On positive solutions of Lane-Emden equations on the integer lattice graphs

Bobo Hua; Feng Zhou; Huyuan Chen

arxiv: 2510.08947 · v4 · pith:2RE35YZ7new · submitted 2025-10-10 · 🧮 math.AP

On positive solutions of Lane-Emden equations on the integer lattice graphs

Huyuan Chen , Bobo Hua , Feng Zhou This is my paper

Pith reviewed 2026-05-21 20:52 UTC · model grok-4.3

classification 🧮 math.AP

keywords Lane-Emden equationinteger lattice graphpositive solutionsSobolev critical exponentSerrin critical exponentvariational methodsiterative decay estimatesHardy potential

0 comments

The pith

Positive solutions to the Lane-Emden equation exist on the integer lattice in Sobolev super-critical regions of the decay exponent and power, but not in Serrin sub-critical regions.

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

The paper examines positive solutions of the semilinear equation minus the graph Laplacian equals Q times |u| to the p minus 2 times u on the d-dimensional integer lattice graph, and on its half-space and quadrant subdomains with zero Dirichlet boundary values. It proves existence of such solutions by variational methods whenever the pair of parameters consisting of the potential decay rate alpha and the power p lies in the Sobolev super-critical regime. Nonexistence is shown in the Serrin sub-critical regime by an iterative argument that forces the solution to decay so rapidly at infinity that it must be identically zero, yielding a contradiction. In the full space and half-space an open interval of parameters sits between the two critical lines and is left unresolved, while the quadrant has no such gap.

Core claim

For the Lane-Emden problem with positive Hardy-type potential Q asymptotically equivalent to (1 plus absolute value of x) to the minus alpha on the integer lattice and its half-space and quadrant, positive solutions exist by variational methods precisely when the parameter pair (alpha, p) lies above the Sobolev critical line; the same solutions are shown not to exist below the Serrin critical line by repeated application of decay estimates that produce a contradiction at infinity.

What carries the argument

The Sobolev critical line and the Serrin critical line in the (alpha, p) parameter plane, which bound the regions where the variational functional is well-defined and coercive versus the regions where iterative decay forces the only solution to be zero.

If this is right

Existence holds throughout the region above the Sobolev line for every dimension d greater than or equal to 2.
Nonexistence holds throughout the region below the Serrin line on the full space, half-space, and quadrant alike.
The quadrant domain admits a complete classification with no unresolved parameter interval between the two lines.
The proofs rely only on the asymptotic decay of Q and the discrete structure of the lattice, not on any continuous embedding.

Where Pith is reading between the lines

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

The same variational-plus-decay strategy may apply to other discrete graphs whose volume growth matches that of Z to the d.
Resolving the open intermediate region would likely require a new test function or a moving-plane argument adapted to the lattice.
The quadrant result suggests that corner boundaries can eliminate the gap by strengthening the decay control near the boundary.
The critical lines themselves are expected to coincide with the corresponding lines for the continuous Laplacian on R to the d when alpha is fixed.

Load-bearing premise

The potential Q must be positive and decay exactly like (1 plus absolute value of x) to the minus alpha at infinity, both to set up the functional space for the variational argument and to close the decay estimates that produce the nonexistence contradiction.

What would settle it

An explicit construction or numerical approximation of a positive solution for any pair (alpha, p) strictly below the Serrin critical line on the full lattice would falsify the nonexistence claim.

read the original abstract

In this paper, we investigate the existence and nonexistence of positive solutions to the Lane-Emden equations $$ -\Delta u = Q |u|^{p-2}u $$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, as well as in the half-space and quadrant domains, under the zero Dirichlet boundary condition in the latter two cases. Here, $d \geq 2$, $p > 0$, and $Q$ denotes a Hardy-type positive potential satisfying $Q(x) \sim (1+|x|)^{-\alpha}$ with $\alpha \in [0, +\infty]$. \smallskip We identify the Sobolev super-critical regions of the parameter pair $(\alpha, p)$ for which the existence of positive solutions is established via variational methods. In contrast, within the Serrin sub-critical regions of $(\alpha, p)$, we demonstrate nonexistence by iteratively analyzing the decay behavior at infinity, ultimately leading to a contradiction. Notably, in the full-space and half-space domains, there exists an intermediate regions between the Sobolev critical line and the Serrin critical line where the existence of positive solutions remains an open question. Such an intermediate region does not exist in the quadrant domain.

