Solutions having boundary layers to a nonlinear elliptic equation on a spherical cap (Nonlinear Evolution Equations and Mathematical Modeling)
Catherine Bandle, Yoshitsugu Kabeya, Hirokazu Ninomiya · Institutional Repositories DataBase (IRDB) · 2008
In this paper, we consider the nonlinear elliptic equation $\Lambda u+\lambda(-u+u_{+}^{p})=0$ in $\Omega\subset S^{n}$ (1.1) under the homogeneous Dirichlet boundary condition.Here A denotes the Laplace-Beltrami operator on the standard unit sphere $S^{n}\subset \mathbb{R}^{n+1}$ .We assume that $n\geq 3,$ $p>1,$ $\lambda>0$ and that $\Omega\subset S^{n}$ is a geodesic open ball, called a "spherical cap", centered at the North Pole $(0, \ldots, 0,1)$ .To start our analysis, we express $\Omega$ in polar coordinates in order to make our setting clear.Let $(y_{1},y_{2}, \ldots , y_{\mathfrak{n}+1})$ be the Cartesian coordinates in $\mathbb{R}^{n+1}$ .We express