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

Abstract

We study the behavior near the origin in ${\mathbb{R}}^n,n\ge 3$, 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 $\alpha \in (0,n),\lambda >0,$ 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.

arXiv comment24 pages, 3 figures