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We investigate the NLS equation with competing Hartree-type and power-typenonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}\psi +\Delta\psi +\gamma (I_{\alpha }\ast |\psi |^{p})|\psi |^{p-2}\psi +\mu |\psi|^{q-2}\psi =0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},%\end{array}% \end{equation*}% where $\gamma \mu <0$. We establish conditionsfor the local well-posedness in the energy space. Under the double$L^{2}$-supercritical case, we prove the existence and multiplicity of standingwaves with prescribed mass by developing a constraint method when $\gamma<0,\mu >0$ and $\gamma >0,\mu <0, $ respectively. Moreover, we prove weakorbital stablility and strong instability of standing waves by considering asuitable local minimization problem and by analyzing the fibering mapping,respectively. A new analysis of the fibering mapping is performed in this work.We believe that it is innovative as it was not discussed at all in any previousresults. The lower bound rate of blow-up solutions for the Cauchy problem isgiven as well. Due to the different \textquotedblleft strength" of the twotypes of nonlinearities, we find some essential differences in our resultsbetween these two competing cases. We will be dealing with two major scenariosthat are totally different from each other due to their diverse geometricstructure. This leads to surprising findings. Additionally, the competing purepower-type nonlinearities case can be derived from our study thanks to a goodchoice of the kernel of the Hartree term.

Standing waves for the NLS equation with competing nonlocal and local nonlinearities: the double $L^{2}$-supercritical case