Estimates for solutions of quasilinear problems with dead cores
Let \(\Omega\subset\mathbb{R}^N\) be an arbitrary domain and \(a,b:\Omega\to \mathbb{R}^+\) be two functions. The authors consider boundary value problems of the form \[ \begin{cases} \sum^N_{i=1} (a(x)|\nabla u|^{p-2} u_{x_i})_{x_i}= b(x) f(u)\quad &\text{in }\Omega,\\ u= 1\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(p\geq 1\) is a real number and \(f(s)\) is a nondecreasing function such that \(f(0)= 0\). The goal of the authors is to derive bounds for (1). To this end they use a Rayleigh-Faber-Krahn type inequality.
scientific article; zbMATH DE number 2021115