Asymptotically radial solutions to an elliptic problem on expanding annular domains in Riemannian manifolds with radial symmetry

Abstract

We consider the boundary value problem {Delta(g)u + u(p) = 0 in Omega(R),u = 0 on partial derivative Omega(R),Omega(R) being a smooth bounded domain diffeomorphic to the expanding domain A(R): = {x is an element of M, R < r(x) < R + 1} in a Riemannian manifold M of dimension n >= 2 endowed with the metric g = dr(2) + S-2(r)g(S)(n-1). After recalling a result about existence, uniqueness, and non-degeneracy of the positive radial solution when Omega(R) = A(R), we prove that there exists a positive non-radial solution to the aforementioned problem on the domain Omega(R). Such a solution is close to the radial solution to the corresponding problem on A(R).