Multiplicity of positive solutions to the boundary value problems for fractional Laplacians

We establish the so-called “multiplicity effect” for the problem $(-\Delta)^su=u^{q-1}$ in the annulus $\Omega_R=B_{R+1}\setminus B_R\in\mathbb R^n$: for each $N\in\mathbb N$ there exists $R_0$ such that for all $R \geq R_0$ this problem has at least $N$ different positive solutions. $(-\Delta)^s$ in this problem stands either for Navier-type or for Dirichlet-type fractional Laplacian. Similar results were proved earlier for the equations with the usual Laplace operator and with the $p$-Laplacian operator.