentration statements not written in the paper.
A natural test of whether (A'2) is close to sharp is to construct potentials whose sublevel sets are infinite but thin, such as a slowly growing sequence of wells; if those fail to concentrate, the finiteness condition is doing the load-bearing work the proof assigns to it.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper fills the last obvious slot in the graph-variational program: p-Laplacian systems with Choquard-type nonlocal coupling on Z^N, under the weaker potential condition (A'2) rather than (A2) or (Ã2). The existence and concentration results are plausible, and the architecture (Nehari manifold, tail estimates, PS compactness, limiting problem on the wells) is standard. The technical core is the tail estimate under (A'2), and it looks right.

What's genuinely new is the combination. Single Choquard equations on graphs, p-Laplacian systems without Choquard terms, and (A'2)-type potential conditions all exist separately, but nobody has put all three together. The 'first work' claim in Remark 1.3(iii) is plausible from the cited literature.

The paper does a lot well. Lemma 3.2's tail estimate is the key step, and the way finiteness of {a≤M1} controls the outer tail as r→∞ while λ controls the other tail is sound. Lemma 3.5's PS argument and Lemma 4.1's m_λ→m_Ω argument interlock correctly. The integration-by-parts lemma for their definition of Δ_p is a useful detail.

Now the soft spots, in order of size.

1. Theorem 1.1's proof has a real gap. After Lemma 2.9, the paper says the mountain-pass geometry 'hence' gives a (PS) sequence at level m_λ. That is not what the mountain-pass theorem gives; it gives a sequence at the minimax level c. The equality c = m_λ is never stated or proved. This is load-bearing: the final step computes J_λ(u_λ,v_λ) = m_λ using that level, which is what makes the critical point a ground state. The gap is repairable by the standard radial-graph argument: N_λ is a graph over the unit sphere and separates 0 from the set {J_λ < 0}, so every admissible path crosses N_λ, giving c ≥ m_λ; radial paths give c ≤ m_λ. But the paper omits this and asserts the stronger conclusion. This needs to be fixed; it is not a typo.

2. Lemma 4.1 uses |u_k−u|^2 (and δ^2) where it should be |u_k−u|^p (and δ^p). This looks like a p=2 leftover in the concentration argument. It is mechanically wrong but repairable, since u(x_k)→0 for |x_k|→∞, so the same lower bound works with exponent p.

3. Lemma 3.2 has a carelessly handled constant: the bound is displayed as (2pc*/(p−1+1)) = 2c*, which doesn't match the (2γpc*/(2γ−p)) from Lemma 3.1. It doesn't change the conclusion because µ(Ω_r^-)→0 controls the limit, but the displayed constant is wrong.

The citation pattern looks fine; the paper leans on the right prior work and doesn't misrepresent it. No data or code, which is normal for this kind of paper.

Who this is for: people working on variational methods on graphs. They should read it. It deserves a serious referee, not a desk reject. I'd send it out, but the referee's first question should be the c_MP = m_λ issue. Conditional accept after revision.

Referee Report

3 major / 5 minor

Summary. The paper studies a p-Laplacian Choquard-type system on the lattice graph Z^N with potentials λa+1 and λb+1. Under assumptions (F1), (A1), and (A'2), it claims two results: Theorem 1.1 asserts existence of a ground state solution for all sufficiently large λ, and Theorem 1.2 asserts that along any sequence λ_k→∞ the ground states converge, up to subsequence, in W^{1,p}(V)×W^{1,p}(V) to a ground state of a limit system posed on the potential wells Ω_a and Ω_b with zero Dirichlet data on the vertex boundary. The proofs are based on the Nehari manifold method, a mountain-pass geometry, tail estimates exploiting the finiteness of the sublevel sets in (A'2), a (PS)_c compactness argument, and a comparison of the Nehari levels m_λ and m_Ω.

