Periodic solutions of a class of nonautonomous second-order Hamiltonian systems with nonsmooth potentials

Abstract

This paper is concerned with nonautonomous second-order Hamiltonian systems with nondifferentiable potentials. By using the nonsmooth critical point theory for locally Lipschitz functionals, we obtain some new existence results for the periodic solutions.

1 Introduction and main results

In this paper we consider the following second-order differential inclusions systems:

$$ \left \{ \begin{array}{l} \ddot{u}(t) + A(t)u(t) \in\partial F(t,u(t)) \quad \mbox{a.e. } t \in[0,T], \\ u(0)-u(T) = \dot{u}(0) - \dot{u}(T) = 0, \end{array} \right . $$

(1)

where $T>0$, $A(t)$ is a continuous symmetric matrix of order N and $F: [0,T]\times{\mathbb{R}}^{N}\rightarrow \mathbb{R}$ is locally Lipschitz continuous in x and $\partial F(t,x)$ denotes the Clarke subdifferential of F for x. There have been a lot of contributions on problem (1); see, for example, [1–3] and the references therein.

When $F(t,x)$ is continuously differentiable in x, problem (1) becomes the second-order Hamiltonian system

$$ \left \{ \begin{array}{l} \ddot{u}(t) + A(t)u(t) = \nabla F(t,u(t))\quad \mbox{a.e. } t \in[0,T], \\ u(0)-u(T) = \dot{u}(0) - \dot{u}(T) = 0. \end{array} \right . $$

(2)

The smooth system (2) has also been studied in the past decades and many excellent results appeared; see [4, 5]. In those works, the following assumption is necessary:

(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

$$\bigl\vert F(t,x)\bigr\vert \leq a\bigl(\vert x\vert \bigr)b(t), \qquad \bigl\vert \nabla F(t,x)\bigr\vert \leq a\bigl(\vert x\vert \bigr)b(t) $$

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

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

(A′):

$F(t,x)$ is integrable in t over $[0,T]$ for each $x\in{\mathbb{R}}^{N}$ and locally Lipschitz continuous in x for each $t\in[0,T]$.

Let $H^{1}_{T}$ be the usual Sobolev space with norm

$$\|u\|=\biggl(\int^{T}_{0}\bigl\vert u(t)\bigr\vert ^{2}\, dt+ \int^{T}_{0}\bigl\vert \dot{u}(t)\bigr\vert ^{2}\, dt \biggr)^{1/2}. $$

Definition 1.1

We call $u\in H^{1}_{T}$ a weak solution of (1) if the following inequality holds:

$$ -\int^{T}_{0}\bigl(\dot{u}(t), \dot{v}(t)\bigr)\, dt+\int^{T}_{0} \bigl(A(t)u(t),v(t)\bigr)\, dt\leq \int^{T}_{0}F^{0} \bigl(t,u(t);v(t)\bigr)\, dt, \quad \forall v\in H^{1}_{T}, $$

(3)

where $F^{0}(t, x; y)$ denotes the generalized directional derivative of F at x along the direction y.

The main results of this paper are as follows.

Theorem 1.2

Suppose $F(t, x)$ satisfies (A′) and the following conditions:

(i₁):

There exist $f, g \in L^{1}(0,T;{\mathbb {R}}^{+})$ and $\alpha\in[0,1)$ such that for all $x \in {\mathbb{R}}^{N}$ and a.e. $t\in[0,T]$,

$$ \xi\in\partial F(t,x) \quad \Rightarrow\quad |\xi|\leq f(t)|x|^{\alpha}+ g(t). $$

(4)

(i₂):

There exists a $\gamma\in L^{1}(0,T; {\mathbb{R}}^{+})$ such that for all $x \in{\mathbb {R}}^{N}$ and a.e. $t\in[0,T]$,

$$ \frac{F(t,x)}{|x|^{2\alpha}}\leq\gamma(t). $$

(5)

(i₃):

There exists a subset E of $[0,T]$ with $\operatorname{meas}(E)>0$ such that for a.e. $t\in E$,

$$ \frac{F(t,x)}{|x|^{2\alpha}}\rightarrow-\infty\quad \textit{as } |x| \rightarrow\infty. $$

(6)

Then problem (1) possesses at least one weak solution.

Theorem 1.3

Suppose $F(t, x)$ satisfies (A′), (i₁) above and the following conditions:

($\mathrm{i}_{2}^{\prime}$):

There exists $\gamma\in L^{1}(0,T;{\mathbb{R}}^{+})$ such that

$$ \frac{F(t,x)}{|x|^{2\alpha}}\geq- \gamma(t) $$

(5a)

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

($\mathrm{i}_{3}^{\prime}$):

There exists a subset E of $[0,T]$ with $\operatorname{meas}(E)>0$ such that

$$ \frac{F(t,x)}{|x|^{2\alpha}}\rightarrow+ \infty \quad \textit{as } |x| \rightarrow\infty $$

(6a)

for a.e. $t\in E$.

Then problem (1) possesses at least one weak solution.

