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Zeitschrift für Analysis und ihre Anwendungen


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Volume 34, Issue 4, 2015, pp. 419–434
DOI: 10.4171/ZAA/1547

Published online: 2015-10-29

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

Bin Ge[1] and Chao Zhang[2]

(1) Harbin Engineering University, China
(2) Harbin Institute of Technology, China

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.

Keywords: Fractional-Laplacian, Variational method, Cut-off function, Pohozaev type identity

Ge Bin, Zhang Chao: Existence of a Positive Solution to Kirchhoff Problems Involving the Fractional Laplacian. Z. Anal. Anwend. 34 (2015), 419-434. doi: 10.4171/ZAA/1547