{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T17:14:17Z","timestamp":1776273257212,"version":"3.50.1"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"5","license":[{"start":{"date-parts":[[2011,1,12]],"date-time":"2011-01-12T00:00:00Z","timestamp":1294790400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"

\n We prove that, for any\n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \\lambda \\in \\mathbb {R}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the system\n \n \n \n \n \n \u2212\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n u<\/mml:mi>\n +<\/mml:mo>\n \n \u03bb\n \n <\/mml:mi>\n u<\/mml:mi>\n =<\/mml:mo>\n \n u<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n u<\/mml:mi>\n \n v<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n -\\Delta u +\\lambda u = u^3-\\beta uv^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \n \u2212\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n v<\/mml:mi>\n +<\/mml:mo>\n \n \u03bb\n \n <\/mml:mi>\n v<\/mml:mi>\n =<\/mml:mo>\n \n v<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n v<\/mml:mi>\n \n u<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n -\\Delta v+\\lambda v =v^3-\\beta vu^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n u<\/mml:mi>\n ,<\/mml:mo>\n v<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n H<\/mml:mi>\n 0<\/mml:mn>\n 1<\/mml:mn>\n <\/mml:msubsup>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n ,<\/mml:mo>\n <\/mml:mrow>\n u,v\\in H^1_0(\\Omega ),<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n where\n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \\Omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a bounded smooth domain of\n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n \\mathbb {R}^3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , admits a bounded family of positive solutions\n \n \n \n \n (<\/mml:mo>\n \n u<\/mml:mi>\n \n \n \u03b2\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n v<\/mml:mi>\n \n \n \u03b2\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n (u_{\\beta }, v_{\\beta })<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as\n \n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n +<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n \\beta \\to +\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . An upper bound on the number of nodal sets of the weak limits of\n \n \n \n \n \n u<\/mml:mi>\n \n \n \u03b2\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2212\n \n <\/mml:mo>\n \n v<\/mml:mi>\n \n \n \u03b2\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n u_{\\beta }-v_{\\beta }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is also provided. Moreover, for any sufficiently large fixed value of\n \n \n \n \n \n \u03b2\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\beta >0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the system admits infinitely many positive solutions.\n <\/p>","DOI":"10.1090\/s0002-9939-10-10231-7","type":"journal-article","created":{"date-parts":[[2010,1,25]],"date-time":"2010-01-25T13:56:56Z","timestamp":1264427816000},"page":"1681-1692","source":"Crossref","is-referenced-by-count":45,"special_numbering":"611","title":["Existence and bounds of positive solutions for a nonlinear Schr\u00f6dinger system"],"prefix":"10.1090","volume":"138","author":[{"given":"Benedetta","family":"Noris","sequence":"first","affiliation":[]},{"given":"Miguel","family":"Ramos","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2010,1,12]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1112\/jlms\/jdl020","article-title":"Standing waves of some coupled nonlinear Schr\u00f6dinger equations","volume":"75","author":"Ambrosetti, Antonio","year":"2007","journal-title":"J. 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