The method in our paper is based on the nonsmooth least action principle and saddle point theorem initiated by Chang [6], different from [1] and [2] based on the nonsmooth mountain pass theorem. Moreover, we consider the case that the growth of the subdifferential $\partial F(t,x)$ in x is sublinear ($\alpha \in[0,1)$), while in [1] and [2] the authors considered the case $\alpha\geq1$.

Barletta and Papageorgiou [3, Theorem 3.4] proved the existence of solutions for problem (1), where they assumed that

• $F(t,x)\geq- \gamma(t)$, $\forall x\in{\mathbb {R}}^{N}$, $t\in[0,T]$,

• $F(t,x)\rightarrow+ \infty$ as $|x|\rightarrow\infty$ and $t\in E$,

• $\dim(\operatorname{span} \{u\in H^{1}_{T} | -\ddot{u} - A(t)u = \lambda u \mbox{ for some } \lambda<0 \})=0$.

By comparison, the first two assumptions are strengthened to ($\mathrm{i}'_{2}$) and ($\mathrm{i}'_{3}$), but the third one is not necessarily needed in the present paper.

Remark 1.4

If $F(t,x)$ is measurable in t for every $x\in{\mathbb{R}}^{N}$ and continuously differentiable in x for a.e. $t\in[0,T]$, the inequality (4) becomes

$$\bigl\vert \nabla F(t,x)\bigr\vert \leq f(t)|x|^{\alpha}+ g(t). $$

Meanwhile, the above inequality (3) takes the form

$$-\int^{T}_{0}\bigl(\dot{u}(t),\dot{v}(t)\bigr)\, dt+\int^{T}_{0}\bigl(A(t)u(t),v(t)\bigr)\, dt = \int^{T}_{0}\bigl(\nabla F\bigl(t,u(t)\bigr);v(t) \bigr)\, dt, \quad \forall v\in H^{1}_{T}. $$

Then Theorem 1.2 and Theorem 1.3 generalize Theorem 2 and Theorem 3 of [5], respectively, without assuming the linear second-order system $\ddot{u}(t) + A(t)u(t)=0$ a.e. $t\in[0,T]$ has a nonzero solution, which is necessary in [5].

Example 1.5

Let

$$F(t,x)= - |\sin\omega t|\bigl(|x|^{1+r}+|x|\bigr) $$

for all $(t,x)\in[0,T]\times{\mathbb{R}}^{N}$, where $r \in[0,1)$, $\omega=\frac{2\pi}{T}$. Then F satisfies the conditions of Theorem 1.2, but it is not covered by the results of [1, 2, 4, 5].

2 Basic definitions and preliminary results

Let $(X,\|\cdot\|)$ be a real Banach space. We 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 correspond a neighborhood $V_{u}$ of u and a constant $L_{u}\geq0$ such that

$$\bigl\vert h(z)-h(w)\bigr\vert \leq L_{u}\|z-w\|,\quad \forall z,w\in V_{u}. $$

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.,

$$h^{0}(u;v):=\limsup_{w\rightarrow u,t\rightarrow0^{+}} \frac {h(w+tv)-h(w)}{t}. $$

It is well known that $h^{0}$ is upper semicontinuous on $X\times X$ (see [7], Proposition 2.1.1).

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

$$(h_{1}+h_{2})^{0}(x; z)\leq h_{1}^{0}(x; z)+h_{2}^{0}(x;z), \quad \forall x,z\in X. $$

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

$$\partial h(u):=\bigl\{ u^{\ast}\in X^{\ast}:\bigl\langle u^{\ast},v\bigr\rangle \leq h^{0}(u;v), \forall v\in X\bigr\} . $$

Proposition 2.1.2 of [7] ensures that $\partial h(u)$ turns out nonempty, convex, weak^∗ compact, thus the function $\lambda(x)= \min_{w\in\partial h(x)}\|w\| _{X^{\ast}}$ exists and is lower semicontinuous, i.e., $\liminf_{x\rightarrow x_{0}}\lambda(x)\geq\lambda(x_{0})$.

If $f, g: X\rightarrow X$ be locally Lipschitz continuous, then

$$\partial(f+g) (x)\subset\partial f(x)+ \partial g(x). $$

A point $u\in X$ is said to be a critical point of h if

$$h^{0}(u;v)\geq0, \quad \forall v\in X. $$

We say the locally Lipschitz functional h satisfies the nonsmooth (PS) condition if any sequence $\{x_{n}\}$ in X such that $\{h(x_{n})\}$ is bounded and $\lambda(x_{n})\rightarrow0$ possesses a convergent subsequence.

For more details, we can refer to [8–11]. To prove Theorem 1.2 and Theorem 1.3 in the next section, first we state the following well-known results.

Lemma 2.1

([7], 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

$$f(y)- f(x) \in\bigl\langle \partial f(u), y-x\bigr\rangle . $$

Lemma 2.2

([6], 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 constants $b_{1}< b_{2}$ and a bounded neighborhood N of θ in $X_{1}$ such that

$$f|_{X_{2}}\geq b_{2}, \qquad f| _{\partial N}\leq b_{1}. $$

Then f has a critical point.

