Open Access Research

On symmetric positive homoclinic solutions of semilinear p-Laplacian differential equations

Stepan Tersian

Author affiliations

Department of Mathematical Analysis, University of Ruse, Ruse, 7017, Bulgaria

Citation and License

Boundary Value Problems 2012, 2012:121  doi:10.1186/1687-2770-2012-121

Published: 24 October 2012

Abstract

In this paper we study the existence of even positive homoclinic solutions for p-Laplacian ordinary differential equations (ODEs) of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3">View MathML</a> and the functions a and b are strictly positive and even. First, we prove a result on symmetry of positive solutions of p-Laplacian ODEs. Then, using the mountain-pass theorem, we prove the existence of symmetric positive homoclinic solutions of the considered equations. Some examples and additional comments are given.

MSC: 34B18, 34B40, 49J40.

Keywords:
p-Laplacian ODEs; homoclinic solution; weak solution; Palais-Smale condition; mountain-pass theorem