Existence and subharmonicity of solutions for nonsmooth $p$-Laplacian systems

1.   Introduction

Consider the nonlinear nonsmooth periodic system with $p$-Laplacian

where $T > 0, \ p > 1$, $F: [0, T]\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ is locally Lipschitz continuous in the vectorial variable $x$ and $\partial F(t, x)$ denotes the generalized subdifferential of $F$ with respect to $x$ in the sense of Clarke (see ^[1]).

When the potential function $F: {\mathbb{R}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ is $T$-period with respect to the first variable, problem (1) becomes the following Hamiltonian system with $p$-Laplacian

A function is called subharmonic solution if it is $kT$-periodic solution for some positive integer $k$ (see ^[2]).

When the potential functional is continuously differentiable (i.e., $F(t, \cdot)\in C^{1}({\mathbb{R}}^{N})$), the existence of periodic solutions and subharmonic solutions of Hamiltonian systems with $p$-Laplacian has been widely concerned by mathematical physicists because of its strong practical significance and theoretical research value (for example ^{[3,4,5,6,7,8,9,10,11,12]} and the references therein). In their papers, the following assumption is always required:

$(A)\ \ F(t, x)$ is measurable in $t$ for every $x\in {\mathbb{R}}^{N}$ and continuously differentiable in $x$ for a.e. $t\in [0, T]$, and there exist $a\in C({\mathbb{R}}^{+}, {\mathbb{R}}^{+}), b\in L^{1}(0, T; {\mathbb{R}}^{+})$ such that

for all $x\in R^{N}$ and a.e. $t\in [0, T]$, where ${\mathbb{R}}^{+}$ is the set of all nonnegative real number.

In recent years, extensive researches on problem (1) have been conducted, for example Gasin'ski and Papageorgiou ^[13] gave the existence and multiplicity of periodic solutions using the nonsmooth mountain lemma and saddle point theorem; Researchers studied the existence and multiplicity of solutions for nonlinear second-order periodic systems with one-dimensional $p$-Laplacian and nonsmooth potentials ^[8,14,15]; Zhang and Liu^[16] obtained the existence of three solutions for the periodic eigenvalue problems driven by the $p$-Laplacian under a bounded interval for the parameter $\lambda$; In ^[17,18], the authors discussed the existence of subharmonic solutions for problem (2).

Inspired by the above papers, we further investigate the periodic solutions and subharmonic solutions for $p$-Hamiltonian systems with nonsmooth potentials. Since the potential is nondifferentiable, the gradient is replaced by the subdifferential and the resulting problem is a quasilinear second order periodic differential inclusion, known as hemivariational inequality. Hemivariational inequalities arise in physical problems, when one wants to consider more realistic models with nonsmooth and non-convex energy functionals. The hemivariational inequalities formalism proved to be an efective analytical tool in the study of many complex mechanical structures, such as multilayered plates, Vonkarman plates in adhesive contact with rigid support, composite structures and others (see ^[7]).

Throughout this paper, we always suppose that $F: [0, T]\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}} \ (N\geq 1)$ satisfies the following assumption $(A^{\prime})$:

$(A^{\prime})$ $F(t, x)$ is measurable in $t$ over $[0, T]$ for each $x\in {\mathbb{R}}^{N}$ and is locally Lipschitz continuous in $x$ for a.e. $t\in [0, T]$, $F(t, 0)\in L^{1}(0, T)$.

We use nonsmooth critical point theories to prove the existence of periodic solutions for problem (1) and subharmonic solutions for problem (2). In the proof of the existence of periodic solutions, we prove that the energy functional satisfies the nonsmooth Cerami condition firstly, and then prove that it has saddle points. Finally we prove that the obtained critical point of the energy function $\varphi$ is the weak solution of the problem (1) (See Theorem 3.1). In particular, in Theorem 4.1, we make use of a weaker condition and prove the existence of the subharmonic solutions for (2), generalizing a result contained in ^[12].

2.   Basic definitions and preliminary results

We start with the subdifferential theory for locally Lipschitz functions. Let $(X, \|\cdot\|)$ be a real Banach space. Denote by $X^{\ast}$ the dual space of $X$, while $\langle\cdot, \cdot\rangle$ stands for the duality pairing between $X$ and $X^{\ast}$. A functional $h:X\rightarrow {\mathbb{R}}$ is called locally Lipschitz continuous if for every $u\in X$ there corresponds a neighborhood $V_{u}$ of $u$ and a constant $L_{u}\geq0$ such that

If $u, v \in X$, we write $h^{0}(u; v)$ for the generalized directional derivative of $h$ at the point $u$ along the direction $v$, i.e.,

It is known that $h^{0}$ is upper semicontinuous on $X\times X$ (see [1,Proposition 2.1.1]).

