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

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.

MULTIPLICITY OF SOLUTIONS OF DIRICHLET'S PROBLEM FOR SECOND-ORDER $p$-LAPLACIAN DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

“…Higher-order equations are studied in [8] using the generalized Clark's theorem. It is applied to fourth-order p-Laplacian equations in [10]. The case p = 2 is considered in the thesis of Kalcheva [5], Chapter 2.…”

“…We will apply variational method and symmetric mountain-pass theorem. Our main result is the following: The case p > r is considered in [10] with application of the generalized Clark's theorem. The periodic and homoclinic solutions are studied in [9] in the case p = 2 and for higher order equations in [8].…”