Nonlinear elliptic problems with critical exponent in shrinking annuli (Q1089505)

Consider the nonlinear boundary value problems \[ \Delta u+u^{(N+2)/(N- 2)}=0\quad in\quad \epsilon <| x| <R,\quad u=0\quad on\quad | x| =\epsilon \quad and\quad on\quad | x| =R \] \[ \Delta v+v^{(N+2)/(N-2)}=0\quad in\quad \epsilon <| x| <R,\quad \partial v/\partial n=0\quad on\quad | x| =\epsilon,\quad v=0\quad on\quad | x| =R. \] Here \(\partial /\partial n\) stands for the outer normal derivative and \(x=(x_ 1,x_ 2,...,x_ N)\), is an arbitrary point in \(R^ N\). Recently it has been shown that both problems have a unique positive radially symmetric solution. In this paper the precise behaviour as \(\epsilon\) \(\to 0\) of \(u_{\max}(\epsilon)\), \(v_{\max}(\epsilon)\) and of the value r(\(\epsilon)\) where u achieves its maximum is computed. Moreover it is shown that v and u tend after suitable normalization to a specific multiple of the Green's function of the Laplace operator in the sphere. The arguments are based on a phase plane analysis. The results are then used to discuss nonlinear Dirichlet problems in more general doubly connected domains.