Singular Choquard elliptic problems involving two nonlocal nonlinearities via the nonlinear Rayleigh quotient

TL;DR

This paper investigates the existence and multiplicity of solutions for a class of nonlocal elliptic singular problems involving two convolution-driven nonlinearities, using variational methods and decay estimates.

Contribution

It introduces a novel approach employing the nonlinear Rayleigh quotient and Nehari method to establish multiple solutions for Choquard problems with two nonlocal nonlinearities.

Findings

Existence of at least two solutions for certain parameter ranges.

Identification of a sharp threshold parameter for solution existence.

Non-existence results based on decay estimates.

Abstract

In the present work we shall consider the existence and multiplicity of solutions for nonlocal elliptic singular problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: \begin{equation*} \left\{\begin{array}{lll} -\Delta u+V(x)u=\lambda(I_{\alpha_1}*a|u|^q)a(x)|u|^{q-2}u+\mu(I_{\alpha_2}*|u|^p)|u|^{p-2}u u\in H^1(\mathbb{R}^N) \end{array}\right. \end{equation*} where $\alpha_2<\alpha_1$; $\alpha_1,\alpha_2\in(0,N)$ and $0<q<1$; $p\in\left(2_{\alpha_2},2^*_{\alpha_2} \right)$. Recall also that $2_{\alpha_j}=(N+\alpha_j)/N$ and $2^*_{\alpha_j}=(N+\alpha_j)/(N-2), j=1,2$. Furthermore, for each $q\in(0,1)$, by using the Hardy-Littlewood-Sobolev inequality we can find a sharp parameter $\lambda^*> 0$ such that our main problem has at least two solutions using the Nehari method. Here we also use the Rayleigh quotient for the following scenarios $\lambda \in (0, \lambda^*)$ and $\lambda = \lambda^*$. Moreover, we consider some decay estimates ensuring a non-existence result for the Choquard type problems in the whole space.