Infinitely Many Homoclinic Solutions for Fourth Order p-Laplacian Differential Equations

Stepan Tersian

doi:10.3390/math8040505

Abstract

The existence of infinitely many homoclinic solutions for the fourth-order differential equation φ p u ″ t ″ + w φ p u ′ t ′ + V ( t ) φ p u t = a ( t ) f ( t , u ( t ) ) , t ∈ R is studied in the paper. Here φ p ( t ) = t p − 2 t , p ≥ 2 , w is a constant, V and a are positive functions, f satisfies some extended growth conditions. Homoclinic solutions u are such that u ( t ) → 0 , | t | → ∞ , u ≠ 0 , known in physical models as ground states or pulses. The variational approach is applied based on multiple critical point theorem due to Liu and Wang.

Highlights

We study the existence of infinitely many nonzero solutions homoclinic solutions for the fourth-order p-Laplacian differential equation φ p u00 (t)
We say that a solution u of (1) is a nontrivial homoclinic solution to zero solution of (1) if u 6= 0, u(t) → 0
The existence of homoclinic and heteroclinic solutions of fourth-order equations is studied by various authors

Summary

Introduction

We study the existence of infinitely many nonzero solutions homoclinic solutions for the fourth-order p-Laplacian differential equation φ p u00 (t). Sun and Wu [4] obtained existence of two homoclinic solutions for a class of fourth-order differential equations: u(4) + wu00 + a(t)u = f (t, u) + λh(t) |u| p−2 u, t ∈ R, where w is a constant, λ > 0, 1 ≤ p < 2, a ∈ C (R, R+ ) and h ∈ L 2− p (R) by using mountain pass theorem. Yang [8] studies the existence of infinitely many homoclinic solutions for a the fourth-order differential equation: u(4) + wu00 + a(t)u = f (t, u), t ∈ R, where w is a constant, a ∈ C (R) and f ∈ C (R × R, R).

Preliminaries

Proof of the Main Result

Conlusions