Some new entire solutions of semilinear elliptic equations on \(\mathbb R^n\) (Q1028350)

The existence of a new type of positive solutions of the semilinear equation \(-\Delta u+u=u^p\) on \({\mathbb R}^n,\) where \(1<p<\frac{n+2}{n-2},\) is proved. These solutions are bounded, but do not tend to zero at infinity. Indeed, they decay to zero away from three half-lines with a common origin, and their asymptotic profile is periodic along these half-lines.