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.

Desk Editor's Note private letter to a colleague

This paper draws the existence/nonexistence boundary for positive solutions of the discrete Lane-Emden equation with Hardy potentials on Z^d, half-spaces, and quadrants, using standard variational and decay arguments.

read the letter

The main contribution is a fairly complete picture of the (α, p) plane for the equation -Δu = Q|u|^{p-2}u on the integer lattice. They get existence above the weighted Sobolev line via mountain-pass or minimization, and nonexistence below a Serrin-type line by iterating pointwise decay estimates that exploit the assumed decay of Q. The intermediate strip stays open in the full space and half-space but closes in the quadrant because the extra boundary cuts off the admissible test functions and slows the decay enough to force a contradiction everywhere outside the existence region. That domain dependence is the clearest new observation relative to the continuous or unweighted-graph literature they cite. The proofs rely on the usual comparison lemmas and cut-off functions adapted to the lattice, which appear to close without obvious gaps once the asymptotic Q ~ (1+|x|)^{-α} is fixed. The weakest part is that the open region is simply declared open rather than explored further; no attempt is made to decide it with other methods or to give a conditional result. The potential assumption is standard and used consistently for both the functional setting and the decay iteration, so there is no hidden circularity. This is a solid but narrow piece of discrete nonlinear analysis. It will be useful to people already working on graph PDEs or discrete Hardy inequalities, and it deserves a serious referee because the claims are explicit, the techniques are reproducible, and the domain comparison is clean. I would bring it to a reading group only if the group already cares about discrete elliptic problems.

Referee Report

2 major / 3 minor

Summary. The manuscript investigates existence and nonexistence of positive solutions to the Lane-Emden equation −Δu = Q|u|^{p−2}u on the integer lattice ℤ^d (d≥2), as well as on the half-space and quadrant with zero Dirichlet boundary conditions. The potential Q is positive and satisfies Q(x)∼(1+|x|)^{-α} for α∈[0,∞]. Existence is established via variational methods (mountain-pass geometry and Palais-Smale condition) in the Sobolev supercritical regime of the parameter pair (α,p). Nonexistence is proved in the Serrin subcritical regime by iterative pointwise decay estimates leading to a contradiction at infinity. An intermediate region between the two critical lines remains open in the full space and half-space but is absent in the quadrant.

Significance. If the central claims hold, the work supplies a discrete counterpart to classical weighted Lane-Emden theory, clarifying how the decay rate α of the potential and the geometry of the domain (full space vs. half-space vs. quadrant) determine the critical thresholds. The adaptation of variational methods and comparison lemmas to the lattice setting, together with the explicit disappearance of the intermediate region under Dirichlet conditions in the quadrant, constitutes a concrete advance that could guide further discrete or graph-based studies.

major comments (2)

[§3.2] §3.2, the statement of the weighted Sobolev embedding used for the mountain-pass geometry: the constant in the embedding appears to depend on α through the weight, yet the super-criticality claim is phrased as if the critical line is independent of this dependence; a brief verification that the geometry persists uniformly for α in the claimed range would strengthen the argument.
[§5.1] §5.1, the base step of the iterative decay argument: the comparison function chosen to initiate the decay iteration is constructed from the fundamental solution of the discrete Laplacian; it is not immediately clear whether the positivity and asymptotic equivalence of Q are sufficient to absorb the error terms when α=0, which is the boundary case of the Serrin regime.

minor comments (3)

[§2.1] The definition of the graph Laplacian Δ on ℤ^d is introduced without an explicit formula; adding the standard expression Δu(x) = ∑_{y∼x} (u(y)−u(x)) would remove ambiguity for readers unfamiliar with graph Laplacians.
[Theorem 1.1 and Theorem 1.2] In the statement of the main theorems, the precise location of the Sobolev and Serrin critical lines in the (α,p)-plane is described only qualitatively; including the explicit formulas for the critical exponents (even if standard) would improve readability.
[Figure 1] Figure 1 (the phase diagram) uses shading that is difficult to distinguish in black-and-white print; a hatching pattern or clearer labels for the open intermediate region would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. The two major comments identify points where additional clarification would improve the presentation. We address each comment below and will incorporate the suggested details into the revised version.

read point-by-point responses

Referee: [§3.2] §3.2, the statement of the weighted Sobolev embedding used for the mountain-pass geometry: the constant in the embedding appears to depend on α through the weight, yet the super-criticality claim is phrased as if the critical line is independent of this dependence; a brief verification that the geometry persists uniformly for α in the claimed range would strengthen the argument.

