About uniqueness for nonlinear boundary value problems (Q792545)

Let A be a general second order elliptic operator in the bounded domain \(\Omega\) and let f:\({\bar \Omega}\times R^+\to R^+\) be Hölder continuous in \(x\in \Omega\) and continuously differentiable in \(\zeta \in R^+\). The author considers the semilinear equation (1) \(Au=\lambda f(x,u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) under some positivity and growth assumptions for f and proves the following: Theorem. There exists a \(\lambda_ 0>0\) such that (1) has a unique solution for \(\lambda>\lambda_ 0\).