Choquard equations via nonlinear Rayleigh quotient for concave-convex nonlinearities

Abstract

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$\begin{array}{rcl}-\Delta u+V(x)u& =& (I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u+\lambda |u{|}^{q-2}u,u\in H^1({\mathbb{R}}^N),\end{array}$$ where $\lambda >0,N\ge 3,\alpha \in (0,N)$. The potential $V$ is a continuous function and $I_{\alpha }$ denotes the standard Riesz potential. Assume also that $1<q<2,2_{\alpha }<p<2_{\alpha }^{\ast }$ 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.