1 Introduction and main results

Consider the second-order damped vibration system:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \ddot{u}(t)+A{\dot{u}}(t)+\nabla _u V(t,u(t))=0, &{} \quad \forall \hbox { t } \in {\mathbb {R}}, \\ u(0)-u(T)={\dot{u}}(0)- {\dot{u}}(T)=0,&{} \quad T>0, \end{array} \right. \end{aligned}$$
(1.1)

where A is a skew-symmetric matrix, \(V(t,u)=-K(t,u)+W(t,u)\) and \(K,W\in C^2({\mathbb {R}}\times {\mathbb {R}}^n,{\mathbb {R}})\) with conditions

$$\begin{aligned} K(t+T,u)=K(t,u),\quad W(t+T,u)=W(t,u),\quad \forall (t,u)\in {\mathbb {R}}\times {\mathbb {R}}^n. \end{aligned}$$

When \(A\equiv 0,\) (1.1) is just the following second order non-autonomous Hamiltonian system:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \ddot{u}(t)+\nabla _u V(t,u(t))=0, &{} \forall \hbox { t }\in {\mathbb {R}}, \\ u(0)-u(T)={\dot{u}}(0)- {\dot{u}}(T)=0,&{} T>0. \end{array} \right. \end{aligned}$$
(1.2)

During the last several decades, the existence and multiplicity of periodic solutions for second-order Hamiltonian systems have been extensively studied via critical point theory, such as [2, 7, 8, 14, 15, 17, 20, 22, 24,25,26] and the references therein. In those papers, V(t, u) was required to satisfy some growth conditions as \(|u|\rightarrow +\infty ,\) such as asymptotically linear, subquadratic, asymptotically quadratic or superquadratic growth. In [8, 20] the authors considered the case that V(t, u) satisfies subquadratic potential condition. In 2009, when \(V(t,u)=-K(t,u)+W(t,u),\) Zhao [27] established the existence result of system (1.2) with conditions that W(t, u) is asymptotically linear and K(t, u) satisfies “pinched” condition:

$$\begin{aligned} a_{1}|u|^{2} \leqslant K(t, u) \leqslant a_{2}|u|^{2}, \end{aligned}$$

where constants \(a_{1}, a_{2}>0 .\)

In 2011, when \(V(t, u) =\frac{1}{2}(L(t)u\cdot u)+W(t,u),\) where L(t) is a \(n\times n\) symmetric matrix, Zhang and Liu [25] considered the multiplicity of periodic solutions for system 1.2 with condition that W(t, u) is asymptotically quadratic or superquadratic. In 2013, Gu and An [7] investigated the multiplicity of periodic solutions for system (1.2) with subquadratic condition. In 2018, Wang and Zhang [22] studied the existence of periodic solutions of system (1.2) with locally asymptotically quadratic condition or locally superquadratic condition. Motivated by [25, 27], in this paper, firstly we generalized the above results by replacing the ‘pinching’ condition by the following conditions:

  • \((H_1)\) There exist constants \(d> 0\) and \(L_1>0\) such that \(K(t,u)\ge -d|u|^{2}\) for all \(t\in [0,T]\) and \(|u|\ge L_1,\)

  • \((H_2)\) There exists a constant \(L_2>0\) such that \((\nabla K(t,u)\cdot u)\le 2K(t,u)\) for all \(t\in [0,T]\) and \(|u|\ge L_2.\)

Secondly, if \(A\not \equiv 0\) and \(V(t,u)=-K(t,u)+W(t,u),\) we discuss the case that W(t, u) satisfies an asymptotically quadratic condition. By using Fountain theorem, we study the existence of infinitely many nontrivial odd T-periodic solutions of (1.1). However, the space in Fountain theorem is not a regular Sobolev space. What’s more, Long (see [10]) introduced the bi-even condition and one space, which is a closed subspace of the commonly used Sobolev space \(H^1_T\) consisting of odd functions, denoted by E in this paper. We would like to remind the readers, if V(t, u) only satisfies \(V(t,-u)=V(t,u)\), then the multiplicity result of periodic solutions but not necessarily odd solutions for problem (1.1) can also be obtained via Fountain theorem.

Now, we will use the following assumptions to prove our results.

  • \((H_3)\) \(\displaystyle \lim \nolimits _{|u|\rightarrow +\infty } [(\nabla W(t,u)\cdot u)-2W(t,u)]=+\infty \) uniformly for \(t\in [0,T].\)

  • \((H_4)\) There exist a function \(b\in L^1([0,T],(0,+\infty ))\) and a constant \(L_3> 0\) such that

    $$\begin{aligned} 0<W(t,u)\le b(t)|u|^2,\quad \forall t\in [0,T],\quad |u|\ge L_3. \end{aligned}$$

  • \((H_5)\) V(t, u) is bi-even, which means that it is even in t and u variables respectively.

  • \((H_6)\) \(\Vert A\Vert \le 1.\)

Now, we are ready to state the main results of this paper as follows.

