{"id":"https://openalex.org/W7212224758","doi":"https://doi.org/10.48550/arxiv.2609.11773","title":"Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation","display_name":"Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212224758","doi":"https://doi.org/10.48550/arxiv.2609.11773"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.11773","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11773","pdf_url":"https://arxiv.org/pdf/2609.11773","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.11773","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062869850","display_name":"Daniele Cassani","orcid":"https://orcid.org/0000-0002-2672-853X","observed_orcids":["https://orcid.org/0000-0002-2672-853X"]},"institutions":[{"id":"https://openalex.org/I115752224","display_name":"University of Insubria","ror":"https://ror.org/00s409261","country_code":"IT","type":"education","lineage":["https://openalex.org/I115752224"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Cassani, Daniele","raw_affiliation_strings":["Dipartimento di Scienza e Alta Tecnologia , Universit\u00e0 degli Studi dell'Insubria and RISM-Riemann International School of Mathematics , Villa Toeplitz , Via G.B. Vico , 46 -21100 Varese , Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Scienza e Alta Tecnologia , Universit\u00e0 degli Studi dell'Insubria and RISM-Riemann International School of Mathematics , Villa Toeplitz , Via G.B. Vico , 46 -21100 Varese , Italy","institution_ids":["https://openalex.org/I115752224"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5130975841","display_name":"Zhisu Liu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I3124059619","display_name":"China University of Geosciences","ror":"https://ror.org/04gcegc37","country_code":"CN","type":"education","lineage":["https://openalex.org/I3124059619"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liu, Zhisu","raw_affiliation_strings":["Center for Mathematical Sciences / School of Mathematics and Physics , China University of Geosciences , Wuhan , Hubei , 430074 , PR China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Mathematical Sciences / School of Mathematics and Physics , China University of Geosciences , Wuhan , Hubei , 430074 , PR China","institution_ids":["https://openalex.org/I3124059619"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056282758","display_name":"Giulio Romani","orcid":"https://orcid.org/0000-0002-1563-3148","observed_orcids":["https://orcid.org/0000-0002-1563-3148"]},"institutions":[{"id":"https://openalex.org/I129043915","display_name":"University of Udine","ror":"https://ror.org/05ht0mh31","country_code":"IT","type":"education","lineage":["https://openalex.org/I129043915"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Romani, Giulio","raw_affiliation_strings":["Dipartimento di Scienze Matematiche , Informatiche e Fisiche , Universit\u00e0 degli Studi di Udine , Via delle Scienze 206 - 33100 Udine , Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Scienze Matematiche , Informatiche e Fisiche , Universit\u00e0 degli Studi di Udine , Via delle Scienze 206 - 33100 Udine , Italy","institution_ids":["https://openalex.org/I129043915"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009274500","display_name":"Antonio Tarsia","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Tarsia, Antonio","raw_affiliation_strings":["Dipartimento di Matematica , Universit\u00e0 di Pisa , Largo B. Pontecorvo , 5 -56127 Pisa"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica , Universit\u00e0 di Pisa , Largo B. Pontecorvo , 5 -56127 Pisa","institution_ids":["https://openalex.org/I108290504"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":1.0,"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":1.0,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.0,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.0,"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/nonlinear-biharmonic-equation","display_name":"nonlinear biharmonic equation","score":0.9819999933242798},{"id":"https://openalex.org/keywords/critical-nonlinearity","display_name":"critical nonlinearity","score":0.8109999895095825},{"id":"https://openalex.org/keywords/subcritical-nonlinearity","display_name":"subcritical nonlinearity","score":0.8489999771118164},{"id":"https://openalex.org/keywords/fourth-order-operators","display_name":"fourth-order operators","score":0.6610000133514404},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.6359999775886536},{"id":"https://openalex.org/keywords/supersolutions","display_name":"supersolutions","score":0.6679999828338623}],"concepts":[{"id":"https://openalex.org/C145784156","wikidata":"https://www.wikidata.org/wiki/Q859808","display_name":"Biharmonic equation","level":3,"score":0.9178000092506409},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8019000291824341},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.6687999963760376},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6417999863624573},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5932000279426575},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5045999884605408},{"id":"https://openalex.org/C69523127","wikidata":"https://www.wikidata.org/wiki/Q4480008","display_name":"Ground state","level":2,"score":0.5020999908447266},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5016000270843506},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4779999852180481},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.4778999984264374},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.46810001134872437},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.4050999879837036},{"id":"https://openalex.org/C40332964","wikidata":"https://www.wikidata.org/wiki/Q6987780","display_name":"Nehari