Existence of homoclinic orbits for unbounded time-dependent   p  -Laplacian systems

Adel Daouas

doi:10.14232/ejqtde.2016.1.88

Abstract

Existence of homoclinic orbits for unbounded time-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math>-Laplacian systems

Highlights

Consider the ordinary p-Laplacian system d |u (t)|p−2u (t) − ∇K(t, u(t)) + ∇W(t, u(t)) = f (t) dt where t ∈ R, p > 1, K, W : R × RN → R are C1-maps and f : R −→ RN is a continuous and bounded function
Homoclinic orbits were introduced by Poincaré more than a century ago, and since they became a fundamental tool in the study of chaos
Motivated by the above mentioned works, in the present paper we study the existence of homoclinic solutions for (HS) under more general conditions which cover the case of unbounded potentials with respect to the variable t

Summary

Introduction

Motivated by the above mentioned works, in the present paper we study the existence of homoclinic solutions for (HS) under more general conditions which cover the case of unbounded potentials with respect to the variable t. The existence of such sequence of solutions is guaranteed through a standard version of the Mountain Pass Theorem. If I satisfies the following conditions: (I1) I(0) = 0, (I2) there exist constants ρ, α > 0 such that I|∂Bρ(0) ≥ α, (I3) there exists e ∈ E\Bρ(0) such that I(e) ≤ 0, where Bρ(0) is an open ball in E of radius ρ centered at 0, I possesses a critical value c ≥ α given by c = inf max I(g(s))

Results

Conclusion