Lemma 2.3

([12], Lemma 2)

Suppose that E is a measurable subset of $[0,T]$ and $G(t,x)$ is continuous in x for a.e. $t\in E$. Assume that

$$G(t,x)\rightarrow-\infty \quad \textit{as } |x|\rightarrow\infty $$

for a.e. $t\in E$. Then for every $\delta> 0$ there exists a subset $E_{\delta}$ of E with $\operatorname{meas}(E / E_{\delta})< \delta$ such that

$$G(t,x)\rightarrow-\infty\quad \textit{as } |x|\rightarrow\infty $$

uniformly for all $t\in E_{\delta}$.

Lemma 2.4

([6], Theorem 3.5)

Suppose a locally Lipschitz functional f, defined on a reflexive Banach space X, satisfies the nonsmooth (PS) condition and is bounded from below. Then $c= \inf_{X} f(x)$ is a critical value of f.

3 Proof of theorems

For every $u\in H_{T}^{1}$, let $\bar{u}=\frac{1}{T}\int^{T}_{0}u(t)\, dt$, $\tilde{u}(t)=u(t)-\bar{u}$. Then the following inequalities hold:

$$\begin{aligned}& \|\tilde{u}\|^{2}_{\infty}\leq\frac{T}{12} \int^{T}_{0}\bigl\vert \dot {u}(t)\bigr\vert ^{2}\, dt \quad \mbox{(Sobolev's inequality)}, \\& \int^{T}_{0}\bigl\vert \tilde{u}(t)\bigr\vert ^{2}\, dt\leq\frac{T^{2}}{4\pi^{2}}\int^{T}_{0} \bigl\vert \dot{u}(t)\bigr\vert ^{2}\, dt\quad \mbox{(Wirtinger's inequality)}, \\& \|u\|_{\infty}\leq C\|u\|, \end{aligned}$$

(7)

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

Define two functionals $\varphi:H_{T}^{1}\rightarrow {\mathbb{R}}$ and $\psi: H_{T}^{1}\rightarrow{\mathbb {R}}$ as follows:

$$\begin{aligned}& \varphi(u)=\frac{1}{2}\int^{T}_{0}\bigl\vert \dot {u}(t)\bigr\vert ^{2}\, dt-\frac{1}{2}\int ^{T}_{0}\bigl(A(t)u(t),u(t)\bigr)\, dt+\int ^{T}_{0}F\bigl(t,u(t)\bigr)\, dt, \\& \psi(u)=\int^{T}_{0}F\bigl(t,u(t)\bigr)\, dt. \end{aligned}$$

It is easy to verify that they are locally Lipschitz continuous on $H_{T}^{1}$, so it makes sense to consider their generalized directional derivatives $\varphi^{0}$ and $\psi^{0}$:

$$\varphi^{0}(u;v)=\int^{T}_{0}\bigl( \dot{u}(t),\dot{v}(t)\bigr)\, dt-\int^{T}_{0} \bigl(A(t)u(t),v(t)\bigr)\, dt+\psi^{0}(u;v), \quad \forall v\in H^{1}_{T}. $$

Equation (9) at p.84 of [7] gives

$$\psi^{0}(u;v)\leq\int^{T}_{0}F^{0} \bigl(t,u(t);v(t)\bigr)\, dt, \quad \forall u, v\in H^{1}_{T}. $$

Moreover, by [7], Theorem 2.7.5, one has

$$\partial\psi(u) \subset\int^{T}_{0}\partial F \bigl(t,u(t)\bigr)\, dt, $$

i.e., to every $\xi\in\partial\psi(u)$ there corresponds a mapping $t \rightarrow q(t)$ from $[0,T]$ to $(H_{T}^{1})^{\ast}$ with $q(t) \in\partial F(t,u(t))$ a.e. $t \in[0,T]$ such that for every $v\in H^{1}_{T}$,

$$\langle\xi,v\rangle= \int_{0}^{T}\bigl( q(t),v(t) \bigr)\, dt. $$

If $u\in H^{1}_{T}$ is a critical point of φ, i.e., $\theta\in\partial\varphi(u)$, then

$$\begin{aligned}& \int^{T}_{0}\bigl(\dot{u}(t),\dot{v}(t)\bigr) \,dt-\int^{T}_{0}\bigl(A(t)u(t),v(t)\bigr)\,dt+ \int^{T}_{0}F^{0}\bigl(t,u(t);v(t)\bigr) \,dt \\& \quad \geq\varphi^{0}(u;v)\geq0 ,\quad \forall v\in H^{1}_{T}. \end{aligned}$$

Thus, the critical points of φ correspond to the solutions of problem (3).

In the same time, there exists $q_{0}(t)\in\partial F(t,u)$ such that for all $v\in H_{T}^{1}$,

$$0=\langle\theta, v\rangle=\int_{0}^{T}\bigl( \dot{u}(t),\dot {v}(t)\bigr)\,dt-\int_{0}^{T} \bigl(A(t)u(t),v(t)\bigr)\,dt+\int^{T}_{0}\bigl( q_{0}(t), v(t)\bigr)\,dt, $$

it follows easily that $q_{0}(t) = \ddot{u}(t)$ a.e. $t\in[0,T]$, thus

$$\ddot{u}(t)+A(t)u(t) \in\partial F\bigl(t,u(t)\bigr) \quad \mbox{a.e. on } [0,T], $$

so that u satisfies the system (1) too.

