On the solutions of nonlocal 1-Laplacian equation with $L^1$-data

Abstract

We study the solutions to a nonlocal 1-Laplacian equation given by $$2\text{P.V.}{\int }_{{\mathbb{R}}^N}\frac{u(x)-u(y)}{|u(x)-u(y)|}\frac{dy}{|x-y{|}^{N+s}}=f(x)\text{for }x\in \hat{I}\text{c◯},$$ with Dirichlet boundary condition $u(x)=0$ in ${\mathbb{R}}^N\backslash \hat{I}\text{c◯}$ and nonnegative $L^1$-data. By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^{+}$, we introduce a suitable definition of solutions and prove that the limit function $u$ of $\{u_p\}$ is a solution of the nonlocal $1$-Laplacian equation above.