On the existence of positive entire solutions of a semilinear elliptic equation (Q1821256)

Existence of positive entire solutions to the equation \(\Delta u- u+Q(x)u^ p=0\) in \({\mathbb{R}}^ n\) is shown, where \(p>1\) and Q(x) is a given potential, which, in general, is not radially symmetric. The approach is by establishing existence of positive solutions in any ball with radius R centered at the origin and then passing to the limit \(R\to \infty\).