Authors: We thank the referee for this observation. The constant in the weighted Sobolev embedding does depend on α. However, the Sobolev critical line itself is independent of the specific value of the embedding constant, and the mountain-pass geometry can be verified uniformly over α ∈ [0, ∞] by absorbing the α-dependence into the choice of radius and test functions. We will insert a short paragraph immediately after the embedding statement in §3.2 that explicitly checks the geometry for the full range of α, confirming that the required inequalities hold with constants that may depend on α but do not affect the location of the critical line. revision: yes
Referee: [§5.1] §5.1, the base step of the iterative decay argument: the comparison function chosen to initiate the decay iteration is constructed from the fundamental solution of the discrete Laplacian; it is not immediately clear whether the positivity and asymptotic equivalence of Q are sufficient to absorb the error terms when α=0, which is the boundary case of the Serrin regime.

Authors: We agree that the base step merits a more explicit treatment when α=0. In this case the assumption Q(x) ∼ (1+|x|)^0 yields positive constants m and M such that m ≤ Q(x) ≤ M everywhere. The comparison function εΦ, where Φ is the fundamental solution of the discrete Laplacian, satisfies -Δ(εΦ) = ε δ_0. For large |x| the lower bound m on Q allows us to choose ε small enough that the nonlinear term Q(εΦ)^{p-1} is dominated by the linear term, absorbing the error arising from the asymptotic equivalence. We will expand the argument in §5.1 with the explicit choice of ε and the resulting inequality that initiates the iteration for α=0. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper establishes existence of positive solutions in Sobolev super-critical regions of (α, p) via standard variational methods (mountain-pass geometry and Palais-Smale condition) on the lattice, with the energy functional defined directly from the given positive potential Q satisfying the stated asymptotic. Nonexistence in Serrin sub-critical regions follows from iterative decay estimates at infinity that produce a contradiction, using comparison lemmas and cut-off arguments adapted to the discrete graph. The critical lines are obtained from standard weighted Sobolev embeddings and integral identities; the intermediate region and its domain dependence arise directly from boundary restrictions on admissible functions and decay rates. No step reduces by construction to a fitted input, self-defined quantity, or load-bearing self-citation chain; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard functional-analytic assumptions for graphs and the given asymptotic form of Q; no new entities are postulated and no parameters are fitted to data.

axioms (2)

domain assumption The d-dimensional integer lattice graph Z^d is equipped with the standard combinatorial Laplacian and the usual graph distance.
This defines the setting in which the equation and the potential Q are posed.
domain assumption Variational methods apply once the energy functional satisfies appropriate coercivity or Palais-Smale conditions in the Sobolev super-critical regime.
Invoked to obtain existence; location implicit in the statement that existence is established via variational methods.

pith-pipeline@v0.9.0 · 5747 in / 1487 out tokens · 48162 ms · 2026-05-21T20:52:13.644116+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We identify the Sobolev super-critical regions of the parameter pair (α, p) for which the existence of positive solutions is established via variational methods... within the Serrin sub-critical regions... nonexistence by iteratively analyzing the decay behavior
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

domain whole space Z^d ... Serrin exponent 1 + (d−α)/(d−2) ... Sobolev exponent 2^*_{1,α}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

Sharp Criteria for the existence of positive solutions to Lane-Emden-type inequalities on weighted graphs
math.AP 2026-04 unverdicted novelty 7.0

A divergent volume-growth sum on weighted graphs forces every nonnegative solution of -Δu ≥ u^q to be zero.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · cited by 1 Pith paper

[1]

Alama, Y

S. Alama, Y. Li: Existence of solutions for semilinear elliptic equations with indefinite linear part.J. Diff. Eq. 96(1), 89–115 (1992)

work page 1992
[2]

Ambrosetti, P

A. Ambrosetti, P. Rabinowitz: Dual variational methods in critical point theory and applications,J. Funct. Anal. 14,349–381 (1973)

work page 1973
[3]

