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Homoclinic and heteroclinic solutions for a class of second-order non-autonomous ordinary differential equations: multiplicity results for stepwise potentials

Elisa Ellero and Fabio Zanolin*

Author Affiliations

Department of Mathematics and Computer Science, University of Udine, Via delle Scienze 206, Udine, 33100, Italy

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Boundary Value Problems 2013, 2013:167  doi:10.1186/1687-2770-2013-167

Published: 15 July 2013

Abstract

We prove some multiplicity results for a class of one-dimensional nonlinear Schrödinger-type equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M2">View MathML</a> and the weight <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/167/mathml/M3">View MathML</a> is a positive stepwise function. Instead of the cubic term, more general nonlinearities can be considered as well.

MSC: 34C37, 34B15.

Keywords:
homoclinic solutions; heteroclinic solutions; multiplicity; nonlinear Schrödinger equation; stepwise potential; topological methods