Multiplicity of the large periodic solutions to a super-linear wave equation with general variable coefficient

1.   Introduction

We consider the nonlinear wave equation

together with time-periodic condition

where $f \in C(\Omega\times \mathbb{R}, \mathbb{R})$ is $T$ periodic with respect to $t$ and the period $T$ is determined by

In addition to (1.2), Equation (1.1) is subject to the boundary condition

where the coefficients $\alpha_i, \beta_i$ for $i = 1, 2$ satisfy

which contains the Dirichlet-Neumann boundary condition (e.g., $\alpha_1\neq 0, \beta_1 = 0, \alpha_2 = 0, \beta_2\neq 0$) and the Dirichlet-Robin boundary condition (e.g., $\alpha_1\neq 0, \beta_1 = 0, \alpha_2 \neq 0, \beta_2\neq 0$).

Equation (1.1) originated from the following equation

which is used to describe the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media (see ^{[1,2,3,4,5,6,7,8,9]}). Here, $u$ represents the vertical displacement of the seismic wave, $\omega(\cdot)$ denotes the rock density, and $\nu(\cdot)$ is the elasticity coefficient. By means of transformation of variables $x = \int_0^z \left(\frac{\omega(s)}{\nu(s)}\right)^{1/2} {\rm{d}}s$, Equation (1.6) is simplified as

where $\rho = (\omega\nu)^{1/2}$ denotes the impedance function.

Equation (1.1) degenerates to the classical wave equation when $\rho(\cdot)\equiv C$. Since the 1960s, much work has focused on periodic solutions of classical wave equations (see ^{[10,11,12,13,14,15,16]}). For recent results on Hamiltonian systems, see ^[17,18], and on higher-dimensional problems, see ^[19,20,21]. For the Euler equation, see ^[22]. In addition, for stability results, see ^[23,24], and for blow-up solutions, see ^[25,26]. Many of the works are based on the spectrum made up of the eigenvalues $n^2-m^2$ with $n\in\mathbb{N}, m\in\mathbb{Z}$ for the frequency $\omega \in \mathbb{Q}$, for example, ^{[27,28,29,30,31]}. This property ensures that the desired compact conditions hold. However, for the frequency $\omega \in \mathbb{R}\backslash\mathbb{Q}$, the "small divisor problem'' raised naturally in realistic models, such as the wave equations and the beam equations. The tools to solve this problem are the Nash-Morse iteration and KAM (Kolmogorov-Arnold-Moser) theory (see ^[16,32]).

In recent decades, the nonlinear wave equations with variable coefficient have attracted broad interests. For the nonlinearity satisfying Lipschitz continuity, Barbu and Pavel in ^[2] used the monotonicity method to establish a periodic solution under the assumption $\text{ess inf}\, \eta_\rho(\cdot) > 0$ with $\eta_\rho(\cdot) = \frac{1}{2}\frac{\rho''}{\rho}-\frac{1}{4}\big(\frac{\rho'}{\rho}\big)^2$. The appearance of the function $\eta_\rho(\cdot)$ is due to the Liouville transformation in investigating the Sturm-Liouville problem. The condition $\text{ess inf} \, \eta_{\rho}(\cdot) > 0$ can make sure that the kernel space of the variable coefficient wave operator has finite dimensions, then the free oscillations can be easily controlled. On the other hand, it is well known that the solvability of the nonlinear problems depend on the properties of nonlinear terms. For the nonlinearity with power-law growth, Rudakov in ^[33] constructed the periodic solutions by the variational method. Ji and his collaborators in ^[6,8,9,34] obtained some interesting results on periodic solutions for several classes of nonlinear problems under various homogeneous boundary conditions via variational methods. With the help of a global inverse function theorem, Chen in ^[35] got an existence and uniqueness theorem for a system with a variable coefficient. Ji et al. in ^[7] found that there is at least two periodic solutions on some subspaces of the $L^2$ space by using the topological degree theory. These works are all focused on the case $\text{ess inf} \, \eta_{\rho}(\cdot) > 0$. However, for the case of $\eta_\rho(\cdot) = 0$, the kernel space becomes infinite dimensional, which, together with the effects of a variable coefficient, provokes further difficulties; hence, this problem was posed by Barbu and Pavel in ^[2] as an open problem. Recently, Ji and his collaborators considered this problem and obtained some interesting results in ^[36,37,38], where $\eta_\rho(\cdot)$ could be equal to zero or even be of sign-changing.

