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

Abstract

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^λ(\mathbb{R}^n ) \end{equation} satisfying the Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq-Δu\leq(|x|^{-α}*u^λ)u^σ\quad\text{ in }B_2 (0)\backslash \{0\} \end{equation} where $α\in(0,n),λ>0,$ and $σ\geq0$ are constants and $*$ is the convolution operation in $\mathbb{R}^n$. We provide optimal conditions on $α,λ$, and $σ$ such that nonnegative solutions $u$ satisfy pointwise bounds near the origin.