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

We study the behavior near the origin in ${\mathbb{R}}^n,n\ge 3$, 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 $\sigma \ge 0$ are constants and $\ast$ 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.