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In present paper, we study the following nonlinear Schr\"{o}dinger equationwith combined power nonlinearities  \begin{align*}  - \Delta u+V(x)u+\lambda u=|u|^{2^*-2}u+\mu |u|^{q-2}u \quad \quad \text{in}\ \mathbb{ R}^N, \ N\geq 3  \end{align*}  having prescribed mass  \begin{align*}  \int_{ \mathbb{ R}^N}u^2dx=a^2,  \end{align*} where $\mu, a>0$, $q\in(2, 2^*)$, $2^*=\frac{2N}{N-2}$ is thecritical Sobolev exponent, $V$ is an external potential vanishing at infinity,and the parameter $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier.Under some mild assumptions on $V$, for the $L^2$-subcritical perturbation$q\in(2, 2+\frac{4}{N})$, we prove that there exists $a_0>0$ such that thenormalized solution with negative energy to the above problem with $\mu>0$ canbe obtained for $a\in (0, a_0)$; for the $L^2$-critical perturbation$q=2+\frac{4}{N}$, by limiting the range of $\mu$, the positive ground statenormalized solution to the above problem for any $a>0$ is also found with theaid of the Poho\v{z}aev constraint; moreover, for the $L^2$-supercriticalperturbation $q\in( 2+\frac{4}{N}, 2^*)$, we get a positive ground statenormalized solution for the above problem with $a>0$ and $\mu>0$ by using thePoho\v{z}aev constraint.  At the same time, the exponential decay property of the positive normalizedsolution is established, which is important for the instability analysis of thestanding waves. Furthermore, we give a description of the ground state set andobtain the strong instability of the standing waves for $q\in[2+\frac{4}{N},2^*)$. This paper can be regarded as a generalization of Soave [J. Funct. Anal.(2020)] in a sense.

Normalized solutions for the nonlinear Schrodinger equation with potential and combined nonlinearities