Uniqueness of positive solutions of nonlinear second order systems

Abstract

In this paper we discuss the uniqueness of positive solutions of the nonlinear second order system $–u^{\prime \prime }=g(v)$, $–v^{\prime \prime }=f(u)$ in $(–R,R)$, $u(\pm R)=v(\pm R)=0$ where $f$ and $g$ satisfy some appropriate conditions. Our result applies, in particular, to $g(v)=v$, $f(u)=u^p$, $p>1$, or $f(u)=\lambda u+a_1{u}^{p1}+\cdots +a_k{u}^{pk}$ with $p_j>1$, $a_j>0$ for $j=1,\dots ,k$ and $0\le \lambda <{\mu }_1^2$ where ${\mu }_1={\pi }^2/4R^2$.