Positive solutions to integral systems with weight and Bessel potentials

In this paper, we consider the integral system with weight and the Bessel potentials: { u ( x ) = ∫ R n g α ( x − y ) u ( y ) p v ( y ) q | y | σ d y , v ( x ) = ∫ R n g α ( x − y ) v ( y ) p u ( y ) q | y | σ d y , where u , v > 0 , σ ⩾ 0 , 0 < α < n , p + q = γ ⩾ 2 and g α ( x ) is the Bessel potential of order α. First, we get the integrability by regularity lifting lemma. Then we also establish the regularity of the positive solutions. Afterwards, by the method of moving planes in integral forms, we show that the positive solutions are radially symmetric and monotone decreasing about the origin. Finally, by an extension of the idea of Lei [14] and analytical techniques, we get the decay rates of solutions when | x | → ∞ .