Fast decay estimates for integrable solutions of the Lane–Emden type integral systems involving the Wolff potentials

Abstract

In this paper, we study the asymptotic estimates of the positive integrable solutions of an integral system involving the Wolff potentials in Rn{u(x)=R1(x)Wβ,γ(vq)(x),v(x)=R2(x)Wβ,γ(up)(x). Here 1<γ⩽2, β>0 and βγ<n. In addition, p,q>1 satisfy the critical condition γ−1p+γ−1+γ−1q+γ−1=n−βγn, and R1(x), R2(x) are double bounded in Rn. For the radial solutions, the decay rates were established recently when |x|→∞. When the solutions have no radial structure, the asymptotic behavior is more complicated. We use an iteration technique to estimate the decay rates of the integrable solutions u and v as |x|→∞. Furthermore, as the corollaries of this result, we also obtain the asymptotic estimates of other Lane–Emden type PDE systems and integral systems, including the γ-Laplace system, the higher-order PDE system, and the integral system involving the Riesz potentials.