Superlinear fractional $\Phi$-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 $\mu \to +\infty$.