Critical boundary value problems in the upper half-space. (November 2021)

• Record Type: Journal Article
• Authors: Furtado, Marcelo F.; da Silva, João Pablo P.
• Is Part Of: Nonlinear analysis. Volume 212(2021)
• Publication Date: 2021-11
• Subjects: primary 35J66 35J20; Weighted trace embedding -- Critical problems -- Nonlinear boundary conditions -- Self-similar solutions -- Half-space; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112441
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the critical problem − Δ u − 1 2 x ⋅ ∇ u = 0 in R + N, ∂ u ∂ ν = λ | u | p − 2 u + | u | 2 ∗ − 2 u on ∂ R + N, where R + N = ( x ′, x N ) : x ′ ∈ R N − 1, x N > 0 is the upper half-space, N ≥ 4, ν is the outward normal vector at the boundary and 2 ≤ p < 2 ∗ : = 2 ( N − 1 ) / ( N − 2 ) . Using a variational approach, we obtain nonnegative nonzero solutions according to the value of the parameter λ > 0 .