Uniqueness of global positive solution branches of nonlinear elliptic problems (Q1340910)

We show that the nonlinear problem \[ \Delta u + \lambda f(u) = 0,\;u>0 \quad \text{in} \quad \Omega \subset \mathbb{R}^ 2, \lambda \in \mathbb{R}, \quad u = 0 \quad \text{on} \quad \partial \Omega, \] has unique global solution branches \(\{(\lambda,u)\}\) which are unbounded \(C^ 1\)-curves parameterized by the amplitude \(\| u \|_ \infty = p\) provided the domain \(\Omega \subset \mathbb{R}^ 2\) enjoys some symmetry and is partially convex. The function \(f:\mathbb{R} \to \mathbb{R}\) is \(C^ 2\) and fulfills \(f(u) \geq 0\) for \(u \geq 0\). The curves \(\{(\lambda,u)\}\) exist only for amplitudes \(p=\| u \|_ \infty\) such that \(f(p)>0\).