Ordered structures and semilinear elliptic equations (Q1202758)

The authors study the problem \(- \Delta u + c^ 2u = f(x,u)\), \(u > 0\) in \(\mathbb{R}^ n\), \(u(x) \to 0\) as \(\| x \| \to 0\), where \(f\) satisfies certain growth conditions and is monotone increasing in the second variable. The fundamental solution of the operator \(L = - \Delta + c^ 2\) is used to transform this problem into an integral equation. Then a fixed point theorem of Tarski is used to obtain a solution of this equation in the ordered set \(M\), consisting of measurable, bounded functions \(u : \mathbb{R}^ n \to [0, \infty)\) with ordering: \(u \leq v\) iff \(u(x) \leq v(x)\) for almost every \(x \in \mathbb{R}^ n\). The results apply for example to nonlinearities \(f\) of the form \(f(x,u) = \beta (x)u^ p\) \((u \geq 0)\) with \(\beta (x) \geq 0\), \(0 < p < 1\).