{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:13:36Z","timestamp":1760058816453,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,30]],"date-time":"2025-04-30T00:00:00Z","timestamp":1745971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work establishes the multiplicity of solutions for the fractional Kazdan\u2013Warner equation on finite graphs for the negative case. Our main focus lies in analyzing the nonlinear equation defined on a finite graph (V,E,\u03bc,w): (\u2212\u0394)su=(K+\u03bb)e2u\u2212\u03bainV, where the fraction s\u2208(0,1) and real parameter \u03bb are given, and the graph functions K and \u03ba satisfy maxx\u2208VK(x)=0, K\u22620 and \u222bV\u03bad\u03bc&lt;0. We derive the solvability characteristics of the above equation with the help of variational theory and the upper and lower solutions method.<\/jats:p>","DOI":"10.3390\/axioms14050345","type":"journal-article","created":{"date-parts":[[2025,4,30]],"date-time":"2025-04-30T05:50:17Z","timestamp":1745992217000},"page":"345","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multiple Solutions of Fractional Kazdan\u2013Warner Equation for Negative Case on Finite Graphs"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-6558-812X","authenticated-orcid":false,"given":"Liang","family":"Shan","sequence":"first","affiliation":[{"name":"Gaoling School of Artificial Intelligence, Renmin University of China, Beijing 100872, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3681-2452","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Renmin University of China, Beijing 100872, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14","DOI":"10.2307\/1971012","article-title":"Curvature functions for compact 2-manifolds","volume":"99","author":"Kazdan","year":"1974","journal-title":"Ann. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"203","DOI":"10.2307\/1970898","article-title":"Curvature functions for open 2-manifolds","volume":"99","author":"Kazdan","year":"1974","journal-title":"Ann. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF02921316","article-title":"Gaussian curvature on singular surfaces","volume":"3","author":"Chen","year":"1993","journal-title":"J. Geom. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"230","DOI":"10.4310\/AJM.1997.v1.n2.a3","article-title":"The differential equation \u0394u = 8\u03c0 \u2212 8\u03c0heu on a compact Riemann Surface","volume":"1","author":"Ding","year":"1997","journal-title":"Asian J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1090\/S0002-9947-1995-1257102-2","article-title":"A note on the problem of prescribing Gaussian curvature on surfaces","volume":"347","author":"Ding","year":"1995","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"407","DOI":"10.4171\/cmh\/358","article-title":"\u201cLarge\u201d conformal metrics of prescribed Gauss curvature on surfaces of higher genus","volume":"90","author":"Borer","year":"2015","journal-title":"Comment. Math. Helv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/BF02392272","article-title":"The existence of surfaces of constant mean curvature with free boundaries","volume":"160","author":"Struwe","year":"1988","journal-title":"Acta Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"167","DOI":"10.5186\/aasfm.2019.4411","article-title":"Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case","volume":"44","author":"Yang","year":"2019","journal-title":"Ann. Acad. Sci. Fenn. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/s00526-016-1042-3","article-title":"Kazdan-Warner equation on graph","volume":"55","author":"Grigor","year":"2016","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1022","DOI":"10.1016\/j.jmaa.2017.04.052","article-title":"Kazdan-Warner equation on graph in the negative case","volume":"453","author":"Ge","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1950052","DOI":"10.1142\/S0219199719500524","article-title":"The p-th Kazdan-Warner equation on graphs","volume":"22","author":"Ge","year":"2020","journal-title":"Commun. Contemp. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.jmaa.2018.05.081","article-title":"p-th Kazdan-Warner equation on graph in the negative case","volume":"466","author":"Zhang","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","first-page":"1091","article-title":"Kazdan-Warner equation on infinite graphs","volume":"55","author":"Ge","year":"2018","journal-title":"J. Korean Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1007\/s00526-018-1329-7","article-title":"The Kazdan-Warner equation on canonically compactifiable graphs","volume":"57","author":"Keller","year":"2018","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"108422","DOI":"10.1016\/j.aim.2022.108422","article-title":"Brouwer degree for Kazdan-Warner equations on a connected finite graph","volume":"404","author":"Sun","year":"2022","journal-title":"Adv. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1007\/s00526-020-01840-3","article-title":"Multiple solutions of Kazdan-Warner equation on graphs in the negative case","volume":"59","author":"Liu","year":"2020","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s10455-024-09960-1","article-title":"Topological degree for Kazdan-Warner equation in the negative case on finite graph","volume":"65","author":"Liu","year":"2024","journal-title":"Ann. Glob. Anal. Geom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4924","DOI":"10.1016\/j.jde.2016.07.011","article-title":"Yamabe type equations on graphs","volume":"261","author":"Grigor","year":"2016","journal-title":"J. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1311","DOI":"10.1007\/s11425-016-0422-y","article-title":"Existence of positive solutions to some nonlinear equations on locally finite graphs","volume":"60","author":"Grigor","year":"2017","journal-title":"Sci. China Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1007\/s10114-021-9523-5","article-title":"p-Laplacian equations on locally finite graphs","volume":"37","author":"Han","year":"2021","journal-title":"Acta Math. Sin. (Engl. Ser.)"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s00526-022-02238-z","article-title":"Existence of solutions to Chern-Simons-Higgs equations on graphs","volume":"61","author":"Hou","year":"2022","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"106997","DOI":"10.1016\/j.aim.2020.106997","article-title":"Dirichlet p-Laplacian eigenvalues and Cheeger constants on symmetric graphs","volume":"364","author":"Hua","year":"2020","journal-title":"Adv. