Critical boundary value problems in the upper half-space. (November 2021) - Furtado, Marcelo F.; da Silva, João Pablo P.
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 .