Significance. If the proof gaps identified below are repaired, the paper would be a meaningful extension of recent single-equation Choquard results on graphs to systems, and it would weaken the compactness assumptions used in prior work from decay or summability conditions to the finiteness condition (A'2). The compactness mechanism built on tail estimates and finite sublevel sets is a useful contribution, and the paper carefully presents the discrete Hardy-Littlewood-Sobolev inequality and a Brezis-Lieb-type splitting lemma adapted to the graph setting. However, the central claims are not fully established as written: the identification of the mountain-pass level with the Nehari level is missing, the non-degeneracy of F is not imposed, and one lower-bound estimate in the convergence proof is not justified. These issues are local and repairable, but they are load-bearing for the main theorems.

major comments (3)

[Section 3, proof of Theorem 1.1] The proof asserts that the mountain-pass geometry of Lemma 2.9 'hence' yields a (PS)_{m_λ} sequence. The mountain-pass theorem only provides a (PS)_c sequence at the minimax level c over paths from 0 to a point of negative energy, and the equality c = m_λ is neither stated nor proved. This equality is load-bearing because the final identification J_λ(u_λ,v_λ)=m_λ uses the (PS) level in place of m_λ. The gap is repairable by the standard radial-graph argument: every admissible path from 0 to a point of negative energy must cross N_λ, giving c ≥ m_λ, while the ray t(u,v) through a point of N_λ gives c ≤ m_λ; however, as written the manuscript omits this argument.
[Lemma 2.9(ii)] The statement that for any nonzero (u,v), J_λ(t(u,v))→−∞ as t→∞ requires ∫_V (R_α*F(u,v))F(u,v) dμ > 0. Under (F1) alone this need not hold: F may vanish on a nontrivial cone, for example F(u,v)=|u|^γ, for which the nonlocal term is zero whenever u≡0, and then J_λ(t(0,v))→+∞ along the v-direction. The mountain-pass geometry only needs one direction with positive nonlocal energy, so the proof can likely be repaired by adding a non-degeneracy assumption such as F(u,v)>0 for all (u,v)≠(0,0), or by explicitly choosing a suitable test direction; but as written Lemma 2.9 and hence Theorem 1.1 rely on an unjustified assertion.
[Lemma 4.1] The displayed lower bound m_{λ_k} ≥ (1/p−1/(2γ)) ∫_{B_r(x_k)∩{a≥M_1}} λ_k a |u_k−u|^2 dμ is not justified: on {a≥M_1} one has u=0, so |u_k|=|u_k−u|, but for p≥2 the pointwise inequality |u_k−u|^p ≥ |u_k−u|^2 is false on points where |u_k−u|<1. Consequently the chain leading to λ_k M_1(δ²/4+o_k(1)) does not follow. The contradiction argument is still recoverable because |u_k(x_k)−u(x_k)|≥δ/2 and a(x_k)≥M_1 for large k imply ∥u_k∥^p_{λ_k} ≥ λ_k M_1(δ/2)^p; however, the proof as written needs this correction.

minor comments (5)

[Lemma 2.5] In the proof, the terms F(w_k,v_k) should be F(w_k,z_k) in the displayed estimates.
[Lemma 3.2] The constant M is introduced although (A'2) quantifies M_1 and M_2; the tail estimate for u should use M_1 and that for v should use M_2.
[Proof of Theorem 1.2] The references to 'the proof of Lemma 3.1' and 'By Lemma 3.1' should refer to Lemma 4.1 for the convergence m_{λ_k}→m_Ω and the ℓ^q convergence.
[Section 1] There is a typo: 'respecct' should be 'respect'.
[Proof of Theorem 1.1] The application of Lemma 3.5 requires a uniform bound c≤c* for the (PS) level; since m_λ≤m_Ω, one should explicitly take c*=m_Ω so that λ_0 is independent of the sequence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence and convergence results are derived from the stated hypotheses via internal variational arguments, with self-citations used only for external standard inequalities and motivation.