Define the subspaces of $H^{1}_{T}$ by

$$\begin{aligned}& W^{-}\triangleq\operatorname{span} \bigl\{ u\in H^{1}_{T} | -\ddot{u} - A(t)u = \lambda u \mbox{ for some } \lambda< 0 \bigr\} , \\& V\triangleq\operatorname{span} \bigl\{ u\in H^{1}_{T} | - \ddot{u} - A(t)u =0\bigr\} , \\& W^{+}\triangleq\operatorname{span} \bigl\{ u\in H^{1}_{T} | -\ddot{u} - A(t)u = \lambda u \mbox{ for some } \lambda>0\bigr\} . \end{aligned}$$

It is easy to verify that $W^{-}$ is finite-dimensional and there exists $\delta>0$ such that

$$\begin{aligned}& \int^{T}_{0}\bigl\vert \dot{u}(t)\bigr\vert ^{2}\,dt - \int^{T}_{0}\bigl(A(t)u(t), u(t)\bigr)\,dt \geq\delta\|u\|^{2}, \quad \forall u\in W^{+}, \\& \int^{T}_{0}\bigl\vert \dot{u}(t)\bigr\vert ^{2}\,dt - \int^{T}_{0}\bigl(A(t)u(t), u(t)\bigr)\,dt \leq- \delta\|u\|^{2}, \quad \forall u\in W^{-}. \end{aligned}$$

Decompose $H^{1}_{T}$ as $H^{1}_{T}=W^{-}\oplus V\oplus W^{+}$, and denote $W=W^{+}+W^{-}$.

Lemma 3.1

Suppose that (5) and (6) hold. Assume that $u_{n}=v_{n}+w_{n}$, $v_{n}\in V$, $w_{n}\in W$ satisfying $\|u_{n}\|\rightarrow\infty$ ($n\rightarrow\infty$) and $\limsup_{n\rightarrow \infty}\frac{\|w_{n}\|}{\|u_{n}\|^{\alpha}}< +\infty$. Then

$$\|u_{n}\|^{-2\alpha}\int^{T}_{0}F \bigl(t,u_{n}(t)\bigr)\,dt\rightarrow-\infty \quad \textit{as } n \rightarrow\infty. $$

Proof

As the proof of Lemma 3 in [5], for every $\beta> 0$, there exists $m_{\beta}> 0$ such that

$$\operatorname{meas}\bigl\{ t\in(0,T) | \bigl\vert v(t)\bigr\vert < m_{\beta}\|v\|\bigr\} <\beta $$

for all $v\in V$.

Let $B=\{t\in(0,T) | |v(t)|\geq m_{\beta}\|v\|\}$ for all $v\in V$, then we have $\operatorname{meas} ((0,T)\setminus B)<\beta$. By (6) and Lemma 2.3, there exists subset $E_{\delta}$ of E with $\operatorname{meas} (E\setminus E_{\delta})<\delta$ such that

$$ \frac{F(t,x)}{|x|^{2\alpha}}\rightarrow-\infty \quad \mbox{as } |x| \rightarrow\infty $$

(8)

uniformly for all $t\in E_{\delta}$. Hence,

$$ \operatorname{meas} (B\cap E_{\delta})\geq \operatorname{meas} (E_{\delta})- \operatorname{meas} \bigl((0,T)\setminus B \bigr)\geq \operatorname{meas} E -\delta-\beta> 0 $$

(9)

for δ and β small enough.

By (8), for every $\eta> 0$, there exists an $M > 0$ such that

$$\frac{F(t,x)}{|x|^{2\alpha}}\leq-\eta $$

for all $|x|\geq M$ and a.e. $t\in E_{\delta}$. Furthermore, it follows from (5) that

$$F(t,x)\leq-\eta|x|^{2\alpha}+ \gamma_{0}(t) $$

for all $x\in {\mathbb{R}}^{N}$, a.e. $t\in E_{\delta}$ and some $\gamma_{0}(t)=M^{2\alpha}(\eta+\gamma(t))$.

From (9) we obtain

$$\int_{B\cap E_{\delta}}F(t,v)\,dt\leq- \eta m^{2\alpha}_{\beta} \|v\| ^{2\alpha}\operatorname{meas}(B\cap E_{\delta}) + \int ^{T}_{0}\gamma_{0}(t)\,dt. $$

By (5) and (7) we have

$$\int_{[0,T]\setminus(B\cap E_{\delta})}F(t,v)\,dt\leq C^{2\alpha}\|v\| ^{2\alpha}\int^{T}_{0}\gamma(t)\,dt. $$

Since $\|u_{n}\|\rightarrow\infty$ ($n\rightarrow\infty$) and $\limsup_{n\rightarrow\infty}\frac{\|w_{n}\|}{\|u_{n}\|^{\alpha}}< +\infty$, by a simple computation we obtain $\|v_{n}\|\rightarrow\infty$ as $n\rightarrow\infty$, and $V \neq\{0\}$.

