# Existence and $L^{\infty}$-estimates for elliptic equations involving convolution

## TL;DR

This paper establishes the existence and boundedness of solutions for elliptic boundary value problems involving nonlocal convolution operators in bounded domains with smooth boundaries.

## Contribution

It introduces new existence and $L^{ty}$-estimates results for elliptic equations with convolution operators, extending classical theory to nonlocal problems.

## Key findings

- Existence of weak solutions under certain conditions.
- Solutions are bounded via boundary-adapted Moser iteration.
- Applicable to nonlocal elliptic equations with convolution terms.

## Abstract

In this paper, with a fixed $p\in (1,+\infty)$ and a bounded domain $\Omega \subset \mathbb{R}^N$ whose boundary $\partial\Omega$ fulfills the $C^1$ regularity, we study a boundary value problem involving a nonlocal operator assigning to $u$ the convolution $\rho \ast E(u)$ of $\rho$ with $E(u)$, where $\rho$ is an integrable function on $\mathbb{R}^N$ and $E$ is an extension operator related to $\Omega$. Under verifiable conditions, we prove the existence of a (weak) solution to our problem by using the surjectivity theorem for pseudomonotone operators. Moreover, through a modified version of Moser iteration up to the boundary, we show that (any) weak solution to our problem is bounded.