full rationale

The paper's central claims, Theorem 1.1 and Theorem 1.2, are obtained by an internal Nehari-manifold and compactness argument: the energy functional and Nehari manifold are defined directly from the system, the boundedness and (PS)_c recovery are proven in the text, and the limit problem (3) is solved independently. The discrete Hardy-Littlewood-Sobolev inequality (Lemma 2.3) is imported as an external benchmark result, and although it is cited to [11,23] where [23] is the author's own work, it is a standard external inequality, not a conclusion of this paper. The author's other self-citations ([23,25,26]) are used only for motivation and comparison, not as load-bearing proof steps. The only notable issue is a proof gap flagged in the proof of Theorem 1.1: the mountain-pass geometry in Lemma 2.9 yields a (PS)_c sequence at the mountain-pass minimax level, while the text asserts a sequence at the Nehari level m_λ without proving c_MP = m_λ. However, this is a missing standard argument (or correctness risk), not circularity: m_λ is defined as the infimum over N_λ independently, and no equation in the paper defines m_λ in terms of the mountain-pass level or vice versa. Similarly, the comparisons m_λ ≤ m_Ω in Lemma 4.1 and the use of (A'2) are substantive compactness hypotheses, not restatements of the conclusions. The derivation chain does not reduce any predicted result to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

Pure existence and convergence paper: no data fitting, no empirical constants. The three 'free parameters' are proof thresholds with arbitrary positive values. The load-bearing inputs are external results (discrete HLS inequality, Hua-Li splitting lemma, standard reflexive-space facts) plus the structural hypotheses (A1), (A'2), (F1), all stated openly. No new entities are invented; the only nonstandard technical object, tail-estimate compactness under (A'2), is a repackaging of the argument in [11].

free parameters (3)

σ = (1/C_{N,α,M_F})^{1/(2γ−p)}
Nehari-manifold norm lower bound (Lemma 2.7(i)); chosen by hand to make the bound uniform, carries no empirical content.
ρ = (3/(4C_{N,α,M_F}))^{1/(2γ−p)}
Mountain-pass radius (Lemma 2.9 and Lemma 3.1(ii)); hand-chosen threshold, its exact value is immaterial.
c0 = (2γ−p)ρ^p/(2γp)
(PS)-level gap constant (Lemma 3.1(ii), Lemma 3.5); chosen so levels below it force the null sequence; not fitted to any data.

assumptions (5)

standard math Discrete Hardy-Littlewood-Sobolev inequality (Lemma 2.3), cited to [11,23]
External result: for 0 < α < N and 1/r + 1/s + (N−α)/N = 2, ∫(R_α∗u)v ≤ C‖u‖_r‖v‖_s on the lattice. Used at r = s = 2N/(N+α) in Corollary 2.4 to bound the Choquard term by ‖(u,v)‖^{2γ}_λ; the variational formulation and every (PS)-estimate depend on it.
standard math Brézis-Lieb type splitting for discrete p-gradients (Corollary 11 in [9])
Cited to Hua-Li (J. Diff. Eq. 305, 2021). Used in Lemma 3.4 and Theorem 1.2 to split ∫|∇u_k|^p as ∫|∇(u_k−u)|^p + ∫|∇u|^p + o(1), a key step in the energy-concentration analysis.
domain assumption Potential-well hypotheses (A1) and (A'2)
(A1): a, b ≥ 0 with non-empty bounded wells Ω_a, Ω_b and Ω_a ∩ Ω_b. (A'2): {a ≤ M1} and {b ≤ M2} finite and non-empty. These define the large-λ limit problem (3) on Ω_a × Ω_b and are the only compactness-replacing input in Lemmas 3.2, 3.5 and 4.1.
domain assumption Homogeneity condition (F1) with γ > (N+α)p/(2N)
F ∈ C^1 and F(tu,tv) = t^γ F(u,v). Gives the Euler identity uF_u + vF_v = γF, the growth bound |F| ≤ M_F(|u|^γ + |v|^γ), and the integrability condition 2Nγ/(N+α) > p; all three are used throughout Sections 2 to 4.
domain assumption R_α is the Green's function of the discrete fractional Laplacian, symmetric and satisfying the HLS bounds
Definition in Section 1 via the heat-kernel integral. Symmetry of R_α is used in computing ⟨J'_λ(u,v),(u,v)⟩ in Section 2; the HLS bounds (Lemma 2.3) make the nonlocal term finite. The kernel's properties are taken as given, not proved here.