Hence, one has

$$\limsup_{v\in V, \|v\|\rightarrow\infty}\|v\|^{-2\alpha}\int^{T}_{0}F(t,v) \,dt\leq-\eta m^{2\alpha}_{\beta} \operatorname{meas}(B\cap E_{\delta})+ C^{2\alpha}\int^{T}_{0} \gamma(t)\,dt, $$

which implies that

$$ \limsup_{n\rightarrow\infty}\|v_{n}\|^{-2\alpha} \int^{T}_{0}F(t,v_{n})\,dt\rightarrow- \infty $$

(10)

by the arbitrariness of η.

By (4), (7), and Lemma 2.1, there exist $s\in[0,1]$ and $\xi_{n}\in\partial F(t,v_{n}+sw_{n})$ such that

$$\begin{aligned}& \biggl\vert \int^{T}_{0}F(t,u_{n}) \,dt -\int^{T}_{0}F(t,v_{n})\,dt\biggr\vert \\& \quad =\biggl\vert \int^{T}_{0} \langle \xi_{n},u_{n}-v_{n}\rangle \,dt\biggr\vert \leq\int^{T}_{0}|\xi_{n}||w_{n}| \,dt \leq\int^{T}_{0}\bigl(f(t)|v_{n}+ sw_{n}|^{\alpha}+ g(t)\bigr)|w_{n}|\,dt \\& \quad \leq\int^{T}_{0}f(t) \bigl(|v_{n}|^{\alpha}+ |w_{n}|^{\alpha }\bigr)|w_{n}|\,dt+\int ^{T}_{0}g(t)|w_{n}|\,dt \\& \quad \leq\|f\|_{L^{1}}\bigl(\|v_{n}\|^{\alpha}_{\infty}+ \|w_{n}\|^{\alpha }_{\infty}\bigr)\|w_{n} \|_{\infty}+ \|g\|_{L^{1}}\|w_{n}\|_{\infty} \\& \quad \leq C^{\alpha+1}\|f\|_{L^{1}}\bigl(\|v_{n} \|^{\alpha}+\|w_{n}\|^{\alpha }\bigr)\|w_{n}\|+ C \|g\|_{L^{1}}\|w_{n}\| \end{aligned}$$

(11)

for all n. Since $\limsup_{n\rightarrow\infty}\frac{\|w_{n}\|}{\| u_{n}\|^{\alpha}}< +\infty$, one has

$$C_{1}\triangleq\limsup_{n\rightarrow\infty} \biggl\vert \|u_{n}\|^{-2\alpha }\int^{T}_{0}F(t,u_{n}) \,dt - \|u_{n}\|^{-2\alpha}\int^{T}_{0}F(t,v_{n}) \,dt \biggr\vert < +\infty. $$

It follows from (10) that

$$\begin{aligned} \limsup_{n\rightarrow\infty} \|u_{n}\|^{-2\alpha}\int ^{T}_{0}F(t,u_{n})\,dt &\leq\limsup _{n\rightarrow\infty} \|u_{n}\|^{-2\alpha}\int ^{T}_{0}F(t,v_{n})\,dt + C_{1} \\ &\leq\limsup_{n\rightarrow\infty} \|v_{n}\|^{-2\alpha}\int ^{T}_{0}F(t,v_{n})\,dt +C_{1} = -\infty. \end{aligned}$$

Hence we obtain

$$\|u_{n}\|^{-2\alpha}\int^{T}_{0}F(t,u_{n}) \,dt\rightarrow-\infty $$

as $n\rightarrow\infty$, which completes the proof. □

Lemma 3.2

Under the conditions (4), (5) and (6), φ satisfies the nonsmooth (PS) condition.

Proof

Let $\{u_{n}\}$ be a sequence in $H_{T}^{1}$ such that $\{\varphi(u_{n})\}$ is bounded and $\lambda(u_{n})= \min_{x^{\ast}\in\partial\varphi(u_{n})}\| x^{\ast}\|\rightarrow0$ as $n\rightarrow\infty$. Put $u_{n}^{\ast}\in\partial\varphi(u_{n})$ such that $\| u_{n}^{\ast}\|= \lambda(u_{n})=o(1)$, then there exists some integer $n_{0}$ such that for each $n\geq n_{0}$, we have

$$\bigl\vert \bigl\langle u_{n}^{\ast}, h\bigr\rangle \bigr\vert \leq\|h\|\quad \mbox{for all } h\in H_{T}^{1}. $$

Let $q_{n}(t) \in\partial(F(t,u_{n}(t)))$ such that

$$\bigl\langle u_{n}^{\ast}, h\bigr\rangle = \int ^{T}_{0}\bigl(\dot{u}_{n}(t), \dot {h}(t)\bigr)\,dt -\int^{T}_{0} \bigl(A(t)u_{n}(t), h(t)\bigr)\,dt + \int^{T}_{0} \bigl( q_{n}(t), h(t)\bigr) \,dt,\quad \forall h \in H_{T}^{1}. $$

