Infinitely many periodic solutions for ordinary $p(t)$-Laplacian differential systems

1.   Introduction and main results

Consider periodic solution of the ordinary $p(t)$-Laplacian differential system

where $T > 0$, $p(t) > 1$ is a $T$-periodic continuous function, $F(t, x)$ is $T$-periodic in $t$ for any $x\in\mathbb{R}^{N}$. Here we always assume that $F(t, x)$ satisfies the following condition.

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

where $\nabla F(t, x)$ denotes the gradient of $F(t, x)$ in $x$.

In the sequel of this paper, we set $p^{-}: = \min\limits_{0\leq t\leq T} p(t)$, $p^{+}: = \max\limits_{0\leq t\leq T} p(t)$ and $q^- > 1$ such that $\frac 1 {p^-}+\frac 1 {q^-} = 1$.

We define the generalized Lebesgue space as

with the norm

$L_{T}^{p(t)}$ is a kind of generalized Orlicz space. We also define the generalized Sobolev space by

with the norm

$W_{T}^{1, p(t)}$ is a kind of generalized Orlicz-Sobolev space. In our case with the condition $p(t) > 1$, the two spaces $L^{p(t)}_T$ and $W_{T}^{1, p(t)}$ are both reflective Banach spaces with the norms defined above. It is known that there is a compact embedding $W_{T}^{1, p(t)}\hookrightarrow C([0, T]; \mathbb{R}^{N})$. One can refer ^[1] for details.

The corresponding functional $\varphi$ of (1.1) on $W_{T}^{1, p(t)}$ is given by

Wang and Yuan in ^[2] verified that the functional $\varphi$ denoted in (1.2) is continuously differentiable, i.e., there holds

for any $u, v\in W_{T}^{1, p(t)}$. So the critical points of $\varphi$ correspond to the solutions of problem (1.1). Then Zhang et al. in ^[3] proved that the functional $\varphi$ defined in (1.2) is weakly lower semicontinuous on $W_{T}^{1, p(t)}$.

When $p(t) = 2$, problem (1.1) reduces to the following Hamiltonian system

which has been extensively investigated in many literatures (see, e.g., ^[4,5,6,7,8] and the references therein). Particularly, under the following conditions:

$(M_{1})$ there exists $g\in L^{1}([0, T]; \mathbb{R}^{+})$ such that $|\nabla F(t, x)|\leq g(t)$ for ${\forall}x\in \mathbb{R}^{N}$ and a.e. $t\in[0, T]$,

$(M_{2})$ $\int_{0}^{T}F(t, x)dt\rightarrow \pm\infty$ as $|x|\rightarrow +\infty$,

Mawhin and Willem in ^[6] established the existence of solutions for problem (1.3).

When $N = 1$, $T = 2\pi$, Habets, Manasevich and Zanolin in ^[4] established the existence of two sequences of different periodic solutions for problem (1.3) when the potential $F(t, x)$ satisfies the following oscillating conditions:

When the nonlinearity $\nabla F(t, x)$ is sublinear, i.e., there exist $f, g\in L^{1}([0, T]; \mathbb{R}^{+})$ and $\alpha\in[0, 1)$ such that

and $F(t, x)$ satisfies oscillating conditions, i.e., there holds

Zhang and Tang in ^[8] obtained the existence of two sequences of different periodic solutions for problem (1.3).

When $p(t) = p$ is a constant for general $p > 1$, problem (1.1) reduces to the following ordinary $p$-Laplacian problem

In recent years, the existence results for problem (1.4) was considered via the variational methods (see, e.g., ^[9,10,11] and references therein). Specially, L$\ddot{u}$, O'Regan and Agarwal in ^[10] generalized the results of ^[4] to the $p$-Laplacian problem (1.4) and established the existence of two sequences of different periodic solutions for problem (1.4). Li, Agarwal and Tang in ^[9] also established the existence of two sequences of different periodic solutions for problem (1.4) for the case with the potential $F(t, x)$ possessing the mixed nonlinearity, that is, $F(t, x) = F_{1}(t, x)+F_{2}(t, x)$ with $F_{1}(t, x)$ being $(\lambda, \mu)$-subconvex in $x$, i.e., function $F_{1}(t, \cdot):\mathbb{R}^{N}\rightarrow \mathbb{R}$ satisfies

for some $\lambda, \mu > 0$ and each $x, y\in\mathbb{R}^{N}$ (see ^[7] for details) and $\nabla F_{2}(t, x)$ being $p$-sublinear, namely, there exist $f, g\in L^{1}([0, T]; \mathbb{R}^{+})$ and $\alpha\in [0, p-1)$ such that