Theorem 1.1

Assume that \((H_1){-}(H_6)\) hold. Then problem (1.1) possesses infinitely many odd T-periodic solutions \(\{u_k\}\) satisfying \(\Vert u_k\Vert _{\infty }\rightarrow +\infty \) as \(k\rightarrow +\infty .\)

Remark 1.1

Conditions \((H_3)\) and \((H_4)\) imply that W(t, u) satisfies the asymptotically quadratic condition, that is

$$\begin{aligned} 0< \displaystyle \liminf _{|u|\rightarrow +\infty }\frac{W(t,u)}{|u|^2}\le \displaystyle \limsup _{|u|\rightarrow +\infty }\frac{W(t,u)}{|u|^2}< +\infty ,\quad \text{ a.e.t }\in [0,T]. \end{aligned}$$

Obviously, if K(t, u) is a quadratic form, then K satisfies \((H_1)\) and \((H_2)\). So functions K in Theorem 1.1 not only can be all quadratic forms, but also may be subquadratic (for example, \(K(t,u)=\frac{|u|^2}{\ln (10+|u|^2)},\;\forall t\in {\mathbb {R}},\forall u\in {\mathbb {R}}^n\) or asymptotically quadratic (for example, \(K(t,u)=\frac{1}{3}|u|^2+\ln (10+|u|^2),\;\forall t\in {\mathbb {R}},\forall u\in {\mathbb {R}}^n\)). The variant fountain theorem requires that the partial functional is nonnegative, which needs \(W(t,u)\ge 0\) for all \(t\in [0,T]\) and \(u\in {\mathbb {R}}^n.\) However, the Fountain theorem used in this paper requires that the functional \(\varphi \) is infinitely large quantity in the subspace of E,  which only needs \(W(t,u)>0\) with |u| large enough. Function V satisfying our assumptions of Theorem 1.1 do really exist, but may not be covered by [25, Theorem 1.1].

Now, we give an example as an application of this result.

Example 1.1

Set \(T = \pi \) and define \(K,G : {\mathbb {R}} \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) with

$$\begin{aligned} K(t,x)=\cos ^2t.\frac{x^2}{ln(e+x^2)} \end{aligned}$$

and

$$\begin{aligned} G(t,x)=\frac{1+\cos ^2t}{2}\left[ 1-\frac{1}{ln(100+x^2)}\right] x^2. \end{aligned}$$

Choose a function \(\lambda \in C^{\infty }({\mathbb {R}}^+ ,[0,1])\) such that \(\lambda (t) = 1\) for \(t \le 1\), \(\lambda (t) = 0\) for \(t \ge 2\), and \(\lambda '(t) \le 0\) for \(t \in [1,2]\). Set

$$\begin{aligned} W(t,x)=-\frac{1}{2}\lambda (|x|)x^2+(1-\lambda (|x|))G(t,x),\quad \forall t,x\in {\mathbb {R}}. \end{aligned}$$

Then functions \(K,W \in C^1 ({\mathbb {R}} \times { {\mathbb {R}}}, {\mathbb {R}})\) hold and are \(\pi \)-periodic with respect to the variable t. Obviously, both K and W satisfy condition (V). For K(t, x), one has \(-|x|^2\le 0\le K(t,x)\le |x|^2\) and

$$\begin{aligned} \frac{\partial K(t,x)}{\partial x}= & {} \cos ^2t.\left[ \frac{2x^2}{ln(e+x^2)}-\frac{2x^4}{(e+x^2)ln^2(e+x^2)}\right] \\= & {} \cos ^2t.\frac{2x^2}{ln(e+x^2)} \\= & {} 2K(t,x),\quad \forall t,x\in {\mathbb {R}}. \end{aligned}$$

Hence, K(t, x) satisfies \((H_1)\) with \(d = 1\) and \((H_2)\) condition. It is evident that W satisfies \((H_4)\) with \(b(t) \equiv 1\) and \(L_3 = 2\). Furthermore,

$$\begin{aligned} \frac{\partial W(t,x)}{\partial x}.x-2W(t,x)= & {} \frac{(1+\cos ^2t)x^4}{(100+x^2)ln^2(100+x^2)} \\\ge & {} \frac{x^4}{(100+x^2)ln^2(100+x^2)}\rightarrow +\infty \quad \text{ as }\quad |x|\rightarrow +\infty . \end{aligned}$$

So

$$\begin{aligned} \frac{\partial W(t,x)}{\partial x}.x-2W(t,x)=+\infty \quad \text{ uniformly } \text{ for }\quad t\in {\mathbb {R}}. \end{aligned}$$

Hence W(t, x) satisfies condition \((H_3)\). By Theorem 1.1, problem (1.1) possesses infinitely many odd \(\pi \)-periodic solutions for above K and W. Obviously, W in Example 1.1 cannot be covered by conditions of [25, Theorem 1.1] because not only K(t, x) is not a quadratic form, but also \(W(t,x) \le 0\) with \(|x| \le 1\).

