Entire solutions of autonomous equations on ${\mathbb{R}}^n$ with nontrivial asymptotics

Abstract

We prove existence of a new type of solutions for the semilinear equation $-\Delta u+u=u^p$ on ${\mathbb{R}}^n$, with $1<p<\frac{n+2}{n-2}$. These solutions are positive, bounded, decay exponentially to zero away from three half-lines with a common origin, and at infinity are asymptotically periodic.