Qualitative properties of singular solutions to nonlocal problems
By using the moving plane method and under suitable assumptions on the semilinear term, the authors prove symmetry and monotonicity properties for the positive weak solutions to the Dirichlet problem for the fractional Laplacian \[ \begin{cases} (-\Delta)^su=f(x,u) & \text{ in }\Omega\setminus\Gamma,\\ u>0 & \text{ in }\Omega\setminus\Gamma,\\ u=0 & \text{ in }\mathbb R^N\setminus\Omega \end{cases} \] in the case when \(\alpha\in(0,1)\), \(N>2s\), \(\Omega\) is either a bounded domain with smooth boundary or the whole space \(\mathbb R^N\) and the compact singular set \(\Gamma\subset\Omega\) has zero \(s\)-capacity.
scientific article; zbMATH DE number 6894904
scientific article