Uniqueness of positive solutions for a class of semilinear elliptic systems (Q1863459)

The author proves existence and uniqueness of positive solutions for the system \[ \begin{cases} \Delta u=-\lambda f(v),\;\Delta v=-\mu g(u) & \text{ in }B,\cr u=v=0 & \text{ on }\partial B,\cr \end{cases} \] where \(B\) is the unit ball in \({\mathbb R}^N;\) \(f,\;g\colon\) \({\mathbb R}^+\to{\mathbb R}^+\) and \(f(x)\sim x^p,\) \(g(x)\sim x^q\) as \(x\to+\infty,\) and \(p,q>0\) with \(pq<1.\)