Reduced limit approach to semilinear partial differential equations (PDEs) involving the fractional Laplacian with measure data

Abstract

We study the following partial differential equation (PDE) \begin{align} \begin{split} (-\Delta)^s u + g(x,u) & = \mu\,\,\mbox{in}\,\,\Omega,\\ u & = 0\,\,\mbox{in}\,\,\mathbb{R}^N\setminus\Omega,\label{eqn_abs} \end{split} \end{align} where $(-\Delta)^s$ is the fractional Laplacian operator, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $\partial\Omega$ being the boundary of $\Omega$, $g(.,.)$ is a nonlinear function defined over $\Omega\times\mathbb{R}$. Let $(\mu_n)_n$ be a sequence of measure in $\Omega$. Assume that there exists a solution $u_n$ with data $\mu_n$, i.e. $u_n$ satisfies the equation (0.1) with $\mu=\mu_n$. We further assume that the sequence of measures weakly converges to $\mu$, while $(u_n)_n$ converges to $u$ in $L^1(\Omega)$. In general, $u$ is not a solution to the partial differential equation in (0.1) with datum $(\mu,0)$. However, there exists a measure $\mu^{\#}$ such that $u$ is a solution of the partial differential equation with this data. $\mu^{\#}$ is called the reduced limit of the sequence $(\mu_n)_n$. We investigate the relation between weak limit $\mu$ and the reduced limit $\mu^{\#}$ and the dependence of $\mu^{\#}$ to the sequence $(\mu_n)_n$. A closely related problem was studied by Bhakta and Marcus [3] and then by Giri and Choudhuri [15] but for the case of a Laplacian and a general second order linear elliptic differential operator, respectively instead of a fractional Laplacian.