Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order

Abstract

We classify all nonnegative nontrivial classical solutions to the equation (−∆) α 2 u = c 1 1 |x| n−β * f (u) g(u) + c 2 h(u) in R n , where 0 < α, β < n, c 1 , c 2 ≥ 0, c 1 + c 2 > 0 and f, g, h ∈ C([0, +∞), [0, +∞)) are increasing functions such that f (t)/t n+β n−α , g(t)/t α+β n−α , h(t)/t n+α n−α are nonincreasing in (0, +∞). We also derive a Liouville type theorem for the equation in the case α ≥ n. The main tool we use is the method of moving spheres in integral forms.