{"id":"https://openalex.org/W7220445155","doi":"https://doi.org/10.48550/arxiv.2610.02751","title":"Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation","display_name":"Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation","publication_year":2026,"publication_date":"2026-10-02","ids":{"openalex":"https://openalex.org/W7220445155","doi":"https://doi.org/10.48550/arxiv.2610.02751"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.02751","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02751","pdf_url":"https://arxiv.org/pdf/2610.02751","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/2610.02751","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045692299","display_name":"Phuong Le","orcid":"https://orcid.org/0000-0003-4724-7118","observed_orcids":["https://orcid.org/0000-0003-4724-7118"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Le, Phuong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5045692299"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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":0.9994999766349792,"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":0.9994999766349792,"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":0.00039999998989515007,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/p-laplacian-equation","display_name":"p-Laplace equation","score":0.8330000042915344},{"id":"https://openalex.org/keywords/lane-emden-equation","display_name":"Lane-Emden equation","score":0.9850000143051147},{"id":"https://openalex.org/keywords/morse-index","display_name":"Morse index","score":0.9620000123977661},{"id":"https://openalex.org/keywords/radial-solutions","display_name":"radial solutions","score":0.656000018119812},{"id":"https://openalex.org/keywords/sign-changing-solutions","display_name":"sign-changing solutions","score":0.6869999766349792},{"id":"https://openalex.org/keywords/supercritical-exponent","display_name":"supercritical exponent","score":0.3889999985694885}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7062000036239624},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6951000094413757},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.5824999809265137},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5472000241279602},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.5396000146865845},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.507099986076355},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.4474000036716461},{"id":"https://openalex.org/C118419359","wikidata":"https://www.wikidata.org/wiki/Q682585","display_name":"Supercritical fluid","level":2,"score":0.44290000200271606},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.4397999942302704},{"id":"https://openalex.org/C140031139","wikidata":"https://www.wikidata.org/wiki/Q79897","display_name":"Morse code","level":2,"score":0.4300999939441681},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.3928999900817871},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.365200012922287},{"id":"https://openalex.org/C196298200","wikidata":"https://www.wikidata.org/wiki/Q577705","display_name":"Critical point (mathematics)","level":2,"score":0.3433000147342682},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.3346000015735626},{"id":"https://openalex.org/C2778722038","wikidata":"https://www.wikidata.org/wiki/Q17030643","display_name":"Homogenization (climate)","level":3,"score":0.3147999942302704},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.290800005197525},{"id":"https://openalex.org/C2775918384","wikidata":"https://www.wikidata.org/wiki/Q4660961","display_name":"A priori estimate","level":2,"score":0.27790001034736633},{"id":"https://openalex.org/C8080151","wikidata":"https://www.wikidata.org/wiki/Q906375","display_name":"Regular solution","level":2,"score":0.2619999945163727},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.2547000050544739},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.2508000135421753}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.02751","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02751","pdf_url":"https://arxiv.org/pdf/2610.02751","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.2610.02751","doi":"https://doi.org/10.48550/arxiv.2610.02751","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.02751","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.02751","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02751","pdf_url":"https://arxiv.org/pdf/2610.02751","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220445155.pdf","grobid_xml":"https://content.openalex.org/works/W7220445155.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7077068840"],"related_works":["https://openalex.org/W1968586023","https://openalex.org/W2041771005","https://openalex.org/W7165662319","https://openalex.org/W3172898918","https://openalex.org/W3043234404","https://openalex.org/W1822181111","https://openalex.org/W6902379396","https://openalex.org/W7098947426","https://openalex.org/W2548684496","https://openalex.org/W2621041136"],"abstract_inverted_index":{"We":[0],"study":[1],"$-\u0394_p":[2],"u":[3],"=":[4],"|u|^{q-1}u$":[5],"in":[6],"$\\mathbb{R}^N$":[7],"with":[8,20],"$N>p\\ge":[9],"2$,":[10],"for":[11,37,71,114],"solutions":[12,50],"that":[13],"are":[14,134],"stable":[15,51,63,125],"outside":[16,52,64,126],"a":[17,53,65,76,115,127],"compact":[18,54,66],"set,":[19],"no":[21],"assumption":[22,80],"of":[23,85,98],"sign,":[24],"boundedness":[25],"or":[26],"symmetry.":[27],"Damascelli,":[28],"Farina,":[29,86],"Sciunzi":[30],"and":[31,39,88,90,129],"Valdinoci":[32],"settled":[33],"the":[34,41,57,75,82,95,99,106,111,131],"subcritical":[35],"range":[36,43],"$p>2$,":[38],"treated":[40],"supercritical":[42],"$p^*-1q_c(N,p)$":[44],"we":[45],"exhibit":[46],"positive":[47],"bounded":[48],"radial":[49],"set.":[55],"At":[56,105],"Sobolev":[58],"endpoint,":[59,108],"every":[60],"$C^1$":[61],"solution":[62,119],"set":[67],"has":[68],"finite":[69],"energy;":[70],"$p>2$":[72],"this":[73],"removes":[74],"priori":[77],"$\\mathcal":[78],"D^{1,p}$":[79],"from":[81],"low-Morse-index":[83],"classification":[84],"Mercuri":[87],"Willem":[89],"answers":[91],"their":[92],"question":[93],"on":[94,120],"Morse":[96],"index":[97],"Aubin--Talenti":[100],"extremals:":[101],"it":[102],"is":[103,110],"one.":[104],"upper":[107],"$q_c(N,p)$":[109],"exact":[112],"threshold":[113,132],"nonzero":[116],"homogeneous":[117],"weak":[118],"$\\mathbb":[121],"R^N\\setminus\\{0\\}$":[122],"to":[123],"be":[124],"ball,":[128],"at":[130],"there":[133],"exactly":[135],"two.":[136]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-06T00:00:00"}