In addition to the effect of the sign of $\eta_{\rho}(\cdot)$ on the dimension of kernel space, the spectrum of the variable coefficient wave operator has the accumulation points (e.g., ^[39]); thus, the existing compactness conditions are not sufficient to deal with super-linear problems. However, for the Dirichlet-Neumann boundary condition and the Dirichlet-Robin boundary condition, when $T$ satisfies (1.3), the compactness can be improved to be good enough. Recently, when the sign of $\eta_{\rho}(\cdot)$ can change, Rudakov in ^[40] assumed $\eta_{\rho}(\cdot) \neq \frac{\rho'(\pi)}{\rho(\pi)}$ to guarantee that the dimension of the kernel is finite, then infinitely many periodic solutions are constructed for the super-linear problem under the Dirichlet-Neumann boundary condition.

This paper aims to establish the multiplicity of large periodic solutions to the problems (1.1), (1.2), and (1.4) with general variable coefficients. The word "general" means that we do not impose any restrictions on $\eta_\rho(\cdot)$. This results in the kernel space being infinitely dimensional. To overcome this difficulty, we make use of the monotonicity method and approximation argument to estimate the component of solutions in the kernel space. To get the discrete spectrum, we assume that the period $T$ satisfies (1.3), which guarantees that the subspace $E^+ \oplus E^-$ of function space $E$ defined in Section 2 is compactly embedded in $L^p$ space for $p > 1$. This compactness condition is sufficient for dealing with the super-linear problem. Compared with the results in ^[40], we remove the restriction on $\eta_\rho(\cdot)$ and consider the Dirichlet-Robin boundary condition. Since the sign of $\eta_\rho(\cdot)$ can change, our results can be applied to the classical wave equation.

In this paper, we make the following assumptions:

(H1) $\rho \in C^2[0, \pi]$ and $\rho(x) > 0$, $\forall x\in [0, \pi]$;

(H2) $-\tilde{f}(\cdot, \cdot, \xi) = \tilde{f}(\cdot, \cdot, -\xi)$ for all $(\cdot, \cdot, \xi) \in \Omega\times \mathbb{R}$, and $\tilde{f}(\cdot, \cdot, \xi)$ is nondecreasing in $\xi$ and $\tilde{f}(\cdot, \cdot, \xi) = 0$ if, and only if, $\xi = 0$, where

(H3) There are $M > 0$, $\mu > 2$, and $a_1, a_2 > 0$ such that

and

where

At the end of this section, we show the outline of this paper. Section 2 gives the main result and some preliminaries and proves that the periodic solutions of problems (1.1), (1.2), and (1.4) are equal to the critical points of the variational problem. We study the restricted functional on a sequence of subspaces with increasing dimension and construct approximate solutions in Section 3. Finally, we obtain the main result by combining uniform boundedness and an approximation argument in Section 4, and we present our conclusions in Section 5.

2.   Notations, definitions and results

Let

and

The inner product on the Hilbert space $L^2(\Omega)$ is defined as

Definition 2.1. A function $u \in L^p(\Omega)$ is a weak solution of the problems (1.1), (1.2), and (1.4) if

The main results are given as follows.

Theorem 2.1. Let $\alpha_i, \beta_i$ satisfy (1.5) for $i = 1, 2$, the period $T$ satisfy (1.3), and let $\rho$ and $f$ satisfy (H1)–(H3), then, there are infinitely many periodic solutions $u_n$ for the problems (1.1), (1.2), and (1.4), satisfying

Furthermore, $u_n \in C(\Omega)\cap H^1(\Omega)$ for the Dirichlet-Neumann boundary condition and $u_n \in C(\Omega)$ for the Dirichlet-Robin boundary condition.

