A local mountain pass type result for a system of nonlinear Schrödinger equations
A local mountain pass type result for a system of nonlinear Schrödinger equations
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DOI：
10.1007/s00526-010-0347-x
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发表时间：
2011
影响因子：
2.1
通讯作者：
N. Ikoma;Kazunaga Tanaka
中科院分区：
文献类型：
--
作者：
N. Ikoma;Kazunaga Tanaka
We consider a singular perturbation problem for a system of nonlinear Schrödinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$whereN= 2, 3,μ1,μ2,β> 0 andV1(x),V2(x):RN→ (0, ∞) are positive continuous functions. We consider the case where the interactionβ> 0 is relatively small and we define forthe least energy levelm(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 setsatisfyingWe show that (*) possesses a family of non-trivial vector positive solutionswhich concentrates-after extracting a subsequenceεn→ 0-to a pointwith. Moreover (v1ε(x),v2ε(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.
We consider a singular perturbation problem for a system of nonlinear Schrödinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$whereN= 2, 3,μ1,μ2,β> 0 andV1(x),V2(x):RN→ (0, ∞) are positive continuous functions. We consider the case where the interactionβ> 0 is relatively small and we define forthe least energy levelm(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 setsatisfyingWe show that (*) possesses a family of non-trivial vector positive solutionswhich concentrates—after extracting a subsequenceεn→ 0—to a pointwith. Moreover (v1ε(x),v2ε(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.