Radial symmetry and monotonicity for an integral equation

In this paper we study radial symmetry and monotonicity of positive solutions of an integral equation arising from some higher-order semilinear elliptic equations in the whole space Rn. Instead of the usual method of moving planes, we use a new Hardy–Littlewood–Sobolev (HLS) type inequality for the Bessel potentials to establish the radial symmetry and monotonicity results.