论文标题
具有电势的非线性Schrödinger系统的最小化序列的紧凑性
The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials
论文作者
论文摘要
在本文中，我们考虑以下最小化问题，并具有两个约束： \ [ \ inf \ left \ {e（u）| u =（u_1，u_2），\ \ | u_1 \ | _ {l^2}^2 =α_1，\ \ | u_2 \ | _ {l^2}^2 =α_2\ right \}， \] 其中$α_1，α_2> 0 $和$ e（u）$由 \ [ 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} {μ_i} {2p_i+2} - \fracβ{p_3+1} | u_1 |^{p_3+1} | u_2 |^{p_3+1} \ right \} \ right \} \ mathrm {d} x。 \] 这里$ n \ geq 1 $，$μ_1，μ_2，β> 0 $和$ v_i（x）$ $（i = 1,2）$具有功能。 对于$ v_i（x）$，我们考虑两种情况：（i）$ v_1 $和$ v_2 $都有限制，（ii）$ v_1 $和$ v_2 $之一。根据$ v_i $和$ p_j $的某些假设，我们讨论了任何最小化序列的紧凑性。
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 $α_1,α_2 > 0$ 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 $N \geq 1$, $ μ_1,μ_2,β> 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.
