On extremal solutions of an elliptic boundary value problem involving discontinuous nonlinearities
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.
extremal solutions between upper and lower solutions
scientific article; zbMATH DE number 68016