Some existence results for a periodic problem with non-smooth potential

Abstract

In this paper, two existence results for a class of second order periodic boundary value problems with non-smooth potential are obtained. We extend the Castro–Lazer–Thews reduction method to non-smooth functionals, the obtained result is then exploited to prove the existence of a nontrivial solution. Furthermore, we prove the existence of multiple solutions by using a multiplicity result based on local linking.

Introduction

In this paper, we consider the following second order periodic boundary value problem with non-smooth potential:

$$\left(P\right)\{\begin{array}{c}-u^{″}\left(t\right)+g\left(t\right)\in \partial j(t,u\left(t\right))\text{,}t\in (0,2\pi )\text{,}\hfill \\ u\left(0\right)=u\left(2\pi \right)\text{,}{u}^{\prime }\left(0\right)=u^{\prime }\left(2\pi \right)\text{,}\hfill \end{array}$$

where $g:[0,2\pi ]\to \mathbb{R}$ is a bounded measurable function, $j(t,\xi )$ is a locally Lipschitz (not necessarily smooth) in the $\xi$-variable integrand and $\partial j(t,\xi )$ stands for the Clarke subdifferential of the potential function $j(t,\xi )$.

Nonlinear second order differential equations with restoring forces describe, among other things, the dynamics of particles under the action of Newtonian type forces caused by compressed gazes. Second order periodic boundary value problems have been receiving a great deal of attention. In the last decades, the problem of existence of periodic solutions of autonomous and non-autonomous second order differential equations, with or without a friction term, has been investigated by many authors. In 1993, Del Pino and Manasevich [1] proved infinitely many positive $T$-periodic solutions for a nonlinear problem which comes from nonlinear elasticity. The proof of this result is based on the version of Poincare–Birkhoff Theorem due to Ding [2]. In 1996, using a direct application of the Leray–Schauder continuation theorem, Dang and Oppenheimer [3] established some existence and uniqueness results for a class of nonlinear boundary value problems. In 2003, Bonheure and Coster [4] obtained the existence of solution for a class of singular second order periodic boundary value problems by using lower and upper solution methods. In particular, Boucherif, Merzagui [5] considered a class of singular non-autonomous second order differential equations and proved the existence of at least one periodic solution by using a variational approach based on the Mountain Pass Lemma.

The study of problems with non-smooth potential is lagging behind. In 2003, Gasinski and Papageorgiou [6] proved the existence of three distinct periodic solutions for nonlinear second order differential equations by using the non-smooth critical point theory of Chang [7] and its extensions due to Kourogenis and Papageorgiou [8]. In 2005, Bonanno and Giovannelli [9] obtained a multiplicity result for an eigenvalue Dirichlet problem by using a three critical points theorem for non-smooth functionals which introduced by Marano and Motreanu [10] and Ricceri [11]. In particular, using an abstract multiplicity result under local linking and an extension of the Castro–Lazer–Thews reduction method to a non-smooth setting, Gasinski et al. [12] considered a class of resonance semilinear elliptic equation, and obtained that multiplicity of nontrivial solutions; Motreanu et al. [13] established the existence of at least two nontrivial solutions for a class of second order periodic systems with a non-smooth potential and an indefinite linear part.

In this paper, we develop and use a non-smooth variant of the so-called “reduction method” (see [12], [13]). This method was first introduced for smooth problems by Castro and Lazer [14] and Thews [15]. We prove the existence of periodic solution for Problem $\left(P\right)$. On the other hand, using non-smooth critical point theorem and a recent multiplicity result based on local linking, we prove the multiplicity results for Problem $\left(P\right)$. Compared with [12], [13], our hypotheses allow the non-smooth potential to cross the two consecutive eigenvalue of high order and Problem $\left(P\right)$ has a friction term $g\left(t\right)$.

Problems with non-smooth potential are known as hemivariational inequalities. So Problem $\left(P\right)$ is a periodic hemivariational inequality. Hemivariational inequalities are a new type of variational expressions which arise naturally in mechanics and engineering when one wants to consider more realistic models with a non-smooth and nonconvex energy functionals. For several concrete applications e refer to the book of Naniewicz and Panagiotopoulos [16].

This paper is arranged as follows. In Section 2, we collect some basic definitions and state the main results. In Section 3, we establish “reduction method” for non-smooth functionals, and prove the existence of periodic solution for Problem $\left(P\right)$. In Section 4, we obtain a multiplicity result for Problem $\left(P\right)$.