Firstly, we show that $\{u_{n}\}$ is bounded. If $\{u_{n}\}$ is unbounded, without loss of generality we may assume that $\|u_{n}\|\rightarrow\infty$ as $n\rightarrow\infty$. Split $u_{n}=v_{n}+w_{n}=v_{n}+w_{n}^{+}+w_{n}^{-}\subset V\oplus W^{+}\oplus W^{-}$. It follows from (4) and (7) that

$$\begin{aligned} \biggl\vert \int^{T}_{0}\bigl\langle q_{n}(t), w_{n}^{+}\bigr\rangle \,dt\biggr\vert &\leq\| f\|_{L^{1}}\|u_{n}\|^{\alpha}_{\infty}\bigl\Vert w_{n}^{+}\bigr\Vert _{\infty}+ \|g \|_{L^{1}}\bigl\Vert w_{n}^{+}\bigr\Vert _{\infty} \\ &\leq C^{\alpha+1}\|f\|_{L^{1}}\|u_{n}\|^{\alpha} \bigl\Vert w_{n}^{+}\bigr\Vert + C\|g\|_{L^{1}} \bigl\Vert w_{n}^{+}\bigr\Vert \end{aligned}$$

for all n. Hence,

$$\begin{aligned} \bigl\Vert w_{n}^{+}\bigr\Vert &\geq\bigl\Vert u_{n}^{\ast}\bigr\Vert \bigl\Vert w_{n}^{+} \bigr\Vert \geq\bigl\langle u_{n}^{\ast}, w_{n}^{+}\bigr\rangle \\ &=\int^{T}_{0}\bigl|\dot{w}_{n}^{+}\bigr|^{2} \,dt -\int^{T}_{0}\bigl(A(t)w_{n}^{+}, w_{n}^{+}\bigr)\,dt + \int^{T}_{0} \bigl( q_{n}(t), w_{n}^{+}(t)\bigr) \,dt \\ &\geq\delta\bigl\Vert w_{n}^{+}\bigr\Vert ^{2} - C^{\alpha+1}\Vert f\Vert _{L^{1}}\Vert u_{n}\Vert ^{\alpha}\bigl\Vert w_{n}^{+} \bigr\Vert - C\Vert g\Vert _{L^{1}}\bigl\Vert w_{n}^{+} \bigr\Vert , \end{aligned}$$

which implies that $\limsup_{n\rightarrow\infty}\frac{\|w_{n}^{+}\| }{\|u_{n}\|^{\alpha}}< +\infty$. In a similar way

$$\biggl\vert \int^{T}_{0}\bigl(q_{n}(t), w_{n}^{-}(t)\bigr) \,dt\biggr\vert \leq C^{\alpha+1} \|f\|_{L^{1}}\|u_{n}\|^{\alpha}\bigl\Vert w_{n}^{-}\bigr\Vert + C\|g\|_{L^{1}}\bigl\Vert w_{n}^{-}\bigr\Vert $$

for all n. Thus one obtains

$$\begin{aligned} - \bigl\Vert u_{n}^{\ast}\bigr\Vert \bigl\Vert w_{n}^{-}\bigr\Vert &\leq\bigl\langle u_{n}^{\ast}, w_{n}^{-}\bigr\rangle \\ &=\int^{T}_{0}\bigl\vert \dot{w}_{n}^{-} \bigr\vert ^{2}\,dt - \int^{T}_{0} \bigl(A(t)w_{n}^{-}, w_{n}^{-}\bigr) \,dt + \int^{T}_{0}\bigl(q_{n}(t), w_{n}^{-}(t)\bigr) \,dt \\ &\leq- \delta\bigl\Vert w_{n}^{-}\bigr\Vert ^{2} + C^{\alpha+1}\Vert f\Vert _{L^{1}}\Vert u_{n}\Vert ^{\alpha}\bigl\Vert w_{n}^{-} \bigr\Vert + C\Vert g\Vert _{L^{1}}\bigl\Vert w_{n}^{-} \bigr\Vert . \end{aligned}$$

This means $\limsup_{n\rightarrow\infty}\frac{\|w_{n}^{-}\|}{\| u_{n}\|^{\alpha}}< +\infty$. Hence we have

$$ \limsup_{n\rightarrow\infty}\frac{\|w_{n}\|}{\|u_{n}\|^{\alpha}}< +\infty. $$

(12)

