A uniqueness theorem for some nonlinear boundary value problem with a large parameter
Let \(-Au=a_{ij}u_{ij}+a_ iu_ i-a_ 0u\) denote a uniformly elliptic operator with Hölder-continuous coefficients and \(a_ 0\geq 0\). A short and simple proof is given, that the nonlinear boundary value problem \(Au=\lambda f(x,u)\) in \(\Omega\), \(u|_{\partial \Omega}=0\) has a unique positive solution for all \(\lambda\) sufficiently large, if f is positive and (roughly) \(f\sim u^{-\gamma}\), \(| \partial f/\partial u| \sim u^{-\gamma -1}\) for \(u\to \infty\) with some \(\gamma \in [0,1)\).
scientific article; zbMATH DE number 3865820