Liouville type theorems for some fractional elliptic problems

Abstract

In this paper, we first establish Liouville type theorems for stable solutions in the whole space RN to the fractional elliptic equation (−Δ)su=f(u),where the nonlinearity is nondecreasing and convex. This extends some result in Dupaigne and Farina (2010) to the fractional setting. Our next purpose is to prove the nonexistence of stable positive solutions to the fractional Lane–Emden system (−Δ)su=vp in RN(−Δ)sv=uq in RN,in the subcritical cases. This is, in particular, the first nonexistence result of stable positive solutions for the fractional Lane–Emden system in literature which extends the result in Cowan (2013) from the local case to the nonlocal one.