An optimal constant for the existence of least energy solutions of a coupled Schrödinger system

Abstract

We study the following coupled Schrodinger system which has appeared as several models from mathematical physics: $$\left\{\begin{array}{ll}-\Delta u + \lambda_1 u = \mu_1 u^3 + \beta uv^2, \quad x \in \mathbb{R}^N,\\-\Delta v + \lambda_2 v = \mu_2 v^3 + \beta vu^2, \quad x \in \mathbb{R}^N,\\u \geq 0, v \geq 0 \,\,{\rm in}\mathbb{R}^N, \quad u, v \in H^1(\mathbb{R}^N).\end{array}\right.$$ Here, N = 2, 3, and λ1, λ2, μ1, μ2 are all positive constants. In [Ambrosetti and Colorado in C R Acad Sci Paris Ser I 342:453–438, 2006], Ambrosetti and Colorado showed that, there exists β0 > 0 such that this system has a nontrivial positive radially symmetric solution for any \({\beta \in (0, \beta_0)}\). Later in [Ikoma and Tanaka in Calc Var 40:449–480, 2011], Ikoma and Tanaka showed that solutions obtained by Ambrosetti and Colorado are indeed least energy solutions for any \({\beta \in (0, {\rm min}\{\beta_0, \sqrt{\mu_1\mu_2}\})}\) . Here, in case λ1 = λ2 and μ1 ≠ μ2, we prove the uniqueness of the positive solutions for min{μ1, μ2} − β > 0 sufficiently small. In case λ1 ≠ λ2 and (λ2 − λ1)(μ2 − μ1) ≤ 0, we prove that \({\beta_0 0 such that this system has no nontrivial nonnegative solutions for any \({\beta \in ({\rm min}\{\mu_1, \mu_2\} - \delta,\, \max\{\mu_1, \mu_2\} + \delta)}\) . This answers an open question of [Sirakov in Commun Math Phys 271:199–221, 2007] partially, and improves a result of [Sirakov in Commun Math Phys 271:199–221, 2007]. The asymptotic behavior of the least energy solutions is also studied as \({\beta \nearrow \beta_0}\) .