1 Introduction

During the past 2 decades there have been numerous applications of methods from the calculus of variations to find periodic, homoclinic and heteroclinic solutions for Hamiltonian systems. Many of the striking results that have been obtained by variational methods can be found in the well-known monographs of Ambrosetti and Coti Zelati [3], Ekeland [8], Hofer and Zehnder [9], Mawhin and Willem [16], as well as in the review articles of Rabinowitz [19, 20].

The aim of this paper is to prove the existence of solutions of the second order Hamiltonian system

$$\begin{aligned} \left\{ \begin{array}{ll} \ddot{q}(t)+\nabla _{q}V(t,q(t))=f(t),\quad t\in \mathbb {R}, \\ \lim \limits _{t\rightarrow \pm \infty }q(t)=\lim \limits _{t\rightarrow \pm \infty }\dot{q}(t)=0, \end{array} \right. \end{aligned}$$
(1)

where the \(C^1\)-smooth potential \(V:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}\) satisfies a relaxed superquadratic growth condition, its gradient \(V_{q}:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is uniformly bounded in the time variable on every compact subset of \(\mathbb {R}^n\), and the norm of the forcing term \(f:\mathbb {R}\rightarrow \mathbb {R}^n\) in the space of square integrable functions is smaller than a bound that we state below in our main theorem.

As homoclinic type solutions are global in time, it is reasonable to use global methods to find them rather than approaches based on their initial value problems. The homogenous systems of (1), i.e. when \(f\equiv 0\), have been studied extensively under the assumption of superquadratic or subquadratic growth of the potential V(t, q) as \(|q|\rightarrow \infty \). Indeed, there are many results on homoclinic solutions for subquadratic Hamiltonian systems (cf. e.g. [20]). The first variational results for homoclinic solutions of first order Hamiltonian systems with superquadratic growth were found by Coti Zelati et al. [6] for time-periodic Hamiltonians. Corresponding results for second order Hamiltonian systems were obtained in [7, 18]. Alama and Li [1] showed that asymptotic periodicity in time actually suffices to get a homoclinic solution, and Serra et al. [21] weakened their periodicity condition to almost periodicity in the sense of Bohr. Finally, Hamiltonian systems with superquadratic non-periodic potentials were investigated for example by Montecchiari and Nolasco [17], Ambrosetti and Badiale [4], and by the second author of this paper in [11, 13, 14].

Our purpose is to generalize Theorem 1.1 of [5], which deals with the existence of solutions of the inhomogeneous systems (1) under the rather restrictive assumption that the potential V is of the special form

$$\begin{aligned} V(t,q)=-\frac{1}{2}|q|^2+a(t)G(q), \end{aligned}$$

where \(a:\mathbb {R}\rightarrow \mathbb {R}\) is a continuous positive bounded function and \(G:\mathbb {R}^n\rightarrow \mathbb {R}\) is of class \(C^1\) and satisfies the Ambrosetti–Rabinowitz superquadratic growth condition. Here, instead, the potential is of the more general form

$$\begin{aligned} V(t,q)=-K(t,q)+W(t,q) \end{aligned}$$

with \(C^1\)-smooth potentials K and W such that

\((C_1)\):

the maps \(\nabla _{q}K\) and \(\nabla _{q}W\) are uniformly bounded in the time variable \(t\in \mathbb {R}\) on every compact subset of \(\mathbb {R}^n\),

\((C_2)\):

there exist two positive constants \(b_1\), \(b_2\) such that for all \(t\in \mathbb {R}\) and \(q\in \mathbb {R}^n\)

$$\begin{aligned} b_{1}|q|^{2}\le K(t,q)\le b_{2}|q|^{2}, \end{aligned}$$
\((C_3)\):

\(K(t,q)\le (q,\nabla _{q}K(t,q))\le 2K(t,q)\) for all \(t\in \mathbb {R}\) and \(q\in \mathbb {R}^n\),

\((C_4)\):

\(\nabla _{q}W(t,q)=o(|q|)\) as \(|q|\rightarrow 0\) uniformly in \(t\in \mathbb {R}\),

\((C_5)\):

there is a constant \(\mu >2\) such that for all \(t\in \mathbb {R}\) and \(q\in \mathbb {R}^n{\setminus }\{0\}\)

$$\begin{aligned} 0<\mu W(t,q)\le (q,\nabla _{q}W(t,q)), \end{aligned}$$
\((C_6)\):

\(m:=\inf \{W(t,q):t\in \mathbb {R}\ \wedge \ |q|=1\}>0.\)

Here and subsequently, we denote by \((\cdot ,\cdot ):\mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}\) the standard inner product in \(\mathbb {R}^n\) and by \(|\cdot |\) its induced norm.

