{"id":"https://openalex.org/W2902344153","doi":"https://doi.org/10.3390/sym10120695","title":"Radial Symmetry for Weak Positive Solutions of Fractional Laplacian with a Singular Nonlinearity","display_name":"Radial Symmetry for Weak Positive Solutions of Fractional Laplacian with a Singular Nonlinearity","publication_year":2018,"publication_date":"2018-12-03","ids":{"openalex":"https://openalex.org/W2902344153","doi":"https://doi.org/10.3390/sym10120695","mag":"2902344153"},"language":"en","primary_location":{"id":"doi:10.3390/sym10120695","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10120695","pdf_url":"https://www.mdpi.com/2073-8994/10/12/695/pdf?version=1543808223","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/10/12/695/pdf?version=1543808223","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101702016","display_name":"Xing Wang","orcid":"https://orcid.org/0000-0001-6130-5789"},"institutions":[{"id":"https://openalex.org/I4210131919","display_name":"Xi'an University of Technology","ror":"https://ror.org/038avdt50","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210131919"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xing Wang","raw_affiliation_strings":["School of Science, Xi\u2019an University of Technology, Xi\u2019an 710054, China","School of Science, Xi'an University of Technology, Xi'an 710054, China"],"raw_orcid":"https://orcid.org/0000-0001-6130-5789","affiliations":[{"raw_affiliation_string":"School of Science, Xi\u2019an University of Technology, Xi\u2019an 710054, China","institution_ids":["https://openalex.org/I4210131919"]},{"raw_affiliation_string":"School of Science, Xi'an University of Technology, Xi'an 710054, China","institution_ids":["https://openalex.org/I4210131919"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100425521","display_name":"Li Zhang","orcid":"https://orcid.org/0000-0002-1999-6819"},"institutions":[{"id":"https://openalex.org/I25355098","display_name":"Chang'an University","ror":"https://ror.org/05mxya461","country_code":"CN","type":"education","lineage":["https://openalex.org/I25355098"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Li Zhang","raw_affiliation_strings":["School of Science, Chang\u2019an University, Xi\u2019an 710064, China","School of Science, Chang'an University, Xi'an 710064, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Chang\u2019an University, Xi\u2019an 710064, China","institution_ids":["https://openalex.org/I25355098"]},{"raw_affiliation_string":"School of Science, Chang'an University, Xi'an 710064, China","institution_ids":["https://openalex.org/I25355098"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101702016"],"corresponding_institution_ids":["https://openalex.org/I4210131919"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.18251975,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"10","issue":"12","first_page":"695","last_page":"695"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/symmetry-in-biology","display_name":"Symmetry in biology","score":0.688882052898407},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6253357529640198},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5941228866577148},{"id":"https://openalex.org/keywords/fractional-laplacian","display_name":"Fractional Laplacian","score":0.5776495933532715},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.5708467960357666},{"id":"https://openalex.org/keywords/multiplicity","display_name":"Multiplicity (mathematics)","score":0.5298371315002441},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.5265042185783386},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.47861069440841675},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4464961290359497},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42829036712646484},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.38612061738967896},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.36717745661735535},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.23479297757148743},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10503602027893066},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10186874866485596},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07805460691452026}],"concepts":[{"id":"https://openalex.org/C64089718","wikidata":"https://www.wikidata.org/wiki/Q598159","display_name":"Symmetry in biology","level":2,"score":0.688882052898407},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6253357529640198},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5941228866577148},{"id":"https://openalex.org/C2775913539","wikidata":"https://www.wikidata.org/wiki/Q30694381","display_name":"Fractional Laplacian","level":2,"score":0.5776495933532715},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.5708467960357666},{"id":"https://openalex.org/C156004811","wikidata":"https://www.wikidata.org/wiki/Q2228257","display_name":"Multiplicity (mathematics)","level":2,"score":0.5298371315002441},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.5265042185783386},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.47861069440841675},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4464961290359497},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42829036712646484},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.38612061738967896},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36717745661735535},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.23479297757148743},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10503602027893066},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10186874866485596},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07805460691452026},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym10120695","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10120695","pdf_url":"https://www.mdpi.com/2073-8994/10/12/695/pdf?version=1543808223","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:6e2f01d9847c403d883165bd083a247a","is_oa":true,"landing_page_url":"https://doaj.org/article/6e2f01d9847c403d883165bd083a247a","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 10, Iss 12, p 695 (2018)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym10120695","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10120695","pdf_url":"https://www.mdpi.com/2073-8994/10/12/695/pdf?version=1543808223","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W1268530400","https://openalex.org/W1481862199","https://openalex.org/W1996591766","https://openalex.org/W2003734192","https://openalex.org/W2007334875","https://openalex.org/W2011095781","https://openalex.org/W2021846512","https://openalex.org/W2024030203","https://openalex.org/W2024483190","https://openalex.org/W2032262114","https://openalex.org/W2036323366","https://openalex.org/W2055831972","https://openalex.org/W2058220480","https://openalex.org/W2063287475","https://openalex.org/W2071132730","https://openalex.org/W2073200116","https://openalex.org/W2168741104","https://openalex.org/W2172034033","https://openalex.org/W2271683060","https://openalex.org/W2278036894","https://openalex.org/W2560436386","https://openalex.org/W2805251394","https://openalex.org/W2962757073","https://openalex.org/W2962788670","https://openalex.org/W3036627574","https://openalex.org/W3103229763","https://openalex.org/W4235609472","https://openalex.org/W4235853207","https://openalex.org/W6632077446","https://openalex.org/W7051806597"],"related_works":["https://openalex.org/W3034228206","https://openalex.org/W2943498627","https://openalex.org/W2315797236","https://openalex.org/W2414337799","https://openalex.org/W2922727309","https://openalex.org/W2326579038","https://openalex.org/W2381983494","https://openalex.org/W2963118341","https://openalex.org/W4386148387","https://openalex.org/W4362597061"],"abstract_inverted_index":{"This":[0],"paper":[1,23],"is":[2,46],"concerned":[3],"with":[4],"the":[5,22,25,54,68,82],"radial":[6,30,95],"symmetry":[7,31,96],"weak":[8,32],"positive":[9,33],"solutions":[10,34,97],"for":[11],"a":[12,64,74],"class":[13],"of":[14,29,57,81,98],"singular":[15],"fractional":[16],"Laplacian.":[17],"The":[18],"main":[19],"results":[20,52,89],"in":[21,66,85],"demonstrate":[24],"existence":[26],"and":[27,41,60],"multiplicity":[28],"by":[35],"Schwarz":[36],"spherical":[37],"rearrangement,":[38],"constrained":[39],"minimization,":[40],"Ekeland\u2019s":[42],"variational":[43],"principle.":[44],"It":[45],"worth":[47],"pointing":[48],"out":[49],"that":[50],"our":[51],"extend":[53],"previous":[55],"works":[56],"T.":[58],"Mukherjee":[59],"K.":[61],"Sreenadh":[62],"to":[63],"setting":[65],"which":[67],"testing":[69],"functions":[70],"need":[71],"not":[72],"have":[73],"compact":[75],"support.":[76],"Moreover,":[77],"we":[78],"weakened":[79],"one":[80],"conditions":[83],"used":[84],"their":[86],"papers.":[87],"Our":[88],"improve":[90],"on":[91,94],"existing":[92],"studies":[93],"nonlocal":[99],"boundary":[100],"value":[101],"problems.":[102]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}