New periodic solutions of singular Hamiltonian systems with fixed energies

Abstract

By using the variational minimizing method with a special constraint and the direct variational minimizing method without constraint, we study second-order Hamiltonian systems with a singular potential $V\in C^2(R^n\mathrm{\setminus }O,R)$ and $V\in C^1(R^2\mathrm{\setminus }O,R)$, which may have an unbounded potential well, and prove the existence of non-trivial periodic solutions with a prescribed energy. Our results can be regarded as complements of the well-known theorems of Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on.

MSC:35A15, 47J30.

1 Introduction

Seifert [1] in 1948 and Rabinowitz [2, 3] in 1978 and 1979 studied classical second-order Hamiltonian systems without singularity, based on their work, Benci [4, 5] and Gluck and Ziller [6] and Hayashi [7] used a Jacobi metric and very complicated geodesic methods and algebraic topology to study the periodic solutions with a fixed energy of the following system:

$$\ddot{q}+V^{\prime }(q)=O,$$

(1.1)

$$\frac{1}{2}{|\dot{q}|}^2+V(q)=h.$$

(1.2)

They proved a very general theorem.

Theorem 1.1 Suppose $V\in C^2(R^n,R)$, if

$$\{x\in R^n|V(x)\le h\}$$

is bounded and non-empty, then (1.1)-(1.2) has a periodic solution with energy h.

Furthermore, if

$$V^{\prime }(x)\ne O,\mathrm{\forall }x\in \{x\in R^n|V(x)=h\},$$

then (1.1)-(1.2) has a nonconstant periodic solution with energy h.

For the existence of multiple periodic solutions for (1.1)-(1.2) with compact energy surfaces, we can refer to Groessen [8] and Long [9] and the references therein.

In the 1987 paper of Ambrosetti and Coti Zelati [10], Clark-Ekeland’s dual action principle, Ambrosetti-Rabinowitz’s mountain pass theorem etc. were used to study the existence of T-periodic solutions of the second-order equation

$$-\ddot{x}=\mathrm{\nabla }U(x),$$

where

$$U=V\in C^2(\mathrm{\Omega };\mathbf{R})$$

is such that

$$U(x)\to \mathrm{\infty },x\to \mathrm{\Gamma }=\partial \mathrm{\Omega };$$

here $\mathrm{\Omega }\subset {\mathbf{R}}^n$ is a bounded and convex domain, and they got the following result.

Theorem 1.2 Suppose that

• (i) $U(O)=0=minU$;

• (ii) $U(x)\le \theta (x,\mathrm{\nabla }U(x))$ for some $\theta \in (0,\frac{1}{2})$ and for all x near Γ (superquadraticity near  Γ);

• (iii) $(U^{″}(x)y,y)\ge k{|y|}^2$ for some $k>0$ and for all $(x,y)\in \mathrm{\Omega }\times {\mathbf{R}}^N$.

Let ${\omega }_N$ be the greatest eigenvalue of $U^{″}(0)$ and $T_0={(2/{\omega }_N)}^{1/2}$. Then $-\ddot{x}=\mathrm{\nabla }U(x)$ has for each $T\in (0,T_0)$ a periodic solution with minimal period T.

For $C^r$ systems, a natural interesting problem is if

$$\{x\in R^n|V(x)\le h\}$$

is unbounded: can we get a nonconstant periodic solution for the system (1.1)-(1.2)?

In 1987, Offin [11] firstly generalized Theorem 1.1 to some non-compact cases under $V\in C^3(R^n,R)$ and complicated geometrical assumptions on potential wells, but it seems to be difficult to verify this for concrete potentials under the geometrical conditions.

In 1988, Rabinowitz [12] studied multiple periodic solutions for classical Hamiltonian systems with potential $V\in C^1(R\times R^n,R)$, where $V(q_1,\dots ,q_n;t)$ is $T_i$-periodic in positions $q_i\in R$ and is T-periodic in t.

In 1990, using Clark-Ekeland’s dual variational principle and Ambrosetti-Rabinowitz’s mountain pass lemma, Coti Zelati et al. [13] studied Hamiltonian systems with convex potential wells, they got the following result.

