On the uniqueness of solutions of a semilinear equation in an annulus

Abstract

We establish the uniqueness of positive radial solutions of $$\{\begin{array}{l}\Delta u+f(u)=0,x\in A\\ u(x)=0x\in \partial A\end{array}$$ where $A:=A_{a,b}=\{x\in {\mathbb{R}}^n:a<|x|<b\}$, $0<a<b\le \infty$. We assume that the nonlinearity $f\in C[0,\infty )\cap C^1(0,\infty )$ is such that $f(0)=0$ and satisfies some convexity and growth conditions, and either $f(s)>0$ for all $s>0$, or has one zero at $B>0$, is non positive and not identically 0 in $(0,B)$ and it is positive in $(B,\infty )$.

20 pages