New classes of entire solutions for semilinear elliptic problems in $R^n$

Abstract

The goal of this paper is to describe some new results concerning entire solutions of semilinear elliptic equations in $R^n$ with non trivial asymptotic behavior at infinity. We describe in particular the (focusing, subcritical) nonlinear Schrödinger equation and the Allen–Cahn equation, which enjoy some common features but also present rather different aspects.