{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:45:34Z","timestamp":1777448734792,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2009,3,1]],"date-time":"2009-03-01T00:00:00Z","timestamp":1235865600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2009,3]]},"abstract":"<jats:p>\n                    Let B\n                    <jats:sub>1<\/jats:sub>\n                    be the unit open ball with center at the origin in R\n                    <jats:sup>N<\/jats:sup>\n                    , N\u22652. We consider the following quasilinear problem depending on a real parameter \u03bb&gt;0:\n                  <\/jats:p>\n                  <jats:p>\n                    \u2212\u0394\n                    <jats:sub>N<\/jats:sub>\n                    u=\u03bb f(u), u&gt;0 in \u03a9,\n                  <\/jats:p>\n                  <jats:p>\n                    u=0 on \u2202\u03a9, \u2003\u2003(P\n                    <jats:sub>\u03bb<\/jats:sub>\n                    )\n                  <\/jats:p>\n                  <jats:p>\n                    where f(t) is a nonlinearity that grows like e\n                    <jats:sup>\n                      t\n                      <jats:sup>N\/N\u22121<\/jats:sup>\n                    <\/jats:sup>\n                    as t\u2192\u221e and behaves like t\n                    <jats:sup>\u03b1<\/jats:sup>\n                    , for some \u03b1\u2208(\u2212\u221e, 0), as t\u21920\n                    <jats:sup>+<\/jats:sup>\n                    . More precisely, we require f to satisfy assumptions (A1) and (A2) listed in Section 1. For such a general nonlinearity we show that if \u03bb&gt;0 is small enough, (P\n                    <jats:sub>\u03bb<\/jats:sub>\n                    ) admits at least one weak solution (in the sense of distributions). We further study the question of uniqueness and multiplicity of solutions to (P\n                    <jats:sub>\u03bb<\/jats:sub>\n                    ) when \u03a9=B\n                    <jats:sub>1<\/jats:sub>\n                    under additional structural conditions on f (see assumptions (A3)\u2013(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)\u2013(A5), we prove uniqueness of solution to (P\n                    <jats:sub>\u03bb<\/jats:sub>\n                    ) for all \u03bb&gt;0 small whereas under (A6), (A7) or (A8), we show multiplicity of solutions to (P\n                    <jats:sub>\u03bb<\/jats:sub>\n                    ) for all \u03bb&gt;0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition lim\u2009inf\u2009\n                    <jats:sub>t\u2192\u221e<\/jats:sub>\n                    h(t)te\n                    <jats:sup>\n                      \u03b5t\n                      <jats:sup>1\/(N\u22121)<\/jats:sup>\n                    <\/jats:sup>\n                    =\u221e \u2200\u03b5&gt;0.\n                  <\/jats:p>","DOI":"10.3233\/asy-2008-0911","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:04:33Z","timestamp":1575054273000},"page":"195-227","source":"Crossref","is-referenced-by-count":1,"title":["Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball"],"prefix":"10.1177","volume":"61","author":[{"given":"J.","family":"Giacomoni","sequence":"first","affiliation":[{"name":"L.M.A., Universit\u00e9 de Pau et des Pays de l'Adour, B.P. 576, 64012 Pau cedex, France. E-mail: jgiacomo@univ-pau.fr"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Prashanth","sequence":"additional","affiliation":[{"name":"T.I.F.R. Center, I.I.Sc. Campus, P.B. No. 1234, Bangalore 560012, India. E-mail: pras@math.tifrbng.res.in"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K.","family":"Sreenadh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology (Delhi), Hauz Khaz, New Delhi-16, India. E-mail: sreenadh@maths.iitd.ernet.in"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2009,3,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2008-0911","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2008-0911","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:38:49Z","timestamp":1777379929000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2008-0911"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,3]]},"references-count":0,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2009,3]]}},"alternative-id":["10.3233\/ASY-2008-0911"],"URL":"https:\/\/doi.org\/10.3233\/asy-2008-0911","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,3]]}}}