Estimating potential functions with applications to elliptic problems in half space

Abstract

We first derive sharp estimates on some potential functions on the half space 0 \\} $ ]]> R + n := { x = ( x 1 , … , x n ) ∈ R n ; x n > 0 } , n ≥ 3 . Then, these estimates are applied to demonstrate the existence of a positive solution with an explicit asymptotic behavior for a semi-linear elliptic problem in the half space. Our approach is founded on Karamata's theory for regularly varying functions, complemented by the use of monotonicity methods.