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Pith. "Pith review of Existence and convergence of ground state solutions for Choquard-type systems on lattice graphs." pith.science (2026). https://pith.science/paper/BORGNQ5S

@misc{pith2026250720464,
  author       = {Pith},
  title        = {Pith review of: Existence and convergence of ground state solutions for Choquard-type systems on lattice graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BORGNQ5S}},
  note         = {Machine review of arXiv:2507.20464}
}

abstract

In this paper, we study the $p$-Laplacian system with Choquard-type nonlinearity $$ \begin{cases}-\Delta_{p} u+(\lambda a+1)|u|^{p-2} u=\frac{1}{\gamma} \left(R_\alpha\ast F(u,v)\right)F_{u}(u, v), \\ -\Delta_{p} v+(\lambda b+1)|v|^{p-2} v=\frac{1}{\gamma} \left(R_\alpha\ast F(u,v)\right)F_{v}(u, v),\end{cases} $$ on lattice graphs $\mathbb{Z}^N$, where $\alpha \in(0,N),\,p\geq 2,\,\gamma> \frac{(N+\alpha)p}{2N},\,\lambda>0$ is a parameter and $R_{\alpha}$ is the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some assumptions on the functions $a,\,b$ and $F$, we prove the existence and asymptotic behavior of ground state solutions by the method of Nehari manifold.

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Works this paper leans on

29 extracted references · 22 canonical work pages

[1]

Alves, Y

C.O. Alves, Y. Ding, Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 265-278

work page 2007
[2]

Bartsch, Z.Q

T. Bartsch, Z.Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN. Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725-1741

work page 1995
[3]

Bartsch, Z.Q

T. Bartsch, Z.Q. Wang, Multiple positive solutions for a nonlinear Schr¨ odinger equation. Z. Angew. Math. Phys. 51 (2000), no. 3, 366-384

work page 2000
[4]

Furtado, Multiple minimal nodal solutions for a quasilinear Schr¨ odinger equation with symmetric potential

M.F. Furtado, Multiple minimal nodal solutions for a quasilinear Schr¨ odinger equation with symmetric potential. J. Math. Anal. Appl. 304 (2005), no. 1, 170-188

work page 2005
[5]

Gheraibia, Solutions for quasilinear elliptic systems with vanishing potentials

B. Gheraibia, Solutions for quasilinear elliptic systems with vanishing potentials. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 153-175

work page 2019
[6]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, and Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci. China Math. 60 (2017), no. 7, 1311-1324

work page 2017
[7]

X. Han, M. Shao, p-Laplacian equations on locally finite graphs. Acta Math. Sin. 37(11) (2021) 1645-1678

work page 2021
[8]

X. Han, M. Shao and L. Zhao, Existence and convergence of solutions for nonlinear biharmonic equations on graphs. J. Differential Equations 268 (2020), no. 7, 3936-3961

work page 2020

Show all 29 references

[9]

B. Hua, R. Li, The existence of extremal functions for discrete Sobolev inequalities on lattice graphs. J. Differential Equations 305 (2021), 224-241

2021
[10]

B. Hua, R. Li and L. Wang, A class of semilinear elliptic equations on groups of polynomial. J. Differential Equations 363 (2023) 327-349

2023
[11]

R. Li, L. Wang, The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth. Accepted by Journal of Partial Differential Equations
[12]

Y. Liu, M. Zhang, Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs. J. Math. Anal. Appl. 534 (2024), no. 2, Paper No. 128079, 18 pp

2024
[13]

Y. Liu, M. Zhang, The ground state solutions to a class of biharmonic Choquard equations on weighted lattice graphs. Bull. Iranian Math. Soc. 50 (2024), no. 1, Paper No. 12, 17 pp

2024
[14]

D. L¨ u, S. Peng, Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type systems. J. Differential Equations 263 (2017), no. 12, 8947-8978. 21

2017
[15]

L¨ u, Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems

D. L¨ u, Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems. Monatsh. Math. 182 (2017), no. 2, 335-358

2017
[16]

D. L¨ u, S. Dai, Existence and asymptotic behavior of solutions for generalized Choquard systems. Bull. Malays. Math. Sci. Soc. 47 (2024), no. 1, Paper No. 29, 18 pp

2024
[17]

D. L¨ u, Q. Liu, Multiplicity of solutions for a class of quasilinear Schr¨ odinger systems inRN. Comput. Math. Appl. 66 (2014), no. 12, 2532-2544

2014
[18]

