Critical conditions of a fully nonlinear inequality of the Hartree type

In this paper, we establish the sharp criteria for the existence and the nonexistence of negative solutions of the following k$k$‐Hessian inequality with a nonlocal term: Fk(D2V)≥[Iα*(−V)p](−V)qinRn,$$\begin{equation*} F_k(D^2V)\ge [I_\alpha \ast (-V)^p](-V)^q \quad \mathrm{in} \; \mathbb {R}^n, \end{equation*}$$and an integral inequality of the Wolff type u≥Wβ,γ[(Iα*up)uq]inRn,$$\begin{equation*} u \ge W_{\beta,\gamma }[(I_\alpha \ast u^p)u^q] \quad \mathrm{in} \; \mathbb {R}^n, \end{equation*}$$where 1≤k 1$\gamma >1$, 0 0$p>0$, q∈R$q\in \mathbb {R}$, and Iα$I_\alpha$ is the Riesz potential of order α∈(0,n)$\alpha \in (0,n)$. The nonlocal term often appears in the Hartree equations. By some priori estimates, we obtain the optimal ranges of exponents p$p$ and q$q$ which describe the existence/nonexistence of negative k$k$‐admissible solutions of the k$k$‐Hessian inequality. In addition, we also give a necessary and sufficient condition for the existence of positive Lloc∞(Rn)$L_{loc}^\infty (\mathbb {R}^n)$‐solutions of the integral inequality of the Wolff type.