A uniqueness result for a singular nonlinear eigenvalue problem (Q2866544)

The authors study the singular nonlinear elliptic eigenvalue problem \(-\Delta u= \mu f(u)/u^\beta\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\mu> 0\) is a real parameter, \(\beta\in(0,1)\) and \(f:[0,\infty)\to(0, \infty)\) is a non-decreasing \(C^1\)-function satisfying the assumption that there exists \(M>0\) such that \(f(u)/u^\beta\) is decreasing if \(u>M\). Under these assumptions it is proved that there is a unique classical solution \(u\in C^2(\Omega)\cap C^1(\overline\Omega)\) of this problem for \(\mu>0\) ``large''. Existence of a minimal positive solution is also proved. The proof is quite tricky and uses different integrations by parts and comparison arguments involving the maximum principle.