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

$$
\left(1+\lambda\int_{\mathbb{R}^N} \left( \big|(-\Delta)^{\frac{\alpha}{2}}u(x) \big|^{2}+V(x)u^2 \right)dx \right)\big[(-\Delta)^\alpha u+V(x)u\big]=f(u) \quad {\rm in} \ \mathbb{R}^N,
$$

where $N\geq 2$, $\lambda\geq 0$ 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.

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

Bin Ge

Chao Zhang

Abstract

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