  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.
