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 ( 1 + λ ∫ RN (∣∣(−∆)α2 u(x)∣∣2 + V (x)u2) dx)[(−∆)αu+ V (x)u] = f(u) in R , where N ≥ 2, λ ≥ 0 is a parameter, α ∈ (0, 1), (−∆)α stands for the fractional Laplacian, f ∈ C(R+,R+). Using a variational method combined with suitable truncation techniques, we obtain the existence of at least one positive solution without compactness conditions.