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

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

In this paper, we consider the following fractional Schrodinger–Poisson system with singularity \begin{equation*}
 \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3,
 \end{array}\right.
 \end{equation*} where 0 0 and 0 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0017089520000099

Jianqing Chen

Papers

Fractional schrödinger–poisson system with singularity: existence, uniqueness, and asymptotic behavior