{"id":"https://openalex.org/W4401541389","doi":"https://doi.org/10.3390/axioms13080548","title":"Existence of the Nontrivial Solution for a p-Kirchhoff Problem with Critical Growth and Logarithmic Nonlinearity","display_name":"Existence of the Nontrivial Solution for a p-Kirchhoff Problem with Critical Growth and Logarithmic Nonlinearity","publication_year":2024,"publication_date":"2024-08-13","ids":{"openalex":"https://openalex.org/W4401541389","doi":"https://doi.org/10.3390/axioms13080548"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13080548","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13080548","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/axioms13080548","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113332181","display_name":"Lixiang Cai","orcid":null},"institutions":[{"id":"https://openalex.org/I145897649","display_name":"Minzu University of China","ror":"https://ror.org/0044e2g62","country_code":"CN","type":"education","lineage":["https://openalex.org/I145897649"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lixiang Cai","raw_affiliation_strings":["School of Mathmatics and Computer Science, Yunnan Minzu University, Kunming 650500, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathmatics and Computer Science, Yunnan Minzu University, Kunming 650500, China","institution_ids":["https://openalex.org/I145897649"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056141466","display_name":"Qing Miao","orcid":"https://orcid.org/0000-0001-8703-3603"},"institutions":[{"id":"https://openalex.org/I145897649","display_name":"Minzu University of China","ror":"https://ror.org/0044e2g62","country_code":"CN","type":"education","lineage":["https://openalex.org/I145897649"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Qing Miao","raw_affiliation_strings":["School of Mathmatics and Computer Science, Yunnan Minzu University, Kunming 650500, China"],"raw_orcid":"https://orcid.org/0000-0001-8703-3603","affiliations":[{"raw_affiliation_string":"School of Mathmatics and Computer Science, Yunnan Minzu University, Kunming 650500, China","institution_ids":["https://openalex.org/I145897649"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5056141466"],"corresponding_institution_ids":["https://openalex.org/I145897649"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":2.4648,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.88462014,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"13","issue":"8","first_page":"548","last_page":"548"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9997000098228455,"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":0.9997000098228455,"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":0.9991999864578247,"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":0.9941999912261963,"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/mathematics","display_name":"Mathematics","score":0.7182369232177734},{"id":"https://openalex.org/keywords/nehari-manifold","display_name":"Nehari manifold","score":0.7176482677459717},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.6953176259994507},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6884896755218506},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6046276092529297},{"id":"https://openalex.org/keywords/mountain-pass-theorem","display_name":"Mountain pass theorem","score":0.5548624396324158},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5538957715034485},{"id":"https://openalex.org/keywords/compact-space","display_name":"Compact space","score":0.5139586329460144},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.491418719291687},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.4797476530075073},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.46660515666007996},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.4491010010242462},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.44881078600883484},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.42990708351135254},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.41677695512771606},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.40314868092536926},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.38980239629745483},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.18357619643211365},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.053283363580703735}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7182369232177734},{"id":"https://openalex.org/C40332964","wikidata":"https://www.wikidata.org/wiki/Q6987780","display_name":"Nehari manifold","level":3,"score":0.7176482677459717},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.6953176259994507},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6884896755218506},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6046276092529297},{"id":"https://openalex.org/C2779779452","wikidata":"https://www.wikidata.org/wiki/Q6925496","display_name":"Mountain pass theorem","level":3,"score":0.5548624396324158},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5538957715034485},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.5139586329460144},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.491418719291687},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.4797476530075073},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.46660515666007996},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4491010010242462},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.44881078600883484},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.42990708351135254},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.41677695512771606},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.40314868092536926},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.38980239629745483},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.18357619643211365},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.053283363580703735},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms13080548","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13080548","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:7c5d673efd8c4d17881658cfeb7b8256","is_oa":false,"landing_page_url":"https://doaj.org/article/7c5d673efd8c4d17881658cfeb7b8256","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 13, Iss 8, p 548 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms13080548","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13080548","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W1973496351","https://openalex.org/W1973748816","https://openalex.org/W1991234149","https://openalex.org/W2007261596","https://openalex.org/W2050458081","https://openalex.org/W2066046051","https://openalex.org/W2074596855","https://openalex.org/W2076421610","https://openalex.org/W2327680971","https://openalex.org/W2507597187","https://openalex.org/W2612037541","https://openalex.org/W2898414676","https://openalex.org/W2905578463","https://openalex.org/W2916974742","https://openalex.org/W2917099245","https://openalex.org/W2963781133","https://openalex.org/W2997737944","https://openalex.org/W3008753896","https://openalex.org/W3033100197","https://openalex.org/W3086135246","https://openalex.org/W3199973928","https://openalex.org/W4212959462","https://openalex.org/W4245298344","https://openalex.org/W4306783431","https://openalex.org/W4319439154","https://openalex.org/W4321456191","https://openalex.org/W4362660255"],"related_works":["https://openalex.org/W2532795237","https://openalex.org/W4295831961","https://openalex.org/W3195700042","https://openalex.org/W4385890035","https://openalex.org/W3202020845","https://openalex.org/W2921722049","https://openalex.org/W4388951543","https://openalex.org/W3134738835","https://openalex.org/W3002023790","https://openalex.org/W2953007861"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"mainly":[4],"study":[5],"the":[6,42,46,51,59,72,76,79,84,94],"p-Kirchhoff":[7,107],"type":[8,108],"equations":[9],"with":[10,27,97],"logarithmic":[11,85,98],"nonlinear":[12,86],"terms":[13],"and":[14,32,35,45,83],"critical":[15,77],"growth:":[16],"\u2212M\u222b\u03a9\u2207updx\u0394pu=up\u2217\u22122u+\u03bbup\u22122u\u2212up\u22122ulnu2":[17],"x\u2208\u03a9,":[18],"u=0":[19],"x\u2208\u2202\u03a9,":[20],"where":[21],"\u03a9\u2282\u211dN":[22],"is":[23],"a":[24,28],"bounded":[25],"domain":[26],"smooth":[29],"boundary,":[30],"2&lt;p&lt;p\u2217&lt;N,":[31],"both":[33],"p":[34,101],"N":[36],"are":[37],"positive":[38],"integers.":[39],"By":[40],"using":[41],"Nehari":[43],"manifold":[44],"Mountain":[47],"Pass":[48],"Theorem":[49],"without":[50],"Palais-Smale":[52],"compactness":[53],"condition,":[54],"it":[55,89],"was":[56],"proved":[57],"that":[58],"equation":[60],"had":[61],"at":[62],"least":[63],"one":[64],"nontrivial":[65],"solution":[66],"under":[67],"appropriate":[68],"conditions.":[69],"It":[70],"addresses":[71],"challenges":[73],"posed":[74],"by":[75],"term,":[78],"Kirchhoff":[80],"nonlocal":[81],"term":[82],"term.":[87],"Additionally,":[88],"extends":[90],"partial":[91],"results":[92],"of":[93],"Br\u00e9zis\u2013Nirenberg":[95],"problem":[96],"perturbation":[99],"from":[100],"=":[102],"2":[103],"to":[104],"more":[105],"general":[106],"problems.":[109]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2024-08-14T00:00:00"}
