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 $\begin{array}{rcl} -Δu +V(x) u &=& (I_α* |u|^p)|u|^{p-2}u+ λ|u|^{q-2}u, \, u \in H^1(\mathbb{R}^{N}), \end{array}$ where $λ> 0, N \geq 3, α\in (0, N)$. The potential $V$ is a continuous function and $I_α$ denotes the standard Riesz potential. Assume also that $1 < q < 2,~2_α < p < 2^*_α$ where $2_α=(N+α)/N$, $2_α=(N+α)/(N-2)$. Our main contribution is to consider a specific condition on the parameter $λ> 0$ taking into account the nonlinear Rayleigh quotient. More precisely, there exists $λ_n > 0$ such that our main problem admits at least two positive solutions for each $λ\in (0, λ_n]$. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter $λ_n > 0$ is optimal in some sense which allow us to apply the Nehari method.

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