Asymptotically radial solutions to an elliptic problem on expanding annular domains in Riemannian manifolds with radial symmetry
Filippo Morabito · Boundary Value Problems · 2016
We consider the boundary value problem $$\left \{ \textstyle\begin{array}{l} \Delta_{\mathbf{g}} u +u^{p}= 0 \quad\mbox{in } \Omega_{R}, \\ u=0 \quad \mbox{on } \partial\Omega_{R}, \end{array}\displaystyle \right . $$ $\Omega_{R}$ being a smooth bounded domain diffeomorphic to the expanding domain $A_{R}:=\{x \in M, R< r(x)< R+1\}$ in a Riemannian manifold M of dimension $n \geq2$ endowed with the metric ${\mathbf{g}}=dr^{2}+S^{2}(r)g_{{\mathbb{S}}^{n-1}}$ . 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 $A_{R}$ .