2 Variational setting and preliminaries

As usually, we denote

$$\begin{aligned} H^1_{T}= & {} \left\{ u:[0,T]\rightarrow {\mathbb {R}}^n,\; u\; is \;absolutely\; continuous, \right. \\&\left. u(0)=u(T)\; and \;{\dot{u}}\in L^2([0,T],{\mathbb {R}}^n)\right\} . \end{aligned}$$

Then \(H^1_{T}\) is a Hilbert space with the norm

$$\begin{aligned} \Vert u\Vert _{H^1_T}=\left[ \int ^{T}_0|u(t)|^2dt+\int ^{T}_0|{\dot{u}}(t)|^2dt\right] ^{1/2},\quad u\;\in H^1_{T} \end{aligned}$$

and the associated inner product

$$\begin{aligned} ( u \cdot v)=\int ^{T}_0(u(t)\cdot v(t))dt+\int ^{T}_0({\dot{u}}(t)\cdot {\dot{v}}(t))dt,\quad u,v\; \in H^1_{T}. \end{aligned}$$

Let E be the subspace of \(H^1_T\) consisting of odd functions. The corresponding norm is

$$\begin{aligned} \Vert u\Vert= & {} \left( \displaystyle \int ^T_0|{\dot{u}}(t)|^2dt\right) ^{\frac{1}{2}}\quad \forall u\in E, \end{aligned}$$
(2.1)

which is equivalent to the norm \(\Vert u\Vert _{H^1_T}\) on E. Note that E is a closed subspace of \(H^1_T\), so it is a reflective Hilbert space.

Lemma 2.1

[12] The space \(H_1^T\) is compactly embedded in \(C([0,T], {\mathbb {R}}^n).\) In addition,

$$\begin{aligned} \Vert u\Vert _{\infty }\le C_{\infty }\Vert u\Vert _{H^1_T},\quad \forall u\in H^1_T, \end{aligned}$$

where \(C_{\infty }\) is a positive constant.

We consider the functional \(\varphi \): \(H^1_{T}\rightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} \varphi (u) =\displaystyle \int ^{T}_{0}\left[ \frac{1}{2}|{\dot{u}}(t)|^2+\frac{1}{2}(Au(t)\cdot {\dot{u}}(t))+K(t,u(t))-W(t,u(t))\right] dt. \end{aligned}$$
(2.2)

It is well know that \(\varphi \) is continuously differentiable on \(H^1_{T}\) and by using the skew-symmetry of A, we have

$$\begin{aligned} (\varphi '(u)\cdot v)= & {} \displaystyle \int ^{T}_{0}({\dot{u}}(t).{\dot{v}}(t))dt-\displaystyle \int ^{T}_{0}(A{\dot{u}}(t)\cdot v(t))dt+\displaystyle \int ^{T}_{0}(\nabla K(t,u(t)).v(t))dt\nonumber \\&- \displaystyle \int ^{T}_{0}\nabla (W(t,u(t))\cdot v(t))dt,\quad \forall u, v \in H^1_{T}. \end{aligned}$$
(2.3)

Furthermore, a point \(u \in H^1_T\) is a T-periodic solution of system (1.1) if and only if u is a critical point to the functional \(\varphi .\)

The following lemmas will be needed in the proof of our results.

Lemma 2.2

[10] For \(T> 0,\) suppose \(V\in C^1({\mathbb {R}}\times {\mathbb {R}}^n, {\mathbb {R}}),\) \(V(t+T,u)=V(t,u)\) for all \((t,u)\in {\mathbb {R}}\times {\mathbb {R}}^n,\) and V is bi-even. Then \(\varphi \in C^1(E,{\mathbb {R}})\) and that \(u\in E\) is a critical point of \(\varphi \) restricted to E if and only if it is an odd \(C^2([0,T],{\mathbb {R}}^n)\)-solution of problem (1.1).

So the odd solutions of problem (1.1) correspond to the critical points of the functional \(\varphi _{|_E}\) via Lemma 2.2. From now on, for simplicity, we use \(\varphi \) for \(\varphi _{|_E}\). By a simple calculation, we obtain that the differential system

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\ddot{u}(t)=\lambda u(t), &{}\forall t\in [0,T], \\ u(0)-u(T)={\dot{u}}(0)-{\dot{u}}(T)=0.&{} \end{array} \right. \end{aligned}$$

has the eigenvalues \( \lambda _j= \frac{4j^2\pi ^2}{T^2}(j\in {\mathbb {N}})\) with \(\lambda _j\rightarrow +\infty \) as \(j\rightarrow +\infty \) and the corresponding normalized eigenfunctions \(\{e_j\}\) \((-\ddot{e}_j=\lambda _je_j,\) and \(e_j\) has the form of \(a_j \cos (\sqrt{\lambda }t)+b_j\sin (\sqrt{\lambda }t),\) \(a_j,\) \(b_j\in {\mathbb {R}}^n\)) with \(\displaystyle \int ^T_0(e_j(t),e_k(t))dt=\delta _{jk}\) and \(\displaystyle \int ^T_0({\dot{e}}_j(t),{\dot{e}}_k(t))dt=\lambda _k.\delta _{jk}\) for every \(j,k\in {\mathbb {N}}^*,\) we define subspaces

$$\begin{aligned} X_j=span\{e_j\}\cap E\ne \varnothing ,&Y_k=\bigoplus _{j=1}^k X_j\quad and\quad Z_k=\overline{\bigoplus _{m\ge k} X_m}, \end{aligned}$$
(2.4)

then \(E=\overline{\bigoplus _{j\in {\mathbb {N}}^*} X_j}=Y_k\oplus Z_{k+1}.\) For every \(u_k\in Z_k\) and \(c_j\in {\mathbb {R}},\) \(u_k=\displaystyle \sum \nolimits ^{\infty }_{j=k}c_je_j,\) then one has

$$\begin{aligned} \Vert u_k\Vert ^2_{L^2}= & {} \displaystyle \sum ^{\infty }_{j=k}c^2_j \quad \text{ and }\quad \Vert {\dot{u}}_k\Vert ^2_{L^2} = \displaystyle \sum ^{\infty }_{j=k}c^2_j\lambda _j. \end{aligned}$$
(2.5)

