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 $\mathrm{\partial }{\mathbb{R}}_{+}^{n+1}$ of half-space ${\mathbb{R}}_{+}^{n+1}=\{(x,t)\in {\mathbb{R}}^{n+1}\mathrm{\mid }t>0\}$ for $n\ge 2$. 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 $f\in \text{weak-}{\mathcal{M}}_p^{n(\rho -1)\mathrm{/}\rho }({\mathbb{R}}^n)$ and potential $V\in \text{week-}{\mathcal{M}}_{\mathrm{\ell }}^n({\mathbb{R}}^n)$ are sufficiently small in the natural $n\mathrm{/}(n-1)<\rho <\mathrm{\infty }$. Our theorems recover the range $(n+1)\mathrm{/}(n-1)\le \rho <\mathrm{\infty }$ 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$f$andpotential$V$ 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}$^{n+1}+})$islocallyH\ddot{o}ldercontinuous,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.