By the boundedness of $\varphi(u_{n})$ and the continuity of $A(\cdot )$, there exists a constant $C_{2}\geq1$ such that

$$\begin{aligned} \int^{T}_{0} F(t,u_{n})\,dt&= \varphi(u_{n})- \frac{1}{2}\int^{T}_{0} \vert \dot{w}_{n}\vert ^{2}\,dt+\frac{1}{2}\int ^{T}_{0}\bigl(A(t)w_{n},w_{n} \bigr)\,dt \\ &\geq-C_{2}-\frac{1}{2}\int^{T}_{0} \vert \dot{w}_{n}\vert ^{2}\,dt-\frac {1}{2}C_{2} \int^{T}_{0}|w_{n}|^{2}\,dt \\ &\geq -C_{2}-\frac{1}{2}C_{2}\|w_{n} \|^{2} \end{aligned}$$

for all n. Furthermore, it follows from (12) that

$$\liminf_{n\rightarrow\infty}\|u_{n}\|^{-2\alpha}\int ^{T}_{0}F(t,u_{n})\,dt> -\infty, $$

which contradicts Lemma 3.1. Hence $\{u_{n}\}$ is bounded in $H_{T}^{1}$, thus there exists an $u\in H^{1}_{T}$ such that $u_{n}\rightharpoonup u$ in $H^{1}_{T}$ and $u_{n}\rightarrow u$ in $C([0,T], {\mathbb{R}}^{N})$, where a subsequence is considered when necessary.

Since $H^{1}_{T}$ is reflexive while $\partial\varphi(u)$ is weak^∗ compact, and the set-valued mapping $u\rightarrow \partial\varphi(u)$ is upper semicontinuous, we can find an $u^{\ast}\in\partial\varphi(u)$ such that

$$\bigl\langle u_{n}^{\ast}-u^{\ast}, u_{n}-u \bigr\rangle \rightarrow0 \quad \mbox{as } n\rightarrow\infty. $$

On the other hand,

$$\begin{aligned} \bigl\langle u_{n}^{\ast}-u^{\ast}, u_{n}-u\bigr\rangle = &\int^{T}_{0} \bigl\vert \dot{u}_{n}(t)-\dot{u}(t)\bigr\vert ^{2}\,dt \\ &{}- \int^{T}_{0}\bigl(A(t) \bigl(u_{n}(t)-u(t) \bigr), u_{n}(t)-u(t)\bigr)\,dt \\ &{}+ \int^{T}_{0}\bigl( q_{n}(t)-q(t), u_{n}(t)-u(t)\bigr) \,dt, \end{aligned}$$

where $q_{n}(t)\in\partial F(t,u_{n}(t))$ and $q(t)\in\partial F(t,u(t))$. From a simple computation we obtain $\int^{T}_{0}|\dot{u}_{n}- \dot{u}|^{2}\,dt\rightarrow0$ as $n\rightarrow\infty$, and hence $u_{n}\rightarrow u$ in $H_{T}^{1}$. Therefore, φ satisfies the nonsmooth (PS) condition. □

Proof of Theorem 1.2

We verify that φ satisfies the other conditions of Lemma 2.2. Firstly we prove that

$$ \varphi(u)\rightarrow+\infty \quad \mbox{as } \|u\| \rightarrow \infty \mbox{ in } W^{+}. $$

(13)

By (4), (7), and Lemma 2.1, there exist $s \in[0,1]$ and $\xi\in \partial F(t,su)$ such that

$$\begin{aligned}& \biggl\vert \int^{T}_{0}F(t,u)\,dt -\int ^{T}_{0}F(t,0)\,dt\biggr\vert \\& \quad =\biggl\vert \int^{T}_{0} \langle\xi,u \rangle \,dt\biggr\vert \leq\|f\|_{L^{1}}\|u\|^{\alpha}_{\infty} \|u\|_{\infty}+ \|g\|_{L^{1}}\|u\|_{\infty} \\& \quad \leq C^{\alpha+1}\|f\|_{L^{1}}\|u\|^{\alpha+1}+ C\|g \|_{L^{1}}\|u\| \end{aligned}$$

for all n. Hence we obtain

$$\begin{aligned} \varphi(u) =&\frac{1}{2}\int^{T}_{0}\bigl\vert \dot {u}(t)\bigr\vert ^{2}\,dt-\frac{1}{2}\int ^{T}_{0}\bigl(A(t)u(t),u(t)\bigr)\,dt+\int ^{T}_{0}F\bigl(t,u(t)\bigr)\,dt \\ \geq&\frac{1}{2}\int^{T}_{0}\bigl\vert \dot{u}(t)\bigr\vert ^{2}\,dt-\frac{1}{2}\int ^{T}_{0}\bigl(A(t)u(t),u(t)\bigr)\,dt +\int ^{T}_{0}F(t,0)\,dt \\ &{}-\biggl\vert \int^{T}_{0}F\bigl(t,u(t) \bigr)\,dt-\int^{T}_{0}F(t,0)\,dt\biggr\vert \\ \geq&\frac{1}{2}\delta\|u\|^{2} - C^{\alpha+1}\|f \|_{L^{1}}\|u\| ^{\alpha+1} - C\|g\|_{L^{1}}\|u\|+\int ^{T}_{0}F(t,0)\,dt \end{aligned}$$

for all $u\in W^{+}$. Since $\alpha\in[0,1)$, it is clear that (13) holds.

