Partially concentrating standing waves for weakly coupled Schrödinger systems

We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in ${\mathbb{R}}^N$, $N=2,3$, $$\{\begin{array}{ll}i\mathrm{\hslash }{\mathrm{\partial }}_t{\psi }_1=-\frac{{\mathrm{\hslash }}^2}{2m_1}\mathrm{\Delta }{\psi }_1+V_1(x){\psi }_1-{\mu }_1\mathrm{\mid }{\psi }_1{\mathrm{\mid }}^2{\psi }_1-\beta \mathrm{\mid }{\psi }_2{\mathrm{\mid }}^2{\psi }_1& \\ i\mathrm{\hslash }{\mathrm{\partial }}_t{\psi }_2=-\frac{{\mathrm{\hslash }}^2}{2m_2}\mathrm{\Delta }{\psi }_2+V_2(x){\psi }_2-{\mu }_2\mathrm{\mid }{\psi }_2{\mathrm{\mid }}^2{\psi }_2-\beta \mathrm{\mid }{\psi }_1{\mathrm{\mid }}^2{\psi }_2,& \end{array}$$ where $V_1$ and $V_2$ are radial potentials bounded from below. We address the case $m_1\sim {\mathrm{\hslash }}^2\to 0$, $m_2$ constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.