The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials

Abstract

In this paper, we consider the following minimizing problem with two constraints: \[ \inf \left\{ E(u) | u=(u_1,u_2), \ \| u_1 \|_{L^2}^2 = α_1, \ \| u_2 \|_{L^2}^2 = α_2 \right\}, \] where ${\alpha }_1,{\alpha }_2>0$ and $E(u)$ is defined by \[ E(u) := \int_{\mathbf{R}^N} \left\{\frac{1}{2} \sum_{i=1}^2 \left( |\nabla u_1|^2 + V_i (x) |u_i|^2 \right) - \sum_{i=1}^2 \frac{μ_i}{2p_i+2} |u_i|^{2p_i+2} - \fracβ{p_3+1} |u_1|^{p_3+1} |u_2|^{p_3+1} \right\} \mathrm{d} x. \] Here $N\ge 1$, ${\mu }_1,{\mu }_2,\beta >0$ and $V_i(x)$ $(i=1,2)$ are given functions. For $V_i(x)$, we consider two cases: (i) both of $V_1$ and $V_2$ are bounded, (ii) one of $V_1$ and $V_2$ is bounded. Under some assumptions on $V_i$ and $p_j$, we discuss the compactness of any minimizing sequence.

31 pages

References in corpus (2)

• Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction
• Existence and stability of standing waves for coupled nonlinear Hartree type equations

Cited by in corpus (1)

• Normalized solutions for nonlinear Schrödinger systems with special mass-mixed terms: The linear couple case