# A local mountain pass type result for a system of nonlinear Schrödinger equations

• Norihisa Ikoma
• Kazunaga Tanaka
Article

## Abstract

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.

## Mathematics Subject Classification (2000)

35B25 35J65 58E05

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