Multiplicity of Positive Solutions to the Boundary-Value Problems for Fractional Laplacians

For the problem (−Δ) s u = u q−1 in the annulus Ω R = B R +1 \ B R ∈ ℝ n , a so-called “multiplicity effect” is established: for each N ∈ ℕ there exists R 0 such that for all R ≥ R 0 this problem has at least N different positive solutions. (−Δ) 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.