A uniqueness principle for up≤(−Δ)α2uup≤(−Δ)α2u in the Euclidean space

This paper establishes such a uniqueness principle that under (p,α,N)∈(1,∞)×(0,2)×Z+(p,α,N)∈(1,∞)×(0,2)×ℤ+ the fractional order differential inequality (†)up≤(−Δ)α2uin RN (†)up≤(−Δ)α2uin ℝN has the property that if N≤αN≤α then a non-negative weak solution to (†)(†) is unique, and if N>αN>α then the uniqueness of a non-negative weak solution to (†)(†) occurs when and only when p≤N/(N−α)p≤N/(N−α), thereby innovatively generalizing Gidas–Spruck’s result for up+Δu≤0up+Δu≤0 in RNℝN discovered in [B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations.