Asymptotic behavior of the unique solution for a fractional Kirchhoff problem with singularity

Abstract

In this paper, we consider the following fractional Kirchhoff problem with singularity $ \left \{\begin{array}{lcl} \Big(1+ b\int_{\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|u(x)-u(y)|^2}{|x-y|^{3+2s}}\mathrm{d}x \mathrm{d}y \Big)(-\Delta)^s u+V(x)u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. $ where $ (-\Delta)^s $ is the fractional Laplacian with $ 0 < s < 1 $, $ b\ge0 $ is a constant and $ 0 < \gamma < 1 $. Under certain assumptions on $ V $ and $ f $, we show the existence and uniqueness of positive solution $ u_b $ by using variational method. We also give a convergence property of $ u_b $ as $ b\rightarrow0 $, where $ b $ is regarded as a positive parameter.

Keywords

• fractional kirchhoff problem
• singularity
• uniqueness
• variational method
• asymptotic behavior