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

TL;DR

This paper proves the uniqueness of positive radial solutions to a semilinear elliptic equation in an annulus under specific conditions on the nonlinearity, extending understanding of solution behavior in bounded and unbounded domains.

Contribution

It establishes the uniqueness of positive radial solutions for a class of semilinear equations in annular regions, under convexity and growth conditions on the nonlinearity.

Findings

Uniqueness of positive radial solutions in annuli.

Conditions on the nonlinearity ensure solution uniqueness.

Applicable to both bounded and unbounded annular domains.

Abstract

We establish the uniqueness of positive radial solutions of $$\begin{cases} \Delta u +f(u)=0,\quad x\in A \\ u(x) =0 \quad x\in \partial A \end{cases} $$ 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 $(B,\infty)$.