Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

Abstract

The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (-Δ)^αu+εg(u)=k\frac{\partial^αν}{\partial \vec{n}^α}\quad &{\rm in}\quad\ \ \barΩ,\\[3mm] \phantom{(-Δ)^α+εg(u)} u=0\quad &{\rm in}\quad\ \ \barΩ^c, \end{array} \end{equation} where $k>0$, $ε=1$ or $-1$, $(-Δ)^α$ with $α\in(0,1)$ is the fractional Laplacian defined in the principle value sense, $Ω$ is a bounded $C^2$ open set in $R^N$ with $N\ge 2$, $ν$ is a bounded Radon measure supported in $\partialΩ$ and $\frac{\partial^αν}{\partial \vec{n}^α}$ is defined in the distribution sense, i.e. $$ \langle\frac{\partial^αν}{\partial \vec{n}^α},ζ\rangle=\int_{\partialΩ}\frac{\partial^αζ(x)}{\partial \vec{n}_x^α}dν(x), \qquad \forallζ\in C^α(R^N), $$ here $\vec{n}_x$ denotes the unit inward normal vector at $x\in\partialΩ$.