Positive supersolutions to (-Delta)^s u >= f(u,x) do not exist in exterior domains when f grows like the critical power near 0 for n>2s, or fast enough at infinity for n<=2s.
Bidaut-Veron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch
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Nonexistence of positive supersolutions for semilinear fractional elliptic equations in exterior domains
Positive supersolutions to (-Delta)^s u >= f(u,x) do not exist in exterior domains when f grows like the critical power near 0 for n>2s, or fast enough at infinity for n<=2s.