Multiple periodic solutions of nonlinear second order differential equations

1.   Introduction and the main result

Let $T > 0$, $a(\cdot)$ be a $T$-periodic continous function defined in ${{\bf R}}$, and ${{\alpha}}\ (\in {{\bf R}})$ be a positive constant. Define a potential function $V$ as follows

where $P > 1$ and $a(t)\geq {{\alpha}} > 0$. In this paper, we investigate the existence of multiple nontrivial $T$-periodic solutions of the following problem

where $x\in{{\bf R}}^N, \ N\geq1$, and $c(\cdot)$ satisfies the following condition, ($H_c$) $0 < c(t)$ is a $T$-periodic continous function and $c \in C({{\bf R}}, {{\bf R}})$.

It's obvious that $x = 0$ is a trivial solution of (1.2). Indeed, we are interested in the multiplicity of nontrivial $T$-periodic solutions of (1.2), and will get that (1.2) has at least two nontrivial $T$-periodic solutions when $c(t)$ is near to any fixed eigenvalue of the linear periodic boundary value problem, for $t\in[0, T]$,

Actually, (1.3) has eigenvalues ${{\lambda}}_k = (\frac{2k\pi}{T})^2, \ k = 0, 1, 2, 3, ...,$ and eigenfunctions

where, $e, f\in {{\bf R}}^N$, as $k > 1$; $e\in {{\bf R}}^ N$, as $k = 0$.

Next, the variational framework for (1.2) will be given. Set

It's obvious that $E$ is a Hilbert space with the inner product and norm listed below

By the compact embeddings

the $T$-periodic solutions of (1.2) correspond to the critical points of the functional

It's obvious that as $c(t)\geq {{\lambda}}_1,$ the trivial solution $x = 0$ is a critical point as a local saddle point of the functional $I$. By assumptions, we know $I\in C^2(E, {{\bf R}})$ and the derivatives

where $x, y, z\in E.$

Our result is the following theorem.

Theorem 1.1. Assume $k\geq1$ and $V(t, x)$ satisfies the (1.1). Then, there exists a ${{\delta}} > 0$ such that as $c(t)\in({{\lambda}}_{k+1}-{{\delta}}, {{\lambda}}_{k+1}+{{\delta}})$, (1.2) has a nontrivial $T$-periodic solution $x_1$, and the critical group satisfies the following

Furthermore, as $c(t)\in({{\lambda}}_{k+1}-{{\delta}}, {{\lambda}}_{k+1})$, there exists another nontrivial $T$-periodic solutions $x_2$ of (1.2), and the critical group satisfies

This paper is directly motivated by ^[1,2]. In ^[1], as $c(t)$ is a positive constant in ${{\bf R}}$, by linking theorem, the existence of one nontrivial $T$-periodic solutions of the problem (1.2) in one dimension is discussed. In ^[2], the authors studied the multiple periodic solutions of superlinear second order ODEs in one dimension by morse theory. Actually, a typical model in the applications of Morse theory and minimax methods is the second order Hamiltonian system. We can refer to ^{[3,4,5,6,7,8,9,10,11,12]} for some historical progress.

Here, applying morse theory and homological linking to look for multiple periodic solutions for ODEs in $N(\geq 1)$ dimensions. On the one hand, in our problem, $c(t)$ is a $T$-periodic vibrating function different from the previous case that $c(t)$ is a positive constant. As $c(t)$ is not the $T$-periodic function, the existence of unbounded solutions of higher order differential equations considered as perturbations of certain linear differential equations can be referred to ^[13]. On the other hand, the dimension in our problem is $N(\geq 1)$, which is more general than the dimension in ^[2]. To solve the problem, we need to construct new critical groups and direct sum decomposition.

