manuscripta math. 138, 273–286 (2012) © Springer-Verlag 2012 Manuel del Pino, Monica Musso, Frank Pacard Solutions of the Allen-Cahn equation which are invariant under screw-motion Received: 25 April 2011 / Revised: 21 August 2011 Published online: 7 February 2012 Abstract. We study entire solutions of the Allen-Cahn equation which are defined in the 3-dimensional Euclidean space and which are invariant under screw-motion. In particular, we discuss the existence and non existence of nontrivial solutions whose nodal set is a helicoid of R 3 . 1. Introduction and statement of the main results In this short note, we are interested in entire solutions of the Allen-Cahn equation u - F ′ (u ) = 0, (1.1) which are defined in R n , with n ≥ 1. Here F ′ is the derivative of the function F which is usually referred to as a double well potential. More precisely, we will assume that t  → F (t ) is an even, positive function which is at least of class C 2 and which has only two distinct nondegenerate absolute minima at the points ±t ∗ ∈ R, where t ∗ > 0. Hence, for all t ∈ R, F (t ) ≥ F (t ∗ ), with equality if and only if t = t ∗ . We further assume that F ′′ (0)< 0 and F ′′ (0) t ≤ F ′ (t ), (1.2) for all t ≥ 0. We define λ ∗ := π √ -F ′′ (0) . M. del Pino: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile. e-mail: delpino@dim.uchile.cl M. Musso: Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna, 4860 Macul, Chile. e-mail: mmusso@mat.puc.cl F. Pacard (B ): Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France and Institut Universitaire de France, Paris, France. e-mail: frank.pacard@math.polytechnique.fr Mathematics Subject Classification (2000): 35J91, 35J20, 47J30 DOI: 10.1007/s00229-011-0492-3