Let us point out that under the above assumptions the Hamiltonian system (1) has the trivial solution when the forcing term f vanishes. Therefore it is reasonable to suppose that homoclinic type solutions exist when f is sufficiently small. Our main result affirms this hypothesis and it also gives an answer to the question how large the forcing term can be.

Theorem 1.1

Set \(M:=\sup \{W(t,q):t\in \mathbb {R}\ \wedge \ |q|=1\}\) and \(\bar{b}_{1}:=\min \{1,2b_{1}\}\). Let us assume that \(M<\frac{1}{2}\bar{b}_{1}\) and \((C_1)-(C_6)\) are satisfied. If the forcing term f is continuous, bounded, and moreover

$$\begin{aligned} \left( \,\int \limits _{-\infty }^{\infty }|f(t)|^{2}dt\,\right) ^\frac{1}{2} <\frac{\sqrt{2}}{4}\left( \bar{b}_{1}-2M\right) , \end{aligned}$$
(2)

then the inhomogenous system (1) possesses at least one solution.

The idea of our proof, which we give in the following second section, is to approximate the original system (1) by time-periodic ones, with larger and larger time-periods. We show that the approximating systems admit periodic solutions of mountain-pass type, and obtain a homoclinic type solution of the original system from them by passing to the limit (in the topology of almost uniform convergence) when the periods go to infinity. Finally, we discuss some examples of Theorem 1.1 in Sect. 3.

2 Proof of Theorem 1.1

For each \(k\in \mathbb {N}\), let \(E_k=W^{1,2}_{2k}(\mathbb {R},\mathbb {R}^n)\) be the Sobolev space of 2k-periodic functions on \(\mathbb {R}\) with values in \(\mathbb {R}^n\) and the standard norm

$$\begin{aligned} \Vert q\Vert _{E_k}=\left( \,\int \limits ^{k}_{-k}\left( |\dot{q}(t)|^{2} + |q(t)|^{2}\right) dt\,\right) ^{\frac{1}{2}}. \end{aligned}$$

We begin with the following estimate that is crucial in the main part of our proof below.

Lemma 2.1

For every \(\zeta \in \mathbb {R}\) and \(q\in E_k\) we have

$$\begin{aligned} \int \limits _{-k}^{k}W(t,\zeta q(t))dt \ge m|\zeta |^{\mu }\int \limits _{-k}^{k}|q(t)|^{\mu }dt-2km. \end{aligned}$$

Proof

Note at first that the assertion is obviously true if \(q=0\) or \(\zeta =0\). Hence we can assume in the rest of the proof that \(\zeta \ne 0\) and \(q\ne 0\). Then it follows from \((C_5)\) that, for every \(q\ne 0\) and \(t\in \mathbb {R}\), the function \(z:(0,+\infty )\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} z(\zeta )=W\left( t,\frac{q}{\zeta }\right) \zeta ^{\mu } \end{aligned}$$

is non-increasing. Hence, for every \(t\in \mathbb {R}\),

$$\begin{aligned} W(t,q)\le W\left( t,\frac{q}{|q|}\right) |q|^{\mu },\quad \ \text {if} \ 0<|q|\le 1 \end{aligned}$$
(3)

and

$$\begin{aligned} W(t,q)\ge W\left( t,\frac{q}{|q|}\right) |q|^{\mu },\quad \ \text {if} \ |q|\ge 1. \end{aligned}$$
(4)

We now fix \(\zeta \in \mathbb {R}{\setminus }\left\{ 0\right\} \), \(q\in E_k{\setminus }\left\{ 0\right\} \) and set

$$\begin{aligned} A_k&=\left\{ t\in [-k,k]:|\zeta q(t)|\le 1\right\} ,\\ B_k&=\left\{ t\in [-k,k]:|\zeta q(t)|\ge 1\right\} . \end{aligned}$$

By (4), we get

$$\begin{aligned} \int _{-k}^{k}W(t,\zeta q(t))dt&\ge \int _{B_k}W(t,\zeta q(t))dt \ge \int _{B_k}W\left( t,\frac{\zeta q(t)}{|\zeta q(t)|}\right) |\zeta q(t)|^{\mu }dt \\&\ge m \int _{B_k}|\zeta q(t)|^{\mu }dt \ge m\int _{-k}^k|\zeta q(t)|^{\mu }dt - m\int _{A_k}|\zeta q(t)|^{\mu }dt \\&\ge m|\zeta |^{\mu }\int _{-k}^k|q(t)|^{\mu }dt - 2km, \end{aligned}$$

which completes the proof. \(\square \)

Further, to prove Theorem 1.1, we need the following approximative method.

