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

Abstract

We are concerned with the following constrained minimization problem: $$e(a_1,a_2,\beta ):=\inf \left\{E_{a_1,a_2,\beta }(u_1,u_2):\Vert u_1{\Vert }_{L^2({\mathbb{R}}^3)}=\Vert u_2{\Vert }_{L^2({\mathbb{R}}^3)}=1\right\},$$ where $E_{a_1,a_2,\beta }$ is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters $a_1,a_2,\beta$ and two trapping potentials $V_1(x)$ and $V_2(x)$. In this paper, we obtain the existence of minimizers of $e(a_1,a_2,\beta )$ for possible $a_1,a_2$ and $\beta$ under suitable conditions on the potentials, which generalizes the results of the papers [16,17,18] in different senses.

32 pages

References in corpus (1)

• Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials