# American Institute of Mathematical Sciences

August  2016, 9(4): 1025-1038. doi: 10.3934/dcdss.2016040

## Multiple homoclinic solutions for a one-dimensional Schrödinger equation

 1 Department of Mathematics - University of Torino, Via Carlo Alberto, 10 - 10123 Torino 2 Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine, via delle Scienze 206, 33100 Udine, Italy

Received  July 2015 Revised  January 2016 Published  August 2016

In this paper we study the problem of the existence of homoclinic solutions to a Schrödinger equation of the form $x''-V(t)x+x^3=0,$ where is a stepwise potential. The technique of proof is based on a topological method, relying on the properties of the transformation of continuous planar paths (the S.A.P. method), together with the application of the classical Conley-Ważewski's method.
Citation: Walter Dambrosio, Duccio Papini. Multiple homoclinic solutions for a one-dimensional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1025-1038. doi: 10.3934/dcdss.2016040
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