Theorem 1.3 Let Ω be a convex open subset of $R^n$ containing the origin O. Let $V\in C^2(\mathrm{\Omega },R)$ be such that

(V1) $V(q)\ge V(O)=0$, $\mathrm{\forall }q\in \mathrm{\Omega }$.

(V2) $\mathrm{\forall }q\ne O$, $V^{″}(q)>0$.

(V3) $\mathrm{\exists }\omega >0$, s.t. $V(q)\le \frac{\omega }{2}{\parallel q\parallel }^2$, $\mathrm{\forall }\parallel q\parallel <ϵ$.

(V4) $V^{″}{(q)}^{-1}\to 0$, $\parallel q\parallel \to 0$, or

(V4)′ $V^{″}{(q)}^{-1}\to 0$, $q\to \partial \mathrm{\Omega }$.

Then, for every $T<\frac{2\pi }{\sqrt{\omega }}$, (1.1) has a solution with minimal period T.

In Theorems 1.2 and 1.3, the authors assumed the convex conditions for potentials and potential wells so that they can apply Clark-Ekeland’s dual variational principle; we notice that Theorems 1.1-1.3 essentially made the following assumption:

$$V(x)\to \mathrm{\infty },x\to \mathrm{\Gamma }=\partial \mathrm{\Omega }.$$

So all the potential wells are bounded.

For singular Hamiltonian systems with a fixed energy $h\in R$, Ambrosetti and Coti Zelati in [14, 15] used Ljusternik-Schnirelmann theory on a $C^1$ manifold to get the following theorem.

Theorem 1.4 (Ambrosetti and Coti Zelati [14])

Suppose $V\in C^2(R^n\mathrm{\setminus }\{O\},R)$ satisfies $V(q)\to -\mathrm{\infty }$, $q\to 0$ and

• (A1) $3V^{\prime }(u)\cdot u+(V^{″}(u)u,u)\ne 0$, $\mathrm{\forall }u\ne 0$;

• (A2) $V^{\prime }(u)\cdot u>0$, $\mathrm{\forall }u\ne 0$;

• (A3) $\mathrm{\exists }\alpha >2$, s.t. $V^{\prime }(u)\cdot u\le -\alpha V(u)$, $\mathrm{\forall }u\ne 0$;

• (A4) $\mathrm{\exists }\beta >2$, $r>0$, s.t. $V^{\prime }(u)\cdot u\ge -\beta V(u)$, $0<|u|<r$;

• (A5) $V(u)+\frac{1}{2}{V}^{\prime }(u)u\le 0$, $\mathrm{\forall }u\ne 0$.

Then (1.1)-(1.2) has at least one nonconstant periodic solution.

Besides Ambrosetti-Coti Zelati, many other mathematicians [16–34] studied singular Hamiltonian systems, here we only mention a related recent paper of Carminati, Sere and Tanaka [16]. They used complex variational and topological methods to generalize Pisani’s results [17], and they got the following theorem.

Theorem 1.5 Suppose $h>0$, $L_0>0$ and $V\in C^{\mathrm{\infty }}(R^n\mathrm{\setminus }\{O\},R)$ satisfies $V(q)\to -\mathrm{\infty }$, $q\to 0$ and

• (B1) $V(q)\le 0$, $\mathrm{\forall }q\ne 0$;

• (B2) $V(q)+\frac{1}{2}{V}^{\prime }(q)q\le h$, $\mathrm{\forall }|q|\ge e^{L_0}$;

• (B3) $V(q)+\frac{1}{2}{V}^{\prime }(q)q\ge h$, $\mathrm{\forall }|q|\le e^{-L_0}$;

• (A4) $\mathrm{\exists }\beta >2$, $r>0$, s.t. $V^{\prime }(q)\cdot q\ge -\beta V(q)$, $0<|q|<r$.

Then (1.1)-(1.2) has at least one periodic solution with the given energy h and whose action is at most $2\pi r_0$ with

$$r_0=max\{{\left[2(h-V(q))\right]}^{\frac{1}{2}};|q|=1\}.$$

Theorem 1.6 Suppose $h>0$, ${\rho }_0>0$, and $V\in C^{\mathrm{\infty }}(R^n\mathrm{\setminus }\{O\},R)$ satisfies $V(q)\to -\mathrm{\infty }$, $q\to 0$ and (B1), (A4) and

(B2)′ ${lim}_{|q|\to +\mathrm{\infty }}{V}^{\prime }(q)=O$;

(B3)′ $V(q)+\frac{1}{2}{V}^{\prime }(q)q\ge h$, $\mathrm{\forall }|q|\le {\rho }_0$.

