Qualitative Analysis for the Static Hartree-Type Equations

This paper is concerned with the positive solutions of a class of static Hartree-type equations. We translate these equations to the equivalent integral systems involving the Riesz potentials. Using the regularity lifting lemma by contracting operators, we obtain the integrability result for the integrable solution u of integral systems. Based on this result, we prove that the integrable solution decays quickly and the nonintegrable bounded solution decays almost slowly. Applying the equivalence, we finally give the corresponding results of the positive solutions of the Hartree-type equations.