Superlinear fractional $Φ$-Laplacian type problems via the nonlinear Rayleigh quotient with two parameters

Abstract: In this work, we establish the existence and multiplicity of weak solutions for nonlocal elliptic problems driven by the fractional $\Phi$-Laplacian operator, in the presence of a sign-indefinite nonlinearity. More specifically, we investigate the following nonlocal elliptic problem: \begin{equation*} \left{\begin{array}{rcl} (-\Delta_\Phi)^s u +V(x)u & = & \mu a(x)|u|^{q-2}u-\lambda |u|^{p-2}u \mbox{ in }\, \mathbb{R}^N, \ u\in W^{s,\Phi}(\mathbb{R}^N),&& \end{array} \right. \end{equation*} where $s \in (0,1), N \geq 2$ and $\mu, \lambda >0$. Here, the potentials $V, a : \mathbb{R}^N \to \mathbb{R}$ satisfy some suitable hypotheses. Our main objective is to determine sharp values for the parameters $\lambda > 0$ and $\mu > 0$ where the Nehari method can be effectively applied. To achieve this, we utilize the nonlinear Rayleigh quotient along with a detailed analysis of the fibering maps associated with the energy functional. Additionally, we study the asymptotic behavior of the weak solutions to the main problem as $\lambda \to 0$ or $\mu \to +\infty$.