Then (1.1)-(1.2) has at least one periodic solution with the given energy h whose action is at most $2\pi r_0$.

By using the variational minimizing method with a special constraint, we obtain the following result.

Theorem 1.7 Suppose $V\in C^2(R^n\mathrm{\setminus }\{O\},R)$ and $V(q)\to -\mathrm{\infty }$, $q\to 0$ and satisfies (A1)-(A3) and

(A4)′ $\mathrm{\exists }\beta >2$, s.t. $V^{\prime }(q)\cdot q\ge -\beta V(q)$, $0<|q|<+\mathrm{\infty }$;

(A5)′ $V(-q)=V(q)$, $\mathrm{\forall }q\ne O$.

Then for any $h>0$, (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.

Using the direct variational minimizing method, we get the following theorem.

Theorem 1.8 Suppose $V\in C^1(R^2\mathrm{\setminus }\{O\},R)$ and $V(q)\to -\mathrm{\infty }$, $q\to 0$ and satisfies

(B1)′ $V(q)<h$, $\mathrm{\forall }q\ne O$;

(P1)′ $V^{\prime }(u)\to O$, $\parallel u\parallel \to +\mathrm{\infty }$;

(A3)′ $\mathrm{\exists }\alpha >2$, ${\mu }_2>0$, s.t. $V^{\prime }(u)\cdot u\le -\alpha V(u)+{\mu }_2$, $\mathrm{\forall }u\ne 0$;

(A4) $\mathrm{\exists }\beta >2$, $r>0$, s.t. $V^{\prime }(u)\cdot u\ge -\beta V(u)$, $0<|u|<r$.

Then for any $h>\frac{{\mu }_2}{\alpha }$, (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.

Corollary 1.9 Suppose $\alpha =\beta >2$ and

$$V(x)=-{|x|}^{-\alpha }.$$

Then for any $h>0$, (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.

Remark In Theorem 1.8, the assumption on regularity for potential V is weaker than Theorems 1.1-1.6. Comparing Theorem 1.5 with Theorem 1.8, our (B1)′ is also weaker than (B1), and (A3)′ is also different from (B2)-(B3) and (B3)′.

2 A few lemmas

Let

$$H^1=W^{1,2}(R/Z,R^n)=\{u:R\to R^n,u\in L^2,\dot{u}\in L^2,u(t+1)=u(t)\}.$$

Then the standard $H^1$ norm is equivalent to

$$\parallel u\parallel ={\parallel u\parallel }_{H^1}={\left({\int }_0^1{|\dot{u}|}^{2}dt\right)}^{1/2}+|u(0)|.$$

Let

$$\mathrm{\Lambda }=\{u\in H^1|u(t)\ne O,\mathrm{\forall }t\}.$$

Lemma 2.1 ([14])

Let

$$F=\{u\in H^1|{\int }_0^1(V(u)+\frac{1}{2}{V}^{\prime }(u)u)dt=h\}.$$

If (A1) holds, then F is a $C^1$ manifold with codimension 1 in $H^1$. Let

$$f(u)=\frac{1}{4}{\int }_0^1{|\dot{u}|}^{2}dt{\int }_0^1{V}^{\prime }(u)udt$$

and let $\tilde{u}\in F$ be such that $f^{\prime }(\tilde{u})=O$ and $f(\tilde{u})>0$. Set

$$\frac{1}{T^2}=\frac{{\int }_0^1{V}^{\prime }(\tilde{u})\tilde{u}dt}{{\int }_0^1{|\dot{\tilde{u}}|}^{2}dt}.$$

If (A2) holds, then $\tilde{q}(t)=\tilde{u}(t/T)$ is a nonconstant T-periodic solution for (1.1)-(1.2). Moreover, if (A2) holds, then $f(u)\ge 0$ on F and $f(u)=0$, $u\in F$ if and only if u is constant.

