{"id":"https://openalex.org/W4415189190","doi":"https://doi.org/10.1137/25m1748846","title":"A New Framework for Ljusternik\u2013Schnirelmann Theory and Its Application to Planar Choquard Equations","display_name":"A New Framework for Ljusternik\u2013Schnirelmann Theory and Its Application to Planar Choquard Equations","publication_year":2026,"publication_date":"2026-04-13","ids":{"openalex":"https://openalex.org/W4415189190","doi":"https://doi.org/10.1137/25m1748846"},"language":"en","primary_location":{"id":"doi:10.1137/25m1748846","is_oa":false,"landing_page_url":"https://doi.org/10.1137/25m1748846","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2502.18421","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120002490","display_name":"Omar Cabrera","orcid":null},"institutions":[{"id":"https://openalex.org/I114090438","display_name":"Goethe University Frankfurt","ror":"https://ror.org/04cvxnb49","country_code":"DE","type":"education","lineage":["https://openalex.org/I114090438"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"O. Cabrera","raw_affiliation_strings":["Institut f\u00fcr Mathematik, Goethe-Universit\u00e4t Frankfurt am Main, D-60629 Frankfurt am Main, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematik, Goethe-Universit\u00e4t Frankfurt am Main, D-60629 Frankfurt am Main, Germany","institution_ids":["https://openalex.org/I114090438"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009404820","display_name":"Silvia Cingolani","orcid":"https://orcid.org/0000-0002-3680-9106"},"institutions":[{"id":"https://openalex.org/I5561750","display_name":"University of Bari Aldo Moro","ror":"https://ror.org/027ynra39","country_code":"IT","type":"education","lineage":["https://openalex.org/I5561750"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"S. Cingolani","raw_affiliation_strings":["Dipartimento di Matematica, Universit\u00e0 degli Studi di Bari Aldo Moro, 70125 Bari, Italy"],"raw_orcid":"https://orcid.org/0000-0002-3680-9106","affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica, Universit\u00e0 degli Studi di Bari Aldo Moro, 70125 Bari, Italy","institution_ids":["https://openalex.org/I5561750"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062208839","display_name":"Tobias Weth","orcid":"https://orcid.org/0000-0001-5347-8057"},"institutions":[{"id":"https://openalex.org/I114090438","display_name":"Goethe University Frankfurt","ror":"https://ror.org/04cvxnb49","country_code":"DE","type":"education","lineage":["https://openalex.org/I114090438"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"T. Weth","raw_affiliation_strings":["Institut f\u00fcr Mathematik, Goethe-Universit\u00e4t Frankfurt am Main, D-60629 Frankfurt am Main, Germany"],"raw_orcid":"https://orcid.org/0000-0001-5347-8057","affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematik, Goethe-Universit\u00e4t Frankfurt am Main, D-60629 Frankfurt am Main, Germany","institution_ids":["https://openalex.org/I114090438"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.04704171,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"58","issue":"2","first_page":"1772","last_page":"1811"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9678999781608582,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11416","display_name":"Numerical methods for differential equations","score":0.9678999781608582,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9650999903678894,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9168999791145325,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/logarithm","display_name":"Logarithm","score":0.48489999771118164},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.4607999920845032},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.4528000056743622},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.42809998989105225},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.40959998965263367},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.39430001378059387},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.3885999917984009},{"id":"https://openalex.org/keywords/energy-functional","display_name":"Energy functional","score":0.3190000057220459}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8029999732971191},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.48489999771118164},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.4607999920845032},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4528000056743622},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4366999864578247},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.42809998989105225},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.40959998965263367},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.39430001378059387},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.3885999917984009},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3614000082015991},{"id":"https://openalex.org/C191640071","wikidata":"https://www.wikidata.org/wiki/Q5377056","display_name":"Energy