Multiplicity of Positive Solutions to the Boundary-Value Problems for Fractional Laplacians
For the problem (−Δ)su=uq−1 in the annulus ΩR = BR+1 \ BR ∈ 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.
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