Lemma 2.2 ([8, 14])

Let $f(u)=\frac{1}{2}{\int }_0^1{|\dot{u}|}^{2}dt{\int }_0^1(h-V(u))dt$ and $\tilde{u}\in \mathrm{\Lambda }$ be such that $f^{\prime }(\tilde{u})=O$ and $f(\tilde{u})>0$. Set

$$\frac{1}{T^2}=\frac{{\int }_0^1(h-V(\tilde{u}))dt}{\frac{1}{2}{\int }_0^1{|\dot{\tilde{u}}|}^{2}dt}.$$

Then $\tilde{q}(t)=\tilde{u}(t/T)$ is a nonconstant T-periodic solution for (1.1)-(1.2). Furthermore, if $V(x)<h$, $\mathrm{\forall }x\ne O$, then $f(u)\ge 0$ on Λ and $f(u)=0$, $u\in \mathrm{\Lambda }$ if and only if u is a nonzero constant.

Lemma 2.3 (Sobolev-Rellich-Kondrachov [35, 36])

$$W^{1,2}(R/Z,R^n)\subset C(R/Z,R^n)$$

and the imbedding is compact.

Lemma 2.4 ([35, 36])

Let $q\in W^{1,2}(R/TZ,R^n)$.

• (1) If $q(0)=q(T)=O$, then we have the Friedrics-Poincaré inequality:

  $${\int }_0^T{|\dot{q}(t)|}^{2}dt\ge {\left(\frac{\pi }{T}\right)}^2{\int }_0^T{|q(t)|}^{2}dt.$$

• (2) If ${\int }_0^{T}q(t)dt=0$, then we have Wirtinger’s inequality:

  $${\int }_0^T{|\dot{q}(t)|}^{2}dt\ge {\left(\frac{2\pi }{T}\right)}^2{\int }_0^T{|q(t)|}^{2}dt$$

  and Sobolev’s inequality:

  $${\int }_0^T{|\dot{q}(t)|}^{2}dt\ge \frac{12}{T}{|q(t)|}_{\mathrm{\infty }}^2.$$

Lemma 2.5 (Eberlein-Shmulyan [37])

A Banach space X is reflexive if and only if any bounded sequence in X has a weakly convergent subsequence.

Definition 2.6 (Tonelli [35])

Let X be a Banach space, $f:X\to R$.

• (i) If for any $\{x_n\}\subset X$ strongly converges to $x_0:x_n\to x_0$, we have

  $$lim\hspace{0.17em}inff(x_n)\ge f(x_0),$$

  then we call $f(x)$ lower semi-continuous at $x_0$.

• (ii) If for any $\{x_n\}\subset X$ weakly converges to $x_0:x_n\rightharpoonup x_0$, we have

  $$lim\hspace{0.17em}inff(x_n)\ge f(x_0),$$

  then we call $f(x)$ weakly lower semi-continuous at $x_0$.

Using the famous Ekeland variational principle, Ekeland proved the following.

Lemma 2.7 (Ekeland [38])

Let X be a Banach space, $F\subset X$ be a closed (weakly closed) subset, let $\delta (x_1,x_2)$ be the geodesic distance between two points $x_1$ and $x_2$ in X, $\delta (x,F)$ be the geodesic distance between x and the set F. Suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous (or weakly lower semi-continuous) and assume $\mathrm{\Phi }{|}_F$ restricted on F is bounded from below. Then there is a sequence $\{x_n\}\subset F$ such that

$$\begin{array}{r}\delta (x_n,F)\to 0,\\ \mathrm{\Phi }(x_n)\to \underset{F}{inf}\mathrm{\Phi },\\ (1+\parallel x_n\parallel )\parallel \mathrm{\Phi }{|}_F^{\prime }(x_n)\parallel \to 0.\end{array}$$

Definition 2.8 ([38, 39])

Let X be a Banach space, $F\subset X$ be a closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence $\{x_n\}\subset F$ is such that

$$\begin{array}{r}\delta (x_n,F)\to 0,\\ \mathrm{\Phi }(x_n)\to c,\\ (1+\parallel x_n\parallel )\parallel \mathrm{\Phi }{|}_F^{\prime }(x_n)\parallel \to 0,\end{array}$$

then $\{x_n\}$ has a strongly convergent subsequence.