For locally Lipschitz continuous functionals $h_{1}, h_{2}:X\rightarrow {\mathbb{R}}$, we have

The generalized gradient of the function $h$ at $u$, denoted by $\partial h(u)$, is the set defined by

In view of [1,Proposition 2.1.2] $\partial h(u)$ is a nonempty, convex in addition to weak$^{\ast}$ compact subset of $X^{\ast}$, thus the function $\lambda (x) = \min_{w\in \partial h(x)}\|w\|_{X^{\ast}}$ is well defined and lower semicontinuous, i.e., $\liminf_{x\rightarrow x_{0}}\lambda (x)\geq \lambda (x_{0})$.

If $h_{1}, h_{2}:X\rightarrow {\mathbb{R}}$ are locally Lipschitz continuous, then

A point $u\in X$ is said to be a critical point of $h$ if $h^{0}(u; v)\geq0, \ \forall v\in X.$ In this framework, the functional $h$ is said to satisfy the nonsmooth (PS) condition if any sequence $\{x_{n}\}$ in $X$ such that $\{h(x_{n})\}$ is bounded and $\lambda (x_{n})\rightarrow 0$ possesses a convergent subsequence. Moreover the locally Lipschitz functional $h$ is said to satisfy the nonsmooth Cerami condition if any sequence $\{x_{n}\}^{\infty}_{n = 1}\subset X$ such that $\{h(x_{n})\}$ is bounded and $(1+\|x_{n}\|)\lambda (x_{n})\rightarrow 0\ (n\rightarrow \infty)$ possesses a convergent subsequence. For more details on this subject one could refer to ^{[1,19,20,21,22,23,24]}. For convenience, in what follows we will denote various positive constants as $c_i, \ i = 1, 2, 3\cdots.$

Finally, we shall make use of the following well known results.

Lemma 2.1. ([1,Theorem 2.3.7]) Let $x$ and $y$ be points in $X$, and suppose that $f$ is Lipschitz on open set containing the line segment $[x, y]$. Then there exists a point $u$ in $(x, y)$ such that

Lemma 2.2. ([25,Theorem 7]) If $X = Y\oplus V$, with dim$Y < \infty$, there exists $r > 0$ such that

and $\phi:X\rightarrow {\mathbb{R}}$ satisfies the nonsmooth $(C)_c$-condition where $c = \inf_{{\gamma}\in{\Gamma}}\max_{x\in E}\phi({\gamma}(x))$ with ${\Gamma} = \{{\gamma}\in C(E, X):{\gamma}|_{\partial E} = identity\}, E = \{x\in Y:\|x\|\leq r\}$ and $\partial E = \{x\in Y:\|x\| = r\}$, then $c\geq\inf_V\phi$ and $c$ is a crucial value of $\phi$. Moreover, if $c = \inf_V\phi$, then

Lemma 2.3. ([22,Theorem 3.3]) Let $X$ be a real Banach space, and let $f$ be a locally Lipschitz function defined on $X$ satisfying the nonsmooth (PS) condition. Suppose $X = X_{1}\oplus X_{2}$ with a finite dimensional subspace $X_{1}$, and there exist contants $b_{1} < b_{2}$ and a bounded neighborhood $N$ of $\theta$ in $X_{1}$ such that

Then $f$ has a critical point.

3.   Existence theorems

In this section, we shall give the existence theorems for problem (1).

Theorem 3.1. Suppose $F(t, x)$ satisfies assumption $(A')$ and the following conditions:

$(l_{1})$ For every $r > 0$ there exists $a_r\in L^1([0, T])_+$ such that for a.e. $t\in[0, T]$, all $|x|\leq r$ and $\xi\in\partial F(t, x)$, we have $|\xi|\leq a_r(t)$;

$(l_{2})$ There exist $\mu\in (0, p)$ and $M > 0$ such that

for almost all $t\in[0, T]$, all $|x|\geq M$;

$(l_{3})$ $\int_{0}^{T}F(t, x)dt \rightarrow +\infty$ as $|x|\rightarrow \infty$ uniformly for a.e. $t\in [0, T]$.

