Ground States for Mass Critical Two Coupled Semi-Relativistic Hartree Equations with Attractive Interactions

We prove the existence and nonexistence of $L^{2}(\mathbb R^3)$-normalized solutions of two coupled semi-relativistic Hartree equations, which arisen from the studies of boson stars and multi-component Bose–Einstein condensates. Under certain condition on the strength of intra-specie and inter-specie interactions, by proving some delicate energy estimates, we give a precise description on the concentration behavior of ground state solutions of the system. Furthermore, an optimal blowing up rate for the ground state solutions of the system is also proved.

Normalized solutions to a Schrödinger–Bopp–Podolsky system under Neumann boundary conditions

In this paper, we study a Schrödinger–Bopp–Podolsky (SBP) system of partial differential equations in a bounded and smooth domain of [Formula: see text] with a nonconstant coupling factor. Under a compatibility condition on the boundary data we deduce existence of solutions by means of the Ljusternik–Schnirelmann theory.

On a system of nonlinear Schrödinger equations with quadratic interaction and L2-critical growth

In this paper, we study the dynamical behavior of solutions of nonlinear Schrödinger equations with quadratic interaction and [Formula: see text]-critical growth. We give sharp conditions under which the existence of global and blow-up solutions are deduced. We also show the existence, stability, and blow-up behavior of normalized solutions of this system.