{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T23:50:36Z","timestamp":1764978636635,"version":"3.46.0"},"reference-count":24,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2016,3,19]],"date-time":"2016-03-19T00:00:00Z","timestamp":1458345600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/00297\/2013","Investigador FCT","PEst-OE\/EEI\/LA0009\/2013"],"award-info":[{"award-number":["UID\/MAT\/00297\/2013","Investigador FCT","PEst-OE\/EEI\/LA0009\/2013"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,5,1]]},"abstract":"Abstract<\/jats:title>\n We study the existence of ground states for the coupled Schr\u00f6dinger system<\/jats:p>\n \n \n \n \n \n {<\/m:mo>\n \n \n \n \n \n \n \n \n -<\/m:mo>\n \n \u0394<\/m:mi>\n \u2062<\/m:mo>\n \n u<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n \n \u03bb<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n u<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mrow>\n =<\/m:mo>\n \n \n \n \u03bc<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n \n |<\/m:mo>\n \n u<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n |<\/m:mo>\n <\/m:mrow>\n \n \n 2<\/m:mn>\n \u2062<\/m:mo>\n q<\/m:mi>\n <\/m:mrow>\n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n u<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n +<\/m:mo>\n \n \n \u2211<\/m:mo>\n \n j<\/m:mi>\n \u2260<\/m:mo>\n i<\/m:mi>\n <\/m:mrow>\n <\/m:munder>\n \n \n b<\/m:mi>\n \n i<\/m:mi>\n \u2062<\/m:mo>\n j<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \u2062<\/m:mo>\n \n \n |<\/m:mo>\n \n u<\/m:mi>\n j<\/m:mi>\n <\/m:msub>\n |<\/m:mo>\n <\/m:mrow>\n q<\/m:mi>\n <\/m:msup>\n \u2062<\/m:mo>\n \n \n |<\/m:mo>\n \n u<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n |<\/m:mo>\n <\/m:mrow>\n \n q<\/m:mi>\n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n u<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n ,<\/m:mo>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n \n \n \n \n \n \n \n u<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n \u2208<\/m:mo>\n \n \n H<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \u211d<\/m:mi>\n n<\/m:mi>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n ,<\/m:mo>\n \n i<\/m:mi>\n =<\/m:mo>\n \n 1<\/m:mn>\n ,<\/m:mo>\n \u2026<\/m:mi>\n ,<\/m:mo>\n d<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n ,<\/m:mo>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n <\/m:mtable>\n <\/m:mrow>\n <\/m:math>\n \n $\\left\\{\\begin{aligned} &\\displaystyle-\\Delta u_{i}+\\lambda_{i}u_{i}=\\mu_{i}|u_% {i}|^{2q-2}u_{i}+\\sum_{j\\neq i}b_{ij}|u_{j}|^{q}|u_{i}|^{q-2}u_{i},\\\\ &\\displaystyle u_{i}\\in H^{1}(\\mathbb{R}^{n}),\\quad i=1,\\ldots,d,\\end{aligned}\\right.$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n <\/jats:p>\n \n \n \n \n \n n<\/m:mi>\n \u2265<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n ${n\\geq 1}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , for\n \n \n \n \n \n \n \u03bb<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n ,<\/m:mo>\n \n \u03bc<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n ><\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n ${\\lambda_{i},\\mu_{i}>0}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n \n \n \n b<\/m:mi>\n \n i<\/m:mi>\n \u2062<\/m:mo>\n j<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n =<\/m:mo>\n \n b<\/m:mi>\n \n j<\/m:mi>\n \u2062<\/m:mo>\n i<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ><\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n ${b_{ij}=b_{ji}>0}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n (the so-called \u201csymmetric attractive case\u201d) and\n \n \n \n \n 1<\/m:mn>\n <<\/m:mo>\n q<\/m:mi>\n <<\/m:mo>\n \n n<\/m:mi>\n \/<\/m:mo>\n \n \n (<\/m:mo>\n \n n<\/m:mi>\n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n +<\/m:mo>\n <\/m:msup>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n ${1<q<n\/(n-2)^{+}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We prove the existence of a nonnegative ground state\n \n \n \n \n (<\/m:mo>\n \n u<\/m:mi>\n 1<\/m:mn>\n *<\/m:mo>\n <\/m:msubsup>\n ,<\/m:mo>\n \u2026<\/m:mi>\n ,<\/m:mo>\n \n u<\/m:mi>\n d<\/m:mi>\n *<\/m:mo>\n <\/m:msubsup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n ${(u_{1}^{*},\\ldots,u_{d}^{*})}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n with\n \n \n \n \n u<\/m:mi>\n i<\/m:mi>\n *<\/m:mo>\n <\/m:msubsup>\n <\/m:math>\n \n ${u_{i}^{*}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n radially decreasing. Moreover, we show that if in addition\n \n \n \n \n q<\/m:mi>\n <<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n ${q<2}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , such ground states are positive in all dimensions and for all values of the parameters.\n <\/jats:p>","DOI":"10.1515\/ans-2015-5029","type":"journal-article","created":{"date-parts":[[2016,3,19]],"date-time":"2016-03-19T16:09:37Z","timestamp":1458403777000},"page":"381-387","source":"Crossref","is-referenced-by-count":8,"title":["Ground States for a Nonlinear Schr\u00f6dinger System with Sublinear Coupling Terms"],"prefix":"10.1515","volume":"16","author":[{"given":"Filipe","family":"Oliveira","sequence":"first","affiliation":[{"name":"Mathematics Department, FCT-UNL, Universidade Nova de Lisboa, Caparica Campus, 2829-516, Portugal"}]},{"given":"Hugo","family":"Tavares","sequence":"additional","affiliation":[{"name":"CAMGSD & Mathematics Department, IST, Universidade de Lisboa, Av. 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