Then we say that f satisfies the ${(CPS)}_{c,F}$ condition at the level c for the closed subset $F\subset X$.

We notice that if $F=X$, then the above condition is the classical Cerami-Palais-Smale condition [40].

We can give a weaker condition than the ${(CPS)}_{c,F}$ condition.

Definition 2.9 Let X be a Banach space, $F\subset X$ be a weakly closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence $\{x_n\}\subset F$ such that

$$\begin{array}{r}\delta (x_n,F)\to 0,\\ \mathrm{\Phi }(x_n)\to c,\\ \parallel \mathrm{\Phi }{|}_F^{\prime }(x_n)\parallel \to 0,\end{array}$$

then $\{x_n\}$ has a weakly convergent subsequence.

Then we say that f satisfies the ${(WCPS)}_{c,F}$ condition.

Lemma 2.10 (Gordon [18])

Let V satisfy the so-called Gordon strong force condition:

There exists a neighborhood $\mathcal{N}$ of O and a function $U\in C^1(\mathrm{\Omega },\mathbb{R})$ such that:

• (i) ${lim}_{s\to 0}U(x)=-\mathrm{\infty }$;

• (ii) $-V(x)\ge {|U^{\prime }(x)|}^2$ for every $x\in \mathcal{N}-\{O\}$.

Let

$$\partial \mathrm{\Lambda }=\{u\in H^1=W^{1,2}(R/Z,R^n),\mathrm{\exists }{t}_0,u(t_0)=O\}.$$

Then we have

$${\int }_0^{1}V(u)dt\to -\mathrm{\infty },\mathrm{\forall }{u}_n\rightharpoonup u\in \partial \mathrm{\Lambda }.$$

Let

$$\partial \mathrm{\Lambda }=\{u\in H^1=W^{1,2}(R/Z,R^n),\mathrm{\exists }{t}_0,u(t_0)=0\}.$$

Then we have

$${\int }_0^{1}V(u)dt\to -\mathrm{\infty },\mathrm{\forall }{u}_n\rightharpoonup u\in \partial \mathrm{\Lambda }.$$

By Lemmas 2.7 and 2.10, it is easy to prove the following.

Lemma 2.11 Let X be a Banach space, let $F\subset X$ be a weakly closed subset. Suppose that Φ defined on F is Gateaux-differentiable and weakly lower semi-continuous and bounded from below on F. If Φ satisfies the ${(CPS)}_{inf\mathrm{\Phi },F}$ condition or the ${(WCPS)}_{inf\mathrm{\Phi },F}$ condition, and suppose that

$$\mathrm{\Phi }(u_n)\to +\mathrm{\infty },u_n\rightharpoonup u\in \partial \mathrm{\Lambda },$$

then Φ attains its infimum on F.

The next lemma is a variant on the classical Tonelli’s theorem, whose proof is easy, so we omit its proof.

Lemma 2.12 Let X be a Banach space, $F\subset X$ be a weakly closed subset. Suppose that $\varphi (u)$ is defined on an open subset $\mathrm{\Lambda }\subset X$ and is Gateaux-differentiable on Λ and weakly lower semi-continuous and bounded from below on $\mathrm{\Lambda }\cap F$, if ϕ is coercive, that is, $\varphi (x)\to +\mathrm{\infty }$ as $\parallel x\parallel \to +\mathrm{\infty }$, and suppose that

$$\varphi (u_n)\to +\mathrm{\infty },u_n\rightharpoonup u\in \partial \mathrm{\Lambda },$$

then ϕ attains its infimum on $\mathrm{\Lambda }\cap F$.

3 The proof of Theorem 1.7

By the symmetrical condition (A5)′, it is easy to prove that the critical point of the functional f on ${\mathrm{\Lambda }}_0$ is also the critical point of the functional f on Λ.

