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).
Asymptotically radial solutions to an elliptic problem on expanding annular domains in Riemannian manifolds with radial symmetry
Morabito F.
2016-01-01
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).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