The sequence of eigenfunctions $\{\phi_j(t)\varphi_k(x) : j\in \mathbb{Z}, k\in \mathbb{N}\}$ forms a completely orthonormal basis of $L^2(\Omega)$ (^[41]), where

and $\lambda_k$, $\varphi_k(x)$ are determined by the following Sturm-Liouville problem

According to Section 4 in ^[42], a direct calculation shows that the eigenvalues $\lambda_k$ have the following asymptotic formula

where

with

The linear operator $L_0$ is defined as

and its extension $L$ is a self-adjoint operator in $L^2(\Omega)$. Furthermore, we have $\lambda_k-\nu_j^2$ as the eigenvalues of $L$. Clearly, $u\in L^2(\Omega)$ is a weak solution of problems (1.1), (1.2), and (1.4) if, and only if,

For any $u, v \in L^2(\Omega)$, we rewrite it as $u(t, x) = \sum\limits_{j, k} u_{jk} \phi_j(t)\varphi_k(x)$ and $v(t, x) = \sum\limits_{j, k} v_{jk} \phi_j(t)\varphi_k(x)$, where $u_{jk}$ and $v_{jk}$ are the Fourier coefficients. Set

then their direct sum $E: = E^+ \oplus E^- \oplus E^0$ is a Hilbert space with the inner product

and its norm is denoted by $\|u\|_E$. For any $u\in E$, split it into $u = u^+ +u^0 + u^-$ with $u^+\in E^+$, $u^0\in E^0$, $u^-\in E^-$. In particular,

For any $m, n\in \mathbb{N}^+$, denote the finite dimensional spaces by

Define the "upper direct sum" spaces and the "under direct sum" spaces by

Let

Obviously, $E^m \subset E^{m+1}$ and $E^m_n$ is a finite dimensional space. Furthermore,

Define the energy functional corresponding to problems (1.1), (1.2), and (1.4) as

Since $\tilde{f}$ is odd, $\Phi$ is an even $C^1$ functional on $E$. In addition,

Therefore, $u$ is a weak solution of problems (1.1), (1.2), and (1.4) if, and only if, $u$ is a critical point of $\Phi$, namely,

In what follows, we first consider the restricted functional $\Phi_m = \Phi|_{E^m}$ and construct its critical points.

3.   Constructing approximate solutions

Proposition 3.1. For any $q > 1$, the embedding

is compact.

Proof. A similar proof as Lemma 2.1 in ^[36] shows that the series

is convergent. Thus,

In consequence, by using the method of Lemma 1 in ^[40], we obtain the result.

Lemma 3.1. For any $m\in \mathbb{N}^+$, let $\{u_i\} \subset E^m$ satisfy $\Phi_m (u_i)\leq \tilde{d}$ (a constant) and $\Phi_m' (u_i) \rightarrow 0$ as $i \rightarrow \infty$, then $\{u_i\}$ has a convergent subsequence, i.e., $\Phi_m$ satisfies the Palais-Smale $(PS)$ condition.

Proof. Split $u_i = u^+_i +\hat{u}_i$ with $u^+_i \in E^+$, $\hat{u}_i \in W_m \cap (E^-\oplus E^0)$. Obviously, $\hat{u}_i = u^-_i + u^0_i$ with $u^-_i \in W_m \cap E^-$, $u^0_i \in W_m \cap E^0$.

Since $\Phi_m (u_i)\leq \tilde{d}$ and $\langle \Phi_m'(u_i), u_i \rangle \rightarrow 0$, by (1.7), we have,

for some constant $M_1 > 0$. Thus,

for some constant $M_2 > 0$. Taking advantage of (1.7) again and the above estimate, we have

for some constant $M_3 > 0$.

According to (1.7), it follows that

for some constants $a_3, a_4 > 0$. By (3.2) and (3.3), we have

for some constant $M_4 > 0$. Moreover, from (1.8), we have

where $\mu' = \mu/(\mu - 1)$.

Noting $\Phi_m'(u_i)\rightarrow 0$, by (3.1), (3.4), and (3.5), it follows that

Therefore, $\{u_i^+\}$ is bounded in $E$.

