Further Applications

Abstract

The linear variational theory developed in the last chapter can be applied to a large variety of problems. Here we show how the weak maximum and comparison principles play a major role in constructing an iterative procedure to solve Fisher’s type semilinear equation. Precisely, we consider the following problem:

$$ \left\{ {\begin{array}{*{20}{c}} { - \Delta u = f\left( u \right)}&{in{\text{ }}\Omega } \\ {u = g}&{on{\text{ }}\partial \Omega ,} \end{array}} \right. $$

((9.1))

where $\Omega \subset {\mathbb{R}^n}$ is a bounded, Lipschitz domain and

$$ f \in {C^1}\left( \mathbb{R} \right),g \in {H^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\left( {\partial \Omega } \right). $$