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

Stepan Tersian

doi:10.1186/1687-2770-2012-121

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 +λb(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.

Highlights

1 Introduction and main results In this paper we prove the existence of positive homoclinic solutions for p-Laplacian ordinary differential equations (ODEs) of the type u u p– – a(x)u|u|p– + λb(x)u|u|q– =, x ∈ R, ( )
We extend the symmetry lemma of Korman and Ouyang [ ] to the p-Laplacian equations
We show that JT satisfies the assumptions of the mountain-pass theorem of Ambrosetti and Rabinowitz [ ]

Summary

The dual space of

Be the duality pairing between Lp (a, b) and Lp(a, b). By the Hölder inequality, | v, u | ≤ |v|p |u|p for any v ∈ Lp (a, b) and u ∈ Lp(a, b). Any positive solution u of ( ) is an even function such that max{u(x) : –T ≤ x ≤ T} = u( ) and u (x) < for x ∈ Remark Let us note that if the function f satisfies ( ), but u is not a positive solution of ( ), u is not necessarily an even function. –π < x < π, The term f (x, u) = u – x + π – satisfies ( ) in the interval (–π, π), but the solution of the problem u(x) = x – π + sin x is negative in (–π, π) and not an even function. UT,λ p + a(x)upT,λ λ q λb(x)uqT ,λ dx pq uT,λ p + a(x)upT,λ dx ≥ a (q – p) pq uT ,λ p T

We get with

Then for