Theorem 2.2

(Approximative Method, [15]) Let \(f:\mathbb {R}\rightarrow \mathbb {R}^n\) be a non-trivial, bounded, continuous and square integrable map. Assume that \(V:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}\) is a \(C^1\)-smooth potential such that \(\nabla _{q}V:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is uniformly bounded in t on every compact subset of \(\mathbb {R}^n\), i.e.

$$\begin{aligned} \forall \ L>0 \ \exists \ C>0 \quad \ \forall \ q\in \mathbb {R}^n \quad \ \forall \ t\in \mathbb {R}\ \quad |q|\le L \Rightarrow |\nabla _{q}V(t,q)|\le C. \end{aligned}$$

Suppose that for each \(k\in \mathbb {N}\) the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} \ddot{q}(t)+\nabla _{q}V_{k}(t,q(t))=f_k(t),\\ q(-k)-q(k)=\dot{q}(-k)-\dot{q}(k)=0, \end{array} \right. \end{aligned}$$

where \(f_k:\mathbb {R}\rightarrow \mathbb {R}^n\) stands for the 2k-periodic extension of \(f|_{[-k,k)}\) to \(\mathbb {R}\) and \(V_k:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}\) denotes the 2k-periodic extension of \(V|_{[-k,k)\times \mathbb {R}^n}\) to \(\mathbb {R}\times \mathbb {R}^n\), has a periodic solution \(q_k\in E_k\) and \(\{\Vert q_k\Vert _{E_k}\}_{k\in \mathbb {N}}\) is a bounded sequence in \(\mathbb {R}\). Then there exists a subsequence \(\{q_{k_j}\}_{j\in \mathbb {N}}\) converging in the topology of \(C^{2}_{loc}(\mathbb {R},\mathbb {R}^n)\) to a function \(q\in W^{1,2}(\mathbb {R},\mathbb {R}^n)\) which is a solution of

$$\begin{aligned} \ddot{q}(t)+\nabla _{q}V(t,q(t))=f(t),\quad \ t\in \mathbb {R}. \end{aligned}$$

The approximative method was introduced by Rabinowitz [18] for homogenous second order Hamiltonian systems with a time-periodic potential. Later, the second author of this paper extended it to inhomogenous time-periodic Hamiltonian systems (see [10, 12]), and more recently, Robert Krawczyk generalized it to the case of aperiodic potentials.

Let us now consider for \(k\in \mathbb {N}\) the boundary value problems

$$\begin{aligned} \left\{ \begin{array}{ll} \ddot{q}(t)-\nabla _{q}K_{k}(t,q(t))+\nabla _{q}W_{k}(t,q(t))=f_k(t),\\ q(-k)-q(k)=\dot{q}(-k)-\dot{q}(k)=0, \end{array} \right. \end{aligned}$$
(5)

where \(f_k:\mathbb {R}\rightarrow \mathbb {R}^n\) stands for the 2k-periodic extension of \(f|_{[-k,k)}\) to \(\mathbb {R}\), and \(K_k:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}\), \(W_k:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}\) are the 2k-periodic extensions of \(K|_{[-k,k)\times \mathbb {R}^n}\) and \(W|_{[-k,k)\times \mathbb {R}^n}\) to \(\mathbb {R}\times \mathbb {R}^n\).

As we have already mentioned in the introduction, our proof consists of two steps. First, we show the existence of solutions of (5), and second, we use Theorem 2.2 to find a solution of (1).

