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In this paper, we are concerned with solutions to the following nonlinearSchr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u+ \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\,\R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$where $N \geq 1$, $\mu =\pm 1$, $2<q<p<{2(N-b)}/{(N-2)^+}$, $0<b<\min\{2, N\}$and the parameter $\lambda \in \R$ appearing as Lagrange multiplier is unknown.In the mass subcritical case, we establish the compactness of any minimizingsequence to the minimization problem given by the underlying energy functionalrestricted on the constraint. As a consequence of the compactness of anyminimizing sequence, orbital stability of minimizers is derived. In the masscritical and supercritical cases, we investigate the existence, radial symmetryand orbital instability of solutions. Meanwhile, we consider the existence,radial symmetry and algebraical decay of ground states to the correspondingzero mass equation with defocusing perturbation. In addition, dynamicalbehaviors of solutions to the Cauchy problem for the associated dispersiveequation are discussed.

Standing waves with prescribed $L^2$-norm to nonlinear Schr\"odinger equations with combined inhomogeneous nonlinearities