$\epsilon=-1$ and $\nu$ is a bounded nonnegative Radon measure supported in $\partial\Omega$. We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that $g$ is a continuous function satisfying $$\int_1^\infty g(s)s^{-1-\frac{N+\alpha}{N-\alpha}}ds<+\infty.$$
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[**Existence, Non-existence, Uniqueness of solutions\
for semilinear elliptic equations involving\
measures concentrated on boundary**]{}

Huyuan Chen[^1] Hichem Hajaiej[^2]

[: Fractional Laplacian; Radon measure; Dirac mass; Green kernel; Schauder’s fixed point theorem.]{}

[: 35R11, 35J61, 35R06. ]{}

Introduction {#sec:intro}
============

Motivation
----------

In 1991, a fundamental contribution of semilinear elliptic equations involving measures as boundary data is due to Gmira and Véron in [@GV], which studied the weak solutions for $$\label{1.1.1}
\arraycolsep=1pt
\begin{array}{lll}
-\Delta  u+g(u)=0\quad &{\rm in}\quad \Omega,
\\[2mm]\phantom{-----}
u=\mu\quad&{\rm on}\quad \partial\Omega,
\end{array}$$ where $\Omega$ is a bounded $C^2$ domain in ${\mathbb{R}}^N$ and $\mu$ is a bounded Radon measure defined in $\partial\Omega$. A function $u$ is said to be a weak solution of (\[1.1.1\]) [*if $u\in
L^1(\Omega)$, $g(u)\in L^1(\Omega,\rho_{\partial\Omega} dx)$ and $$\label{1.1.1.0}
\int_\Omega [u(-\Delta)\xi+ g(u)\xi]dx=\int_{\partial\Omega}\frac{\partial\xi(x)}{\partial\vec{n}_x}d\mu(x),\quad \forall\xi\in C^{1.1}_0(\Omega),$$ where $\rho_{\partial\Omega}(x)={\rm dist}(x,\partial\Omega)$ and $\vec{n}_x$ denotes the unit inward normal vector at point $x$.*]{} Gmira and V´eron proved that problem (\[1.1.1\]) admits a unique weak solution when $g$ is a continuous and nondecreasing function satisfying $$\label{14.04}
\int_1^\infty [g(s)-g(-s)]s^{-1-\frac{N+1}{N-1}}ds<+\infty.$$ 