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

We consider the boundary value problem (Formula presented.) Ω_R being a smooth bounded domain diffeomorphic to the expanding domain A_R: = { x∈ 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 Ω_R= A_R, we prove that there exists a positive non-radial solution to the aforementioned problem on the domain Ω_R. Such a solution is close to the radial solution to the corresponding problem on A_R.