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 $1 0$ 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.