Open Access Research

On the solvability of a boundary value problem on the real line

Giovanni Cupini1, Cristina Marcelli2 and Francesca Papalini2*

Author Affiliations

1 Dipartimento di Matematica - Università di Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy

2 Dipartimento di Scienze Matematiche - Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy

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Boundary Value Problems 2011, 2011:26 doi:10.1186/1687-2770-2011-26

Published: 23 September 2011

Abstract

We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M1">View MathML</a>

governed by a nonlinear differential operator Φ extending the classical p-Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M2">View MathML</a>. We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x') = b(t, x)c(x, x').

Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.

Keywords:
boundary value problems; unbounded domains; heteroclinic solutions; nonlinear differential operators; p-Laplacian operator; Φ-Laplacian operator