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"109218","DOI":"10.1016\/j.jfa.2021.109218","article-title":"Mean field equation and relativistic Abelian Chern-Simons model on finite graphs","volume":"281","author":"Huang","year":"2021","journal-title":"J. Funct. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, J., Sun, L., and Yang, Y. (2023). Topological degree for Chern-Simons Higgs models on finite graphs. arXiv.","DOI":"10.1007\/s00526-024-02706-8"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"153","DOI":"10.24033\/bsmf.1309","article-title":"L\u2019int\u00e9grale de Riemann-Liouville et le probl\u00e8me de Cauchy pour l\u2019\u00e9quation des ondes","volume":"67","author":"Riesz","year":"1939","journal-title":"Bull. Soc. Math. Fr."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1016\/s0294-1449(01)00080-4","article-title":"Critical nonlinearity exponent and self-similar asymptotics for L\u00e9vy conservation laws","volume":"18","author":"Biler","year":"2001","journal-title":"Ann. Inst. Henri Poincar\u00e8 C Anal. Non Lin\u00e9aire"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s00526-010-0359-6","article-title":"Uniform estimates and limiting arguments for nonlocal minimal surfaces","volume":"41","author":"Caffarelli","year":"2011","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2140003","DOI":"10.1142\/S0217732321400034","article-title":"Fractional Schr\u00f6dinger equation in gravitational optics","volume":"36","author":"Iomin","year":"2021","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1007\/s12220-008-9064-5","article-title":"Elliptic PDEs with fibered nonlinearities","volume":"19","author":"Savin","year":"2009","journal-title":"J. Geom. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"317","DOI":"10.5802\/aif.2020","article-title":"The form boundedness criterion for the relativistic Schr\u00f6dinger operator","volume":"54","author":"Verbitsky","year":"2004","journal-title":"Ann. Inst. Fourier"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Meerschaert, M. (2012). Fractional Calculus, Anomalous Diffusion, and Probability, World Scientific Publishing Co. Pte. Ltd.","DOI":"10.1142\/9789814340595_0011"},{"key":"ref_32","unstructured":"Adams, R. (1975). Sobolev Spaces, Pure and Applied Mathematics, Academic Press."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/PL00001378","article-title":"Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces","volume":"1","author":"Brezis","year":"2001","journal-title":"J. Evol. Equ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.bulsci.2011.12.004","article-title":"Hitchhiker\u2019s guide to the fractional Sobolev spaces","volume":"136","author":"Palatucci","year":"2012","journal-title":"Bull. Sci. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1515\/fca-2017-0002","article-title":"Ten equivalent definitions of the fractional laplace operator","volume":"20","year":"2017","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Runst, T., and Sickel, W. (1996). Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter & Co.","DOI":"10.1515\/9783110812411"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1016\/j.aim.2018.03.023","article-title":"Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications","volume":"330","author":"Ciaurri","year":"2018","journal-title":"Adv. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2729","DOI":"10.1007\/s00023-023-01307-z","article-title":"Optimal Hardy inequality for fractional Laplacians on the integers","volume":"24","author":"Keller","year":"2023","journal-title":"Ann. Henri Poincar\u00e9"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.3934\/dcds.2018056","article-title":"H\u00f6lder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian","volume":"38","author":"Lizama","year":"2018","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_40","unstructured":"Wang, J. (2024). Eigenvalue estimates for the fractional Laplacian on lattice subgraphs. arXiv."},{"key":"ref_41","unstructured":"Zhang, M., Lin, Y., and Yang, Y. (2024). Fractional Laplace operator on finite graphs. arXiv."},{"key":"ref_42","unstructured":"Zhang, M., Lin, Y., and Yang, Y. (2024). Fractional Laplace operator and related Schr\u00f6dinger equations on locally finite graphs. arXiv."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s11854-017-0015-6","article-title":"Harmonic analysis associated with a discrete Laplacian","volume":"132","author":"Ciaurri","year":"2017","journal-title":"J. Anal. Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.aml.2015.05.007","article-title":"On a connection between the discrete fractional Laplacian and superdiffusion","volume":"49","author":"Ciaurri","year":"2015","journal-title":"Appl. Math. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"127443","DOI":"10.1016\/j.amc.2022.127443","article-title":"On fractional discrete p-Laplacian equations via Clark\u2019s theorem","volume":"434","author":"Ju","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"126051","DOI":"10.1016\/j.jmaa.2022.126051","article-title":"On a connection between the N-dimensional fractional Laplacian and 1-D operators on lattices","volume":"511","author":"Lizama","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1016\/j.camwa.2017.01.012","article-title":"Exact discretization of fractional Laplacian","volume":"73","author":"Tarasov","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/0022-1236(73)90051-7","article-title":"Dual variational methods in critical point theory and applications","volume":"14","author":"Ambrosetti","year":"1973","journal-title":"J. Funct. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/345\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:24:43Z","timestamp":1760030683000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/345"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,30]]},"references-count":48,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050345"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050345","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,4,30]]}}}