Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity

• Main Authors: Shengbin Yu, Jianqing Chen
• Format: Article
• Language: English
• Published: SpringerOpen 2021-03-01
• Series: Boundary Value Problems
• Subjects: Fractional Kirchhoff problem; Strong singularity; Uniqueness; Variational method; Concentration
• Online Access: https://doi.org/10.1186/s13661-021-01507-8

Abstract In this paper, we consider the following fractional Kirchhoff problem with strong singularity: { ( 1 + b ∫ R 3 ∫ R 3 | u ( x ) − u ( y ) | 2 | x − y | 3 + 2 s d x d y ) ( − Δ ) s u + V ( x ) u = f ( x ) u − γ , x ∈ R 3 , u > 0 , x ∈ R 3 , $$ \textstyle\begin{cases} (1+ b\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}} \frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3+2s}}\,\mathrm{d}x \,\mathrm{d}y )(-\Delta )^{s} u+V(x)u = f(x)u^{-\gamma }, & x \in \mathbb{R}^{3}, \\ u>0,& x\in \mathbb{R}^{3}, \end{cases} $$ where ( − Δ ) s $(-\Delta )^{s}$ is the fractional Laplacian with 0 < s < 1 $0< s<1$ , b > 0 $b>0$ is a constant, and γ > 1 $\gamma >1$ . Since γ > 1 $\gamma >1$ , the energy functional is not well defined on the work space, which is quite different with the situation of 0 < γ < 1 $0<\gamma <1$ and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution u b $u_{b}$ by using variational methods and the Nehari manifold method. We also give a convergence property of u b $u_{b}$ as b → 0 $b\rightarrow 0$ , where b is regarded as a positive parameter.