Secondly we show that

$$ \varphi(u)\rightarrow-\infty \quad \mbox{as } \|u\| \rightarrow \infty \mbox{ in } W^{-}\oplus V. $$

(14)

Arguing by contradiction, assume that there exist $M\in {\mathbb{R}}$ and a sequence $\{u_{n}\}\subset W^{-}\oplus V$ such that $\|u_{n}\|\rightarrow\infty$ as $n\rightarrow\infty$ and

$$ \varphi(u_{n})\geq M $$

(15)

for all n. Write $u_{n}=v_{n}+w_{n}^{-}$, $v_{n}\in V$, $w_{n}^{-}\in W^{-}$. We consider the case that $\{u_{n}\}$ has a subsequence, say $\{u_{n}\}$, such that $\limsup_{n\rightarrow \infty}\frac{\|w_{n}^{-}\|}{\|u_{n}\|^{\alpha}}< +\infty$. By Lemma 3.1, one has

$$\limsup_{n\rightarrow\infty} \|u_{n}\|^{-2\alpha}\int ^{T}_{0}F(t,u_{n})\,dt = -\infty. $$

Hence

$$\limsup_{n\rightarrow\infty}\|u_{n}\|^{-2\alpha} \varphi(u_{n})\leq \limsup_{n\rightarrow\infty}\|u_{n} \|^{-2\alpha}\int^{T}_{0}F(t,u_{n}) \,dt=-\infty, $$

which contradicts (15).

Now we consider the case that $\frac{\|w_{n}^{-}\|}{\|u_{n}\|^{\alpha }}\rightarrow+\infty$ as $n\rightarrow\infty$, in this case, $W^{-}\neq\{0\}$ and $\|w_{n}^{-}\|\rightarrow\infty$ as $n\rightarrow\infty$. From (11), one obtains that

$$\begin{aligned} \varphi(u) =&\frac{1}{2}\int^{T}_{0}\bigl\vert \dot {w}_{n}^{-}\bigr\vert ^{2}\,dt- \frac{1}{2}\int^{T}_{0}\bigl(A(t)w_{n}^{-},w_{n}^{-} \bigr)\,dt +\int^{T}_{0}F(t,u_{n})\,dt \\ \leq&-\frac{1}{2}\delta\bigl\Vert w_{n}^{-}\bigr\Vert ^{2} +\int^{T}_{0}F(t,v_{n}) \,dt+\biggl\vert \int^{T}_{0}F(t,u_{n}) \,dt-\int^{T}_{0}F(t,v_{n})\,dt\biggr\vert \\ \leq&-\frac{1}{2}\delta\bigl\Vert w_{n}^{-}\bigr\Vert ^{2} + C^{\alpha+1}\|f\| _{L^{1}}\bigl( \|v_{n}\|^{\alpha}+\bigl\Vert w_{n}^{-} \bigr\Vert ^{\alpha}\bigr)\bigl\Vert w_{n}^{-}\bigr\Vert \\ &{}+ C\|g\|_{L^{1}}\bigl\Vert w_{n}^{-}\bigr\Vert +\int^{T}_{0}F(t,v_{n})\,dt \end{aligned}$$

for all n. It follows from (5) and (7) that

$$\int^{T}_{0}F(t,v_{n})\,dt\leq C^{2\alpha}\|v_{n}\|^{2\alpha}\int^{T}_{0} \gamma(t)\,dt \leq C^{2\alpha}\|u_{n}\|^{2\alpha}\int ^{T}_{0}\gamma(t)\,dt, $$

which implies that

$$\limsup_{n\rightarrow\infty}\|u_{n}\|^{-\alpha}\bigl\Vert w_{n}^{-}\bigr\Vert ^{-1}\int ^{T}_{0} F(t,v_{n})\,dt \leq \limsup _{n\rightarrow\infty} C^{2\alpha}\frac{\|u_{n}\|^{\alpha }}{\|w_{n}^{-}\|}\int ^{T}_{0}\gamma(t)\,dt =0, $$

hence we obtain

$$\limsup_{n\rightarrow\infty}\|u_{n}\|^{-\alpha}\bigl\Vert w_{n}^{-}\bigr\Vert ^{-1}\varphi(u_{n}) \leq- \frac{1}{2}\delta\liminf_{n\rightarrow\infty}\frac{\|w_{n}^{-}\| }{\|u_{n}\|^{\alpha}}+ 2C^{2\alpha+1}\|f\|_{L^{1}}=-\infty, $$

which contradicts (15) too.

By Lemma 2.2, $\varphi(u)$ has a critical point $u\in H_{T}^{1}$. The proof is completed.

If $W^{-}\oplus V = \{0\}$, the solution of system (1) can be obtained according to Lemma 2.4. □

Proof of Theorem 1.3

Similar to the proof of Lemma 3.2 and Theorem 1.2, we can prove that φ satisfies the nonsmooth (PS) condition, and

$$\varphi(u)\rightarrow+\infty \quad \mbox{as } \|u\| \rightarrow\infty \mbox{ in } W^{+}\oplus V. $$

If $W^{-}=\{0\}$, φ has a minimum by Lemma 2.4. In the case of $W^{-}\neq\{0\}$, we have

$$\varphi(u)\rightarrow-\infty \quad \mbox{as } \|u\| \rightarrow\infty \mbox{ in } W^{-}. $$

According to Lemma 2.2, φ has a critical point. □