Superharmonic functions in the upper half space with a nonlocal boundary condition

We discuss the existence of positive superharmonic functions $u$ in ${\mathbb{R}}_{+}^N={\mathbb{R}}^{N-1}\times (0,\mathrm{\infty })$, $N\ge 3$, in the sense $-\mathrm{\Delta }u=\mu$ for some Radon measure $\mu$, so that $u$ satisfies the nonlocal boundary condition $$\frac{\mathrm{\partial }u}{\mathrm{\partial }n}(x^{\mathrm{\prime }},0)=\lambda \underset{{\mathbb{R}}^{N-1}}{\int }\frac{u(y^{\mathrm{\prime }},0{)}^p}{\mathrm{\mid }{x}^{\mathrm{\prime }}-y^{\mathrm{\prime }}{\mathrm{\mid }}^k}d{y}^{\mathrm{\prime }}on\mathrm{\partial }{\mathbb{R}}_{+}^N,$$ where $p,\lambda >0$ and $k\in (0,N-1)$. First, we show that no solutions exist if $0<k\le 1$. Next, if $1<k<N-1$, we obtain a new critical exponent given by $p^{\ast }=\frac{N-1}{k-1}$ for the existence of such solutions. If $\mu \equiv 0$ we construct an exact solution for $p>p^{\ast }$ and discuss the existence of regular solutions, case in which we identify a second critical exponent given by $p^{\ast \ast }=2\cdot \frac{N-1}{k-1}-1$. Our approach combines various integral estimates with the properties of the newly introduced $\alpha$-lifting operator and fixed point theorems.