Nonlocal elliptic systems via nonlinear Rayleigh quotient with general concave and coupling nonlinearities

In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of nonlocal elliptic systems: \begin{equation*} \left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u = \lambda|u|^{p - 2}u+ \frac{\alpha}{\alpha+\beta}\theta |u|^{\alpha - 2}u|v|^{\beta}, \;\;\; \mbox{in}\;\;\; \mathbb{R}^N, (-\Delta)^sv +V_2(x)v= \lambda|v|^{q - 2}v+ \frac{\beta}{\alpha+\beta}\theta |u|^{\alpha}|v|^{\beta-2}v, \;\;\; \mbox{in}\;\;\; \mathbb{R}^N, (u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N). \end{array}\right. \end{equation*} Here we mention that $\alpha > 1, \beta > 1, 1 \leq p \leq q < 2 < \alpha + \beta < 2^*_s$, $\theta > 0, \lambda > 0, N > 2s$, and $s \in (0,1)$. Notice also that continuous potentials $V_1, V_2: \mathbb{R}^N \to \mathbb{R}$ satisfy some extra assumptions. Furthermore, we find the largest positive number $\lambda^* > 0$ such that our main problem admits at least two positive solutions for each $ \lambda \in (0, \lambda^*)$. This can be done by using the nonlinear Rayleigh quotient together with the Nehari method. The main feature here is to minimize the energy functional in Nehari manifold which allows us to prove our main results without any restriction on size of parameter $\theta > 0$.
Comment: In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of nonlocal elliptic systems