Lemma 2.3

[23, Theorem 3.6] Assume that \( \varphi \in C^1(E,{\mathbb {R}})\) satisfies \(\varphi (u)=\varphi (-u)\) and the subspace \(X_j,\) \(Y_k,\) \(Z_k\) defined in (2.4). For every \(k\in {\mathbb {N}}^*\), there exists \(\rho _k> r_k>0\) such that

  • \((A_1)\) \(\displaystyle \inf \nolimits _{u \in Z_k, \Vert u\Vert =r_k}\varphi (u)\rightarrow +\infty \) as \(k\rightarrow +\infty .\)

  • \((A_2)\) \(\displaystyle \max \nolimits _{u\in Y_k, \Vert u\Vert =\rho _k} \varphi (u)\le 0,\)

  • \((A_3)\) \(\varphi \) satisfies the \((PS)_c\) condition for every \(c>0.\)

If k is large enough, and set \(c_k =\displaystyle \inf \nolimits _{h\in \Gamma _k}\max \nolimits _{u\in B_k}\varphi (h(u))\), where \(B_k :=\{u\in Y_k :\Vert u\Vert \le \rho _k\},\) \(\Gamma _k:=\{h\in C(B_k,E): \text{ h } \text{ is } \text{ odd } \text{ and } h_{\mid {\partial B_k}}=id\}\) then \(c_k\ge \displaystyle \inf \nolimits _{u\in Z_k,\Vert u\Vert =r_k} \varphi (u),\) furthermore, \({c_k}\) is an unbounded sequence of critical values of \(\varphi \) .

Remark 2.1

As shown in [1], a deformation lemma can be proved with the weaker (C) condition which is due to Cerami (see [3]) replacing the usual \((PS)_c\) condition, and it turns out that Lemma 2.3 holds under (C) condition.

Lemma 2.4

Set \(\beta _k:=\sup _{u\in Z_k,\Vert u\Vert =1}\Vert u\Vert _{\infty }\) \(k\in {\mathbb {N}}^*,\) one has

$$\begin{aligned} \beta _k\rightarrow 0\quad \text{ as }\quad k\rightarrow +\infty . \end{aligned}$$

Proof

The main idea comes from [11]. The definition of \(\beta _k\) implies that \(0\le \beta _{k+1}\le \beta _k,\) so \(\displaystyle \lim \nolimits _{k\rightarrow +\infty }\beta _k\) does really exist. For every \(k\in {\mathbb {N}}^*,\) there exists \(u_k\in Z_k\subseteq E=Z_1\) such that \(\Vert u_k\Vert =1\) and

$$\begin{aligned} \Vert u_k\Vert _{\infty }> & {} \frac{1}{2}\beta _k. \end{aligned}$$
(2.6)

For the above \(u_k(t)=(u^1_k(t),u_k^2(t),\ldots ,u_k^n(t))\in Z_k\subseteq E,\) one has that \(\displaystyle \int ^T_0u^i_k(t)dt=0\; (i=1,2,\ldots ,n).\) From the mean value theorem, there exists a \(\xi _i\in (0,T)\) such that \(u_k^i(\xi _i)=0\; (i=1,2,\ldots ,n).\) Then one has

$$\begin{aligned} |u_k^i(t)|^2=2\int ^ t_{\xi }{\dot{u}}^i_k(s).u^i_k(s)ds\le 2\Vert {\dot{u}}^i_k\Vert _{L^2}\Vert u^i_k\Vert _{L^2},\quad i=1,2,\ldots ,n,\quad \forall \;\;t\in [0,T], \end{aligned}$$

which implies that

$$\begin{aligned} |u_k(t)|^2= & {} \displaystyle \sum ^n_{i=1}|u_k^i(t)|^2\nonumber \\\le & {} 2\displaystyle \sum ^n_{i=1}\Vert {\dot{u}}^i_k\Vert _{L^2} \Vert u^i_k\Vert _{L^2}\nonumber \\\le & {} 2\left( \displaystyle \sum ^n_{i=1}\Vert {\dot{u}}^i_k\Vert _{L^2}\right) ^{\frac{1}{2}}\left( \displaystyle \sum ^n_{i=1} \Vert u^i_k\Vert _{L^2}\right) ^{\frac{1}{2}}\nonumber \\= & {} 2\Vert {\dot{u}}_k\Vert _{L^2}\Vert u_k\Vert _{L^2}. \end{aligned}$$
(2.7)

