caa a22 4500 445843411 CHVBK 20180317145351.0 cr unu---uuuuu 170323e20110301xx s 000 0 eng 10.1007/s00526-010-0347-x doi (NATIONALLICENCE)springer-10.1007/s00526-010-0347-x A local mountain pass type result for a system of nonlinear Schrödinger equations [Elektronische Daten] [Norihisa Ikoma, Kazunaga Tanaka] We consider a singular perturbation problem for a system of nonlinear Schrödinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$where N=2, 3, μ 1, μ 2, β > 0 and V 1(x), V 2(x): R N → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for $${P\in{\bf R}^N}$$ the least energy level m(P) for non-trivial vector solutions of the rescaled "limit” problem:$$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$We assume that there exists an open bounded set $${\Lambda\subset{\bf R}^N}$$ satisfying$$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$We show that (*) possesses a family of non-trivial vector positive solutions $${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$$ which concentrates—after extracting a subsequence ε n → 0—to a point $${P_0\in\Lambda}$$ with $${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$$. Moreover (v 1ε (x), v 2ε (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling. Springer-Verlag, 2010 Ikoma Norihisa Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, 169-8555, Tokyo, Japan aut Tanaka Kazunaga Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, 169-8555, Tokyo, Japan aut Calculus of Variations and Partial Differential Equations Springer-Verlag 40/3-4(2011-03-01), 449-480 0944-2669 40:3-4<449 2011 40 526 https://doi.org/10.1007/s00526-010-0347-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00526-010-0347-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ikoma Norihisa Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, 169-8555, Tokyo, Japan aut NATIONALLICENCE 700 1- Tanaka Kazunaga Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, 169-8555, Tokyo, Japan aut NATIONALLICENCE 773 0- Calculus of Variations and Partial Differential Equations Springer-Verlag 40/3-4(2011-03-01), 449-480 0944-2669 40:3-4<449 2011 40 526 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer