  We investigate the NLS equation with competing Hartree-type and power-type
nonlinearities \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 conditions
for the local well-posedness in the energy space. Under the double
$L^{2}$-supercritical case, we prove the existence and multiplicity of standing
waves with prescribed mass by developing a constraint method when $\gamma
<0,\mu >0$ and $\gamma >0,\mu <0, $ respectively. Moreover, we prove weak
orbital stablility and strong instability of standing waves by considering a
suitable 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 previous
results. The lower bound rate of blow-up solutions for the Cauchy problem is
given as well. Due to the different \textquotedblleft strength" of the two
types of nonlinearities, we find some essential differences in our results
between these two competing cases. We will be dealing with two major scenarios
that are totally different from each other due to their diverse geometric
structure. This leads to surprising findings. Additionally, the competing pure
power-type nonlinearities case can be derived from our study thanks to a good
choice of the kernel of the Hartree term.