For our first step, let us consider the functionals \(I_{k}:E_k\rightarrow \mathbb {R}\) given by

$$\begin{aligned} I_k(q)=\int \limits ^{k}_{-k}\left( \frac{1}{2}|\dot{q}(t)|^2+K_k(t,q(t)) -W_k(t,q(t))\right) dt+\int \limits ^{k}_{-k}(f_k(t),q(t))dt. \end{aligned}$$
(6)

Standard arguments show that \(I_k\in C^1(E_k,\mathbb {R})\), and

$$\begin{aligned} I'_{k}(q)v=\int \limits ^{k}_{-k}\left( (\dot{q}(t),\dot{v}(t)) +(\nabla _{q}K_k(t,q(t))-\nabla _{q}W_k(t,q(t)),v(t))\right) dt +\int \limits ^{k}_{-k}(f_k(t),v(t))dt. \end{aligned}$$
(7)

Moreover, the critical points of the functional \(I_k\) are classical 2k-periodic solutions of (5), and we now show their existence by using the Mountain Pass Theorem. Let us recall the latter result before proceeding with our proof.

Theorem 2.3

(Mountain Pass Theorem, [2]) Let E be a real Banach space and \(I:E\rightarrow \mathbb {R}\) a \(C^1\)-smooth functional. If I satisfies the following conditions:

  1. (i)

    \(I(0)=0\),

  2. (ii)

    every sequence \(\left\{ u_j\right\} _{j\in \mathbb {N}}\) in E such that \(\left\{ I(u_j)\right\} _{j\in \mathbb {N}}\) is bounded in \(\mathbb {R}\) and \(I'(u_j)\rightarrow 0\) in \(E^{*}\) as \(j\rightarrow +\infty \) contains a convergent subsequence (the Palais–Smale condition),

  3. (iii)

    there exist constants \(\rho ,\alpha >0\) such that \(I|_{\partial B_{\rho }(0)}\ge \alpha \),

  4. (iv)

    there exists \(e\in E{\setminus }\bar{B}_{\rho }(0)\) such that \(I(e)\le 0\),

where \(B_{\rho }(0)\) is the open ball of radius \(\rho \) about 0 in E, then I possesses a critical value \(c\ge \alpha \) given by

$$\begin{aligned} c=\inf _{g\in \Gamma }\max _{s\in [0,1]}I(g(s)), \end{aligned}$$
(8)

where

$$\begin{aligned} \Gamma =\left\{ g\in C([0,1],E):\ g(0)=0,\ g(1)=e \right\} . \end{aligned}$$

We now denote by \(L^{\infty }_{2k}(\mathbb {R},\mathbb {R}^n)\) the space of 2k-periodic essentially bounded functions from \(\mathbb {R}\) into \(\mathbb {R}^n\) equipped with the norm

$$\begin{aligned} \Vert q\Vert _{L^{\infty }_{2k}}=\text {ess}\sup \left\{ |q(t)|:t\in [-k,k]\right\} . \end{aligned}$$

It is well known that for each \(k\in \mathbb {N}\) and \(q\in E_k\)

$$\begin{aligned} \Vert q\Vert _{L_{2k}^{\infty }}\le \sqrt{2}\Vert q\Vert _{E_k}. \end{aligned}$$
(9)

The proof of (9) can be found for example in [10] (see Fact 2.8, p. 385).

Furthermore, we will write \(L^2_{2k}(\mathbb {R},\mathbb {R}^n)\) for the Hilbert space of 2k-periodic functions on \(\mathbb {R}\) with values in \(\mathbb {R}^n\) and with the norm

$$\begin{aligned} \Vert q\Vert _{L^{2}_{2k}} =\left( \,\int \limits ^{k}_{-k}|q(t)|^{2}dt\,\right) ^{\frac{1}{2}}. \end{aligned}$$

Note that by (2),

$$\begin{aligned} \Vert f_k\Vert _{L^{2}_{2k}}<\frac{\sqrt{2}}{4}\left( \bar{b}_{1}-2M\right) . \end{aligned}$$
(10)

The following lemma shows the existence of a solution of (5) and is the main part of the first step of our proof.

Lemma 2.4

For each \(k\in \mathbb {N}\), the functional \(I_k\) has a critical value of mountain pass type.

Proof

We let \(k\in \mathbb {N}\) be fixed and note at first that it is evident by \((C_2)\) and \((C_5)\) that \(I_k(0)=0\), which shows (i) in Theorem 2.3.