Man, Existence results for some nonlinear elliptic systems on graphs

S. Man, Existence results for some nonlinear elliptic systems on graphs. J. Math. Anal. Appl. 543 (2025), no. 2, part 2, Paper No. 128973, 33 pp

2025
[19]

Shao, Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs

M. Shao, Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs. J. Fixed Point Theory Appl. 25 (2023), no. 2, Paper No. 50, 21 pp

2023
[20]

M. Shao, Y. Yang and L. Zhao, Multiplicity and limit of solutions for logarithmic Schr¨ odinger equations on graphs. J. Math. Phys. 65 (2024), no. 4, Paper No. 041508, 17 pp

2024
[21]

L. Shen, M. Squassina, Existence and concentration of positive solutions to generalized Chern-Simons-Schr¨ odinger system with critical exponential growth. J. Math. Anal. Appl. 543 (2025), no. 2, part 2, Paper No. 128926, 29 pp

2025
[22]

J. Wang, Y. Zhu and K. Wang, Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs. Electron. Res. Arch. 31 (2023), no. 2, 812-839

2023
[23]

Wang, The ground state solutions to discrete nonlinear Choquard equations with Hardy weights

L. Wang, The ground state solutions to discrete nonlinear Choquard equations with Hardy weights. Bull. Iranian Math. Soc. 49 (2023), no. 3, Paper No. 30, 29 pp

2023
[24]

Wang, Solutions to discrete nonlinear Kirchhoff-Choquard equations

L. Wang, Solutions to discrete nonlinear Kirchhoff-Choquard equations. Bull. Malays. Math. Sci. Soc. 47 (2024), no. 5, Paper No. 138

2024
[25]

Wang, p-Laplacian equations with general Choquard nonlinearity on lattice graphs

L. Wang, p-Laplacian equations with general Choquard nonlinearity on lattice graphs. arXiv:2408.10584

arXiv
[26]

Wang, A class of p-Laplacian equations on lattice graphs

L. Wang, A class of p-Laplacian equations on lattice graphs. Accepted by Acta Mathematica Sinica (English sysries)
[27]

J. Xu, L. Zhao, Existence and convergence of solutions for nonlinear elliptic systems on graphs. Commun. Math. Stat. 12 (2024), no. 4, 735-754

2024
[28]

Zhang, L

N. Zhang, L. Zhao, Convergence of ground state solutions for nonlinear Schr¨ odinger equations on graphs. Sci. China Math. 61 (2018), no. 8, 1481-1494

2018
[29]

Zhang, X

X. Zhang, X. Zhang, Existence and convergence of ground state solutions for a ( p,q )-Laplacian system on weighted graphs. arXiv:2403.02048 Email address : wanglidan@ujs.edu.cn Lidan W ang: School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China 22

arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Alves, Y

C.O. Alves, Y. Ding, Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 265-278

work page 2007

[2] [2]

Bartsch, Z.Q

T. Bartsch, Z.Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN. Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725-1741

work page 1995

[3] [3]

Bartsch, Z.Q

T. Bartsch, Z.Q. Wang, Multiple positive solutions for a nonlinear Schr¨ odinger equation. Z. Angew. Math. Phys. 51 (2000), no. 3, 366-384

work page 2000

[4] [4]

Furtado, Multiple minimal nodal solutions for a quasilinear Schr¨ odinger equation with symmetric potential

M.F. Furtado, Multiple minimal nodal solutions for a quasilinear Schr¨ odinger equation with symmetric potential. J. Math. Anal. Appl. 304 (2005), no. 1, 170-188

work page 2005

[5] [5]

Gheraibia, Solutions for quasilinear elliptic systems with vanishing potentials

B. Gheraibia, Solutions for quasilinear elliptic systems with vanishing potentials. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 153-175

work page 2019

[6] [6]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, and Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci. China Math. 60 (2017), no. 7, 1311-1324

work page 2017

[7] [7]

X. Han, M. Shao, p-Laplacian equations on locally finite graphs. Acta Math. Sin. 37(11) (2021) 1645-1678

work page 2021

[8] [8]

X. Han, M. Shao and L. Zhao, Existence and convergence of solutions for nonlinear biharmonic equations on graphs. J. Differential Equations 268 (2020), no. 7, 3936-3961

work page 2020