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

Abstract

We study the following coupled Schrödinger 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 λ ₁, λ ₂, μ ₁, μ ₂ 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 β ₀ > 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 λ ₁ = λ ₂ and μ ₁ ≠ μ ₂, we prove the uniqueness of the positive solutions for min{μ ₁, μ ₂} − β > 0 sufficiently small. In case λ ₁ ≠ λ ₂ and (λ ₂ − λ ₁)(μ ₂ − μ ₁) ≤ 0, we prove that ${\beta_0 < \sqrt{\mu_1\mu_2}}$ and β ₀ is optimal, in the sense that this system has no nontrivial least energy solutions for ${\beta \in (\beta_0, \sqrt{\mu_1\mu_2})}$ . Moreover, there exists δ > 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 ${\beta \nearrow \beta_0}$ .