Precisely, in Theorem 1.1, as $c(t)$ is near to ${{\lambda}}_{k+1}$, we can construct the homological linking with respect to $E_{k+1}\oplus E_{k+1}^\perp$. And we get at least two nontrivial $T$-periodic solutions, which is different from the result there is at least one nontrivial $T$-periodic solution in ^[1]. Furthermore, as $c(t)\in ({{\lambda}}_{k+1}-{{\delta}}, {{\lambda}}_{k+1})$, we can also construct the homological linking w.r.t. $E_{k}\oplus E_{k}^\perp$ and investigate the existence of the nontrivial $T$-periodic solutions. In the remarkable paper ^[14], the author obtained one nonconstant periodic solution as $c(t) = 0$ and the potential $V$ was of class $C^1$, using a critical point theorem, which is now famous as the generalized mountain pass theorem. In ^[6], the author extended the existence result in ^[14] investigating (1.2) as $c(t)$ is a constant symmetric matrix by local linking argument, and got one nontrivial periodic solution as the potential $V$ was of class $C^1$ and satisfied local sign condition ^[6] near the origin. The fundamental idea here is sources from ^[15], where the authors studied the superlinear elliptic problem with a saddle structure near zero by bifurcation methods, Morse theory, and topological linking.

The structure of the paper is arranged as follows. In Section 2, we recall the basic Morse theories and give a lemma on the ($PS$) condition. In Section 3, we prove the main result.

2.   Preliminaries

In this section, we recall some preliminaries on Morse theory and homological linking in ^[7,8,16].

Assume the functional $I\in C^2(E, {{\bf R}})$, where $E$ is a Hilbert space. Set $K = \{x\in E|I'(x) = 0\}, I^c = \{x\in E|I(x)\leq c\},$ and $K_c = \{x\in K|I(x) = c\}$. We recall that $I$ satisfies $(PS)_c$ condition at the level $c\in{{\bf R}}$, if any sequence $\{x_n\}\subset E$ satisfying $I(x_n)\rightarrow c, \ I'(x_n)\rightarrow 0$ as $n\rightarrow \infty$, has a convergent subsequence. $I$ satisfies $(PS)$ if $I$ satisfies $(PS)_c$ at any $c\in{{\bf R}}.$

Assume that the functional $I$ satisfies ($PS$) and # $K < \infty$ in this paper. Let $x_0\in K$ with $I(x_0) = c\in{{\bf R}}$, and $U$ be a neighborhood of $x_0$ such that $U\cap K = \{x_0\}$. Then, $q$-th critical group of $I$ at $x_0$ is defined below

where $H_*(A, B)$ denotes the singular relative homology group of the pair $(A, B)$ with coefficient field $\mathbb{F}$ (^[7,8,16]). We can distinguish critical points by critical groups. The multiplicity of critical points and critical groups can be referred to ^[16].

For the critical groups of $I$ at an isolated critical point, the following basic conclusions hold.

Proposition 2.1. Assume that $x$ is an isolated critical point of $I\in C^2(E, {{\bf R}})$ with finite Morse index $i(x)$ and nullity $\nu(x)$. Then

(1) $C_q(I, x)\cong{{\delta}}_{q, i(x)}\mathbb{F}$, if $\nu(x) = 0$;

(2) $C_q(I, x)\cong0$ for $q\ \not \in [i(x), i(x)+\nu(x)]$ (Gromoll and Meyer ^[17]).

From Theorems $1.1'$ and 1.5 of Chapter II in ^[8], the following abstract linking theorem is easily obtained (See also ^[3,18]).