Then problem (1) has at least one solution $u\in W^{1, p}_{T}$, where

is the reflexive Banach space with the norm

Proof. We start by observing that, because of hypotheses $(l_{1})$, using the mean value theorem, we have that for a.e. $t\in [0, T]$ and all $x\in {\mathbb{R}}^N$ with $|x|\leq r$,

where $b_r(t) = |F(t, 0)|+a_r(t)$ and

Let $\tilde{W}^{1, p}_T = \{u\in W^{1, p}_T :\ \int^{T}_0 u(t)dt = 0\}$, then $W^{1, p}_{T} = \tilde{W}^{1, p}_T\oplus {\mathbb{R}}^N$, for every $u\in W^{1, p}_{T}$. Put $\bar{u} = \frac{1}{T}\int^{T}_{0}u(t)dt, \, \, \, \tilde{u}(t) = u(t)-\bar{u}$, then $\bar{u}\in {\mathbb{R}}^N, \ \tilde{u}\in \tilde{W}^{1, p}_T$ and the following inequalities hold:

where $C > 0$ is a constant and $\|u\|_{\infty} = \max\limits_{t\in [0, T]}|u(t)|$.

Consider the energy functional $\varphi:W^{1, p}_{T}\rightarrow \mathbb{R}$ for problem (1) defined by

for all $u\in W^{1, p}_{T}$.

It is straightforward to verify that $\varphi$ is well defined and locally Lipschitz continuous on $W^{1, p}_{T}$ (see [1,page 83]) under assumption $(A')$.

Claim 1. $\varphi$ satisfies the nonsmooth Cerami condition.

Let$\{u_n\}\subset W^{1, p}_{T}$ be a sequence such that $|\varphi (u_{n})|\leq M_1$ for some $M_1 > 0$, $\forall\ n\geq 1$ and $(1+\|u_{n}\|)\lambda(u_n)\rightarrow 0$ as $n\rightarrow \infty$.

Since $\partial\varphi(u_n)\subset(W^{1, p}_{T})^*$ is nonempty, weakly compact and the norm functional in a Banach space is weakly lower semicontinuous, by Weierstrass theorem we can find $u^*_n\in\partial\varphi(u_n)$ such that $\lambda(u_n) = \|u^*_n\|_{X^*}, \ n\geq1$, then (see [1,page 76]) there exist $\xi_{n} \in L^{1}(0, T), \ \xi_{n}(t)\in \partial F(t, u_n(t))$ a.e. on $[0, T]$ such that

From the choice of the sequence $\{u_n\}\subset W^{1, p}_{T}$, we have

where $\epsilon_n\downarrow 0$. Since

we deduce

and since $|\varphi (u_{n})|\leq M_1$, we obtain

for all $n\geq 1$. Adding (4) and (5), we have then

Now set $A_n = \Big\{ t\in [0, T]| \ |u_n(t)| < M\Big\}$ and $B_n = \Big\{ t\in [0, T]| \ |u_n(t)|\geq M\Big\}$. From $(l_1), \ (l_2)$ and the properties of $F^{0}$ (see [7,page 545]) we obtain

while

It follows that

then by Poincare-Wirtinger inequality, $\{\tilde{u}_n\}$ is bounded in $W_{T}^{1, p}$ and by Sobolev inequality, we get ${\|\tilde{u}_{n}\|_{\infty}}$ is bounded.

We claim that the sequence $\{\bar{u}_n\}$ is bounded, otherwise, there is a subsequence, again denoted by $\{\bar{u}_n\}$, such that $|\bar{u}_n|\rightarrow \infty \ \hbox{as} \ n\rightarrow \infty$.

Thus

for all $t\in[0, T]$. From the condition $(l_3)$, we have

which contradicts the choice of $\{u_n\}$. Hence $\{u_n\}$ is bounded in $W^{1, p}_T$. By the compactness of the embedding $W^{1, p}_T\subset C(0, T; {\mathbb{R}}^N)$, the sequence $\{u_n\}$ has a subsequence, denoted by $\{u_n\}$ again, such that $u_n \rightharpoonup u$ weakly in $W^{1, p}_T$ and $u_n \rightarrow u$ strongly in $C(0, T; {\mathbb{R}}^N)$.

Note that

and

Since $\{u_n\}$ is bounded and

there exists $M_2 > 0$ such that $|u_n(t)|\leq M_2$ for a.e. $t\in [0, T]$ and $n\geq 1.$ By $(l_2)$, we have

for some positive constants $c_3$.