By $\dim \Big(W_m \cap (E^-\oplus E^0)\Big) < \infty$, from (3.4), we have that $\{\hat{u}_i\}$ is bounded in $E$.

Consequently, $\{u_i\}$ is bounded in $E$, thus $u_i \rightharpoonup u$ in $E$ as $i\rightarrow \infty$ for some $u \in E$. Let $u^+$, $\hat{u}$ denote weak limits of $\{u^+_i\}$, $\{\hat{u}_i\}$, respectively, where $u^+_i, u^+ \in E^+$, $\hat{u}_i, \hat{u} \in W_m \cap (E^-\oplus E^0)$.

Since $\dim \Big(W_m \cap (E^-\oplus E^0)\Big) < \infty$, then

For $u^+_i \in E^+$, from (2.3), it follows that

By (3.1), $u_i^+$ weakly converges to $u^+$, and it follows that

Therefore,

which completes the proof.

Proposition 3.2. Set

then $\zeta_{n}\rightarrow 0$ as $n\rightarrow \infty$.

Proof. Noting that $(E_{n-1})^\perp \subset E^+$ and the embedding $E^+ \hookrightarrow L^\mu(\Omega)$ is compact, a similar proof as ^[43] yields the result.

Lemma 3.2. For any $n\in \mathbb{N}^+$, there exist the constants $\sigma_n$, $r_n > 0$ satisfying

and

Proof. From (3.6), (1.8), and (2.2), for $u\in (E_{n-1})^\perp \subset E^{+}$, it follows that

Taking $r_n = (\mu a_1 \zeta^{\mu}_n)^{\frac{1}{2-\mu}}$ in the above estimate, with the help of $\mu > 2$ and $\zeta_{n}\rightarrow 0$ as $n\rightarrow \infty$, for $n$ large enough, it follows that

For $n$ large enough, $\sigma_n: = (\frac{1}{2}-\frac{1}{\mu})r_n^2 - a_2T\pi > 0$ and $\sigma_n\rightarrow \infty$ as $n\rightarrow \infty$, and the proof is complete.

Lemma 3.3. For any $n\in \mathbb{N}^+$, there exist the constants $R_n$, $\varrho_n > 0$ satisfying

Proof. By (3.3), it follows that

for some positive constants $M_7, M_8$.

Since $u^+ \in W_n\cap E^+$ and $\dim (W_n\cap E^+) < \infty$, then $\|u^+\|^{\mu}_{L^{\mu}(\Omega)} \geq C_1 \|u^+\|^{\mu}_E$. Moreover, since $\|u^0\|_E = \|u^0\|_{L^{2}(\Omega)}$ and $L^{\mu}(\Omega) \hookrightarrow L^{2}(\Omega)$, then $C_2\|u^0\|_E \leq \|u^0\|_{L^{\mu}(\Omega)}$. Therefore,

Thus, noting $\mu > 2$, we arrive at the result.

Let

where $B^m_n = \{u\in E^m_n \mid \|u\|_E \leq R_n\}$, $\partial B^m_n$ denotes the boundary of $B^m_n$, the constant $R_n$ is given in Lemma 3.3, and $id$ is the identity map.

Define $A_{c} = \{u\in E^m \mid \Phi_m(u) \leq c\}$, $\forall c\in \mathbb{R}$, and

Define

Lemma 3.4. For $n$ large, $c_{mn}$ are the critical values of $\Phi_m$ and satisfy

Proof. First, it is proved by contradiction. In virtue of $\Phi_m$ satisfying the $(PS)$ condition, suppose that $c_{mn}$ are not the critical values of $\Phi_m$ and there is $\bar{\varepsilon} > 0$ satisfying $\Phi_m^{-1} [c_{mn}-\bar{\varepsilon}, c_{mn}+\bar{\varepsilon}]\cap K = \emptyset$. By the definition of $c_{mn}$ and taking $\gamma_0 \in \mathscr{F}_{mn}$ such that $\max_{u\in B^m_n} \Phi_m (\gamma _0(u)) \leq c_{mn} + \bar{\varepsilon}$, we have