Proposition 2.2. (^[3,8,18]) Let $E$ be a real Banach space with $E = X\oplus Y$ and suppose that $l = \dim X$ is finite. Suppose that $I\in C^1(E, {{\bf R}})$ satisfies ($PS$)condition and

($H_1$) there exist $\rho > 0, {{\alpha}}_0 > 0$ such that

where $B_\rho = \{x\in E|\ \|x\|\leq\rho\},$

($H_2$) there exist $R > \rho > 0,$ and $e\in Y$ with $\|e\| = 1$ such that

where $Q = \{x = x_1+se| \ \ \|x\|\leq R, x_1\in X, \ 0\leq s\leq R\}.$

Then $I$ has a critical point $x_*$ with $I(x_*)\geq {{\alpha}}_0$ and

Remark 2.1. In proposition above, $S_\rho$ and $\partial Q$ are homotopically linked with respect to direct sum decomposition $E = X\oplus Y.$

Lemma 2.1. Assume that the $T$-periodic function $c(\cdot)$ satisfies ($H_c$), and the potential function $V$ satisfies (1.1). Then, I satisfy the (PS) condition.

Proof. Let $\{x_n\}\subset E = H^1_{per}((0, T), {{\bf R}}^N)$ satisfy the following,

where $\theta$ is a zero vector, $C > 0$ is a constant. In fact, $\forall {{\varphi}}\in E = H^1_{\text{per}}((0, T), {{\bf R}}^N)$, we have

Taking ${{\varphi}} = x_n$ in (2.2), we have

and by (2.1), the following holds

i.e.,

Combining (2.3) and (2.4) (or (2.5)), we have

Since $P > 1$, we have $\frac{1}{2}-\frac{1}{P+1} > 0$. So by (2.6), the following holds

where $C_1 > 0$ is a constant. Then, by (2.7) and (2.3), we have

where $C_2 > 0$ is a constant. Furthermore, there exists a $C_M > 0$ such that $0 < c(t)\leq C_M, \forall\ t\in [0, T]$. By Hölder inequality, we get

and

Next, by (2.7), (2.8), and (2.10), we have

where $C_3 > 0$ is a constant. So, $\|x_n\|\leq C_4$, where $C_4 > 0$ is a constant, i.e., $\{x_n\}$ is bounded in $E$. Without loss of generality, up to a subsequence, assume that there exists a point $x_0\in E$ such that as $n\rightarrow \infty, \ x_n\rightharpoonup x_0$ in $E$ and $x_n\rightarrow x_0$ in $C([0, T], {{\bf R}}^N)$, and furthermore $x_n\rightarrow x_0$ in $L^2([0, T], {{\bf R}}^N)$. So

Hence, $x_n\rightarrow x_0$ in $E$. The proof is completed.

We need some notations to construct the linking. For $j\in {{\bf N}}$, set

For each $j\geq 1$, by (1.4), we have $\nu_j: = \dim E({{\lambda}}_j) = 2N, \ i_j: = \dim E_j = \sum\limits_{k = 0}^j \nu_k = 2jN+N.$ It's obvious that in $E_j^\bot$,

and in $E_j$,

3.   Proof of the main result

At the beginning of this section, to verify the conditions ($H_1$) and ($H_2$) in Proposition 2.2, we give some preliminary lemmas below.

Lemma 3.1. For any $0 < c(t) < {{\lambda}}_{j+1}, \ t\in [0, T]$, there exist $r_j, {{\alpha}}_j > 0$ such that

Proof. By embedding theorem (Chapter 10 in ^[1], Chapter 1 in ^[16]), we have

where $C > 0$ is a constant, $\|x\|_{P+1} = (\int_0^T|x|^{P+1}dt)^{\frac{1}{P+1}}$, $\|x\| = (\int_0^T|\dot{x}|^{2}+|x|^{2}dt)^{\frac{1}{2}}$. Meanwhile, since $a(t)$ is a $T$-periodic continuous function. There exist $m, M$ such that $0 < {{\alpha}}\leq m\leq a(t)\leq M$. So by (3.1), the following holds

In addition, since $c(t)$ is a $T$-periodic continuous function, for any $0 < c(t) < {{\lambda}}_{j+1},$ there exists a constant $c_M > 0$, which is the maximum of $c(t), \ t\in[0, T]$, such that $0 < c(t)\leq c_M < {{\lambda}}_{j+1}$. So for $x\in E_j^\bot$, by (2.11), we have

