Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces
Year: 2018
Type: preprint
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}\,\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 $f\in\text{weak-}\mathcal{M}_{p}^{n{(ρ-1)}/ρ}(\mathbb{R}^{n})$ and potential $V\in \text{week-}\mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ are sufficiently small in the natural $n/(n-1)
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2848975175