Then they also considered the other case with the nonlinearity $\nabla F(t, x)$ being $p$-linear, i.e., there exist $f, g\in L^{1}([0, T]; \mathbb{R}^{+})$ such that

Many scholars are interested in the nonlinear problems including the $p(t)$-Laplacian problem, this kind of problems can be used to model the dynamical phenomena arising from the field of elastic mechanics. One can refer ^[12,13] and the references therein for more application of the $p(t)$-Laplacian problem to various practical issues. The $p(t)$-Laplacian problem has more complicated nonlinearity than that of the $p$-Laplacian problem, for example, the second order differential operator is not homogeneous, this may causes some troubles such as some classical theories and methods, for instance, the Sobolev spaces theory are not applicable.

In recent years, the study of variational problems and elliptic partial differential equations with nonstandard growth conditions has been an interesting topic (see, e.g., ^[13,14] and references therein). In 2001, Fan et al. in ^[14,15] established some basic theory of space $L^{p(x)}(\Omega)$ and $W^{k, p(x)}(\Omega)$. In 2003, Fan et al. in ^[1] first studied the ordinary $p(t)$-Laplacian problem (1.1). Subsequently, Wang and Ruan in ^[2] established the existence and multiplicity of periodic solutions for the ordinary $p(t)$-Laplacian problem (1.1) under the generalized Ambrosetti-Rabinowitz conditions. Recently, many scholars considered the existence of periodic solutions of the ordinary $p(t)$-Laplacian problem (1.1). Some existence theorems of the ordinary $p(t)$-Laplacian problem (1.1) are established by using the least action principle and minimax methods in critical point theory (see, e.g., ^[1,3,16] and the references therein).

In this paper, motivated by the results of ^[4,5,8,9,10], we first aim to consider the ordinary $p(t)$-Laplacian problem (1.1) with a mixed nonlinearity, that is, $F(t, x) = F_{1}(t, x)+F_{2}(t, x)$ with $F_{1}(t, x)$ being $(\lambda, \mu)$-subconvex and $\nabla F_{2}(t, x)$ being $p^{-}$-sublinear. Then we consider the other case with the nonlinearity $\nabla F(t, x)$ being $p^{-}$-linear. Following the idea of paper ^[8,9], we obtain the existence of two sequences of different periodic solutions for problem (1.1). The main results of this paper are stated as follows.

Theorem 1.1 Assume that $F(t, x) = F_{1}(t, x)+F_{2}(t, x)$ where $F_{1}(t, x)$ and $F_{2}(t, x)$ satisfy condition $(A)$ and the following conditions.

$(H_{1})$ $F_{1}(t, \cdot)$ is $(\lambda, \mu)$-subconvex defined in (1.5) with $\lambda > \frac{1}{2}$ and $1\leq2\mu < (2\lambda)^{p^{-}}$ for a.e. $t\in[0, T]$, and there exist $\alpha\in[0, p^{-}-1), f, g\in L^{1}([0, T]; \mathbb{R}^{+})$ such that

$(H_{2})$ There exists a positive sequence $\{R_n\}$ with $\lim\limits_{n\to +\infty}R_n = +\infty$, such that

and there exists a positive sequence $\{r_m\}$ with $\lim\limits_{m\to +\infty}r_m = +\infty$, such that

Then problem (1.1) possesses two sequences of solutions $\{u^{\pm}_{n}\}$ with $\varphi(u^{\pm}_{n})\rightarrow \pm\infty$ as $n\rightarrow +\infty$.

Remark 1.2 Theorem 1.1 above generalizes Theorem 1.1 of ^[10], Theorem 1.1 of ^[8], Theorem 1 of ^[5] and Theorem 1.1 of ^[9]. There are functions $F(t, x)$ satisfying the conditions in Theorem 1.1 but not satisfying the conditions in ^[5,8,9,10]. For example, for $p(t) = 4+cos\omega t$ with $\omega = \frac{2\pi}{T}$, we take

where $e\in L^{1}([0, T]; \mathbb{R}^{N})$ and $\int_{0}^{T}e(t)dt = 0$. Then $F(t, x) = F_1(t, x)+F_2(t, x)$ is such a function.

Theorem 1.3 Assume that $F(t, x)$ satisfies condition $(A)$ and the following conditions.

$(H_{3})$ There exist $f, g\in L^{1}([0, T]; \mathbb{R}^{+})$ with $\|f\|_{L^{1}} < \alpha(p)$ such that

where $\alpha(p)$ is a positive constant depending on the function $p(t)$, and so on the constants $p^{\pm}$.

