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

Abstract

For the problem (−Δ)^su=u^q−1 in the annulus Ω_R = B_R+1 \ B_R ∈ ℝⁿ, a so-called “multiplicity effect” is established: for each N ∈ ℕ there exists R₀ such that for all R ≥ R₀ 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.