Get quick answers to your questions about the article from our AI researcher chatbot
{'id': 'https://openalex.org/W1964016061', 'doi': 'https://doi.org/10.1016/s0022-1236(03)00138-1', 'title': 'Nonlinear Liouville theorems for some critical problems on H-type groups', 'display_name': 'Nonlinear Liouville theorems for some critical problems on H-type groups', 'publication_year': 2004, 'publication_date': '2004-02-01', 'ids': {'openalex': 'https://openalex.org/W1964016061', 'doi': 'https://doi.org/10.1016/s0022-1236(03)00138-1', 'mag': '1964016061'}, 'language': 'en', 'primary_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1016/s0022-1236(03)00138-1', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S54953690', 'display_name': 'Journal of Functional Analysis', 'issn_l': '0022-1236', 'issn': ['0022-1236', '1096-0783'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310320990', 'host_organization_name': 'Elsevier BV', 'host_organization_lineage': ['https://openalex.org/P4310320990'], 'host_organization_lineage_names': ['Elsevier BV'], 'type': 'journal'}, 'license': 'publisher-specific-oa', 'license_id': 'https://openalex.org/licenses/publisher-specific-oa', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'type': 'article', 'type_crossref': 'journal-article', 'indexed_in': ['crossref'], 'open_access': {'is_oa': True, 'oa_status': 'green', 'oa_url': 'https://doi.org/10.1016/s0022-1236(03)00138-1', 'any_repository_has_fulltext': True}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5004451209', 'display_name': 'Andrea Bonfiglioli', 'orcid': 'https://orcid.org/0000-0001-7463-7803'}, 'institutions': [{'id': 'https://openalex.org/I9360294', 'display_name': 'University of Bologna', 'ror': 'https://ror.org/01111rn36', 'country_code': 'IT', 'type': 'education', 'lineage': ['https://openalex.org/I9360294']}], 'countries': ['IT'], 'is_corresponding': False, 'raw_author_name': 'Andrea Bonfiglioli', 'raw_affiliation_strings': ['Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5-40126 Bologna, Italy'], 'affiliations': [{'raw_affiliation_string': 'Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5-40126 Bologna, Italy', 'institution_ids': ['https://openalex.org/I9360294']}]}, {'author_position': 'last', 'author': {'id': 'https://openalex.org/A5079460580', 'display_name': 'Francesco Uguzzoni', 'orcid': None}, 'institutions': [{'id': 'https://openalex.org/I9360294', 'display_name': 'University of Bologna', 'ror': 'https://ror.org/01111rn36', 'country_code': 'IT', 'type': 'education', 'lineage': ['https://openalex.org/I9360294']}], 'countries': ['IT'], 'is_corresponding': False, 'raw_author_name': 'Francesco Uguzzoni', 'raw_affiliation_strings': ['Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5-40126 Bologna, Italy'], 'affiliations': [{'raw_affiliation_string': 'Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5-40126 Bologna, Italy', 'institution_ids': ['https://openalex.org/I9360294']}]}], 'institution_assertions': [], 'countries_distinct_count': 1, 'institutions_distinct_count': 1, 'corresponding_author_ids': [], 'corresponding_institution_ids': [], 'apc_list': {'value': 2560, 'currency': 'USD', 'value_usd': 2560, 'provenance': 'doaj'}, 'apc_paid': None, 'fwci': 1.439, 'has_fulltext': True, 'fulltext_origin': 'pdf', 'cited_by_count': 56, 'citation_normalized_percentile': {'value': 0.96266, 'is_in_top_1_percent': False, 'is_in_top_10_percent': True}, 'cited_by_percentile_year': {'min': 93, 'max': 94}, 'biblio': {'volume': '207', 'issue': '1', 'first_page': '161', 'last_page': '215'}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T10194', 'display_name': 'Fractional Laplacian Operators', 'score': 0.9995, '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': 'Fractional Laplacian Operators', 'score': 0.9995, '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': 'Multiscale Methods for Heterogeneous Systems', 'score': 0.9986, '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/T11110', 'display_name': 'Analysis and Control of Distributed Parameter Systems', 'score': 0.9834, 'subfield': {'id': 'https://openalex.org/subfields/2207', 'display_name': 'Control and Systems Engineering'}, 