Symmetry and monotonicity of solutions for equations involving the fractional Laplacian of higher order
The paper under review deals with symmetry and monotonicity of solutions to the problem \[ \begin{cases} (-\Delta)^{\alpha+1}u(x)=f(u(x))+g(x) & x\in B_1,\\ (-\Delta)u(x)>0 & x\in B_1,\\ u(x)=0 & x\in \mathbb{R}^n\setminus B_1, \end{cases} \] where \(B_1\) is the open unit ball in \(\mathbb{R}^n,\) \((-\Delta)^{\alpha+1},\) \(\alpha\in(0,1),\) is the higher order fractional Laplacian \[ (-\Delta)^{\alpha+1}u(x):=C(n,\alpha)\text{P.V.} \int_{\mathbb{R}^n} \dfrac{u(x)+(1/2n)\Delta u(x)|x-y|^2-u(y)}{|x-y|^{n+2\alpha+2}}\;dy, \] \(f\) is increasing and locally Lipschitz continuous, while \(g\) is radially symmetric and decreasing in \(|x|.\) The approach is based on reduction of the equation into a system via composition of lower fractional Laplacians, and then getting symmetry and monotonicity of solutions to the system by means of the moving planes method.
scientific article; zbMATH DE number 6883477
scientific article