{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T15:43:09Z","timestamp":1775835789056,"version":"3.50.1"},"reference-count":40,"publisher":"American Mathematical Society (AMS)","issue":"1","license":[{"start":{"date-parts":[[2012,8,9]],"date-time":"2012-08-09T00:00:00Z","timestamp":1344470400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>\n                    We study an elliptic system of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L u equals StartAbsoluteValue v EndAbsoluteValue Superscript p minus 1 Baseline v\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>|<\/mml:mo>\n                                <mml:mi>v<\/mml:mi>\n                                <mml:mo>|<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>v<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Lu = \\left | v\\right |^{p-1} v<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L v equals StartAbsoluteValue u EndAbsoluteValue Superscript q minus 1 Baseline u\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>|<\/mml:mo>\n                                <mml:mi>u<\/mml:mi>\n                                <mml:mo>|<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>u<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Lv=\\left | u\\right |^{q-1} u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u03a9\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with homogeneous Dirichlet boundary condition, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L u colon equals minus normal upper Delta u\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Lu:=-\\Delta u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the case of a bounded domain and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L u colon equals minus normal upper Delta u plus u\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Lu:=-\\Delta u + u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the cases of an exterior domain or the whole space\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R Superscript upper N\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.\n                  <\/p>","DOI":"10.1090\/s0002-9947-2011-05452-8","type":"journal-article","created":{"date-parts":[[2011,8,9]],"date-time":"2011-08-09T08:23:16Z","timestamp":1312878196000},"page":"447-491","source":"Crossref","is-referenced-by-count":48,"special_numbering":"920","title":["Ground state and non-ground state solutions of some strongly coupled elliptic systems"],"prefix":"10.1090","volume":"364","author":[{"given":"Denis","family":"Bonheure","sequence":"first","affiliation":[]},{"given":"Ederson","family":"Moreira dos Santos","sequence":"additional","affiliation":[]},{"given":"Miguel","family":"Ramos","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2011,8,9]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1016\/S0022-247X(02)00413-4","article-title":"On the existence of positive solutions of a perturbed Hamiltonian system in \u211d^{\u2115}","volume":"276","author":"Alves, Claudianor Oliveira","year":"2002","journal-title":"J. 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