Hence one has

Besides it is easy to drive from $u_n \rightarrow u$ strongly in $C(0, T; {\mathbb{R}}^N)$ that

Since $\|u\|^p = \|u\|^p_p+\|\dot{u}\|^p_p$ and the norm in a Banach space is weakly lower semicontinuous, we have

Using the H$\ddot{o}$lder inequality, we have

which yields $\|u_n\|\rightarrow \|u\|$. Since $u_n\rightharpoonup u$ weakly in $W^{1, p}_T$ and $\dot{u}_n\rightharpoonup \dot{u}$ in $L^p(T, {\mathbb{R}}^N)$ and the latter space is uniformly convex, by the Kadec-Klee property, we have $\dot{u}_n\rightarrow \dot{u}$ in $L^p(T, {\mathbb{R}}^N)$. Therefore $u_n\rightarrow u$ in $W^{1, p}_T$ and $\varphi$ satisfies the nonsmooth Cerami condition.

Claim 2. $\varphi$ is coercive on $\tilde{W}^{1, p}_T$.

For every $u\in \tilde{W}^{1, p}_T$ we have

where $A = \{ t\in [0, T]: \ |u(t)| < M\}$ and $B = \{ t\in [0, T]: \ |u(t)|\geq M\}$. Note that from the mean value theorem and $(l_1)$, for $|x|\leq M$ and a.e. $t\in [0, T]$, it is possible to find $r\in [0, 1]$ and $\xi \in \partial F(t, rx)$ such that

Therefore we can see that for all $|x|\leq M$ and a.e. $t\in [0, T]$

where $\beta_{M}(t)\in L^1(0, T)_+$. Immediately we have

By $(l_2)$, for a.e. $t\in [0, T]$, all $|x|\geq M$ and all $s\geq 1$, one has $F(t, sx)\leq s^{\mu}F(t, x)$ (see [26,Theorem 3.14]), then

and so

Now from $(6)$ and (7), using the Poincar$\acute{e}$-Wirtinger inequality again, we obtain

Since $\mu < p$, we conclude that $\varphi$ is coercive on $\tilde{W}^{1, p}_T$ as claimed.

Claim 3. $\varphi$ is anticoercive on ${\mathbb{R}}^N$.

Since for $x\in {\mathbb{R}}^N$, $\varphi (x) = -\int ^T_0F(t, x)dt$, the claim is a direct consequence of hypothesis $(l_3)$.

From the claims proved we are in the position of applying Lemma 2.2 and obtaining the existence of a $u\in W^{1, p}_T$ such that $\theta \in \partial \varphi (u)$. Moreover, there exists $\xi(t) \in \partial F(t, u(t))$ a.e. $t\in[0, T]$ such that

which implies

thus

So $u\in W^{1, p}_T$ is a solution of problem (1).

Example 3.2. Let $F: [0, T]\times {\mathbb{R}}\rightarrow {\mathbb{R}}$ be defined as

where $\mu \in (0, p)$. Then

It is easy to verify that $F(t, x)$ satisfies the condition of theorem 3.1.

4.   Subharmonic solutions

Consider problem (2)

where $p > 1$ and $F: {\mathbb{R}}\times {\mathbb{R}}^N \rightarrow {\mathbb{R}}$ is $T$-periodic $(T > 0)$ in its first variable for all $x\in {\mathbb{R}}^N.$

Theorem 4.1. Suppose $F(t, x)$ satisfies the assumption $(A')$ and the following conditions:

$(h_{1})$ There exist $a\in C({\mathbb{R}}^{+}, {\mathbb{R}}^{+}), b\in L^{1}(0, T; {\mathbb{R}}^{+})$ such that

for all $x\in R^{N}$ and a.e. $t\in [0, T]$;

$(h_{2})$ There exist constants $C^{*} > 0, \ K_1 > 0, \ K_2 > 0, \ \alpha \in [0, p-1)$ and a positive function $h\in C({\mathbb{R}}^+, {\mathbb{R}}^+)$ with the properties:

(i) $h(s)\leq h(t)$ for all $s\leq t, \ s, t \in {\mathbb{R}}^+$,

(ii) $h(s+t)\leq C^{*}(h(s)+h(t))$ for all $s, t \in {\mathbb{R}}^+$,

(iii) $0 < h(t)\leq K_1 t^{\alpha}+K_2$ for all $t\in {\mathbb{R}}^+$,

(iv) $h(t)\rightarrow +\infty \ \hbox{as}\ t\rightarrow + \infty.$

Moreover, there exist $f, g\in L^{1}(0, T; {\mathbb{R}}^+)$ such that for a.e. $t\in [0, T]$, all $x\in {\mathbb{R}}^N$ and $\xi \in \partial F(t, x)$, one has

$(h_{3})$

uniformly for a.e. $t\in [0, T]$, where $\frac{1}{p}+\frac{1}{q} = 1$.

