Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

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The purpose of this paper is to study the weak solutions of the fractional elliptic problem equation000 arraylll (-Δ)αu+εg(u)=k∂αν∂ nα & inΩ,3mm] (-Δ)α+εg(u) u=0 & inΩᶜ, array equation where k>0, ε=1 or -1, (-Δ)α with α∈(0,1) is the fractional Laplacian defined in the principle value sense, Ω is a bounded C² open set in Rⁿ with N≥ 2, ν is a bounded Radon measure supported in ∂Ω and ∂αν∂ nα is defined in the distribution sense, i.e. ∂αν∂ nα,ζ=∫∂Ω∂αζ(x)∂ nₓαdν(x), ∀ζ∈ Cα(Rⁿ), here nₓ denotes the unit inward normal vector at x∈∂Ω.