Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form\begin{document}$ \begin{equation*} \begin{cases} -\Delta u +V(x) u = (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u \, {\rm{\;in\;}}\, \mathbb{R}^N, \\ \ u\in H^1( \mathbb{R}^N) \end{cases} \end{equation*} $\end{document}where \begin{document}$ \lambda > 0, N \geq 3, \alpha \in (0, N) $\end{document} . The potential \begin{document}$ V $\end{document} is a continuous function and \begin{document}$ I_\alpha $\end{document} denotes the standard Riesz potential. Assume also that \begin{document}$ 1 < q < 2 $\end{document} , \begin{document}$ 2_\alpha < p < 2^*_\alpha $\end{document} where \begin{document}$ 2_\alpha = (N+\alpha)/N $\end{document} , \begin{document}$ 2_\alpha = (N+\alpha)/(N-2) $\end{document} . Our main contribution is to consider a specific condition on the parameter \begin{document}$ \lambda > 0 $\end{document} taking into account the nonlinear Rayleigh quotient. More precisely, there exists \begin{document}$ \lambda^* > 0 $\end{document} such that our main problem admits at least two positive solutions for each \begin{document}$ \lambda \in (0, \lambda^*] $\end{document} . In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter \begin{document}$ \lambda^*> 0 $\end{document} is optimal in some sense which allow us to apply the Nehari method.