Then problem (2) has kT-periodic solution $u_k\in W^{1, p}_{kT}$ for every positive integer k such that $\|u_k\|_{\infty}\rightarrow +\infty$ as $k\rightarrow +\infty$, where $\|u_k\|_{\infty} = max_{0\leq t\leq kT}|u_k(t)|$ and

is the reflexive Banach space with the norm

Proof. For $u\in W^{1, p}_{kT}$, set $\bar{u} = \frac{1}{kT}\int^{kT}_{0}u(t)dt, \, \, \, \tilde{u}(t) = u(t)-\bar{u}$ and $\tilde{W}^{1, p}_{kT} = \{u\in W^{1, p}_{kT} |\ \int^{kT}_0 u(t)dt = 0\}$, then $W^{1, p}_{kT} = \tilde{W}^{1, p}_{kT}\oplus {\mathbb{R}}^N$. By [2,Proposition 1.1], there exists a constant $C_k > 0$ such that

and

for every $u\in W^{1, p}_{kT}$. Hence

By assumption $(A')$, the corresponding energy functional $\varphi_{k}:W^{1, p}_{kT}\rightarrow \mathbb{R}$ of problem (2) defined by

is locally Lipschitz continuous on $W^{1, p}_{kT}$ and for every $u\in W^{1, p}_{kT}$ and $u^*\in\partial\varphi_k(u)$ there exists $\xi \in L^{1}(0, kT), \ \xi(t)\in \partial F(t, u(t))$ a.e. on $[0, kT]$ such that

First, we prove that $\varphi_k$ satisfies the nonsmooth (PS) condition on $W^{1, p}_{kT}$.

Let$\{u_n\}\subset W^{1, p}_{kT}$ be a sequence such that $\{\varphi_k (u_{n})\}$ is bounded and $\lambda_k(u_n)\rightarrow 0$ as $n\rightarrow \infty$, where $\lambda_k(x) = \min_{w\in \partial\varphi_k(x)}\|w\|$.

Since $\partial\varphi_k(u_n)\subset(W^{1, p}_{kT})^*$ is nonempty, weakly compact and the norm functional in a Banach space is weakly lower semicontinuous, by Weierstrass theorem we can find $u^*_n\in\partial\varphi_k(u_n)$ such that $\lambda_k(u_n) = \|u^*_n\|_{X^*}, \ n\geq1$, then there exists $\xi_{n}(t)\in \partial F(t, u_n(t))$ such that

Since ${\varphi_k (u_{n})}$ is bounded and $\lambda_k(u_n)\rightarrow 0$ as $n\rightarrow \infty$, there exists $M_3 > 0$ such that $|\varphi_k (u_{n})|\leq M_3$ and $\|u^*_n\|\leq 1$ when $n$ is large enough, hence $|\langle u^{*}_n, v\rangle|\leq \|v\|$ for large $n$.

By condition $(h_2)$, Sobolev inequality and Young inequality, we have

Hence,

for large $n$.

Since $\alpha < p-1$ and by Wirtinger inequality

we obtain

for all large $n$, which implies that

Then

for all large $n$ and every $t\in [0, kT]$.

We claim that ${|\bar{u}_n|}$ is bounded, if not, without loss of generality we may assume that

Since $0\leq\alpha < p-1, \frac{1}{p}+\frac{1}{q} = 1$, we have $\frac{{\alpha} q}{p} < 1$. From (9), one has

for all large $n$ and every $t\in [0, kT]$. Then we have

In virtue of $(h_3)$ and the T-periodicity of $F(t, x)$, for every $\beta > 0$, there exists $M_4\geq 1$ such that

for all $|x|\geq M_4$. By (9) and (10), when $n$ is large enough, one has

Thus

for all large $n$. So by the arbitrariness of $\beta$, one has

Since $|\bar{u}_n|\rightarrow \infty$, by $(iv)$ of $(h_2)$ and $(h_3)$, $h(|\bar{u}_n|)\rightarrow + \infty\ \ \hbox{as} \ n\rightarrow \infty,$ thus $\varphi_k(u_n) = -\infty,$ which contradicts the boundedness of $\varphi_k(u_n)$. Hence $\{|\bar{u}_n|\}$ is bounded. Furthermore, by (8) and $(iii)$ of $(h_2)$, we know $\{u_n\}$ is bounded. Arguing then as the proof of Theorem 3.1, we conclude that $\varphi_k$ satisfies nonsmooth (PS) condition.

