Bound states for a coupled Schrödinger system

Abstract.

We consider the existence of bound states for the coupled elliptic system

$$ \begin{array}{*{20}c} {\Delta u_1 - \lambda _1 u_1 + \mu _1 u_1^3 + \beta u_2^2 u_1 = 0\quad \text{in}\, {\mathbb{R}}^n,} \\ {\Delta u_2 - \lambda _2 u_2 + \mu _2 u_2^3 + \beta u_1^2 u_2 = 0\quad \text{in}\, {\mathbb{R}}^n,} \\ {u_1 > 0,\quad u_2 > 0,\quad u_1 ,u_2 \in {\mathbb{H}}^1({\mathbb{R}}^n),} \\ \end{array} $$

where n ≤ 3. Using the fixed point index in cones we prove the existence of a five-dimensional continuum ${\mathcal{C}}\subset {\mathbb{R}}_+^5 \times {\mathbb{H}}^1 ({\mathbb{R}}^n)\times {\mathbb{H}}^1 ({\mathbb{R}}^n)$ of solutions (λ₁, λ₂, μ ₁, μ ₂, β, u ₁, u ₂) bifurcating from the set of semipositive solutions (where u ₁ = 0 or u ₂ = 0) and investigate the parameter range covered by ${\mathcal{C}}$.