For checking the Palais–Smale condition (ii), we consider a sequence \(\{u_j\}_{j\in \mathbb {N}}\subset E_k\) such that \(\{I_k(u_j)\}_{j\in \mathbb {N}}\) is bounded in \(\mathbb {R}\) and \(I'_k(u_j)\rightarrow 0\) in \(E^{*}_{k}\) as \(j\rightarrow \infty \). Then there exists a constant \(C_k>0\) such that for all \(j\in \mathbb {N}\)

$$\begin{aligned} |I_k(u_j)|\le C_k \end{aligned}$$
(11)

and

$$\begin{aligned} \Vert I'_k(u_j)\Vert _{E^{*}_{k}}\le C_k. \end{aligned}$$
(12)

Now, we will first show that \(\{u_j\}_{j\in \mathbb {N}}\) is bounded in the Hilbert space \(E_k\). Using (6) and \((C_5)\) we get

$$\begin{aligned} 2I_k(u_j)&\ge \int _{-k}^{k}\left( |\dot{u}_{j}(t)|^2+2K_k(t,u_j(t))\right) dt - \frac{2}{\mu }\int _{-k}^{k}(\nabla _{q}W_k(t,u_j(t)),u_j(t))dt \\&\quad + 2\int _{-k}^{k}(f_k(t),u_j(t))dt. \end{aligned}$$

From (7) and \((C_3)\) it follows that

$$\begin{aligned} I'_{k}(u_j)u_j&\le \int _{-k}^{k}\left( |\dot{u}_{j}(t)|^2+2K_k(t,u_j(t))\right) dt - \int _{-k}^{k}(\nabla _{q}W_k(t,u_j(t)),u_j(t))dt \\&\quad + \int _{-k}^{k}(f_k(t),u_j(t))dt. \end{aligned}$$

Thus

$$\begin{aligned} 2I_k(u_j)-\frac{2}{\mu }I'_{k}(u_j)u_j&\ge \left( 1-\frac{2}{\mu }\right) \int _{-k}^{k}\left( |\dot{u}_{j}(t)|^2 +2K_k(t,u_j(t))\right) dt\\&\quad + \left( 2-\frac{2}{\mu }\right) \int _{-k}^{k}(f_k(t),u_j(t))dt, \end{aligned}$$

and by \((C_2)\) we have

$$\begin{aligned} 2I_k(u_j)-\frac{2}{\mu }I'_{k}(u_j)u_j \ge \left( 1-\frac{2}{\mu }\right) \bar{b}_{1}\Vert u_j\Vert _{E_k}^{2} + \left( 2-\frac{2}{\mu }\right) \int _{-k}^{k}(f_k(t),u_j(t))dt. \end{aligned}$$

Finally, aplying the Hölder inequality, as well as (10), (11) and (12), we obtain

$$\begin{aligned} \left( 1-\frac{2}{\mu }\right) \bar{b}_{1}\Vert u_j\Vert _{E_k}^{2} -\frac{2C_k}{\mu }\Vert u_j\Vert _{E_k} -\frac{\sqrt{2}}{4}(\bar{b}_{1}-2M)\left( 2-\frac{2}{\mu }\right) \Vert u_j\Vert _{E_k} -2C_k \le 0. \end{aligned}$$

Since \(\mu >2\) we conclude that \(\{u_j\}\) is bounded.

Going to a subsequence if necessary, we can assume that there exists a function \(u\in E_k\) such that \(u_j\rightharpoonup u\) weakly in \(E_k\) as \(j\rightarrow +\infty \). Hence \(u_j\rightarrow u\) uniformly on \([-k,k]\), which implies that

$$\begin{aligned} \begin{aligned}&\left( I'_k(u_j)-I'_k(u)\right) (u_j-u) \rightarrow 0,\\&\Vert u_j-u\Vert _{L^{2}_{2k}} \rightarrow 0 \end{aligned} \end{aligned}$$
(13)

and

$$\begin{aligned}&\int ^{k}_{-k}(\nabla _{q}K_k(t,u_j(t)) -\nabla _{q}W_k(t,u_j(t)),u_j(t)-u(t))dt\\&\quad -\int ^{k}_{-k}(\nabla _{q}K_k(t,u(t))-\nabla _{q}W_k(t,u(t)),u_j(t)-u(t))dt \rightarrow 0 \end{aligned}$$