'field': {'id': 'https://openalex.org/fields/22', 'display_name': 'Engineering'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}], 'keywords': [{'id': 'https://openalex.org/keywords/heisenberg-group', 'display_name': 'Heisenberg group', 'score': 0.75995284}, {'id': 'https://openalex.org/keywords/elliptic-problems', 'display_name': 'Elliptic Problems', 'score': 0.51561}, {'id': 'https://openalex.org/keywords/mountain-pass-theorem', 'display_name': 'Mountain pass theorem', 'score': 0.4526321}], 'concepts': [{'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.94821465}, {'id': 'https://openalex.org/C127519595', 'wikidata': 'https://www.wikidata.org/wiki/Q1601337', 'display_name': 'Heisenberg group', 'level': 2, 'score': 0.75995284}, {'id': 'https://openalex.org/C2777299769', 'wikidata': 'https://www.wikidata.org/wiki/Q3707858', 'display_name': 'Type (biology)', 'level': 2, 'score': 0.7161872}, {'id': 'https://openalex.org/C158622935', 'wikidata': 'https://www.wikidata.org/wiki/Q660848', 'display_name': 'Nonlinear system', 'level': 2, 'score': 0.5731619}, {'id': 'https://openalex.org/C202444582', 'wikidata': 'https://www.wikidata.org/wiki/Q837863', 'display_name': 'Pure mathematics', 'level': 1, 'score': 0.5486291}, {'id': 'https://openalex.org/C169214877', 'wikidata': 'https://www.wikidata.org/wiki/Q981016', 'display_name': 'Dirichlet distribution', 'level': 3, 'score': 0.45682552}, {'id': 'https://openalex.org/C2779779452', 'wikidata': 'https://www.wikidata.org/wiki/Q6925496', 'display_name': 'Mountain pass theorem', 'level': 3, 'score': 0.4526321}, {'id': 'https://openalex.org/C2779560616', 'wikidata': 'https://www.wikidata.org/wiki/Q1192869', 'display_name': 'Dirichlet problem', 'level': 3, 'score': 0.4514878}, {'id': 'https://openalex.org/C2781311116', 'wikidata': 'https://www.wikidata.org/wiki/Q83306', 'display_name': 'Group (periodic table)', 'level': 2, 'score': 0.43462417}, {'id': 'https://openalex.org/C134306372', 'wikidata': 'https://www.wikidata.org/wiki/Q7754', 'display_name': 'Mathematical analysis', 'level': 1, 'score': 0.3290211}, {'id': 'https://openalex.org/C182310444', 'wikidata': 'https://www.wikidata.org/wiki/Q1332643', 'display_name': 'Boundary value problem', 'level': 2, 'score': 0.05301577}, {'id': 'https://openalex.org/C62520636', 'wikidata': 'https://www.wikidata.org/wiki/Q944', 'display_name': 'Quantum mechanics', '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/C185592680', 'wikidata': 'https://www.wikidata.org/wiki/Q2329', 'display_name': 'Chemistry', 'level': 0, 'score': 0.0}, {'id': 'https://openalex.org/C178790620', 'wikidata': 'https://www.wikidata.org/wiki/Q11351', 'display_name': 'Organic chemistry', '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/C121332964', 'wikidata': 'https://www.wikidata.org/wiki/Q413', 'display_name': 'Physics', 'level': 0, 'score': 0.0}], 'mesh': [], 'locations_count': 2, 'locations': [{'is_oa': True, 'landing_page_url': 'https://doi.org/10.1016/s0022-1236(03)00138-1', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S54953690', 'display_name': 'Journal of Functional Analysis', 'issn_l': '0022-1236', 'issn': ['0022-1236', '1096-0783'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310320990', 'host_organization_name': 'Elsevier BV', 'host_organization_lineage': ['https://openalex.org/P4310320990'], 'host_organization_lineage_names': ['Elsevier BV'], 'type': 'journal'}, 'license': 'publisher-specific-oa', 'license_id': 'https://openalex.org/licenses/publisher-specific-oa', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, {'is_oa': True, 'landing_page_url': 'https://www.openaccessrepository.it/record/163338', 'pdf_url': 'https://www.openaccessrepository.it/record/163338/files/fulltext.pdf', 'source': {'id': 'https://openalex.org/S4306402478', 'display_name': 'INFM-OAR (INFN Catania)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I4210116497', 'host_organization_name': 'INFN Sezione di Catania', 'host_organization_lineage': ['https://openalex.org/I4210116497'], 'host_organization_lineage_names': ['INFN Sezione di Catania'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}], 