functional","level":2,"score":0.3190000057220459},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.31619998812675476},{"id":"https://openalex.org/C196298200","wikidata":"https://www.wikidata.org/wiki/Q577705","display_name":"Critical point (mathematics)","level":2,"score":0.31610000133514404},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.28850001096725464},{"id":"https://openalex.org/C2777059694","wikidata":"https://www.wikidata.org/wiki/Q2706744","display_name":"Invariant subspace","level":3,"score":0.288100004196167},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.27410000562667847},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.26499998569488525},{"id":"https://openalex.org/C153576954","wikidata":"https://www.wikidata.org/wiki/Q594452","display_name":"Functional equation","level":3,"score":0.26440000534057617},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.26339998841285706},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.2597000002861023},{"id":"https://openalex.org/C149364088","wikidata":"https://www.wikidata.org/wiki/Q185917","display_name":"Translation (biology)","level":4,"score":0.25949999690055847}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/25m1748846","is_oa":false,"landing_page_url":"https://doi.org/10.1137/25m1748846","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2502.18421","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2502.18421","pdf_url":"https://arxiv.org/pdf/2502.18421","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},{"id":"pmh:oai:ricerca.uniba.it:11586/555160","is_oa":false,"landing_page_url":"https://hdl.handle.net/11586/555160","pdf_url":null,"source":{"id":"https://openalex.org/S4377196296","display_name":"CINECA IRIS Institutional Research Information System (University of Bari Aldo Moro)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I5561750","host_organization_name":"University of Bari Aldo Moro","host_organization_lineage":["https://openalex.org/I5561750"],"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":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.2502.18421","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2502.18421","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2502.18421","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2502.18421","pdf_url":"https://arxiv.org/pdf/2502.18421","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2433072319","display_name":null,"funder_award_id":"57507871","funder_id":"https://openalex.org/F4320320875","funder_display_name":"Deutscher Akademischer Austauschdienst"}],"funders":[{"id":"https://openalex.org/F4320320875","display_name":"Deutscher Akademischer Austauschdienst","ror":"https://ror.org/039djdh30"},{"id":"https://openalex.org/F4320333369","display_name":"Gruppo Nazionale per l'Analisi Matematica, la Probabilit\u00e0 e le loro Applicazioni","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0],"consider":[1],"the":[2,21,35,46,59,99,102,108,122,141,165],"planar":[3],"logarithmic":[4,109],"Choquard":[5,110],"equation":[6,111],"$$-":[7],"\u0394u":[8],"+":[9,11],"a(x)u":[10],"(\\log|\\cdot|":[12],"\\ast":[13,170,174,177],"u^2)u":[14],"=":[15,179],"0,\\qquad":[16],"\\text{in":[17],"}":[18],"\\mathbb{R}^2$$":[19],"in":[20,58],"strongly":[22],"indefinite":[23],"and":[24,144],"possibly":[25],"degenerate":[26],"setting":[27],"where":[28,61],"no":[29],"sign":[30],"condition":[31,143],"is":[32,63,112,126,133],"imposed":[33],"on":[34,115,135,145],"linear":[36],"potential":[37],"$a":[38],"\\in":[39,162],"L^\\infty(\\mathbb{R}^2)$.":[40],"In":[41,94],"particular,":[42,95],"we":[43,78],"shall":[44],"prove":[45],"existence":[47],"of":[48,51,91,140,152],"a":[49,71,80,116,136,150],"sequence":[50],"high":[52],"energy":[53,103],"solutions":[54],"to":[55,70,87,149],"this":[56],"problem":[57,100],"case":[60],"$a$":[62],"invariant":[64],"under":[65],"$\\mathbb{Z}^2$-translations.":[66],"The":[67,130],"result":[68],"extends":[69],"more":[72],"general":[73],"$G$-equivariant":[74,138],"setting,":[75],"for":[76],"which":[77,84],"develop":[79],"new":[81,131,137],"variational":[82],"approach":[83,132],"allows":[85],"us":[86],"find":[88],"critical":[89],"points":[90],"Ljusternik-Schnirelmann":[92],"type.":[93],"our":[96],"method":[97],"resolves":[98],"that":[101,124],"functional":[104],"$\u03a6$":[105],"associated":[106],"with":[107,121,164],"only":[113],"defined":[114],"subspace":[117],"$X":[118],"\\subset":[119],"H^1(\\mathbb{R}^2)$":[120],"property":[123,167],"$\\|\\cdot\\|_X$":[125],"not":[127],"translation":[128],"invariant.":[129],"based":[134],"version":[139],"Cerami":[142],"deformation":[146],"arguments":[147],"adapted":[148],"family":[151],"suitably":[153],"constructed":[154],"scalar":[155],"products":[156],"$\\langle":[157,168],"\\cdot,":[158],"\\cdot":[159],"\\rangle_u$,":[160],"$u":[161],"X$":[163],"$G$-equivariance":[166],"g":[169,173],"v":[171,181],",":[172,182],"w":[175,183],"\\rangle_{g":[176],"u}":[178],"\\langle":[180],"\\rangle_u.$":[184]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-15T00:00:00"}