By (2.5), there holds

$$\begin{aligned} 1=\Vert u_k\Vert ^2=\displaystyle \int ^T_0|{\dot{u}}_k(t)|^2dt\ge \lambda _k\Vert u_k\Vert ^2_{L^2},\quad \forall u_k\in Z_k, \end{aligned}$$

which implies that \(\Vert u_k\Vert _{L^2}\rightarrow 0\) as \(k\rightarrow +\infty .\) By (2.7), one has \(\Vert u_k\Vert ^2_{\infty }\le 2\Vert {\dot{u}}_k\Vert _{L^2}\Vert u_k\Vert _{L^2}\le 2\Vert u_k\Vert _{L^2}\rightarrow 0\) as \(k\rightarrow +\infty .\) So (2.6) implies that \(\beta _k\rightarrow 0\) as \(k\rightarrow +\infty .\) \(\square \)

Lemma 2.5

[16, Lemma 1] Suppose that \(\Omega \) is a Lebesgue measurable subset of \({\mathbb {R}}\) with meas(\(\Omega )<+\infty \) (‘meas’ denotes the Lebesgue measure) and \(\{f_n(t)\}\) is a sequence of Lebesgue measurable functions such that \(f_n(t)\rightarrow +\infty \) as \(n\rightarrow +\infty \) for a.e. \(t\in \Omega \). Then for every \(\delta > 0\), there exists a subset \(\Omega _{\delta }\) with meas(\(\Omega \backslash \Omega _{\delta }) < \delta \) such that \(f_n(t)\rightarrow +\infty \) as \(n\rightarrow \infty \) uniformly for all \(t\in \Omega _{\delta }\).

3 Proof of Theorem 1.1

Before the proof of Theorem 1.1, we need the following lemmas.

Lemma 3.1

[4, Lemma 1] Assume that W(t, x) satisfies \((H_3)\), K(t, x) satisfies \((H_2),\) then there exists a constant \(M>L_2\) large enough such that

$$\begin{aligned} W(t,x)\ge \frac{|x|^2}{M^2}.\displaystyle \min _{|x|=M}W(t,x),&\text{ if }\quad |x|\ge M\quad \text{ and }\quad t\in [0,T], \\ K(t,x)\le \frac{|x|^2}{L^2_2}.\displaystyle \max \nolimits _{|x|=L_2} K(t,x),&\text{ if }\quad |x|\ge L_2\quad \text{ and }\quad t\in [0,T]. \end{aligned}$$

Remark 3.1

Lemma 3.1 implies that there exists a function \(c(t)> 0\) such that

$$\begin{aligned} W(t,x)\ge & {} c(t)|x|^2,\quad |x|\ge M. \end{aligned}$$
(3.1)

Condition \((H_4)\) and inequality (3.1) imply that

$$\begin{aligned} c(t)\le \frac{W(t,x)}{|x|^2}\le b(t). \end{aligned}$$

Then W(t, x) is an asymptotically quadratic function.

Lemma 3.2

Assume that W(t, x) satisfies \((H_3)\) and \((H_4)\), K(t, x) satisfies \((H_1)\) and \((H_2)\), then the functional \(\varphi \) satisfies the (C) condition.

Proof

Let \(\{u_m\}\subset E\) be a C-sequence, that is, \(\displaystyle \sup \nolimits _{m\in {\mathbb {N}}^*}\{|\varphi (u_m)|\}< +\infty \) and \((1+\Vert u_m\Vert )\Vert \varphi '(u_m)\Vert \rightarrow 0\) as \(m\rightarrow +\infty .\) Then there exists a constant \(L>0\) such that

$$\begin{aligned} |\varphi (u_m)|\le L,\;\;(1+\Vert u_m\Vert )\Vert \varphi '(u_m)\Vert \le L&\forall \;m\in {\mathbb {N}}^*. \end{aligned}$$
(3.2)

We claim that \(\{u_m\}\) is bounded. Otherwise, there exists a subsequence of \(\{u_{m_k}\}\) such that \(\Vert u_{m_k}\Vert \rightarrow +\infty \) as \(k\rightarrow +\infty ,\) and we still denote \(\{u_{m_k}\}\) by \(\{u_m\}\). Set \(z_m(t)=\frac{u_m(t)}{\Vert u_m\Vert },\) then \(\Vert z_m\Vert =1.\) So there exists a \(z\in E\) with \(\Vert z\Vert \le 1\) such that \(z_m\rightharpoonup z\) in E. By Lemma 2.1, one has \(z_m\rightarrow z\) in \(C([0, T ],{\mathbb {R}}^n)\) as \(m\rightarrow +\infty .\) We consider the following cases \(z(t)\not \equiv 0\) and \(z(t)\equiv 0\) respectively.

Case 1: \(z(t)\not \equiv 0.\) Set \(\Omega :=\{t\in [0,T]/ |z(t)|> 0\},\) then meas \((\Omega )>0,\) By lemma 2.5 and \(\Vert u_m\Vert \rightarrow +\infty \) as \(m\rightarrow +\infty ,\) there exists a subset \(\Omega _{\delta _0}\subseteq \Omega \) with meas \((\Omega _{\delta _0})> 0\) and meas \((\Omega \backslash \Omega _{\delta _0})< \delta _0\) such that

$$\begin{aligned} |u_m(t)|=\Vert u_m\Vert .|z_m(t)|\rightarrow & {} +\infty \;\text{ uniformly }\; \forall t\in \Omega _{\delta _0}\; \text{ as }\; m\rightarrow +\infty . \end{aligned}$$
(3.3)

