Critical conditions of a fully nonlinear inequality of the Hartree type

Abstract In this paper, we establish the sharp criteria for the existence and the nonexistence of negative solutions of the following ‐Hessian inequality with a nonlocal term: and an integral inequality of the Wolff type where , , , , , and is the Riesz potential of order . The nonlocal term often appears in the Hartree equations. By some priori estimates, we obtain the optimal ranges of exponents and which describe the existence/nonexistence of negative ‐admissible solutions of the ‐Hessian inequality. In addition, we also give a necessary and sufficient condition for the existence of positive ‐solutions of the integral inequality of the Wolff type.