Let

$$\partial {\mathrm{\Lambda }}_0=\{u\in H^1=W^{1,2}(R/Z,R^n),u(t+1/2)=-u(t),\mathrm{\exists }{t}_0,u(t_0)=0\}.$$

Lemma 3.1 Assume (A4)′ holds, then for any weakly convergent sequence $u_n\rightharpoonup u\in \partial {\mathrm{\Lambda }}_0$, we have

$$f(u_n)\to +\mathrm{\infty }.$$

Proof Similar to the proof of Zhang [19]. □

Lemma 3.2 $F\cap \mathrm{\Lambda }$ is a weakly closed subset in $H^1$.

Proof Let $\{u_n\}\subset F\cap \mathrm{\Lambda }$ be a weakly convergent sequence, we use the embedding theorem to find which uniformly converges to $u\in H^1$.

Now we claim $u\in \mathrm{\Lambda }$, and then it is obvious that $u\in F$. In fact, if $u\in \partial \mathrm{\Lambda }$, by $V(q)\to -\mathrm{\infty }$, $q\to 0$ and the condition (A4)′ we have

$$-V(u)\ge C_1{|u|}^{-\beta },0<|u|<r^{\prime }<r.$$

So $V(u)$ satisfies Gordon’s strong force condition, and by his lemma, we have

$${\int }_0^1-V(u_n)dt\to +\mathrm{\infty },\mathrm{\forall }{u}_n\rightharpoonup u\in \partial \mathrm{\Lambda }.$$

The condition (A4)′ implies

$$V(u_n)+\frac{1}{2}〈V^{\prime }(u_n),u_n〉\ge (1-\frac{\beta }{2})V(u_n).$$

Hence

$$h={\int }_0^1[V(u_n)+\frac{1}{2}〈V^{\prime }(u_n),u_n〉]dt\to +\mathrm{\infty }.$$

This is a contradiction. □

Lemma 3.3 $f(u)$ is weakly lower semi-continuous on $F\cap {\mathrm{\Lambda }}_0$

Proof For any $\{u_n\}\subset F:u_n\rightharpoonup u$, then by Sobolev’s embedding theorem and functional analysis, we have uniform convergence:

$${|u_n(t)-u(t)|}_{\mathrm{\infty }}\to 0.$$

• (i) If $u\in {\mathrm{\Lambda }}_0$, then by $V\in C^1(R^n\mathrm{\setminus }\{0\},R)$, we have

  $${|V(u_n(t))-V(u(t)|}_{\mathrm{\infty }}\to 0.$$

  It’s well known that the norm is weakly lower semi-continuous, we have

  $$lim\hspace{0.17em}inf\parallel u_n\parallel \ge \parallel u\parallel .$$

  Hence

  $$\begin{array}{rl}lim\hspace{0.17em}inff(u_n)& =lim\hspace{0.17em}inf\left(\frac{1}{2}{\int }_0^1{|{\dot{u}}_n|}^{2}dt\right){\int }_0^1(h-V(u_n))dt,\\ \ge \frac{1}{2}{\int }_0^1{|\dot{u}|}^{2}dt{\int }_0^1(h-V(u))dt=f(u).\end{array}$$

• (ii) If $u\in \partial {\mathrm{\Lambda }}_0$, then by our assumption on V which satisfies Gordon’s strong force condition, we have

  $${\int }_0^1-V(u_n)dt\to +\mathrm{\infty },\mathrm{\forall }{u}_n\rightharpoonup u\in \partial {\mathrm{\Lambda }}_0.$$

  • (1) (1) If $u\equiv 0$, then

    $${|u_n|}_{\mathrm{\infty }}\to 0,n\to +\mathrm{\infty }.$$

    Then similar to the proof in [19], we have

    $$f(u_n)\ge 6{|u_n|}_{\mathrm{\infty }}^{2-\beta }\to +\mathrm{\infty },n\to +\mathrm{\infty }.$$

    So in this case we have

    $$lim\hspace{0.17em}inff(u_n)=+\mathrm{\infty }\ge f(u).$$
    $$lim\hspace{0.17em}inf\parallel u_n\parallel \ge \parallel u\parallel >0.$$

  • (2) (2) If $u\ne 0$, then by the weakly lower semi-continuity for norm, we have

    So by Gordon’s lemma, we have

    $$\begin{array}{rcl}lim\hspace{0.17em}inff(u_n)& =& lim\hspace{0.17em}inf\left(\frac{1}{2}{\int }_0^1{|{\dot{u}}_n|}^{2}dt\right){\int }_0^1(h-V(u_n))dt=+\mathrm{\infty }\\ \ge & \frac{1}{2}{\int }_0^1{|\dot{u}|}^{2}dt{\int }_0^1(h-V(u))dt=f(u).\end{array}$$

     □

Lemma 3.4 The functional $f(u)$ has a positive lower bound on F.

