In this paper we study the existence of even positive homoclinic solutions for p -Laplacian ordinary differential equations (ODEs) of the type ( u ′ | u ′ | p − 2 ) ′ − a ( x ) u | u | p − 2 + λ b ( x ) u | u | q − 2 = 0 , where 2 ≤ p < q , λ > 0 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.
Tersian Boundary Value Problems 2012, 2012 :121 http://www.boundaryvalueproblems.com/content/2012/1/121
R E S E A R C H Open Access On symmetric positive homoclinic solutions of semilinear p -Laplacian differential equations Stepan Tersian * * Correspondence: sterzian@uni-ruse.bg Department of Mathematical Analysis, University of Ruse, Ruse, 7017, Bulgaria
Abstract In this paper we study the existence of even positive homoclinic solutions for p -Laplacian ordinary differential equations (ODEs) of the type ( u | u | p –2 ) – a ( x ) u | u | p –2 + λ b ( x ) u | u | q –2 = 0, where 2 ≤ p < q , λ > 0 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