Next, taking ${{\alpha}}_* = \frac{{{\lambda}}_{j+1}-c_M}{{{\lambda}}_{j+1}+1} > 0, \ \beta = M'C^{P+1}$, we have

As a result of $P > 1$, the function $g(s) = \frac{{{\alpha}}_*}{2}s^2-\beta s^{P+1}$ on $(0, \infty)$ has its maximum

Set $r_j = [\frac{{{\alpha}}_*}{(P+1)\beta}]^\frac{1}{P-1}$. By the process above, we therefore have

The proof is completed.

Remark 3.1. It's obvious that $\alpha_j$ and $r_j$ decreasingly approach to zero as $c_M\rightarrow {{\lambda}}^-_{j+1}$.

Now, for the eigenvalue ${{\lambda}}_{j+1}$ of $(1.3)$, we take the corresponding unit eigenfunction $\varphi_{j+1}$, i.e., $\|\varphi_{j+1}\| = 1$. For $j = k, k+1$, define

where $r_j$ is defined in Lemma 3.1, $j = k, k+1$.

Lemma 3.2. Assume $V(t, x)$ satisfies $(1.1)$ and $c(t)\in ({{\lambda}}_k, {{\lambda}}_{k+2}).$There exist ${{\sigma}}\in{{\bf R}},$ $\delta > 0$, and $R > 0$ independent $c(t)$ and ${{\delta}}$, such that as $c(t)\in({{\lambda}}_{k+1}-{{\delta}}, {{\lambda}}_{k+1}+{{\delta}})$, the following holds

where $Q_{k+1} = \{x\in V_{k+1}|\ \|x\|\leq R, x = x_1+s\varphi_{k+2}, x_1\in E_{k+1}, s\geq0\}.$ Furthermore, as $c(t)\in({{\lambda}}_k, {{\lambda}}_{k+1})$, there exists a constant $R > 0$ independent of $c(t)$ such that

where $Q_{k} = \{x\in V_{k}|\ \|x\|\leq R, x = x_1+s\varphi_{k+1}, x_1\in E_{k}, s\geq0\}.$

Proof. Since $a(\cdot) > 0$ is a $T$-periodic continuous function and $a(t)\geq {{\alpha}}, \ t\in [0, T],$ by (2.12) and $c(t)\in({{\lambda}}_{k}, {{\lambda}}_{k+2})$, we easily get, for $x = x_1+x_2+x_3\in V_{k+1}$, where $x_1\in E_k, \ x_2\in E({{\lambda}}_{k+1}), \ x_3\in {{{\rm{span}}}} \{\varphi_{k+2}\}$,

Since $P > 1, \dim V_{k+1} < \infty$, and all norms are equivalent in a space with a finite dimension, by (3.3), we have $I(x)\rightarrow -\infty, \ \text{as }\ \|x\|\rightarrow \infty.$ So, there exists a constant $R > 0$ independent of $c(t)$ such that

Now, choosing $R > \max\{r_k, r_{k+1}\} > 0$. For $E_{k+1}\ni y = y_1+y_2$ with $\|y\|\leq R,$ where $y_1\in E_k, y_2\in E({{\lambda}}_{k+1})$.

Next, for $c(t)\in({{\lambda}}_k, {{\lambda}}_{k+2})$, by (2.12), we have

Now, taking ${{\sigma}} = \frac{{{\alpha}}_{k+1}}{2}, {{\delta}} = \frac{2{{\sigma}}}{R^2},$ where ${{\alpha}}_{k+1}$ is defined in Lemma 3.1, we get as $|{{\lambda}}_{k+1}-c(t)| < {{\delta}},$

Meanwhile, we have

Then, by (3.4) and (3.5), we obtain

Furthermore, as a matter of fact, we have

On the one hand, since $V_{k}\subset V_{k+1}$, for $c(t)\in({{\lambda}}_k, {{\lambda}}_{k+1})\subset({{\lambda}}_k, {{\lambda}}_{k+2}), \ R > 0$ mentioned above, by (3.4), it follows that

On the other hand, it's obvious that, for $\forall\ x_1\in E_k,$

So, by (3.6), (3.7), and (3.8), we get

The proof is completed.

