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

TL;DR

This paper investigates the behavior of nonnegative solutions to a class of Choquard-Pekar inequalities near an isolated singularity, establishing optimal conditions for their pointwise bounds based on parameters.

Contribution

It derives sharp conditions on parameters ensuring solutions are bounded near the singularity, advancing understanding of these nonlinear inequalities.

Findings

Identifies parameter regimes for boundedness of solutions.

Establishes blow-up conditions when parameters fall outside bounds.

Provides optimal bounds for solutions near the origin.

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^\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.