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 $- \\D u + u = u^p$ on $\\R^n$, 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.