Pointwise Bounds and Blow-up for Choquard-Pekar Inequalities at an Isolated Singularity

We study the behavior near the origin in $\mathbb{R}^n ,n\geq3$, of nonnegative functions \begin{equation}\label{0.1} u\in C^2 (\mathbb{R}^n \backslash \{0\})\cap L^\lambda (\mathbb{R}^n ) \end{equation} satisfying the Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq-\Delta u\leq(|x|^{-\alpha}*u^\lambda )u^\sigma \quad\text{ in }B_2 (0)\backslash \{0\} \end{equation} where $\alpha\in(0,n),\lambda>0,$ and $\sigma\geq0$ are constants and $*$ is the convolution operation in $\mathbb{R}^n$. We provide optimal conditions on $\alpha,\lambda$, and $\sigma$ such that nonnegative solutions $u$ satisfy pointwise bounds near the origin.