It follows from \((H_3)\) that there exists a constant \(M_1>0\) large enough such that

$$\begin{aligned} (\nabla W(t,x).x)-2W(t,x)\ge & {} -M_1,\; \forall (t,x)\in [0,T]\times {\mathbb {R}}^n. \end{aligned}$$
(3.4)

By \((H_2)\), there exists a constant \(M_2>0 \) large enough such that

$$\begin{aligned} 2K(t,x)-(\nabla K(t,x).x)\ge -M_2,&\forall (t,x)\in [0,T]\times {\mathbb {R}}^n. \end{aligned}$$
(3.5)

By \((H_3)\) and (3.3), one has

$$\begin{aligned} \displaystyle \int _{\Omega _{\delta _0}}\left[ (\nabla W(t,u_m(t)).u_m(t))-2W(t,u_m(t))\right] dt\rightarrow & {} +\infty ,m\rightarrow +\infty . \end{aligned}$$
(3.6)

According to (2.3), (3.2), (3.4), (3.5) and (3.6), there holds

$$\begin{aligned} 3L\ge & {} 2\varphi (u_m)-(\varphi '(u_m).u_m) \\= & {} \displaystyle \int ^T_0\left[ (\nabla W(t,u_m(t)).u_m(t))-2W(t,u_m(t))\right] dt \\&+ \displaystyle \int ^T_0\left[ 2K(t,u_m(t))-(\nabla K(t,u_m(t)).u_m(t))\right] dt\\= & {} \displaystyle \int _{\Omega _{\delta _0}}\left[ (\nabla W(t,u_m(t)).u_m(t))-2W(t,u_m(t))\right] dt \\&+ \displaystyle \int _{[0,T]\backslash \Omega _{\delta _0}}\left[ (\nabla W(t,u_m(t)).u_m(t))-2W(t,u_m(t))\right] dt\\&+ \displaystyle \int ^T_0\left[ 2K(t,u_m(t))-(\nabla K(t,u_m(t)).u_m(t))\right] dt \\\ge & {} \displaystyle \int _{\Omega _{\delta _0}}\left[ (\nabla W(t,u_m(t)).u_m(t))-2W(t,u_m(t))\right] dt-M_1T-M_2T \\\rightarrow & {} +\infty , m\rightarrow +\infty , \end{aligned}$$

which yields a contradiction.

Case 2: \(z(t)\equiv 0.\) From \((H_6),\) (2.1) and (2.2), one has

$$\begin{aligned}&\displaystyle \int ^T_0W(t,u_m(t))dt-\displaystyle \int ^T_0K(t,u_m(t))dt\\&\quad =\frac{1}{2}\Vert u_m\Vert ^2+\frac{1}{2}\displaystyle \int ^T_0(Au_m(t).{\dot{u}}_m(t))dt-\varphi (u_m). \end{aligned}$$

Divided by \(\Vert u_m\Vert ^2\) on both sides, together with (3.2), one has

$$\begin{aligned} \frac{1}{2}(1-\Vert A\Vert ) \le \displaystyle \int ^T_0\frac{W(t,u_m(t))-K(t,u_m(t))dt}{\Vert u_m\Vert ^2}dt\le & {} \frac{1}{2}(1+\Vert A\Vert ). \end{aligned}$$
(3.7)

in the meantime, condition \((H_1)\) implies that

$$\begin{aligned} K(t,x)\ge & {} -d|x|^2-{\widetilde{M}}, \forall (t,x)\in [0,T]\times {\mathbb {R}}^n. \end{aligned}$$
(3.8)

where \({\widetilde{M}}=\displaystyle \max \nolimits _{t\in [0,T]}\{\displaystyle \max \nolimits _{|x|\le L_1}|K(t,x)|\}> 0.\) By \((H_4),\) one has

$$\begin{aligned} W(t,x)\le & {} b(t)|x|^2+{\overline{M}}, \forall (t,x)\in [0,T]\times {\mathbb {R}}^n, \end{aligned}$$
(3.9)

where \({\overline{M}}=\displaystyle \max \nolimits _{t\in [0,T]}\{\displaystyle \max \nolimits _{|x|\le L_3}|W(t,x)|\}> 0.\)

According to (3.8), (3.9) and \(z(t)\equiv 0,\) there holds

$$\begin{aligned} \displaystyle \int ^T_0\frac{W(t,u_m(t))-K(t,u_m(t))}{\Vert u_m\Vert ^2}dt= & {} \displaystyle \int ^T_0\frac{b(t)|u_m(t)|^2+d|u_m(t)|^2+{\overline{M}}+{\widetilde{M}}}{\Vert u_m\Vert ^2}dt \\\le & {} \Vert z_m\Vert ^2_{\infty }.\displaystyle \int ^T_0(b(t)+d)dt+\frac{({\overline{M}}+{\widetilde{M}})T}{\Vert u_m\Vert ^2} \\\rightarrow & {} 0 , m\rightarrow +\infty . \end{aligned}$$

which contradicts (3.7). Hence \((u_m)_{m\in {\mathbb {N}}}\) is bounded in E. By Proposition 4.3 in [12] we can assume that \(\{u_m\}_{m\in {\mathbb {N}}}\) has a convergent subsequence in E. Hence \(\varphi \) satisfies the C condition. The proof is complete. \(\square \)

Lemma 3.3

If W(t, x) satisfies \((H_3)\) and \((H_4)\), K(t, x) satisfies \((H_1)\) and \((H_2)\) and \((H_6)\) satisfied, then the functional \(\varphi \) satisfies \((A_1)\) in Lemma 2.3.