as \(j\rightarrow +\infty \). On the other hand, it is readily seen that

$$\begin{aligned} \Vert \dot{u}_j-\dot{u}\Vert ^{2}_{L^{2}_{2k}}&=(I'_{k}(u_j)-I'_{k}(u))(u_j-u)\\&\quad -\int ^{k}_{-k}(\nabla _{q}K_k(t,u_j(t))-\nabla _{q}W_k(t,u_j(t)),u_j(t)-u(t))dt\\&\quad +\int ^{k}_{-k}(\nabla _{q}K_k(t,u(t))-\nabla _{q}W_k(t,u(t)),u_j(t)-u(t))dt, \end{aligned}$$

and consequently

$$\begin{aligned} \Vert \dot{u}_j-\dot{u}\Vert _{L^{2}_{2k}}\rightarrow 0. \end{aligned}$$
(14)

By (13) and (14), we see that \(\Vert u_j-u\Vert _{E_k}\rightarrow 0\), and thus \(I_k\) satisfies the Palais–Smale condition.

To show (iii), we set

$$\begin{aligned} \rho =\frac{\sqrt{2}}{2}. \end{aligned}$$

Assume that \(q\in \partial B_{\rho }(0)\subset E_{k}\). Then \(\Vert q\Vert _{L^{\infty }_{2k}}>0\) and \(\Vert q\Vert _{L^{\infty }_{2k}}\le 1\) by (9). Let

$$\begin{aligned} D_{k}=\{t\in [-k,k]:0<|q(t)|\le 1\}. \end{aligned}$$

Thus, we can apply (3) to obtain

$$\begin{aligned} \begin{aligned} \int ^{k}_{-k}W(t,q(t))dt&= \int _{D_{k}}W(t,q(t))dt \le \int _{D_{k}}W\left( t,\frac{q(t)}{|q(t)|}\right) |q(t)|^{\mu }dt \\&\le M\int _{D_{k}}|q(t)|^{2}dt = M\int ^{k}_{-k}|q(t)|^{2}dt \le M\Vert q\Vert ^{2}_{E_k} = \frac{1}{2}M. \end{aligned} \end{aligned}$$

From this, \((C_2)\) and (2), we get

$$\begin{aligned} \begin{aligned} I_k(q)&\ge \frac{1}{2}\bar{b}_{1}\Vert q\Vert _{E_k}^{2}-\frac{1}{2}M-\Vert f_k\Vert _{L^{2}_{2k}}\Vert q\Vert _{E_k} \\&\ge \frac{1}{4}(\bar{b}_{1}-2M)-\frac{\sqrt{2}}{2}\Vert f\Vert _{L^2} \\&= \frac{\sqrt{2}}{2}\left( \frac{\sqrt{2}}{4}(\bar{b}_{1}-2M)-\Vert f\Vert _{L^2}\right) \equiv \alpha >0. \end{aligned} \end{aligned}$$
(15)

To complete the proof, we have to show (iv), i.e. we need to find \(e_k\in E_k\) such that \(\Vert e_k\Vert _{E_k}>\rho \) and \(I_k(e_k)\le 0\).

Let

$$\begin{aligned} \bar{b}_{2}=\max \{1,2b_2\}. \end{aligned}$$

Combining (6) and Lemma 2.1 gives

$$\begin{aligned} I_k(\zeta q) \le \frac{\bar{b}_{2}\zeta ^2}{2}\Vert q\Vert ^{2}_{E_k} -m|\zeta |^{\mu }\int ^{k}_{-k}|q(t)|^{\mu }dt +|\zeta |\cdot \Vert f_k\Vert _{L^{2}_{2k}}\Vert q\Vert _{E_k}+2km \end{aligned}$$
(16)

for all \(\zeta \in \mathbb {R}{\setminus }\{0\}\) and \(q\in E_k{\setminus }\{0\}\).

We now let \(Q\in E_1\) be such that \(Q\ne 0\) and \(Q(-1)=Q(1)=0\). It follows from (16) that \(\Vert \zeta Q\Vert _{E_1}>\rho \) and \(I_1(\zeta Q)<0\) for \(\zeta \in \mathbb {R}{\setminus }\{0\}\) large enough. Hence, if we define \(e_1(t)=\zeta Q(t)\) and for each \(k\ge 2\),

$$\begin{aligned} e_k(t)=\left\{ \begin{array}{ll} e_1(t) &{} \quad \text {for}\, \, t\in [-1,1],\\ 0 &{} \quad \text {for}\, \, t\in [-k,-1)\cup (1,k], \end{array} \right. \end{aligned}$$
(17)

then \(e_k\in E_k\), and \(\Vert e_k\Vert _{E_k}=\Vert e_1\Vert _{E_1}>\rho \) as well as \(I_k(e_k)=I_1(e_1)<0\).

