Existence of a Positive Solution to Kirchhoff Problems Involving the Fractional Laplacian

Abstract

The goal of this paper is to establish the existence of a positive solution to the following fractional Kirchhoff-type problem

where $N\ge 2$, $\lambda \ge 0$ is a parameter, $\alpha \in (0,1)$, $(-\Delta {)}^{\alpha }$ stands for the fractional Laplacian, $f\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$. Using a variational method combined with suitable truncation techniques, we obtain the existence of at least one positive solution without compactness conditions.