Existence of Positive and Symmetric Solutions for Some Elliptic Equation on Spherical Cap of S³

Abstract

In this paper, we study existence of positive and symmetric solutions for some boundary value problem on a spherical cap of unit ball in four-dimensional space. It seems that this problem has been studied for the equation containing critical Sobolev exponent for three-dimensional space; so that we consider non-critical case in this paper. Then for subcritical case, we can show the existence and non-existence result for such solution completely if the spherical cap is contained in the hemisphere. In addition, we get some partial result if the spherical cap contains the hemisphere.