manifold","level":3,"score":0.4000000059604645},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.3953000009059906},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.39480000734329224},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.38589999079704285},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.35260000824928284},{"id":"https://openalex.org/C31010330","wikidata":"https://www.wikidata.org/wiki/Q1914255","display_name":"Maximum principle","level":3,"score":0.3334999978542328},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.3140000104904175},{"id":"https://openalex.org/C64618476","wikidata":"https://www.wikidata.org/wiki/Q164296","display_name":"Variational method","level":2,"score":0.30160000920295715},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.28929999470710754},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.2892000079154968},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C8171440","wikidata":"https://www.wikidata.org/wiki/Q903414","display_name":"Steady state (chemistry)","level":2,"score":0.27079999446868896}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.11773","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11773","pdf_url":"https://arxiv.org/pdf/2609.11773","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.11773","doi":"https://doi.org/10.48550/arxiv.2609.11773","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.11773","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.11773","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11773","pdf_url":"https://arxiv.org/pdf/2609.11773","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1902850319","display_name":null,"funder_award_id":"CUGST2","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G2934962698","display_name":null,"funder_award_id":"CUG2106211","funder_id":"https://openalex.org/F4320328899","funder_display_name":"China University of Geosciences"},{"id":"https://openalex.org/G2971504390","display_name":null,"funder_award_id":"E53C25002010001","funder_id":"https://openalex.org/F4320311030","funder_display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\""},{"id":"https://openalex.org/G37756095","display_name":null,"funder_award_id":"CUG2106211","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G4024343793","display_name":null,"funder_award_id":"2023A1515011679","funder_id":"https://openalex.org/F4320337111","funder_display_name":"Basic and Applied Basic Research Foundation of Guangdong Province"},{"id":"https://openalex.org/G7784579292","display_name":null,"funder_award_id":"2024A1515012704","funder_id":"https://openalex.org/F4320337111","funder_display_name":"Basic and Applied Basic Research Foundation of Guangdong Province"},{"id":"https://openalex.org/G8674206558","display_name":null,"funder_award_id":"CUGST2","funder_id":"https://openalex.org/F4320328899","funder_display_name":"China University of Geosciences"}],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320328899","display_name":"China University of Geosciences","ror":"https://ror.org/04gcegc37"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null},{"id":"https://openalex.org/F4320337111","display_name":"Basic and Applied Basic Research Foundation of Guangdong Province","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212224758.pdf","grobid_xml":"https://content.openalex.org/works/W7212224758.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2111215635","https://openalex.org/W2121826827","https://openalex.org/W2065759067","https://openalex.org/W2899791873","https://openalex.org/W2548374574","https://openalex.org/W1455034684","https://openalex.org/W4311696520","https://openalex.org/W2315885074","https://openalex.org/W4390231844","https://openalex.org/W2362938380"],"abstract_inverted_index":{"We":[0,108],"prove":[1],"the":[2,53,61,74,84,96,106,116,120],"existence":[3],"of":[4,12,30,70,119],"nonnegative":[5],"radial":[6,62],"ground":[7,32,76],"states":[8,33,55],"for":[9],"a":[10],"class":[11],"nonlinear":[13],"biharmonic":[14],"equations":[15],"in":[16,42],"$\\mathbb":[17,81,102],"R^N$,":[18,103],"covering":[19],"subcritical":[20],"and":[21],"critical":[22],"nonlinearities.":[23],"The":[24,37],"solutions":[25],"are":[26],"obtained":[27],"as":[28],"limits":[29],"clamped":[31],"on":[34,60,80],"expanding":[35],"balls.":[36],"Cassani-Tarsia":[38],"homogeneous":[39,121],"maximum":[40,122],"principle":[41,123],"[D.":[43],"Cassani.":[44],"A.":[45],"Tarsia,":[46],"Adv.":[47],"Nonlinear":[48],"Anal.":[49],"11":[50],"(2022)]":[51],"makes":[52],"approximating":[54],"positive.":[56],"A":[57],"variational":[58],"decomposition":[59],"Nehari":[63],"manifold":[64],"then":[65],"excludes":[66],"every":[67],"zero":[68],"sphere":[69],"positive":[71,79,94,138],"radius,":[72],"so":[73],"limiting":[75],"state":[77],"is":[78,131],"R^N\\setminus\\{0\\}$.":[82],"When":[83],"fourth":[85],"order":[86],"operator":[87],"factorises":[88],"into":[89],"two":[90],"second-order":[91],"operators":[92],"with":[93,126,134],"resolvents,":[95],"conclusion":[97],"upgrades":[98],"to":[99,124],"$u>0$":[100],"throughout":[101],"including":[104],"at":[105],"origin.":[107],"also":[109],"give":[110],"an":[111,135],"explicit":[112],"counterexample":[113],"showing":[114],"that":[115],"tempting":[117],"extension":[118],"super-solutions":[125],"arbitrary":[127],"nonhomogeneous":[128],"boundary":[129],"data":[130],"false,":[132],"even":[133],"arbitrarily":[136],"large":[137],"zeroth-order":[139],"coefficient.":[140]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-12T00:00:00"}