{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:33:59Z","timestamp":1760060039910,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,27]],"date-time":"2025-07-27T00:00:00Z","timestamp":1753574400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Yunnan Fundamental Research Projects in China","award":["202301AT070465"],"award-info":[{"award-number":["202301AT070465"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we investigate the existence of multiple solutions for a Kirchhoff-type equation with Dirichlet boundary conditions defined on locally finite graphs. Our study extends some previous results on nonlinear Laplacian equations to the more complex Kirchhoff equation which incorporates a nonlocal term. By employing an abstract three critical points theorem that is based on Morse theory, we provide sufficient conditions that guarantee the existence of at least three distinct solutions, including two strictly positive solutions. We also present an example to verify our results.<\/jats:p>","DOI":"10.3390\/axioms14080585","type":"journal-article","created":{"date-parts":[[2025,7,28]],"date-time":"2025-07-28T09:53:02Z","timestamp":1753696382000},"page":"585","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Existence of Strictly Positive Solutions for a Kirchhoff-Type Equation with the Dirichlet Boundary on Locally Finite Graphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Yanhong","family":"Li","sequence":"first","affiliation":[{"name":"City College, Kunming University of Science and Technology, Kunming 650500, China"}]},{"given":"Xingyong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China"},{"name":"Research Center for Mathematics and Interdisciplinary Sciences, Kunming University of Science and Technology, Kunming 650500, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,27]]},"reference":[{"key":"ref_1","unstructured":"Kirchhoff, G. 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