$(H_{4})$ There exists a positive sequence $\{R_n\}$ with $\lim\limits_{n\to +\infty}R_n = +\infty$, such that

and there exists a positive sequence $\{r_m\}$ with $\lim\limits_{m\to +\infty}r_m = +\infty$, such that

where $\beta(p)$ is a positive constant depending on the function $p(t)$, and so on the constants $p^{\pm}$. Then problem (1.1) possesses two sequences of solutions $\{u^{\pm}_{n}\}$ such that $\varphi(u^{\pm}_{n})\rightarrow \pm\infty$ as $n\rightarrow +\infty$.

Remark 1.4 In the process of the proof of Theorem 1.3, the constants $\alpha(p)$ and $\beta(p)$ will be defined clearly (see (3.20) and (3.26)). Theorem 1.3 above generalizes Theorem 1.3 of ^[9] since here the range of the function $p(t)$ is wider than theirs. There exist functions satisfying the conditions of Theorem 1.3. For example, for $p(t) = \frac{5}{4}+\frac{1}{8}cos\omega t$, one can take

Then $F(t, x)$ satisfies the conditions of Theorem 1.3 with $p^{-} = \frac{9}{8}, p^{+} = \frac{11}{8}, q^- = 9, \theta_{p^{-}} = 1, f(t) = \frac{3}{T}$.

2.   Preliminaries

For the convenience of readers, we first introduce two important properties of the variable exponent Lebesgue-Sobolev spaces $L^{p(t)}$ and $W_{T}^{1, p(t)}$.

We define

For $u\in W_{T}^{1, p(t)}$, let

then $\quad\tilde{u}(t) = u(t)-\bar{u}\in \widetilde{W}_{T}^{1, p(t)}$. So we get

Lemma 2.1 ^[16] For all $u\in \widetilde{W}_{T}^{1, p(t)}$, there exist constants $C_{0}^{\prime}$, $C_{0}$ such that

where $C_{0}^{\prime}$, $C_{0} > 0$.

Lemma 2.2 ^[1] Let $u = \bar{u}+\tilde{u}\in W_{T}^{1, p(t)}$ with $\bar{u}\in \mathbb{R}^{N}$ and $\tilde{u}\in \widetilde{W}_{T}^{1, p(t)}$, then the norm $\|\dot{\tilde{u}}\|_{L^{p(t)}}$ is an equivalent norm on $\widetilde{W}_{T}^{1, p(t)}$ and $|\bar{u}|+\|\dot{u}\|_{L^{p(t)}}$ is an equivalent norm on $W_{T}^{1, p(t)}$. Therefore, for $u\in W_{T}^{1, p(t)}$,

Next we give two important properties of minimax methods as follows.

Lemma 2.3^[6] Let $K$ be a compact metric space, $X$ a Banach space and $K_{0}\subset K$ is a closed subset, $\chi\in C(K_{0};X)$. We define a complete metric space $M$ as

with the usual norm and metric. Let $\varphi\in C^{1}(X; R)$ and

If $c > c_{1}$, then for each $\varepsilon > 0$ and each $\gamma\in M$ such that

there exists $v\in X$ such that

Corollary 2.4^[6] Under the conditions of Lemma 2.3, for each sequence $(\gamma_{k})$ in $M$ such that

there exists a sequence $(v_{k})$ in $X$ such that

3.   Proofs of theorems

Proof of Theorem 1.1. The proof is divided into four steps.

Step 1. We demonstrate that $\varphi(u)\rightarrow +\infty$ as $\|u\|\rightarrow \infty$ for $u\in\widetilde{W}_{T}^{1, p(t)}$.

Let $\beta = \log_{2\lambda}(2\mu)$, then one gets $0\leq \beta < p^{-}$. For $|x| > 1$, there exists a positive integer $n$ such that

Therefore one has $|x|^{\beta} > (2\lambda)^{(n-1)\beta} = (2\mu)^{n-1}$ and $|x|\leq(2\lambda)^{n}$. Combining the conditions $(A)$ and $(H_{1})$, one obtains

for any $|x| > 1$ and a.e. $t\in[0, T]$, where $a_{0} = \max\limits_{0\leq s\leq1}a(s)$. Therefore one gets

for any $|x| > 1$ and a.e. $t\in[0, T]$. In the sequel, we denote by $C$ a suitable positive constant which may take different value in various estimations.

