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{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*} $

where $ \lambda > 0, N \geq 3, \alpha \in (0, N) $. The potential $ V $ is a continuous function and $ I_\alpha $ denotes the standard Riesz potential. Assume also that $ 1 < q < 2 $, $ 2_\alpha < p < 2^*_\alpha $ where $ 2_\alpha = (N+\alpha)/N $, $ 2_\alpha = (N+\alpha)/(N-2) $. Our main contribution is to consider a specific condition on the parameter $ \lambda > 0 $ taking into account the nonlinear Rayleigh quotient. More precisely, there exists $ \lambda^* > 0 $ such that our main problem admits at least two positive solutions for each $ \lambda \in (0, \lambda^*] $. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter $ \lambda^*> 0 $ is optimal in some sense which allow us to apply the Nehari method.