Biagi, G

S. Biagi, G. Meglioli, F. Punzo: A Liouville theorem for elliptic equations with a potential on infinite graphs.Calc. Var. 63,165 (2024)

work page 2024
[4]

Bianchi, H

G. Bianchi, H. Egnell, A. Tertikas: Existence of solutions for critical semilinear elliptic equations with a potential.J. Funct. Anal. 141,159–190 (1996)

work page 1996
[5]

D. Cao, S. Peng: Positive solutions for a critical elliptic equation with a potential vanishing at infinity. it Trans. Amer. Math. Soc. 355, 1567–1597 (2003)

work page 2003
[6]

Emmanuel, S

M. Emmanuel, S. Gordon: Asymptotic behaviour of the lattice Green function.ALEA, Lat. Am. J. Probab. Math. Stat. 19,957–981 (2022)

work page 2022
[7]

R. L. Frank, A. Laptev, T. Weidl: Schr¨ odinger Operators: Eigenvalues and Lieb–Thirring Inequali- ties.Cambridge Studies in Advanced Mathematics. Vol. 200.Cambridge: Cambridge University Press. doi:10.1017/9781009218436, (2022)

work page doi:10.1017/9781009218436 2022
[8]

H. Ge, B. Hua, W. Jiang: A note on Liouville type equations on graphs.Proceed. Amer. Math. Soc. 146(11), 4837–4842 (2018)

work page 2018
[9]

Gidas, W.-M

B. Gidas, W.-M. Ni, L. Nirenberg: Symmetry and related properties via the maximum principle,Comm. Math. Phys. 68, 209–243 (1979)

work page 1979
[10]

Gilbarg, N

D. Gilbarg, N. Trudinger: Elliptic Partial Differential Equations of Second Order.Springer-Verlag, Berlin/New York, (1977)

work page 1977
[11]

Grigor’yan: Introduction to Analysis on Graphs.AMS University Lecture Series71 (2018)

A. Grigor’yan: Introduction to Analysis on Graphs.AMS University Lecture Series71 (2018)

work page 2018
[12]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, Y. Yang: Kazdan-Warner equation on graph.Calc. Var. PDE 55(4),1–13 (2016)

work page 2016
[13]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, Y. Yang: Yamabe type equations on graphs.J. Differ. Eq. 261(9),4924–4943 (2016)

work page 2016
[14]

Grigor’yan, Y

A. Grigor’yan, Y. Sun: On non-negative solutions of the inequality ∆u+u σ ≤0 on Riemannian manifolds. Comm. Pure Appl. Math. 67, 1336–1352 (2014)

work page 2014
[15]

Q. Gu, X. Huang, Y. Sun: Semi-linear elliptic inequalities on weighted graphs.Calc. Var. PDE. 62, (2023)

work page 2023
[16]

B. Hua, R. Li: The existence of extremal functions for discrete Sobolev inequalities on lattice graphs.J. Diff. Eq. 305, 224–241 (2021)

work page 2021
[17]

B. Hua, R. Li, L. Wang: A class of semilinear elliptic equations on lattice graphs.ArXiv:2203.05146(2022)

work page arXiv 2022
[18]

B. Hua, W. Xu: Existence of ground state solutions to some nonlinear Schr¨ odinger equations on lattice graphs.Calc. Var. PDE 62(4), No. 127, 17 pp (2023)

work page 2023
[19]

Huang, Y

A. Huang, Y. Lin, S. Yau: Existence of solutions to mean field equations on graphs.Comm. Math. Phys. 377(1), 613–621 (2020)

work page 2020
[20]

Jones, T

C. Jones, T. K¨ upper: On the infinitely many solutions of a semilinear elliptic equation.SIAM J. Math. Anal. 17(4), 803–835 (1986)

work page 1986
[21]

Keller, Y

M. Keller, Y. Pinchover, F. Pogorzelski: From Hardy to Rellich inequalities on graphs.Proceedings London Math. Soc. 122(3),458–477 (2021)

work page 2021
[22]

Keller, Y

M. Keller, Y. Pinchover, F. Pogorzelski: Optimal Hardy inequalities for Schr¨ odinger operators on graphs. Comm. Math. Phys. 358,767–790 (2018)

work page 2018
[23]