By the standard deformation lemma, there is an odd mapping $\eta_t(\cdot): = \eta(t, \cdot) \in C([0, 1]\times E^m, E^m)$ such that $\eta_1(A_{c_{mn} + \bar{\varepsilon}}) \subset A_{c_{mn} - \bar{\varepsilon}}$, which implies

Consequently, $\eta_1\circ \gamma_0\mid_{\partial B^m_n} = id$ and $\eta_1\circ \gamma_0$ is odd, i.e., $\eta_1\circ\gamma_0 \in \mathscr{F}_{mn}$. Therefore,

This is a contradiction.

Now, we prove $\sigma_n\leq c_{mn} \leq \varrho_n$. According to Lemma 3.3 and the definitions of $c_{mn}$, we have $c_{mn} \leq \varrho_n$.

On the other hand, for each $\gamma \in \mathscr{F}_{mn}$, let

where the constant $r_n$ is present in Lemma 3.2. Since $\gamma$ is odd continuous, then $B_{r_n}$ is a symmetrically bounded open ball and $0\in B_{r_n}$. Moreover, from Lemmas 3.2 and 3.3, it is easy to see $R_n > r_n$. The combining of $R_n > r_n$ and $\gamma|_{\partial B^m_n} = id$ yields $B_{r_n} \cap \partial B^m_n = \emptyset$. Let $P: E^m \rightarrow E^m_{n-1}$ be the natural projection. Therefore, by the Borsuk-Ulam theorem ^[44], there exists $u_0\in \partial B_{r_n}$ satisfying $P\gamma(u_0) = 0$, then we have $\|\gamma(u_0)\|_E = r_n$ and $\gamma(u_0) \in (E_{n-1})^\perp$. Thus, by Lemma 3.2, we have

We arrive at the conclusion.

For each $n$ large, suppose that $u_{mn}$ are the critical points of $\Phi_m$ corresponding to $c_{mn}$. In what follows, to obtain Theorem 2.1, we are going to prove the uniform boundedness of $\{u_{mn}^{\pm}\}$ for any $n\in\mathbb{N}^+$, then we use the approximation argument to get the desired results.

4.   Proof of Theorem 2.1

Let $M_i > 0$ ($i = 9, 10, 11, 12$) denote the constants that are independent of $m$.

We have

Moreover, since $u_{mn}$ are the critical points of $\Phi_m$, then

Thus, for any $v\in E^m$, it follows that

Lemma 4.1. The sequence $\{u_{mn}^{\pm}\}$ is uniformly bounded for any $n\in\mathbb{N}^+$.

Proof. Since

from (1.7), (2.2), (4.2), and Lemma 3.4, there exists $M_9 > 0$ such that

Thus, by (1.7), we have that $\int_\Omega \widetilde{F}(t, x, u_{mn}) \rho {\rm{d}}t {\rm{d}}x$ is uniformly bounded.

From (3.3), it follows that

From (1.8), it follows that

where $\mu' = \mu/(\mu - 1)$. Since

we have that $\{u_{mn}^+\}$ is uniformly bounded for any $n\in\mathbb{N}^+$. The similar conclusion holds for $\{u_{mn}^-\}$. We arrive at the conclusion.

Since $L^{p}(\Omega)$ and $E$ are reflexive and the embedding $E^-\oplus E^+ \hookrightarrow L^q(\Omega)$ is compact for $q > 1$, then by the above lemma and (4.3), without loss of generality, we have

Thanks to the above lemmas, now let's prove Theorem 2.1.

