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

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.

14 pages, comments are welcome