Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert u\vert^{\rho-1}u+f \; & \text{on} \;\;\partial\mathbb{R}^{n+1}_+\;\;\;\;\;\;\;\;\,\, \end{cases} \end{align*} %Laplace equation in the upper half-space with nonlinear Neumann boundary with high singular data $f$ and potential $V$ on boundary $\partial\mathbb{R}^{n+1}_+$ of half-space $\mathbb{R}^{n+1}_{+}=\{(x,t)\in\mathbb{R}^{n+1}\,\vert\, t>0\}$ for $n\geq 2$. More precisely, inspired at \cite{deAlmeida1} and \cite{Quittner} we introduce a new functional space based in weak-Morrey spaces and we shown existence of positive solutions $u$ to the above problem when inhomogeneous term $f\in\text{weak-}\mathcal{M}_{p}^{n{(\rho-1)}/{\rho}}(\mathbb{R}^{n})$ and potential $V\in \text{week-}\mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ are sufficiently small in the natural $n/(n-1)<\rho<\infty$. Our theorems recover the range $(n+1)/(n-1)\leq \rho<\infty$ and immediately imply in solvability of the equivalent nonlocal half-Laplacian problem $(-\Delta)^{{1}/{2}}u=Vu+b\vert u\vert^{\rho-1}u+ f (x)$ for $f$ and potential $V$ rough than previous ones, in view of strictly inclusions $L^\lambda \varsubsetneq\mathcal{M}^{\lambda}_{p} \varsubsetneq \text{week-}\mathcal{M}^{\lambda}_{p}$ for $1<p<\lambda<\infty$. Also, from Campanato's lemma we conclude that $u\in C^{0,\alpha}_{loc}( \overline{\mathbb{R}^{n+1}_+})$ is locally H\"older continuous, for $f\in\mathcal{M}_{p}^{n{(\rho-1)}/{\rho}}(\mathbb{R}^{n})$ and $V\in \mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ in Morrey spaces.