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

TL;DR

This paper proves the existence and multiplicity of solutions for a class of nonlocal fractional elliptic problems involving the $rac{ ext{fractional } ext{ extPhi}- extLaplacian$ operator, using the nonlinear Rayleigh quotient and fibering maps.

Contribution

It introduces a novel approach combining the nonlinear Rayleigh quotient with fibering maps to analyze parameter ranges for solution existence in fractional $ extPhi$-Laplacian problems.

Findings

Established parameter ranges for solution existence.

Analyzed asymptotic behavior of solutions as parameters vary.

Applied Nehari method effectively to nonlocal elliptic problems.

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 $\mu \to +\infty$.