The properties of positive solutions to semilinear equations involving the fractional Laplacian
Rongrong Yang, Zhongxue Lü · Communications on Pure & Applied Analysis · 2018
Let $Ω$ be either a unit ball or a half space. Consider the following Dirichlet problem involving the fractional Laplacian \begin{document}$\left\{ \begin{array}{*{35}{l}} \begin{align} & {{(-\Delta )}^{\frac{\alpha }{2}}}u=f(u),\ \ \text{in}\ \ \Omega , \\ & u=0, ~~~~~~~~~~~~~~~~~~~~ \text{in}\ \ {{\Omega }^{c}},\ \\ \end{align} & \ & {} \\\end{array} \right.~~~~(1)$ \end{document} where $α$ is any real number between $zhongwenzy$ and $$. Under some conditions on $f$, we study the equivalent integral equation \begin{document} $ \begin{align}u(x) \ = \ \int{{}}_{ Ω}G(x, y)f(u(y))dy, \end{align}~~~~(2) $ \end{document} here $G(x, y)$ is the Green's function associated with the fractional Laplacian in the domain $Ω$. We apply the method of moving planes in integral forms to investigate the radial symmetry, monotonicity and regularity for positive solutions in the unit ball. Liouville type theorems-non-existence of positive solutions in the half space are also deduced.