Uniqueness of the positive solution of \(\Delta u+f(u)=0\) in an annulus. (English) Zbl 0724.34023

The uniqueness of the positive radial solution to the problem \[ \Delta u+f(u)=0\text{ in } \Omega;\quad \partial u/\partial n=0\quad if\quad | x| =a>0;\quad u=0\quad if\quad | x| =b \] is proved (under suitable assumptions on f), where \(0\leq a<b\leq \infty\) and \(\Omega\) is an “annular” domain in \({\mathbb{R}}^ n\) \((n>2)\), i.e., \(\Omega =\{a<| x| <b\}\) if \(a>0\) and \(\Omega =\{| x| <b\}\) if \(a=0\). In fact, the problem reduces to the uniqueness of the positive solution for the corresponding ODE: \[ u''(r)+r^{-1}(n- 1)u'(r)+f(u)=0,\quad a<r<b;\quad u'(a)=0\quad if\quad a>0,\quad u(b)=0. \]


34B15 Nonlinear boundary value problems for ordinary differential equations
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations