A constrained minimization problem related to two coupled pseudo-relativistic Hartree equations

We are concerned with the following constrained minimization problem: $e(a_{1},a_{2},β) := \inf\left\{E_{a_{1},a_{2},β}(u_{1},u_{2}): \|u_{1}\|_{L^{2}(\mathbb{R}^{3})} = \|u_{2}\|_{L^{2}(\mathbb{R}^{3})} = 1\right\},$ where $E_{a_{1},a_{2},β}$ is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters $a_{1}, a_{2}, β$ and two trapping potentials $V_1(x)$ and $V_2(x)$. In this paper, we obtain the existence of minimizers of $e(a_{1},a_{2},β)$ for possible $a_{1}, a_{2}$ and $β$ under suitable conditions on the potentials, which generalizes the results of the papers [16,17,18] in different senses.

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