Stein-Weiss problems via nonlinear Rayleigh quotient for concave-convex nonlinearities

In the present work, we consider existence and multiplicity of positive solutions for nonlocal elliptic problems driven by the Stein-Weiss problem with concave-convex nonlinearities defined in the whole space ${\mathbb{R}}^N$. More precisely, we consider the following nonlocal elliptic problem: \begin{equation*} - Δu + V(x)u = λa(x) |u|^{q-2} u + \displaystyle \int \limits_{$\mathbb{R}$^N}\frac{b(y)\vert u(y) \vert^p dy}{\vert x\vert^α\vert x-y\vert^μ\vert y\vert^α} b(x)\vert u\vert^{p-2}u, ,, \hbox{in}\ $\mathbb{R}$^N, ,, u\in H^1($\mathbb{R}$^N), \end{equation*} where $\lambda >0,\alpha \in (0,N),N\ge 3,0<\mu <N,0<\mu +2\alpha <N$. Furthermore, we assume also that $V:{\mathbb{R}}^N\to \mathbb{R}$ is a bounded potential, $a\in L^r({\mathbb{R}}^N),a>0$ in ${\mathbb{R}}^N$ and $b\in L^t({\mathbb{R}}^N),b>0$ in ${\mathbb{R}}^N$ for some specific $r,t>1$. We assume also that $1\le q<2$ and $2_{\alpha ,\mu }<p<2_{\alpha ,\mu }^{\ast }$ where $2_{\alpha ,\mu }=(2N-2\alpha -\mu )\mathrm{/}N$ and $2_{\alpha ,\mu }^{\ast }=(2N-2\alpha -\mu )\mathrm{/}(N-2)$. Our main contribution is to find the largest ${\lambda }^{\ast }>0$ in such way that our main problem admits at least two positive solutions for each $\lambda \in (0,{\lambda }^{\ast })$. In order to do that we apply the nonlinear Rayleigh quotient together with the Nehari method. Moreover, we prove a Brezis-Lieb type Lemma and a regularity result taking into account our setting due to the potentials $a,b:{\mathbb{R}}^N\to \mathbb{R}$.