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

2018/07/11 by Marcelo F. de Almeida, de Almeida, Marcelo F., Lidiane S. M. Lima +1
Mathematics · #35J05 #35J65 #35J67 #49J52 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1807.04122

openalex publication_date 2018/07/11 · openalex created_date 2018/07/19 · openalex updated_date 2026/07/28

Abstract

In this paper we are interested on solvability of the problem \begincases -Δu=0 amp; in ℝⁿ⁺¹₊
(∂ u)/(∂ ν) = V(x)u+b \vert u\vert^ρ-1u+f amp; on ∂ℝⁿ⁺¹₊ \endcases %Laplace equation in the upper half-space with nonlinear Neumann boundary with high singular data f and potential V on boundary ∂ℝⁿ⁺¹₊ of half-space ℝⁿ⁺¹₊=\(x,t)∈ℝⁿ⁺¹ \vert t>0\ for n≥ 2. More precisely, inspired at \citedeAlmeida1 and \citeQuittner 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\inweak-M_p^n(ρ-1)/ρ(ℝⁿ) and potential V∈ week-Mⁿ_ℓ(ℝⁿ) are sufficiently small in the natural n/(n-1)

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