'best_oa_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1016/s0022-1236(03)00138-1', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S54953690', 'display_name': 'Journal of Functional Analysis', 'issn_l': '0022-1236', 'issn': ['0022-1236', '1096-0783'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310320990', 'host_organization_name': 'Elsevier BV', 'host_organization_lineage': ['https://openalex.org/P4310320990'], 'host_organization_lineage_names': ['Elsevier BV'], 'type': 'journal'}, 'license': 'publisher-specific-oa', 'license_id': 'https://openalex.org/licenses/publisher-specific-oa', 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'sustainable_development_goals': [], 'grants': [], 'datasets': [], 'versions': [], 'referenced_works_count': 51, 'referenced_works': ['https://openalex.org/W115020753', 'https://openalex.org/W1509786127', 'https://openalex.org/W1510601398', 'https://openalex.org/W1519669307', 'https://openalex.org/W1522020530', 'https://openalex.org/W1565401649', 'https://openalex.org/W1570120176', 'https://openalex.org/W1605472651', 'https://openalex.org/W1709307408', 'https://openalex.org/W1866311589', 'https://openalex.org/W188050773', 'https://openalex.org/W1968014615', 'https://openalex.org/W1968580541', 'https://openalex.org/W1976284538', 'https://openalex.org/W197818049', 'https://openalex.org/W1979129279', 'https://openalex.org/W1981352904', 'https://openalex.org/W1981479249', 'https://openalex.org/W1985879458', 'https://openalex.org/W1987071998', 'https://openalex.org/W1988967439', 'https://openalex.org/W1992974193', 'https://openalex.org/W2011655994', 'https://openalex.org/W2022757729', 'https://openalex.org/W2030113972', 'https://openalex.org/W2030921258', 'https://openalex.org/W2035173346', 'https://openalex.org/W2038928461', 'https://openalex.org/W2043956755', 'https://openalex.org/W2051089144', 'https://openalex.org/W2061091809', 'https://openalex.org/W2062728483', 'https://openalex.org/W2072078658', 'https://openalex.org/W2080842686', 'https://openalex.org/W2082101282', 'https://openalex.org/W2112046334', 'https://openalex.org/W2148302699', 'https://openalex.org/W2166950238', 'https://openalex.org/W2243251959', 'https://openalex.org/W2322757099', 'https://openalex.org/W2529655271', 'https://openalex.org/W2610857016', 'https://openalex.org/W2731197558', 'https://openalex.org/W2732843230', 'https://openalex.org/W2804870597', 'https://openalex.org/W3021722416', 'https://openalex.org/W3089772770', 'https://openalex.org/W3140963071', 'https://openalex.org/W4210663128', 'https://openalex.org/W4231701177', 'https://openalex.org/W4292360407'], 'related_works': ['https://openalex.org/W4386565746', 'https://openalex.org/W4300200796', 'https://openalex.org/W4297793631', 'https://openalex.org/W4224992894', 'https://openalex.org/W3097902004', 'https://openalex.org/W2887716537', 'https://openalex.org/W2184854647', 'https://openalex.org/W2089536263', 'https://openalex.org/W1892228711', 'https://openalex.org/W1005908918'], 'abstract_inverted_index': {'In': [0], 'this': [1], 'paper,': [2], 'we': [3], 'provide': [4], 'a': [5, 9, 28], 'non-existence': [6], 'result': [7, 26], 'for': [8], 'semilinear': [10], 'sub-elliptic': [11], 'Dirichlet': [12], 'problem': [13], 'with': [14], 'critical': [15], 'growth': [16], 'on': [17], 'the': [18], 'half-spaces': [19], 'of': [20, 23], 'any': [21], 'group': [22], 'Heisenberg-type.': [24], 'Our': [25], 'improves': [27], 'recent': [29], 'theorem': [30], 'in': [31], '(Math.': [32], 'Ann.': [33], '315': [34], '(3)': [35], '(2000)': [36], '453).': [37]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W1964016061', 'counts_by_year': [{'year': 2024, 'cited_by_count': 4}, {'year': 2023, 'cited_by_count': 2}, {'year': 2022, 'cited_by_count': 2}, {'year': 2021, 'cited_by_count': 6}, {'year': 2020, 'cited_by_count': 3}, {'year': 2019, 'cited_by_count': 2}, {'year': 2018, 'cited_by_count': 1}, {'year': 2017, 'cited_by_count': 1}, {'year': 2016, 'cited_by_count': 3}, {'year': 2015, 'cited_by_count': 1}, {'year': 2014, 'cited_by_count': 5}, {'year': 2013, 'cited_by_count': 3}, {'year': 2012, 'cited_by_count': 2}], 'updated_date': '2024-09-15T04:35:35.586567', 'created_date': '2016-06-24'}