Proof

Take \(r_k = \beta ^{-1}_k\), then Lemma 2.4 implies that \(r_k\rightarrow +\infty \) as \(k\rightarrow +\infty \). By Lemma 3.1, one has

$$\begin{aligned} K(t,x)\le M_3|x|^2+M_4,\quad \forall (t,x)\in [0,T]\times {\mathbb {R}}^n, \end{aligned}$$
(3.10)

where constants \(M_3=\frac{\displaystyle \max \nolimits _{t\in [0,T]}\left\{ \displaystyle \max \nolimits _{|x|=L_2}|K(t,x)|\right\} }{L_2^2}>0,\quad M_4=\displaystyle \max \nolimits _{t\in [0,T]}\left\{ \displaystyle \max \nolimits _{|x|\le L_2}|K(t,x)|\right\} >0\).

By Lemma 3.1, for a certain constant \(\sigma > \max \{L_2 ,L_3\}\) large enough, one has

$$\begin{aligned} W(t,x)\ge \frac{|x|^2}{\sigma ^2}.\min _{|x|=\sigma }W(t,x),\quad |x|\ge \sigma \quad \forall t\in [0,T]. \end{aligned}$$
(3.11)

Hence, by \((H_4)\) and (3.11), one has

$$\begin{aligned} W(t,x)\ge B|x|^2-C,\quad \forall (t,x)\in [0,T]\times {\mathbb {R}}^n, \end{aligned}$$
(3.12)

where constants \(B=\frac{\displaystyle \min \nolimits _{t\in [0,T]}\left\{ \displaystyle \min \nolimits _{|x|=\sigma }W(t,x)\right\} }{\sigma ^2}>0,\quad C=B\sigma ^2+\displaystyle \max \nolimits _{t\in [0,T]}\left\{ \displaystyle \max \nolimits _{|x|\le \sigma }|W(t,x)|\right\} >0.\)

From (3.8), (3.9), (3.10) and (3.12), there exists a function \({\widetilde{b}}\in L^1([0,T], {\mathbb {R}}_+)\) such that

$$\begin{aligned} |K(t,x)|+|W(t,x)|\le {\widetilde{b}}(t)|x|^2+M_0,\quad \forall (t,x)\in [0,T]\times {\mathbb {R}}^n, \end{aligned}$$
(3.13)

where \({\widetilde{b}}(t)= b(t)+ B + \max \{M_3 ,d\}> 0,~~M_0 = \max \{M_4 , {\widetilde{M}}\} + \max \{C,{\overline{M}}\} > 0.\) For \(u_k\in Z_k\subseteq E\) with \(\Vert u_k\Vert =r_k\), set \(z_k(t)=\frac{u_k(t)}{\Vert u_k\Vert }\), then \(\Vert z_k\Vert =1\). By the definition of \(\beta _k\), one has \(\Vert z_k\Vert _{\infty }\le \beta _k\), which implies that \(\Vert u_k\Vert _{\infty }\le \beta _k\Vert u_k\Vert =\beta _k.r_k=1\). It follows from \((H_6),\) (2.1) and (3.13) that

$$\begin{aligned} \varphi (u_k)= & {} \frac{1}{2}\int _{0}^{T}|{\dot{u}}_k(t)|^2dt+\frac{1}{2}\displaystyle \int ^T_0Au_k(t).{\dot{u}}_k(t)dt+\int _{0}^{T}K(t,u_k(t))dt\\&-\int _{0}^{T} W(t,u_k(t))dt \\\ge & {} \frac{1}{2}\Vert u_k\Vert ^2-\frac{1}{2}\displaystyle \int ^T_0A{\dot{u}}_k(t).u_k(t)dt-\int _{0}^{T}[|K(t,u_k(t))|+|W(t,u_k(t))|]dt\\\ge & {} \frac{1}{2}\Vert u_k\Vert ^2(1-\Vert A\Vert )- \Vert u_k\Vert ^2_{\infty }.\int _{0}^{T}{\widetilde{b}}(t)dt-M_0T\\\ge & {} \frac{1}{2}(1-\Vert A\Vert )r_k^2-\int _{0}^{T}{\widetilde{b}}(t)dt-M_0T, \end{aligned}$$

which implies that \(\inf _{u\in Z_k,\Vert u\Vert =r_k}\varphi (u)\rightarrow +\infty \) as \(k\rightarrow +\infty \). \(\square \)

Lemma 3.4

If W(t, x) satisfies \((H_3)\) and \((H_4)\), K(t, x) satisfies \((H_2)\) and \((H_6)\) satisfied, then the functional \(\varphi \) satisfies \((A_2)\) in Lemma 2.3.