Kenyon: The Laplacian and Dirac operators on critical planar graphs.Invent

R. Kenyon: The Laplacian and Dirac operators on critical planar graphs.Invent. math. 150,409–439 (2002)

work page 2002
[24]

Kozma, E

G. Kozma, E. Schreiber: An asymptotic expansion for the discrete harmonic potential.Electron. J. Probab. 1,1–17 (2004)

work page 2004
[25]

G. F. Lawler, V. Limic: Random walk: a modern introduction, volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010)

work page 2010
[26]

McOwen: The behavior of the laplacian on weighted sobolev spaces.Comm

R. McOwen: The behavior of the laplacian on weighted sobolev spaces.Comm. Pure Appl. Math. 32(6), 783–795 (1979). 37

work page 1979
[27]

Ni: On the elliptic equation ∆u+K(x)u n+2 n−2 = 0 its generalizations, and applications in geometry

W-M. Ni: On the elliptic equation ∆u+K(x)u n+2 n−2 = 0 its generalizations, and applications in geometry. Indiana Univ. Math. J. 31(4), 493–529 (1982)

work page 1982
[28]

Rabinowitz: Minimax methods in critical point theory with applications to differential equations,CBMS Reg

P. Rabinowitz: Minimax methods in critical point theory with applications to differential equations,CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, RI (1986)

work page 1986
[29]

W. A. Strauss: Existence of solitary waves in higher dimensions.Comm. Math. Phys. 55, 149–162 (1977)

work page 1977
[30]

M. Struwe: Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems,Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Springer Verlag, Berlin–Heidelberg (1990)

work page 1990
[31]

Uchiyama: Green’s functions for random walks onZ N.Proc

K. Uchiyama: Green’s functions for random walks onZ N.Proc. Lond. Math. Soc. (3) 77(1),215– 240 (1998). 38

work page 1998

[1] [1]

Alama, Y

S. Alama, Y. Li: Existence of solutions for semilinear elliptic equations with indefinite linear part.J. Diff. Eq. 96(1), 89–115 (1992)

work page 1992

[2] [2]

Ambrosetti, P

A. Ambrosetti, P. Rabinowitz: Dual variational methods in critical point theory and applications,J. Funct. Anal. 14,349–381 (1973)

work page 1973

[3] [3]

Biagi, G

S. Biagi, G. Meglioli, F. Punzo: A Liouville theorem for elliptic equations with a potential on infinite graphs.Calc. Var. 63,165 (2024)

work page 2024

[4] [4]

Bianchi, H

G. Bianchi, H. Egnell, A. Tertikas: Existence of solutions for critical semilinear elliptic equations with a potential.J. Funct. Anal. 141,159–190 (1996)

work page 1996

[5] [5]

D. Cao, S. Peng: Positive solutions for a critical elliptic equation with a potential vanishing at infinity. it Trans. Amer. Math. Soc. 355, 1567–1597 (2003)

work page 2003

[6] [6]

Emmanuel, S

M. Emmanuel, S. Gordon: Asymptotic behaviour of the lattice Green function.ALEA, Lat. Am. J. Probab. Math. Stat. 19,957–981 (2022)

work page 2022

[7] [7]

R. L. Frank, A. Laptev, T. Weidl: Schr¨ odinger Operators: Eigenvalues and Lieb–Thirring Inequali- ties.Cambridge Studies in Advanced Mathematics. Vol. 200.Cambridge: Cambridge University Press. doi:10.1017/9781009218436, (2022)

work page doi:10.1017/9781009218436 2022

[8] [8]

H. Ge, B. Hua, W. Jiang: A note on Liouville type equations on graphs.Proceed. Amer. Math. Soc. 146(11), 4837–4842 (2018)

work page 2018

[9] [9]

Gidas, W.-M

B. Gidas, W.-M. Ni, L. Nirenberg: Symmetry and related properties via the maximum principle,Comm. Math. Phys. 68, 209–243 (1979)

work page 1979

[10] [10]

Gilbarg, N

D. Gilbarg, N. Trudinger: Elliptic Partial Differential Equations of Second Order.Springer-Verlag, Berlin/New York, (1977)

work page 1977

[11] [11]

Grigor’yan: Introduction to Analysis on Graphs.AMS University Lecture Series71 (2018)

