Classification of solutions to equations involving higher-order fractional Laplacian

We study the following equation involving higher-order fractional Laplacian: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Under either one of the two integral constraints [Formula: see text] and [Formula: see text] we establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. From this, we prove that these solutions are radially symmetric about some point in [Formula: see text] and monotone decreasing in the radial direction via method of moving planes in integral forms. We also establish a Moser–Trudinger-type inequality associated to the equation [Formula: see text] in bounded domain for [Formula: see text].