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*} - \Delta u + V(x)u = \lambda a(x) |u|^{q-2} u + \displaystyle \int \limits_{\mathbb{R}^N}\frac{b(y)\vert u(y) \vert^p dy}{\vert x\vert^\alpha\vert x-y\vert^\mu \vert y\vert^\alpha} b(x)\vert u\vert^{p-2}u, \,\, \hbox{in}\ \mathbb{R}^N, \,\, u\in H^1(\mathbb{R}^N), \end{equation*} where $\lambda >0, \alpha \in (0,N), N\geq3, 0<\mu<N, 0 < \mu + 2 \alpha < N$. 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 $1\leq q<2$ and $2_{\alpha,\mu} < p<2_{\alpha,\mu}^*$ where $2_{\alpha ,\mu}=(2N-2\alpha-\mu)/N$ and $2_{\alpha,\mu}^*= (2N-2\alpha-\mu)/(N-2)$. Our main contribution is to find the largest $\lambda^* > 0$ in such way that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda^*)$. 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 $a, b : \mathbb{R}^N \to \mathbb{R}$.