Liouville theorem for the fractional Lane–Emden equation in an unbounded domain

Abstract

Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of(1)(−Δ)αu=up+νinΩ,u=ginRN∖Ω, where α∈(0,1), p>0, Ω is a unbounded C2 domain in RN with N>2α, g∈L1(RN∖Ω,dx1+|x|N+2α) nonnegative and ν is a nonnegative Radon measure. We obtain that(i) if Ω⊇(RN∖Br0(0)‾) for some r0>0 and p<NN−2α, then problem (1) has no weak solutions;(ii) if Ω⊇{x∈RN:x⋅a>r0} for some r0≥0, a∈RN and p<N+αN−α, then problem (1) has no weak solutions. Here N+αN−α is sharp for the nonexistence in the half space.The above Liouville theorem could be applied to obtain nonexistence of classical solution of the fractional Lane–Emden equations(−Δ)αu=upinΩ,u≥0inRN∖Ω, where Ω=RN∖Br0(0) with r0>0 or Ω=RN−1×(0,+∞).