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 $–u'' = g(v)$, $–v'' = f (u)$ in $(–R,R)$, $u(±R) = v(±R) = 0$ where $f$ and $g$ satisfy some appropriate conditions. Our result applies, in particular, to $g(v) = v$, $f(u) = u^p$, $p>1$, or $f(u) = \lambda u + a_1u^{p1} + \cdots + a_ku^{pk}$ with $p_j > 1$, $a_j > 0$ for $j = 1,\dots , k$ and $0 ≤ \lambda < \mu_1^2$ where $\mu_1 = \pi^2 / 4R^2$.