In summary, it follows from Theorem 2.3 that the action functional \(I_k\) has a critical value \(c_k\ge \alpha \) given by

$$\begin{aligned} c_k=\inf _{g\in \Gamma _k}\max _{s\in [0,1]}I_k(g(s)), \end{aligned}$$
(18)

where

$$\begin{aligned} \Gamma _k=\left\{ g\in C([0,1],E_k):g(0)=0,\ g(1)=e_k \right\} . \end{aligned}$$

\(\square \)

In what follows, we let \(q_k\) be a critical point for the corresponding critical value \(c_k\) that we have found in Lemma 2.4. The functions \(q_k\), \(k\in \mathbb {N}\), are solutions of (5) and as second step of our proof of Theorem 1.1, we now want to apply Theorem 2.2 to this sequence of functions.

Lemma 2.5

The sequence \(\{\Vert q_k\Vert _{E_k}\}_{k\in \mathbb {N}}\subset \mathbb {R}\) is bounded.

Proof

We set

$$\begin{aligned} M_0=\max _{s\in [0,1]}I_1(se_1). \end{aligned}$$

and conclude from (17) and (18) that

$$\begin{aligned} c_k\le M_0 \end{aligned}$$
(19)

for each \(k\in \mathbb {N}\). By assumption,

$$\begin{aligned} c_k&=I_k(q_k)=I_k(q_k)-\frac{1}{2}I'_{k}(q_k)q_k = \int _{-k}^{k}\left( K_k(t,q_k(t))-\frac{1}{2}(\nabla _{q}K_k(t,q_k(t)),q_k(t))\,\right) dt \\&\quad +\int _{-k}^{k}\left( \frac{1}{2}(\nabla _{q}W_k(t,q_k(t)),q_k(t))-W_k(t,q_k(t))\,\right) dt +\frac{1}{2}\int _{-k}^{k}(f_k(t),q_k(t))dt. \end{aligned}$$

Applying \((C_3)\) and \((C_5)\) we obtain

$$\begin{aligned} c_k\ge \left( \frac{\mu }{2}-1\right) \int ^k_{-k}W_k(t,q_k(t))dt + \frac{1}{2}\int ^k_{-k}(f_k(t),q_k(t))dt. \end{aligned}$$

Furthermore, it follows from (6) and \((C_2)\) that

$$\begin{aligned} \int ^k_{-k}{W_{k}(t,q_k(t))dt}\ge \frac{1}{2}\bar{b}_{1}\Vert q_k\Vert _{E_k}^{2} + \int _{-k}^{k}(f_k(t),q_k(t))dt - I_k(q_k). \end{aligned}$$

Using that \(I_k(q_k)=c_k\), the previous two inequalities give

$$\begin{aligned} \frac{1}{2}\bar{b}_{1}\Vert q_k\Vert _{E_k}^{2} -\frac{\mu -1}{\mu -2}\Vert f_k\Vert _{L^{2}_{2k}}\Vert q_k\Vert _{E_k} -\frac{\mu }{\mu -2}c_k \le 0, \end{aligned}$$

which implies by (2) and (19) that

$$\begin{aligned} \frac{1}{2}\bar{b}_{1}\Vert q_k\Vert _{E_k}^{2} -\frac{\sqrt{2}}{4}\left( \bar{b}_{1}-2M\right) \frac{\mu -1}{\mu -2}\Vert q_k\Vert _{E_k} -\frac{\mu }{\mu -2}M_0 \le 0 . \end{aligned}$$

Hence there is \(M_1>0\) such that for each \(k\in \mathbb {N}\),

$$\begin{aligned} \Vert q_k\Vert _{E_k}\le M_1. \end{aligned}$$

\(\square \)

Now, using Theorem 2.2 we see that there exists a solution \(q:\mathbb {R}\rightarrow \mathbb {R}^n\) of (1) such that \(q(t)\rightarrow 0\) as \(|t|\rightarrow \infty \).

