Regularity and symmetry for solutions to a system of weighted integral equations

We study positive solutions of the following system of integral equations related to the weighted Hardy–Littlewood–Sobolev inequality: { u ( x ) = 1 | x | α ∫ R n v q ( y ) | y | β | x − y | λ d y , v ( x ) = 1 | x | β ∫ R n u p ( y ) | y | α | x − y | λ d y , where u , v ⩾ 0 , 0 < p , q < ∞ , α , β ⩾ 0 , 0 < λ < n , α n < 1 p + 1 < λ + α n and 1 p + 1 + 1 q + 1 = λ + α + β n . We obtain regularity of the solutions by regularity-lifting-method, which has been extensively used by many authors. Moreover, using the method of moving planes in integral forms, we also establish symmetry of the solutions under the weaker condition that u and v are only locally integrable.