Next we verify the following condition:

$(\Pi _1)$ $\varphi_k(u)\rightarrow +\infty$ as $\|u\|\rightarrow \infty$ in $\tilde{W}^{1, p}_{kT}$;

$(\Pi _2)$ $\varphi_k(x+e_k(t))\rightarrow -\infty$ as $|x|\rightarrow \infty$ in ${\mathbb{R}}^N$, where $e_k(t) = kcos(k^{-1}\omega t)x_0\in\tilde{W}^{1, p}_{kT}, x_0\in {\mathbb{R}}^N$, $|x_0| = 1$ and $\omega = \frac{2\pi}{T}$.

For every $u\in\tilde{W}^{1, p}_{kT}$, it follows from the Sobolev inequality that there exist $s\in[0, 1]$ and $\xi(t)\in\partial F(t, su(t))$ such that

Since $p > 1$ and ${\alpha} < p-1$, then $\varphi_k(u)\rightarrow +\infty$ as $\|u\|\rightarrow \infty$ in $\tilde{W}^{1, p}_{kT}$, which proves $(\Pi _1)$.

For all $x\in {\mathbb{R}}^N$, it follows from $(11)$ that

for all $|x|\geq M_3+k$. By the arbitrariness of ${\beta}$, one has

Thus $(\Pi_2)$ is satisfied. By $(\Pi_1)$, $(\Pi_2)$ and the nonsmooth saddle point theorem, there exists a critical point $u_k\in\tilde{W}^{1, p}_{kT}$ for $\varphi_k$ such that

For fixed $x\in {\mathbb{R}}^N$, set

Then we have $meas A_k\leq\frac{kT}{2}$ for all large $k$. In fact if $meas A_k > \frac{kT}{2}$, there exists $t_1\in A_k$ such that

or

Moreover, there exists $t_2\in A_k$ such that

and

It follows that

and

From (13) and (14) we obtain

Furthermore, by (12) we have

But due to $t_1, \ t_2 \in A_k$, one has

which is a contradiction for large $k$. Hence

for large $k$. From $(h_1)$ and $(h_3)$, we have

for every $x\in {\mathbb{R}}^N$ and all large $k$, which implies that

By the arbitrariness of $\beta$, we obtain

which follows that

Now we prove that $\|u_k\|_{\infty}\rightarrow +\infty$ as $k\rightarrow \infty$. If not, going to a subsequence if necessary, we may assume that

for all $k\in N$. Hence, by $(h_1)$ we have

it follows that

which contradicts (15), therefore by Lemma 2.3 the proof is completed.

Remark 4.2. Theorem $4.1$ generalizes [12,Theorem 1.2] and the conclusion in the document ^[17]. There exists function $F$ satisfying the conditions in Theorem 4.1 but not satisfying conditions in ^[12,17]. For example, let

for all $x\in {\mathbb{R}}^N$ and $t\in {\mathbb{R}}$, where $\omega = \frac{2\pi}{T}$. It is clear that $F$ is locally Lipschitz continuous in $x$ and

Moreover, one has

for any $\alpha \in (\frac{1}{2}, 1)$ and $t\in {\mathbb{R}}$. Hence this example can not be solved by the results in ^{[11,15,27,28]} even when $p = 2$.

5.   Conclusions

In this paper we investigate the existence and subharmonicity of solutions for two nonsmooth $p$-Laplacian systems. We use nonsmooth critical point theories to prove the existence of periodic solutions for problem (1) and subharmonic solutions for problem (2). Since the potential is nondifferentiable, the gradient is replaced by the subdifferential and the resulting problem is a quasilinear second order periodic differential inclusion, known as hemivariational inequality. In particular, we make use of a weaker condition and prove the existence of the subharmonic solutions for (2), generalizing the results of the reference. Thus the results we obtain could be applied more widely.

Acknowledgments

The authors are grateful to the reviewers for their valuable comments. The work was supported by the NSF of Shandong Province (No. ZR2018PA006), SNSFC (No. ZR2020MA005) and the Research Start-up Foundation of Jining University (Nos. 2018BSZX01, 2017BSZX01).

Conflict of interest

All authors declare no conflicts of interest in this paper.