Proof. Let $P_m :E \rightarrow E^m$ be the natural projection. According to $u_{mn}^{\pm} \in E^m$ and $u_{n}^{\pm} \in E = \bigcup_{m\in\mathbb{N}^+} E^m$, it follows that

In virtue of (3.1) and $\|(P_m - id) u_{n}^+\|_E \rightarrow 0$ as $m \rightarrow \infty$, we have

Replacing $v$ with $u_{mn}^+ - P_m u_{n}^+$ in (4.1), with the aid of (4.4)–(4.7), we have

as $m\rightarrow \infty$. Therefore, by (4.5), we have

By using (4.5) again, a similar proof shows

Consequently,

To continue discussion, for any $v \in E$ and since $(id - P_m)v\in (E^m)^\perp$ and $u_{mn} - P_mu_n \in E^m$, $u_{mn}$ is the critical point of $\Phi_m$, then from (4.1), we have

Since $\tilde f$ is monotone in $u$, a simple calculation yields

Moreover, according to $\|(id - P_m) v\|_{L^{\mu}(\Omega)}\rightarrow 0$ and (4.4), we have

as $m \rightarrow \infty$. In virtue of the embedding $L^{\mu}(\Omega) \hookrightarrow L^{2}(\Omega)$ and $u_{mn}\rightharpoonup u_{n}$ in $L^{\mu}(\Omega)$, it follows that $u_{mn}\rightharpoonup u_{n}$ in $L^{2}(\Omega)$ as $m \rightarrow \infty$. From (4.8) and (4.9), with the help of $u_{mn}^\pm \rightarrow u_n^\pm$ in $E$, we have

For $s > 0$ and $\psi \in E$, taking $v = u_n -s \psi$ and dividing by $s$ in (4.10) shows

then letting $s \rightarrow 0$ gets

By using the arbitrariness of $\psi$, it follows that

Therefore, $u_n$ is the critical point of $\Phi$ for $n$ large enough.

Moreover, since $\tilde{f}$ is a nondecreasing function with respect to $u$, then its primitive function $\widetilde{F}$ is convex with respect to $u$. Thus, according to $u_{mn}^\pm \rightarrow u_n^\pm$ in $E$, we have

In consequence, by (1.8), Lemma 3.2 and the embedding $L^\mu(\Omega) \hookrightarrow L^1(\Omega)$, we have

where the constant $C_3$ is independent of $n$. Thus, taking into account $\sigma_n \rightarrow \infty$ as $n \rightarrow \infty$, we have

Moreover, we have

Replacing $v$ with $\phi_j(t)\varphi_k(x)$ in the above equation, we obtain

where $u_{jk}$, $\tilde{f}_{jk}$, respectively, denote the Fourier coefficients of $u_{n}$, $\tilde{f}$. Noting that the series $\sum_{\lambda_k\neq \nu_j^2} \frac{1}{|\lambda_k-\nu_j^2|^\mu}$ is convergent and the combination of the Hausdorff-Young and Hölder inequalities yields

Therefore, recalling $\dim E^0 < \infty$ (see ^[36]), we have $u_n \in C(\Omega)$.

Furthermore, under the Dirichlet-Neumann boundary condition, the system

forms an orthonormal basis of $L^2(0, \pi)$. By the methods in ^[40], we have

By the above estimate and (2.1), the sequences $\{\frac{|j|}{|\lambda_k-\nu_j^2|}\}$ and $\{\frac{\sqrt{\lambda_k}}{|\lambda_k-\nu_j^2|}\}$ are bounded. Therefore, we have $u_n \in H^1(\Omega)$, and the proof is complete.

5.   Conclusion

In this paper, we established the multiplicity of large periodic solutions for the super-linear problem under the Dirichlet-Neumann boundary condition and the Dirichlet-Robin boundary condition. We remove the only restrict condition $\eta_{\rho}(\cdot) \neq \frac{\rho'(\pi)}{\rho(\pi)}$ on $\eta_\rho(\cdot)$ in ^[40]; thus, we do not impose any restrictions on $\eta_\rho(\cdot)$. To get better compactness conditions, we assume the period $T$ satisfies (1.3). Finally, since the sign of $\eta_\rho(\cdot)$ can change, our results can be applied to the classical wave equation.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by NSFC Grants (Nos. 12071065, 12226330 and 12226310), the Natural Science Foundation of Jilin Province (No. 20210101142JC) and the Key Scientific Research Projects of Colleges and Universities in Henan Province (No. 22A110016).

Conflict of interest

The authors declare there is no conflict of interest.