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

Abstract

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -Δu=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial ν}} = V(x)u+b \vert u\vert^{ρ-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 ${\mathbb{R}}_{+}^{n+1}=\{(x,t)\in {\mathbb{R}}^{n+1}|t>0\}$ 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 $f\in \text{weak-}{\mathcal{M}}_p^{n(\rho -1)/\rho }({\mathbb{R}}^n)$ and potential $V\in \text{week-}{\mathcal{M}}_{\ell }^n({\mathbb{R}}^n)$ are sufficiently small in the natural $n/(n-1)<\rho <\infty$. Our theorems recover the range $(n+1)/(n-1)\le \rho <\infty$ and immediately imply in solvability of the equivalent nonlocal half-Laplacian problem $(-\Delta {)}^{1/2}u=Vu+b|u{|}^{\rho -1}u+f(x)$ for $f$ and potential $V$ rough than previous ones, in view of strictly inclusions $L^{\lambda }{\mathcal{M}}_p^{\lambda }\text{week-}{\mathcal{M}}_p^{\lambda }$ for $1<p<\lambda <\infty$. Also, from Campanato's lemma we conclude that $u\in C_{loc}^{0,\alpha }(\overline{{\mathbb{R}}_{+}^{n+1}})$ is locally Hölder continuous, for $f\in {\mathcal{M}}_p^{n(\rho -1)/\rho }({\mathbb{R}}^n)$ and $V\in {\mathcal{M}}_{\ell }^n({\mathbb{R}}^n)$ in Morrey spaces.

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