Superharmonic functions in the upper half space with a nonlocal boundary condition
Marius Ghergu
TL;DR
This work analyzes positive superharmonic functions in the upper half-space $\mathbb{R}^N_+$ that satisfy a nonlocal boundary condition linking $\partial u/\partial n$ to a convolution of the boundary trace $u(\cdot,0)^p$ with the kernel $|x'-y'|^{-k}$. The authors introduce the $\alpha$-lifting operator $J_{\alpha}$ to reformulate the problem in integral form and employ fixed-point methods to obtain existence results, identifying a critical exponent $p^*=(N-1)/(k-1)$ that separates nonexistence from existence for nonzero data $\mu$. They prove nonexistence when $0<k\le 1$ or when $1<k<N-1$ and $p\le p^*$ (including the endpoint $p=p^*$ under a trace condition), and show existence for $p>p^*$ when $\mu$ satisfies a decay bound. In the special case $\mu\equiv 0$, the paper provides an explicit regular solution for $p>p^*$ and uncovers a second threshold $p^{**}=2(N-1)/(k-1)-1$ governing regularity, with explicit forms at the threshold. The results illuminate how nonlocal boundary coupling induces new critical phenomena in half-space problems and extend the theory of boundary-value problems with measure data via integral representations and lifting operators.
Abstract
We discuss the existence of positive superharmonic functions $u$ in $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times (0, \infty)$, $N\geq 3$, in the sense $-Δu=μ$ for some Radon measure $μ$, so that $u$ satisfies the nonlocal boundary condition $$ \frac{\partial u}{\partial n}(x',0)=λ\int\limits_{\mathbb{R}^{N-1}}\frac{u(y',0)^p}{|x'-y'|^k}dy' \quad\mbox{ on }\partial \mathbb{R}^N_+, $$ where $p,λ>0$ and $k\in (0, N-1)$. First, we show that no solutions exist if $0<k\leq 1$. Next, if $1<k<N-1$, we obtain a new critical exponent given by $p^*=\frac{N-1}{k-1}$ for the existence of such solutions. If $μ\equiv 0$ we construct an exact solution for $p>p^*$ and discuss the existence of regular solutions, case in which we identify a second critical exponent given by $p^{**}=2\cdot \frac{N-1}{k-1}-1$. Our approach combines various integral estimates with the properties of the newly introduced $α$-lifting operator and fixed point theorems.