A solution between sub- and supersolutions for semilinear elliptic equations with a nonlocal term in a continuous setting

In the 1987 publication [5] we showed that for second order semilinear elliptic problems a solution exists between a subsolution and a supersolution without using monotone iteration and under minimal assumptions. The alternative is based on a Schauder fixed point argument. Filling some missing details of that paper we noticed that even nonlocal operators are allowed on the right hand side as long as these are quasimonotone. The setting uses ‘weak’-solutions which are continuous on the closure of the domain and solve the equation in distributional sense. As such they differ from the standard weak solutions. We conclude with some examples concerning the setting.