Choquard equations via nonlinear Rayleigh quotient for concave-convex nonlinearities

TL;DR

This paper proves the existence of multiple solutions for a class of Choquard equations with concave-convex nonlinearities by employing a nonlinear Rayleigh quotient and Nehari method, identifying an optimal parameter range.

Contribution

It introduces a novel approach using a nonlinear Rayleigh quotient to analyze Choquard equations, establishing existence and multiplicity of solutions under specific parameter conditions.

Findings

Existence of at least two positive solutions for certain parameter ranges.

Development of a method combining Nehari approach with nonlinear Rayleigh quotient analysis.

Identification of an optimal parameter threshold for solution existence.

Abstract

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ \begin{array}{rcl} -\Delta u +V(x) u &=& (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u, \, u \in H^1(\mathbb{R}^{N}), \end{array} $$ 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.