Proof

For every \(k \in {\mathbb {N}}^*\) , \(Y_k\) is a finite dimensional space, so there exists a constant \(d_k > 0\) such that

$$\begin{aligned} \Vert u_k\Vert _{L^2}\ge d_k\Vert u_k\Vert ,\quad \forall u_k\in Y_k. \end{aligned}$$
(3.14)

By (3.12), \(\forall \alpha \in (0,2)\), one has

$$\begin{aligned} \frac{W(t,x)}{|x|^{\alpha }}\ge B|x|^{2-\alpha }-C|x|^{-\alpha }\rightarrow +\infty \text{ uniformly } \forall t \in [0,T] \text{ as } |x|\rightarrow +\infty . \end{aligned}$$
(3.15)

Then (3.15) implies that there exists a certain constant \(L_4 \ge \sigma ^{2-\alpha }(M_3+\frac{2}{d_k^2}\}\) large enough such that

$$\begin{aligned} \min _{|x|=\sigma }W(t,x)>L_4\sigma ^{\alpha },\quad \forall t\in [0,T]. \end{aligned}$$
(3.16)

By (3.11) and (3.16), there exists a constant \(M_5 > 0\) such that

$$\begin{aligned} W(t,x)>L_4\sigma ^{\alpha -2}|x|^2-M_5,\quad \forall (t,x)\in [0,T]\times {\mathbb {R}}^n. \end{aligned}$$
(3.17)

where \(M_5 = L_4 \sigma ^{\alpha } + \max \nolimits _{t\in [0,T]}\{\max \nolimits _{|x|\le \sigma } |W(t,x)|\} > 0\). For every \(u_k\in Y_k\) with \(\Vert u_k\Vert = \rho _k (\rho _k > r_k\) is determined later), by \((H_6),\) (3.10), (3.14) and (3.17,) there holds

$$\begin{aligned} \varphi (u_k)= & {} \frac{1}{2}\int _{0}^{T}|{\dot{u}}_k(t)|^2dt+\frac{1}{2}\displaystyle \int ^T_0Au_k(t).{\dot{u}}_k(t)dt+\int _{0}^{T}K(t,u_k(t))dt\end{aligned}$$
(3.18)
$$\begin{aligned}&-\int _{0}^{T} W(t,u_k(t))dt \nonumber \\\le & {} \frac{1}{2}\Vert u_k\Vert ^2 +\frac{1}{2}\Vert A\Vert \Vert u_k\Vert ^2+M_3\Vert u_k\Vert ^2_{L^2}-L_4\sigma ^{\alpha -2}\Vert u_k\Vert ^2_{L^2}+M_6 \nonumber \\\le & {} \left[ \frac{1}{2}(1+\Vert A\Vert )-(L_4\sigma ^{\alpha -2}-M_3)d_k^2\right] \Vert u_k\Vert ^2+M_6 \nonumber \\\le & {} -\rho _k^2+M_6, \end{aligned}$$
(3.19)

where \(M_6 = (M_4 + M_5 )T > 0\). Therefore, if \(\rho _k > \max \{r_k, \sqrt{2M_6}\}\) large enough, then (3.18) implies that \(\max \nolimits _{u\in Y_k ,\Vert u\Vert =\rho _k} \varphi (u) < 0\). \(\square \)

Proof of Theorem 1.1

In view of Lemma 2.2, \(\varphi \in C^1(E,{\mathbb {R}})\) holds. Condition \((H_5)\) shows that \(\varphi (-u) = \varphi (u)\). Lemma 2.3 and Lemmas 3.2–3.4 imply that \(\varphi \) possesses a sequence of critical points \(\{u_k\}\) such that

$$\begin{aligned} \varphi '(u_k)=0 \quad \text{ and }\quad c_k=\varphi (u_k)\rightarrow +\infty \quad \text{ as }\quad k\rightarrow +\infty . \end{aligned}$$
(3.20)

As is well known, \(u \in E\) is a weak solution of problem (1.1) which corresponds to the critical points of the functional \(\varphi \). Hence by Lemma 2.2, u is an odd classical solution of problem (1.1). Next, we claim that \(\Vert u_k\Vert _{\infty }\rightarrow +\infty \) as \(k\rightarrow +\infty \). If not, then there exists a constant \(M_7 > 0\) such that

$$\begin{aligned} \varphi '(u_k)=0\quad \text{ and }\quad \Vert u_k\Vert _{\infty }\le M_7,\quad \forall k\in {\mathbb {N}}^*. \end{aligned}$$
(3.21)

By a simple calculation, \(K,W \in C^1({\mathbb {R}} \times {\mathbb {R}}^ n ,{\mathbb {R}})\) and (3.21), there exists a constant \(M_8 > 0\) independent of k such that

$$\begin{aligned} \varphi (u_k) -\frac{1}{2}(\varphi '(u_k),u_k)= & {} \int _{0}^{T}\left[ K(t,u_k(t))-\frac{1}{2}(\nabla K(t,u_k(t)),u_k(t))\right] dt \\&- \int _{0}^{T}\left[ W(t,u_k(t))-\frac{1}{2}(\nabla W(t,u_k(t)),u_k(t))\right] dt \\\le & {} M_8,\quad \forall k\in {\mathbb {N}}^*, \end{aligned}$$

which contradicts \(\varphi (u_k) -\frac{1}{2}(\varphi '(u_k),u_k) =c_k \rightarrow +\infty \) via (3.20). The proof is complete. \(\square \)