CLASSIFICATION OF NONNEGATIVE SOLUTIONS TO AN EQUATION INVOLVING THE LAPLACIAN OF ARBITRARY ORDER

We classify all nonnegative nontrivial classical solutions to the equation (-Delta)(alpha/2) u = c(1) (1/vertical bar x vertical bar(n-beta) * f(u))g(u) + c(2)h(u) in R-n, where 0 < alpha, beta < n, c(1), c(2) >= 0, c(1) + c(2) > 0 and f, g, h is an element of C([0,-infinity), [0, +infinity)) are increasing functions such that f(t)/t(n+beta/n-alpha), g(t)/(t)/t(alpha+beta/n-alpha), h(t)/t(n+alpha/n-alpha)are non-increasing in (0, +infinity). We also derive a Liouville type theorem for the equation in the case alpha >= n. The main tool we use is the method of moving spheres in integral forms.

Automated summary

This study classifies all nonnegative nontrivial classical solutions to the given equation in R^n, with additional considerations for the case when alpha >= n, resulting in a Liouville type theorem. The method of moving spheres in integral forms is the main tool utilized in deriving these results.