Decay Rates for Solutions of an Integral System of Wolff Type

Abstract

In this paper, we study the asymptotic behavior of the positive solutions of the following system of involving Wolff potentials in Rn $$ \begin{array}{rll} u(x) &=& W_{\beta,\gamma}\left(v^q\right)(x),\\ v(x) &=& W_{\beta,\gamma}\left(u^p\right)(x). \end{array} $$ Applying the integrability intervals of u and v which were established recently by Chen et al., we obtain the decay rates of the solutions near infinity. In the special case of γ = 2 and β = α/2, the Wolff type system becomes an integral system of Hardy-Littlewood-Sobolev type. Thus, we also establish the decay rates of the positive solutions of the Hardy-Littlewood-Sobolev type integral system.