On the uniqueness of positive solution of an elliptic equation (Q812663)

The authors consider the following semilinear elliptic boundary value problem: \[ \begin{cases} -\Delta u = f(x,u) & \text{ in }\Omega,\\ u = 0 & \text{ on }\partial \Omega, \end{cases}\tag{1} \] with \(\Omega\) regular domain in \(\mathbb R^N\) and \(f : \Omega \times \mathbb R \to \mathbb R\) measurable function. The main result is the following: Assume that there exists \(g \in C([0, +\infty))\cap C^1(0, +\infty)\), such that \(g(t) > 0\) for \(t > 0\), \(g'\) is non-increasing and the map \(u \to f(x,u)/g(u)\) is non-increasing in \((0, +\infty)\) for almost every \(x \in \Omega\). Then problem (1) has a unique nonnegative solution. It is shown also that \(u(x) > 0\) \(\forall x \in \Omega\) and \(\partial u/\partial n < 0\) on \(\partial \Omega\).