We consider the existence of normalized solutions to the following coupled Schrödinger system: \(\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V_1(x)u+\lambda _1 u-\omega (x) v=\mu _{1}|u|^{p-2}u+\beta |u|^{\frac{r}{2}-2}|v|^{\frac{r}{2}}u, \quad & \text {in}\quad \mathbb {R}^{N}, \\ \displaystyle -\Delta v+V_2(x)v+\lambda _2v-\omega (x) u=\mu _{2}|v|^{q-2}v+\beta |u|^{\frac{r}{2}}|v|^{\frac{r}{2}-2}v, \quad & \text {in}\quad \mathbb {R}^{N}, \\ \displaystyle \int _{\mathbb {R}^{N}}u^2\textrm{d}x=a, \int _{\mathbb {R}^{N}}v^2\textrm{d}x=b, \end{array} \right. \end{aligned}\) where \(N\ge 1\) , \(a,b,\mu _1,\mu _2,\beta >0\) , \(\omega (x)>0\) , \(2<p,q,r<2+\frac{4}{N}\) , \(V_1(x),V_2(x)\le 0\) and \(\lambda _1,\lambda _2\) both appear as Lagrange multipliers. We study three cases: the case \(V_1(x),V_2(x)\equiv 0\) , \(\omega (x)\equiv \omega \) is a constant; the case \(V_1(x),V_2(x)\ne 0\) , \(\omega (x)\equiv \omega \) is a constant ; the case \(V_1(x),V_2(x)\ne 0\) , \(\omega (x)\) is a function. For the first case, we use Schwarz symmetric decreasing rearrangement and energy estimate method to prove the existence of positive normalized solution \((u,v,\lambda _1,\lambda _2)\) , and we also prove a strictly sub-additive inequality of the minimizing energy value. For the second case, we use Lions concentrate-compactness principle to derive a splitting Lemma, and then use a delicate energy estimate method to prove the existence of positive normalized solution \((u,v,\lambda _1,\lambda _2)\) . For the third case, we investigate two subcases: when \(\omega (x)\) is continuous and bounded, we show the existence results by also Lions concentrate-compactness principle; while when \(\omega (x)\in L^{s}(\mathbb {R}^{N})\cap L^{\infty }(\mathbb {R}^{N})\) , we prove the compactness of minimizing sequence by a Liouville type result.