Combining Lemma 2.1, $(H_{1})$ and (3.1), for the functional $\varphi$ defined in (1.2), we obtain

for any $u\in \widetilde{W}_{T}^{1, p(t)}$. From Lemma 2.2, estimate (3.2) implies that

Step 2. We verify that for the positive sequence $\{r_{m}\}$ defined in $(H_2)$, there holds

where $W_{r_{m}} = \{u\in\mathbb{R}^{N}||u| = r_{m}\}\oplus\widetilde{W}_{T}^{1, p(t)}$.

By Lemma 2.1, $(H_{1})$ and Young inequality, we obtain

for any $u\in W_{T}^{1, p(t)}$, where $C_{0}$ is the constant defined in Lemma 2.1.

Hence from $(H_{1})$, (3.1) and (3.4), for the functional $\varphi$ defined in (1.2), we have

One can check that

and

Thus the above inequality and $(H_{2})$ imply (3.3).

Step 3. Prove (ⅰ) of Theorem 1.1. For the positive sequence $\{R_n\}$ defined in the condition ($H_2$), we define

and

where

One can verify that $\gamma(B_{R_{n}})\cap \widetilde{W}_{T}^{1, p(t)}\neq\emptyset$ (for more details see ^[10]).

From Step 1, we know that $\varphi$ is coercive on $\widetilde{W}_{T}^{1, p(t)}$. Since $W_T^{1, p(t)}$ is a reflective Banach space, hence there is a constant $M$ such that

Combining (3.6) and (3.7), one obtains that $c_{n}\geq M$.

As in (3.3), we can easily verify that

From (3.8), for large enough $n$, we obtain

Now in Lemma 2.3 and Corollary 2.4, we take the compact set $K = B_{R_n}$ and $K_0 = \partial B_{R_n}$, then for any sequence $\{\gamma_{k}\}\subset\Gamma_{n}$ such that

there exists a sequence $\{v_{k}\}\subset W_{T}^{1, p(t)}$ satisfying

Combining (3.6), (3.9) and (3.10), for $k$ large enough, we have

Define $\pi:W_{T}^{1, p(t)}\rightarrow \mathbb{R}^{N}$ as

For fixed $n$, we obtain

for some positive constant $K_{n}$, here $|S|: = \sup_{x\in S}\|x\|$ for a set $S\subset \mathbb{R}^{N}$.

We now take $\omega_{k}\in\gamma_{k}(B_{R_{n}})$ such that

We write $\omega_{k} = \bar{\omega}_{k}+\tilde{\omega}_{k}$ with $\bar{\omega}_{k}\in\mathbb{R}^{N}$ and $\tilde{\omega}_{k}\in \widetilde{W}_{T}^{1, p(t)}$. So there holds $|\bar{\omega}_{k}|\leq K_{n}$, which implies that

is bounded. Combining $(H_{1})$, (3.1), (3.4), (3.11) and Lemma 2.1, we obtain

Therefore one can check that $\{\tilde{\omega}_{k}\}$ and $\{\omega_{k}\}$ is bounded from Lemma 2.2 and (3.13). By (3.12), we obtain that $\{v_{k}\}$ is bounded. By a standard argument, one can easy to verify that $\{v_{k}\}$ possesses a convergent subsequence. It is still denoted by $\{v_{k}\}$. One can refer ^[2] for more details about the proof of (PS) condition.

Let

Then from (3.10), we obtain

This means $u^+_{n}$ is a solution of problem (1.1) with $\varphi(u^+_{n}) = c_{n}$.

For all $\gamma\in\Gamma_{n}$, by definition, there holds $\gamma(B_{R_{n}})\cap W_{r_{m}}\neq\emptyset$. Then we get

Now combining (3.3), (3.6) and (3.14), we obtain

Hence there exists a sequence of periodic solutions $\{u^+_{n}\}$ which are minimax-type critical points of the functional $\varphi$ and $\varphi(u^+_{n})\rightarrow +\infty$ as $n\rightarrow +\infty$. So (ⅰ) of Theorem 1.1 is proved.

Step 4. Prove (ⅱ) of Theorem 1.1. For the sequence $\{r_n\}$ defined in condition ($H_2$) and as the same from (3.3), we define $P_{n}$ as

For $u\in P_{n}$, it is easy to see that

is bounded.

Combining $(H_{1})$, (3.1) and (3.4), for $u\in P_{n}$, by the same process as in (3.13), we get

Thus the functional $\varphi$ is bounded below in $P_{n}$.

