On the existence of a maximal weak solution for a semilinear elliptic equation
The authors consider the problem \(-\nabla u=f(x,u)\) in \(\Omega\) ; \(u=0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\) and f is a continuous function. They then define weak sub- and supersolutions as follows: \(\bar u\) is a supersolution if (a) \(\bar u\in C({\bar \Omega})\), (b) \(\int_{\Omega}(\bar u-\Delta \phi -f(x,\bar u)\phi)dx\geq 0\) for all \(\phi \in C^{\infty}_ 0(\Omega)\), \(\phi \geq 0\), (c) \(\bar u\geq 0\) on \(\partial \Omega\). A weak subsolution is defined by reversing the inequality signs and a C-solution is defined by taking equality signs. They then show that if there exist a subsolution \b{u} and a supersolution \(\bar u\) with \b{u}\(\leq \bar u\) in \(\Omega\) then there exist a maximal and a minimal solution \(u_ 2\) and \(u_ 1\) respectively, such that \b{u}\(\leq u_ 1\leq u\leq u_ 2\leq \bar u\) in \(\Omega\). Then they extend this result to appropriate solutions in \(W_ 0^{1,2}(\Omega)\) if \(f(\cdot,u)\in L^ 1(\Omega)\).
scientific article; zbMATH DE number 4210638