On extremal solutions of an elliptic boundary value problem involving discontinuous nonlinearities (Q1194549)

Let \(\Omega\subset\mathbb{R}^ n\) be a bounded domain with a boundary \(\partial\Omega\) of class \(C^ 1\). We consider the following boundary value problem \[ -Lu=F(u,u)\text{ in }\Omega,\quad u=0\text{ on }\partial\Omega, \tag{1.1} \] where \(F\) is the superposition operator associated with a function \(f:\Omega\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) by \(F(u,v)(x)=f(x,u(x),v(x))\), and \(L\) is assumed to be a uniformly elliptic linear operator of the form \[ Lu={\partial\over\partial x_ i}\left(a_{ij}{\partial u\over\partial x_ j}\right)-b_ i{\partial u\over\partial x_ i}, \] with coefficients \(a_{ij}\), \(b_ i\in L^ \infty(\Omega)\), \(i,j=1,\dots,n\). The aim of this paper is to show the existence of the greatest and the smallest solutions of the BVP (1.1) lying between upper and lower solutions.