Now, by two lemmas proved above, using the Proposition 2.2, we could prove the main result.

Proof. Proof of Theorem 1.1. To give a clear proof, we shall take several steps to finish it.

Firstly, by Lemma 2.1, $I$ satisfies the ($PS$) condition.

Secondly, by Lemmas 3.1 and 3.2, $I$ satisfies the $(H_1)$ and $(H_2)$ in Proposition 2.2. So the following holds

Next, for the $R$ mentioned in Lemma 3.2, it satisfies the following

So we have $S_{k+1}$ and $\partial Q_{k+1}$ homotopically link w.r.t. the decomposition $E = E_{k+1}\bigoplus E_{k+1}^\bot$. Furthermore, since $\dim V_{k+1} = i_{k+1}+1$, by Proposition 2.2, we have a critical point $x_1$ of the functional $I$ with $I(x_1)\geq{{\alpha}}_{k+1} > 0$ and

Thirdly, for the situation $c(t)\in({{\lambda}}_{k+1}-\delta, {{\lambda}}_{k+1}) \subset({{\lambda}}_{k}, {{\lambda}}_{k+1}), \ I$ satisfies ($H_1$) and ($H_2$). What's more, the following holds

By (3.9), $S_{k}$ and $\partial Q_{k}$ homotopically link w.r.t. the decomposition $E = E_{k}\bigoplus E_{k}^\perp$. Furthermore, since $\dim V_{k} = i_{k}+1$, by Proposition 2.2, we have a critical point $x_2$ of the functional $I$ with $I(x_2)\geq{{\alpha}}_{k} > 0$ and

Finally, as a matter of fact, we have

Combining the following fact,

where $x_*$ is a critical point of the functional $I$, and

by Proposition 2.1(2), we obtain $x_1\neq x_2$. The proof is completed.

Right now, we can directly have the following conclusion by Theorem 1.1.

Corollary 3.1. Assume $V(t, x)$ satisfies (1.1) and $k(\in{{\bf N}})\geq1$. Then, there exists a ${{\delta}} > 0$ such that as $c(t)\in ({{\lambda}}_{k+1}, {{\lambda}}_{k+1}+{{\delta}})$, (1.2) has at least one nontrivial $T$-periodic solution.

4.   Conclusions

In this paper, we prove the existence of multiple nontrivial $T$-periodic solutions of the equation $\ddot{x}+V_x(t, x) = 0$ in ${{\bf R}}^N$, $N(\geq 1)$, where $V_x(t, x)$ is the derivative of $V(t, x)$ satisfying (1.1) with respect to $x$. Since the nontrivial $T$-periodic solutions correspond to the critical points of the functional of the problem, to get the multiple $T$-periodic solutions, we prove the existence and multiplicity of critical points of the functional. Employing the homological linking and morse theory, we get at least two nontrivial $T$-periodic solutions and distinguish them by critical groups. By the way, as the range of the $T$-periodic function $c(t)$ satisfying ($H_c$) in (1.1), a right small neighborhood of the $k$-th eigenvalue of (1.3), there exists a nontrivial $T$-periodic solution. Here, our main result is different from the previous works (see ^[1,2] and the references therein).

Acknowledgments

This work is partially supported by China Scholarship Council (No.202108410329), Shaoguan Science and Technology Project (No.210726224533614, 210726214533591), Natural Science Foundation of Guangdong Province (No.2021A1515010292, 2023A1515010825), and National College Students Innovation and Entrepreneurship Training Program (No.202210576012).

Conflict of interest

The authors declare no conflicts of interest.