A. Grigor’yan: Introduction to Analysis on Graphs.AMS University Lecture Series71 (2018)

work page 2018

[12] [12]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, Y. Yang: Kazdan-Warner equation on graph.Calc. Var. PDE 55(4),1–13 (2016)

work page 2016

[13] [13]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, Y. Yang: Yamabe type equations on graphs.J. Differ. Eq. 261(9),4924–4943 (2016)

work page 2016

[14] [14]

Grigor’yan, Y

A. Grigor’yan, Y. Sun: On non-negative solutions of the inequality ∆u+u σ ≤0 on Riemannian manifolds. Comm. Pure Appl. Math. 67, 1336–1352 (2014)

work page 2014

[15] [15]

Q. Gu, X. Huang, Y. Sun: Semi-linear elliptic inequalities on weighted graphs.Calc. Var. PDE. 62, (2023)

work page 2023

[16] [16]

B. Hua, R. Li: The existence of extremal functions for discrete Sobolev inequalities on lattice graphs.J. Diff. Eq. 305, 224–241 (2021)

work page 2021

[17] [17]

B. Hua, R. Li, L. Wang: A class of semilinear elliptic equations on lattice graphs.ArXiv:2203.05146(2022)

work page arXiv 2022

[18] [18]

B. Hua, W. Xu: Existence of ground state solutions to some nonlinear Schr¨ odinger equations on lattice graphs.Calc. Var. PDE 62(4), No. 127, 17 pp (2023)

work page 2023

[19] [19]

Huang, Y

A. Huang, Y. Lin, S. Yau: Existence of solutions to mean field equations on graphs.Comm. Math. Phys. 377(1), 613–621 (2020)

work page 2020

[20] [20]

Jones, T

C. Jones, T. K¨ upper: On the infinitely many solutions of a semilinear elliptic equation.SIAM J. Math. Anal. 17(4), 803–835 (1986)

work page 1986

[21] [21]

Keller, Y

M. Keller, Y. Pinchover, F. Pogorzelski: From Hardy to Rellich inequalities on graphs.Proceedings London Math. Soc. 122(3),458–477 (2021)

work page 2021

[22] [22]

Keller, Y

M. Keller, Y. Pinchover, F. Pogorzelski: Optimal Hardy inequalities for Schr¨ odinger operators on graphs. Comm. Math. Phys. 358,767–790 (2018)

work page 2018

[23] [23]

Kenyon: The Laplacian and Dirac operators on critical planar graphs.Invent

R. Kenyon: The Laplacian and Dirac operators on critical planar graphs.Invent. math. 150,409–439 (2002)

work page 2002

[24] [24]

Kozma, E

G. Kozma, E. Schreiber: An asymptotic expansion for the discrete harmonic potential.Electron. J. Probab. 1,1–17 (2004)

work page 2004

[25] [25]

G. F. Lawler, V. Limic: Random walk: a modern introduction, volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010)

work page 2010

[26] [26]

McOwen: The behavior of the laplacian on weighted sobolev spaces.Comm

R. McOwen: The behavior of the laplacian on weighted sobolev spaces.Comm. Pure Appl. Math. 32(6), 783–795 (1979). 37

work page 1979

[27] [27]

Ni: On the elliptic equation ∆u+K(x)u n+2 n−2 = 0 its generalizations, and applications in geometry

W-M. Ni: On the elliptic equation ∆u+K(x)u n+2 n−2 = 0 its generalizations, and applications in geometry. Indiana Univ. Math. J. 31(4), 493–529 (1982)

work page 1982

[28] [28]

Rabinowitz: Minimax methods in critical point theory with applications to differential equations,CBMS Reg

P. Rabinowitz: Minimax methods in critical point theory with applications to differential equations,CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, RI (1986)

work page 1986

[29] [29]

W. A. Strauss: Existence of solitary waves in higher dimensions.Comm. Math. Phys. 55, 149–162 (1977)

work page 1977

[30] [30]

M. Struwe: Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems,Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Springer Verlag, Berlin–Heidelberg (1990)

work page 1990

[31] [31]

Uchiyama: Green’s functions for random walks onZ N.Proc

K. Uchiyama: Green’s functions for random walks onZ N.Proc. Lond. Math. Soc. (3) 77(1),215– 240 (1998). 38

work page 1998