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

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.