{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T16:10:40Z","timestamp":1760199040020,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,12,3]],"date-time":"2018-12-03T00:00:00Z","timestamp":1543795200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Li Zhang","award":["National Natural Science Foundation of China (No.11801038)","Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China( No.2018JQ1023)"],"award-info":[{"award-number":["National Natural Science Foundation of China (No.11801038)","Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China( No.2018JQ1023)"]}]},{"name":"Xing Wang","award":["National Natural Science Foundation of China (No.11626185)","Natural Science Foundation of Shaanxi Provincial Department of Education (No.16KJ1558)","Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China(No.2017JQ1011)"],"award-info":[{"award-number":["National Natural Science Foundation of China (No.11626185)","Natural Science Foundation of Shaanxi Provincial Department of Education (No.16KJ1558)","Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China(No.2017JQ1011)"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian. The main results in the paper demonstrate the existence and multiplicity of radial symmetry weak positive solutions by Schwarz spherical rearrangement, constrained minimization, and Ekeland\u2019s variational principle. It is worth pointing out that our results extend the previous works of T. Mukherjee and K. Sreenadh to a setting in which the testing functions need not have a compact support. Moreover, we weakened one of the conditions used in their papers. Our results improve on existing studies on radial symmetry solutions of nonlocal boundary value problems.<\/jats:p>","DOI":"10.3390\/sym10120695","type":"journal-article","created":{"date-parts":[[2018,12,3]],"date-time":"2018-12-03T06:02:09Z","timestamp":1543816929000},"page":"695","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Radial Symmetry for Weak Positive Solutions of Fractional Laplacian with a Singular Nonlinearity"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6130-5789","authenticated-orcid":false,"given":"Xing","family":"Wang","sequence":"first","affiliation":[{"name":"School of Science, Xi\u2019an University of Technology, Xi\u2019an 710054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Science, Chang\u2019an University, Xi\u2019an 710064, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,3]]},"reference":[{"key":"ref_1","first-page":"1336","article-title":"L\u00e9vy process-from probability to finance and quantum groups","volume":"51","author":"Applebaum","year":"2004","journal-title":"N. 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