The effect of the domain topology on the number of solutions of fractional Laplace problems
Boundary value problems for elliptic equations involving fractional operators represent a very active field of research, see the survey of \textit{G. Grubb} [in: Mathematical analysis and applications -- plenary lectures. Papers based on the presentations at the 11th international ISAAC congress, Växjö, Sweden, August 14--18, 2017. Cham: Springer. 51--81 (2018; Zbl 1414.35099)] for a precise setting of the boundary conditions. Here the authors consider the nonlinear equation \[(-\Delta)^s u= (\lambda|u|^p+|u|^q)\,u,\] with $0<s<1$, $\lambda>0$, in a bounded domain $\Omega\subset\mathbb{R}^n$ and discuss in detail multiplicity of solutions under differential assumptions on $p$ an $q$. In particular, the number of the nontrivial solutions is related to the geometric properties of the domain $\Omega$.
scientific article; zbMATH DE number 6916248
scientific article