Set

and let $\{u_{k}\}$ be a sequence on $P_{n}$ such that

We write

From (3.15) and (3.16), it is easy to see that $\{u_{k}\}$ is bounded sequence in $W_{T}^{1, p(t)}$. Then $\{u_{k}\}$ possesses a subsequence, which is still denoted by $\{u_{k}\}$ such that

As $P_{n}$ is a closed convex subset of $W_{T}^{1, p(t)}$, one has $u^{-}_{n}\in P_{n}.$

Since the functional $\varphi$ is weakly lower semicontinuous, we obtain

Therefore we get

From (3.3), for large $n$, we get that $u^{-}_{n}\notin\{\bar{u}+\tilde{u}, \bar{u}\in\mathbb{R}^{N}, |\bar{u}| = r_{n}\}$, and hence $u^{-}_{n}\in Int P_{n} = \{u\in W_{T}^{1, p(t)}|u = \bar{u}+\tilde{u}, |\bar{u}| < r_{n}\}$. Thus combining (3.16) and (3.17), one obtains

and $u^{-}_{n}$ is a solution of problem (1.1).

Since $r_n\to +\infty$ as $n\to +\infty$, we can choose the sequence $\{R_n\}$ in the condition ($H_2$) and as the same in (3.8) satisfying $0 < R_n < r_n$. By (3.16), we obtain

From (3.8) and (3.18), we have

So (ⅱ) of Theorem 1.1 is true. The proof of Theorem 1.1 is complete.

Proof of Theorem 1.3. The proof is divided into two steps.

Step 1. We show that $\varphi(u)\rightarrow +\infty$ as $\|u\|\rightarrow \infty$ for $u\in\widetilde{W}_{T}^{1, p(t)}$.

Combining Lemma 2.1 and $(H_{3})$, we obtain

for any $u\in \widetilde{W}_{T}^{1, p(t)}$, where $C_0$ is the positive constant defined in Lemma 2.1. We now define the constant $\alpha(p)$ in Theorem 1.3 as

where $\theta_{p^{-}}$ is a positive constant defined by $\theta_{p^{-}} = 1$ for $p^-\in (1, 2]$, and $\theta_{p^{-}} = 2^{p^–2}$ for $p^- > 2$. Now if $\|f\|_{L^1} < \alpha(p)$, in view of Lemma 2.2, the estimate (3.19) implies that

Step 2. We verify that for the positive sequence $\{r_{m}\}$ defined in condition ($H_4$), there holds

where $W_{r_{m}} = \{u\in\mathbb{R}^{N}||u| = r_{m}\}\oplus\widetilde{W}_{T}^{1, p(t)}$.

For $u\in W_{T}^{1, p(t)}$ with $u = \bar{u}+\tilde{u}$ where $|\bar{u}| = r_{m}$ and $\tilde{u}\in\widetilde{W}_{T}^{1, p(t)}$. Here we recall that $r_m$ is defined in condition $(H_{4})$, namely we have

Combining $(H_{3})$, we obtain

for any $u\in W_{T}^{1, p(t)}$, where $\frac{1}{p^-}+\frac{1}{q^-} = 1$. Where in (3.23), in the second estimate, we have used the inequality $(a+b)^c\le a^c+b^c$ for $a, b\ge 0$, $c\in (0, 1]$, $(a+b)^c\le 2^{c-1}(a^c+b^c)$ for $a, b\ge 0$, $c\in (1, +\infty)$, and in the last estimate, we have used the Young inequality $ab\le \frac{a^{p^-}}{p^-}+ \frac{b^{q^-}}{q^-}$, $a, b\ge 0$. In the Lemma 2.1 of ^[11], we know that for a fixed $\rho > 1$ and any $\varepsilon > 0$, there is a number $B(\varepsilon) > 0$ such that

Now we take $\varepsilon > 0$ such that $\alpha(p)\ge (1+\varepsilon)\|f\|_{L^1}+\varepsilon$, and $\rho = p^-$, from Lemma 2.1, there holds

Thus we get

Therefore, from $(H_{3})$ and (3.24), we have

for all $u\in W_{T}^{1, p(t)}$. From (3.20), we can check that

Hence we obtain

Now define the constant $\beta(p)$ in Theorem 1.3 as

Therefore combining the above expression and (3.22), we obtain (3.21).

The remainder is similar to that as in the proof of Theorem 1.1, we omit the detail here.

Acknowledgments

The first author is Partially supported by the NSF of China (11790271, 12171108), Guangdong Basic and Applied basic Research Foundation(2020A1515011019), Innovation and Development Project of Guangzhou University.

Conflict of interest

All authors declare no conflicts of interest in this paper.