All what is left to show for the proof of Theorem 1.1 is that actually \(\dot{q}(t)\rightarrow 0\) as \(|t|\rightarrow \infty \). This, however, follows from the inequality

$$\begin{aligned} |\dot{q}(t)|\le \sqrt{2}\left( \int _{t-\frac{1}{2}}^{t+\frac{1}{2}} \left( |\dot{q}(s)|^2 + |\ddot{q}(s)|^2 \right) ds\right) ^{\frac{1}{2}},\quad t\in \mathbb {R}, \end{aligned}$$
(20)

which can be found in [10] [Inequality (28), p. 385]. Indeed, we just need to note that by (1), \((C_2)\), \((C_4)\) and (2)

$$\begin{aligned} \int \limits _{t-\frac{1}{2}}^{t+\frac{1}{2}}|\ddot{q}(s)|^{2}ds\rightarrow 0,\quad |t|\rightarrow \infty . \end{aligned}$$

If now |t| goes to \(\infty \) in (20) we see that \(|\dot{q}(t)|\rightarrow 0\) as \(|t|\rightarrow \infty \). Consequently, q is a solution of (1) and the proof of Theorem 1.1 is complete.

3 One-Dimensional Examples

In this section we present examples for \(n=1\) satisfying the assumptions of Theorem 1.1, and the graphs of their approximating solutions \(q_k\) of (5) for increasing values of k.

Example 3.1

Consider \(K:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\), \(W:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) and \(f:\mathbb {R}\rightarrow \mathbb {R}\) given by

$$\begin{aligned} K(t,q)= & {} \frac{t^2+1}{t^2+2}q^2,\\ W(t,q)= & {} \frac{t^2+12}{3t^2+27}q^4 \end{aligned}$$

and

$$\begin{aligned} f(t)=\frac{1}{36}e^{-t^2}, \end{aligned}$$

where \(t,q\in \mathbb {R}\). One can easily check that K, W and f satisfy the assumptions of Theorem 1.1. The Figs. 1, 2 and 3 show the graphs of numerical solutions \(q_k\) of (5) for \(k = 57, 100, 250\).

Fig. 1
figure 1

A numerical solution of (5) for \(k=57\) in Example 1

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Fig. 2
figure 2

A numerical solution of (5) for \(k=100\) in Example 1

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Fig. 3
figure 3

A numerical solution of (5) for \(k=250\) in Example 1

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Example 3.2

Let \(K:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\), \(W:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) and \(f:\mathbb {R}\rightarrow \mathbb {R}\) be given by

$$\begin{aligned} K(t,q)= & {} \left( \frac{1}{8}sin(t)+\frac{1}{8}sin(\sqrt{2}t)+\frac{3}{4}\right) q^2,\\ W(t,q)= & {} \frac{1}{4}q^4 \end{aligned}$$

and

$$\begin{aligned} f(t)=\frac{1}{32}e^{-t^2}, \end{aligned}$$

where \(t,q\in \mathbb {R}\). It is immediate that K, W and f satisfy the assumptions of Theorem 1.1. The Figs. 4, 5 and 6 show the graphs of numerical solutions \(q_k\) of (5) for \(k = 10, 40, 160\).

Fig. 4
figure 4

A numerical solution of (5) for \(k=10\) in Example 2

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Fig. 5
figure 5

A numerical solution of (5) for \(k=40\) in Example 2

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Fig. 6
figure 6

A numerical solution of (5) for \(k=160\) in Example 2

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Fig. 7
figure 7

A numerical solution of (5) for \(k=100\) in Example 3

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Fig. 8
figure 8

A numerical solution of (5) for \(k=140\) in Example 3

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Fig. 9
figure 9

A numerical solution of (5) for \(k=180\) in Example 3

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Example 3.3

Consider \(K:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\), \(W:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) and \(f:\mathbb {R}\rightarrow \mathbb {R}\) given by

$$\begin{aligned} K(t,q)= & {} q^2,\\ W(t,q)= & {} \frac{10}{33}q^4\left( arctg^2\left( \frac{q^2}{t^2+1}\right) +1\right) \end{aligned}$$

and

$$\begin{aligned} f(t)=\frac{1+t^2}{10}e^{-t^2}, \end{aligned}$$

where \(t,q\in \mathbb {R}\). Again, it is readily seen that K, W and f satisfy the assumptions of Theorem 1.1. The Figs. 7, 8 and 9 show the graphs of numerical solutions \(q_k\) of (5) for \(k = 100, 140, 180\).