{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T02:19:14Z","timestamp":1781057954465,"version":"3.54.1"},"reference-count":72,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T00:00:00Z","timestamp":1625184000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["PRIN 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs"],"award-info":[{"award-number":["PRIN 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica &quot;Francesco Severi&quot;","doi-asserted-by":"publisher","award":["GNAMPA"],"award-info":[{"award-number":["GNAMPA"]}],"id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["Grant-in-Aid for Scientific Research (19H00644, 18KK0073, 17H02855, 16K13771)"],"award-info":[{"award-number":["Grant-in-Aid for Scientific Research (19H00644, 18KK0073, 17H02855, 16K13771)"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We prove the existence of a spherically symmetric solution for a Schr\u00f6dinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem.<\/jats:p>","DOI":"10.3390\/sym13071199","type":"journal-article","created":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T10:06:34Z","timestamp":1625220394000},"page":"1199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Symmetric Ground States for Doubly Nonlocal Equations with Mass Constraint"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3680-9106","authenticated-orcid":false,"given":"Silvia","family":"Cingolani","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 degli Studi di Bari Aldo Moro\u2014Via E. Orabona 4, 70125 Bari, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3141-9598","authenticated-orcid":false,"given":"Marco","family":"Gallo","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 degli Studi di Bari Aldo Moro\u2014Via E. Orabona 4, 70125 Bari, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1144-1536","authenticated-orcid":false,"given":"Kazunaga","family":"Tanaka","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Science and Engineering, Waseda University\u20143-4-1 Ohkubo, Shijuku-ku, Tokyo 169-8555, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Pekar, S. (1954). Untersuchung \u00fcber die Elektronentheorie der Kristalle, Akademie.","DOI":"10.1515\/9783112649305"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1002\/sapm197757293","article-title":"Existence and uniqueness of the minimizing solution of Choquard\u2019s nonlinear equation","volume":"57","author":"Lieb","year":"1977","journal-title":"Stud. Appl. Math."},{"key":"ref_3","unstructured":"Fr\u00f6hlich, J., and Lenzmann, E. (2004). Mean-field limit of quantum Bose gases and nonlinear Hartree equation. S\u00e9min. \u00c9qu. D\u00e9riv. Partielles, \u00c9cole Polytech.. Talk No. 18."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s002200100579","article-title":"On the point-particle (Newtonian) limit of the non-linear Hartree equation","volume":"225","author":"Tsai","year":"2002","journal-title":"Comm. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1017\/S0305004100011919","article-title":"The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods","volume":"24","author":"Hartree","year":"1928","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_6","first-page":"2733","article-title":"Spherically-symmetric solutions of the Schr\u00f6dinger-Newton equations","volume":"Volume 15","author":"Moroz","year":"1998","journal-title":"Proceedings of the Topology of the Universe Conference, Cleveland, OH, USA, 17\u201319 October 1997"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1007\/BF02105068","article-title":"On gravity\u2019s role in quantum state reduction","volume":"28","author":"Penrose","year":"1996","journal-title":"Gen. Rel. Grav."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1927","DOI":"10.1098\/rsta.1998.0256","article-title":"Quantum computation, entanglement and state reduction","volume":"356","author":"Penrose","year":"1998","journal-title":"R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_9","unstructured":"Penrose, R. (2005). The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0375-9601(01)00059-7","article-title":"The ground state energy of the Schr\u00f6dinger-Newton equation","volume":"280","author":"Tod","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1016\/0362-546X(80)90016-4","article-title":"The Choquard equation and related questions","volume":"4","author":"Lions","year":"1980","journal-title":"Nonlinear Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s00033-011-0166-8","article-title":"Multiple solutions to a magnetic nonlinear Choquard equation","volume":"63","author":"Cingolani","year":"2012","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_13","first-page":"891","article-title":"Intertwining semiclassical solutions to a Schr\u00f6dinger-Newton system","volume":"6","author":"Cingolani","year":"2013","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2013.04.081","article-title":"Positive and sign changing solutions to a nonlinear Choquard equation","volume":"407","author":"Clapp","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/apde.2009.2.1","article-title":"Uniqueness of ground states for pseudorelativistic Hartree equations","volume":"2","author":"Lenzmann","year":"2009","journal-title":"Anal. PDE"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/s00205-008-0208-3","article-title":"Classification of positive solitary solutions of the nonlinear Choquard equation","volume":"195","author":"Ma","year":"2010","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.jfa.2013.04.007","article-title":"Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics","volume":"265","author":"Moroz","year":"2013","journal-title":"J. Funct. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"6557","DOI":"10.1090\/S0002-9947-2014-06289-2","article-title":"Existence of groundstates for a class of nonlinear Choquard equations","volume":"367","author":"Moroz","year":"2015","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1007\/s11784-016-0373-1","article-title":"A guide to the Choquard equation","volume":"19","author":"Moroz","year":"2017","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_20","first-page":"530","article-title":"Existence and stability waves for nonlinear fractional Schr\u00f6dinger equation with Hartree type nonlinearity","volume":"4112","author":"Ye","year":"2014","journal-title":"J. Math. Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s42985-020-00036-w","article-title":"Normalized solutions for a class of nonlinear Choquard equations","volume":"1","author":"Bartsch","year":"2020","journal-title":"SN Partial. Differ. Equ. Appl."},{"key":"ref_22","unstructured":"Cingolani, S., and Tanaka, K. (2019). Ground State Solutions for the Nonlinear Choquard Equation with Prescribed Mass, Springer."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3533","DOI":"10.1137\/19M1243907","article-title":"Stationary solutions with prescribed L2-norm for the planar Schr\u00f6dinger-Poisson system","volume":"51","author":"Cingolani","year":"2019","journal-title":"SIAM J. Math. Anal."},{"key":"ref_24","first-page":"56","article-title":"Fractional quantum mechanics and L\u00e9vy path integrals","volume":"268","author":"Laskin","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Bucur, C., and Valdinoci, E. (2016). Nonlocal Diffusion and Applications, Springer.","DOI":"10.1007\/978-3-319-28739-3"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.bulsci.2011.12.004","article-title":"Hitchhiker\u2019s guide to the fractional Sobolev spaces","volume":"136","author":"Palatucci","year":"2012","journal-title":"Bull. Sci. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00220-007-0272-9","article-title":"Boson stars as solitary waves","volume":"274","author":"Jonsson","year":"2007","journal-title":"Comm. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s00220-012-1621-x","article-title":"On the continuum limit for discrete NLS with long-range lattice interactions","volume":"317","author":"Kirkpatrick","year":"2013","journal-title":"Comm. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"20140364","DOI":"10.1098\/rspa.2014.0364","article-title":"Numerical study of fractional nonlinear Schr\u00f6dinger equations","volume":"470","author":"Klein","year":"2014","journal-title":"Proc. R. Soc. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1364\/OL.40.001117","article-title":"Fractional Schr\u00f6dinger equation in optics","volume":"40","author":"Longhi","year":"2015","journal-title":"Optics Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walks guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"V\u00e1zquez, J.L. (2017). The mathematical theories of diffusion: Nonlinear and fractional diffusion. Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, Springer.","DOI":"10.1007\/978-3-319-61494-6_5"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1186\/s13661-020-01361-0","article-title":"A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions","volume":"2020","author":"Baleanu","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"109705","DOI":"10.1016\/j.chaos.2020.109705","article-title":"A new study on the mathematical modelling of human liver with Caputo\u2013Fabrizio fractional derivative","volume":"134","author":"Baleanu","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.anihpc.2013.02.001","article-title":"Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates","volume":"31","author":"Sire","year":"2014","journal-title":"Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1245","DOI":"10.1080\/03605300600987306","article-title":"An extension problem related to the fractional Laplacian","volume":"32","author":"Caffarelli","year":"2007","journal-title":"Comm. Partial Differ. Equ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1002\/cpa.21591","article-title":"Uniqueness of radial solutions for the fractional Laplacian","volume":"69","author":"Frank","year":"2016","journal-title":"Comm. Pure Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1088\/1361-6544\/aa60b4","article-title":"Nonlinear scalar field equations involving the fractional Laplacian","volume":"30","author":"Byeon","year":"2017","journal-title":"Nonlinearity"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1088\/0951-7715\/26\/2\/479","article-title":"Ground state of scalar field equations involving a fractional Laplacian with general nonlinearities","volume":"26","author":"Chang","year":"2013","journal-title":"Nonlinearity"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1017\/S0308210511000746","article-title":"Positive solutions of the nonlinear Schr\u00f6dinger equations with the fractional Laplacian","volume":"142","author":"Felmer","year":"2012","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/s11784-016-0369-x","article-title":"Existence of solutions of scalar field equations with fractional operator","volume":"19","author":"Ikoma","year":"2017","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4017","DOI":"10.1088\/1361-6544\/ac0166","article-title":"Normalized solutions for fractional nonlinear scalar field equation via Lagrangian formulation","volume":"34","author":"Cingolani","year":"2021","journal-title":"Nonlinearity"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s00526-020-01814-5","article-title":"Normalized solutions to the fractional Schr\u00f6dinger equations with combined nonlinearities","volume":"59","author":"Luo","year":"2020","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01217684","article-title":"The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics","volume":"1121","author":"Lieb","year":"1987","journal-title":"Comm. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.na.2011.08.038","article-title":"Solutions to quasi-relativistic multi-configurative Hartree\u2013Fock equations in quantum chemistry","volume":"75","author":"Argaez","year":"2012","journal-title":"Nonlinear Anal."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1007\/s00023-008-0370-z","article-title":"Hartree\u2013Fock Theory for Pseudorelativistic Atoms","volume":"9","author":"Stockmeyer","year":"2008","journal-title":"Ann. Henri Poincar\u00e9"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1142\/S0219530516500056","article-title":"Orbital stability of standing waves of a class of fractional Schr\u00f6dinger equations with Hartree-type nonlinearity","volume":"15","author":"Cho","year":"2017","journal-title":"Anal. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s00332-015-9259-4","article-title":"Orbital-free density functional theory of out-of-plane charge screening in graphene","volume":"25","author":"Lu","year":"2015","journal-title":"J. Nonlinear. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1142\/S0218202515500384","article-title":"On the fractional Choquard equations","volume":"25","author":"Siciliano","year":"2015","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"3197","DOI":"10.1016\/j.jde.2017.04.034","article-title":"On fractional Schr\u00f6dinger systems of Choquard type","volume":"263","author":"Bhattara","year":"2017","journal-title":"J. Diff. Equ."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.1007\/s13398-018-0598-5","article-title":"Ground state solutions of fractional Choquard equations with general potentials and nonlinearities","volume":"113","author":"Gao","year":"2019","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat. RACSAM"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1016\/j.camwa.2018.10.024","article-title":"Ground state solutions of Pohozaev type for fractional Choquard equations with general nonlinearities","volume":"77","author":"Luo","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"4082","DOI":"10.1002\/mma.3849","article-title":"Ground state for nonlinear fractional Choquard equations with general nonlinearities","volume":"39","author":"Shen","year":"2016","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF00250555","article-title":"Nonlinear scalar field equations. I. Existence of a ground state","volume":"82","author":"Berestycki","year":"1983","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1515\/ans-2018-2039","article-title":"Nonlinear scalar field equations with L2 constraint: Mountain pass and symmetric mountain pass approaches","volume":"19","author":"Hirata","year":"2019","journal-title":"Adv. Nonlinear Stud."},{"key":"ref_56","first-page":"609","article-title":"A note on deformation argument for L2 constraint problems","volume":"24","author":"Ikoma","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2399","DOI":"10.1090\/S0002-9939-02-06821-1","article-title":"A remark on least energy solutions in \u211dN","volume":"131","author":"Jeanjean","year":"2003","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.jmaa.2017.11.060","article-title":"Stability of standing waves for the fractional Schr\u00f6dinger-Hartree equation","volume":"460","author":"Feng","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_59","unstructured":"Lieb, E.H., Seiringer, R., Solovej, J.P., and Yngvason, J. (2005). The mathematics of the Bose gas and its condensation. Oberwolfach Seminars, Birk\u00e4user."},{"key":"ref_60","unstructured":"Pitaevskii, L., and Stringari, S. (2003). Bose-Einstein Condensation, Oxford University Press."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"349","DOI":"10.2307\/2007032","article-title":"Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities","volume":"118","author":"Lieb","year":"1983","journal-title":"Ann. Math."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/0022-1236(82)90072-6","article-title":"Sym\u00e9trie et compacit\u00e9 dans les espaces de Sobolev","volume":"49","author":"Lions","year":"1982","journal-title":"J. Funct. Anal."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1142\/S0219199709003399","article-title":"Sobolev inequalities with symmetry","volume":"11","author":"Cho","year":"2009","journal-title":"Commun. Contemp. Math."},{"key":"ref_64","first-page":"583","article-title":"Fractional Gagliardo-Nirenberg inequality","volume":"24","author":"Park","year":"2011","journal-title":"J. Chungcheong Math. Soc."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/BF01941322","article-title":"The principle of symmetric criticality","volume":"69","author":"Palais","year":"1979","journal-title":"Comm. Math. Phys."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Willem, M. (1996). Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, Birk\u00e4user.","DOI":"10.1007\/978-1-4612-4146-1"},{"key":"ref_67","first-page":"253","article-title":"Nonlinear scalar field equations in \u211dN: Mountain pass and symmetric mountain pass approaches","volume":"35","author":"Hirata","year":"2010","journal-title":"Topol. Methods Nonlinear Anal."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1016\/S0362-546X(96)00021-1","article-title":"Existence of solutions with prescribed norm for semilinear elliptic equations","volume":"28","author":"Jeanjean","year":"1997","journal-title":"Nonlinear Anal."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"105840","DOI":"10.1016\/j.cnsns.2021.105840","article-title":"Necklace beams carrying fractional angular momentum in fractional systems with a saturable nonlinearity","volume":"99","author":"Dong","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_70","unstructured":"Frank, R.L., and Lenzmann, E. (2010). On ground states for the L2-critical boson star equation. arXiv."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.na.2013.11.023","article-title":"The Boson star equation with initial data of low regularity","volume":"97","author":"Herr","year":"2014","journal-title":"Nonlinear Anal."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"3515","DOI":"10.1088\/0951-7715\/24\/12\/009","article-title":"On singularity formation for the L2-critical Boson star equation","volume":"24","author":"Lenzmann","year":"2011","journal-title":"Nonlinearity"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1199\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:25:13Z","timestamp":1760163913000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,2]]},"references-count":72,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071199"],"URL":"https:\/\/doi.org\/10.3390\/sym13071199","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,2]]}}}