Liouville theorems involving the fractional Laplacian

The fractional Laplacian in Euclidean space is a nonlocal operator. We established Liouville type theorems, nonexistence of positive solutions of Dirichlet problem involving fractional Laplacian. We obtained the equivalence between partial differential equations and integral equations from the uniqueness harmonic functions on half space. Applying the method of moving planes in integral forms, we proved the nonexistence of positive solutions of integral equations.

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