Quaas, Alexander

We study existence of principal eigenvalues of a fully nonlinear integro-differential elliptic equations with a drift term via the Krein–Rutman theorem and regularity estimates up to boundary of viscosity solutions. We also show simplicity of eigenfunctions in the viscosity sense by using a nonlocal version of the ABP estimate and a “sweeping lemma”.

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: (−∆)αu = λa(x)u − b(x)u p in R N , where α ∈ (0, 1), N ≥ 2, λ > 0, a and b are positive smooth function in R N satisfying a(x) → a ∞ > 0 and b(x) → b ∞ > 0 as |x| → ∞. Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.