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 $$ begin{array}{rcl} -Delta u +V(x) u &=& (I_alpha* |u|^p)|u|^{p-2}u+ lambda |u|^{q-2}u, , u in H^1(mathbb{R}^{N}), end{array} $$ 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 $lambda_n > 0$ such that our main problem admits at least two positive solutions for each $lambda in (0, lambda_n]$. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter $lambda_n > 0$ is optimal in some sense which allow us to apply the Nehari method.