# zbMATH — the first resource for mathematics

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: $u''(r)+r^{-1}(n- 1)u'(r)+f(u)=0,\quad a<r<b;\quad u'(a)=0\quad if\quad a>0,\quad u(b)=0.$

##### MSC:
 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