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} (-\Delta)^\alpha u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{(-\Delta)^\alpha +\epsilon g(u)} u=0\quad &{\rm in}\quad\ \ \bar\Omega^c, \end{array} \end{equation} where $k>0$, $\epsilon=1$ or $-1$, $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian defined in the principle value sense, $\Omega$ is a bounded $C^2$ open set in $R^N$ with $N\ge 2$, $\nu$ is a bounded Radon measure supported in $\partial\Omega$ and $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ is defined in the distribution sense, i.e. $$ \langle\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha},\zeta\rangle=\int_{\partial\Omega}\frac{\partial^\alpha\zeta(x)}{\partial \vec{n}_x^\alpha}d\nu(x), \qquad \forall\zeta\in C^\alpha(R^N), $$ here $\vec{n}_x$ denotes the unit inward normal vector at $x\in\partial\Omega$.