Proof By the definitions of $f(u)$ and F and the assumption (A2), we have

$$f(u)=\frac{1}{4}{\int }_0^1{|\dot{u}|}^{2}dt{\int }_0^1(V^{\prime }(u)u)dt\ge 0,\mathrm{\forall }u\in F.$$

 □

By the definitions of the functional $f(u)$ and its domain ${\mathrm{\Lambda }}_0$, and the conditions on the energy $h>0$ and the potential $V(u)<0$, it is easy to prove the following lemma.

Lemma 3.5 The functional $f(u)$ is coercive.

Furthermore, we claim that

$$c=\underset{F\cap {\mathrm{\Lambda }}_0}{inf}f(u)>0,$$

since otherwise, $u_0(t)=\text{const}$ attains the infimum 0, then by the symmetry of ${\mathrm{\Lambda }}_0$, we have $u_0(t)\equiv o$, which contradicts the definition of ${\mathrm{\Lambda }}_0$. Now by Lemmas 3.1-3.4 and Lemmas 2.11 and 2.12, we know $f(u)$ attains the infimum on F, furthermore we know that the minimizer is nonconstant.

4 The proof of Theorem 1.8

In order to prove the Cerami-Palais-Smale type condition and get a nonconstant periodic solution in non-symmetrical case, we need to add a topological condition, we know that there are winding numbers (degrees) in the planar case, so we define

$${\mathrm{\Lambda }}_1=\{u\in \mathrm{\Lambda },deg(u)\ne 0\}.$$

Lemma 4.1 If $u_n\rightharpoonup u\in \partial {\mathrm{\Lambda }}_1$, then $f(u_n)\to +\mathrm{\infty }$.

Proof By V satisfying Gordon’s strong force condition, we have

$${\int }_0^1-V(u_n)dt\to +\mathrm{\infty },\mathrm{\forall }{u}_n\rightharpoonup u\in \partial {\mathrm{\Lambda }}_1.$$

• (1) If $u\equiv 0$, then by Sobolev’s embedding theorem, we have

  $${|u_n|}_{\mathrm{\infty }}\to 0,n\to +\mathrm{\infty }.$$

  Then by $deg(u_n)\ne 0$, we have $c>0$ such that

  $$c{|u_n|}_{\mathrm{\infty }}\le {\parallel {\dot{u}}_n\parallel }_{L^2}$$

  and ${\parallel {\dot{u}}_n\parallel }_{L^2}$ is an equivalent norm of $W^{1,2}$ and

  $$f(u_n)\ge c{|u_n|}_{\mathrm{\infty }}^{2-\beta }\to +\mathrm{\infty },n\to +\mathrm{\infty }.$$

  So in this case, we have

  $$lim\hspace{0.17em}inff(u_n)=+\mathrm{\infty }\ge f(u).$$

• (2) If $u\ne 0$, then by the weakly lower semi-continuity for the norm, we have

  $$lim\hspace{0.17em}inf\parallel u_n\parallel \ge \parallel u\parallel >0.$$

  So by Gordon’s lemma, we have

  $$\begin{array}{rcl}lim\hspace{0.17em}inff(u_n)& =& lim\hspace{0.17em}inf\left(\frac{1}{2}{\int }_0^1{|{\dot{u}}_n|}^{2}dt\right){\int }_0^1(h-V(u_n))dt=+\mathrm{\infty }\\ =& \frac{1}{2}{\int }_0^1{|\dot{u}|}^{2}dt{\int }_0^1(h-V(u))dt=f(u).\end{array}$$

   □

Lemma 4.2 Under the assumptions of Theorem  1.8,

$$f(u)=\frac{1}{2}{\int }_0^1{|\dot{u}|}^{2}dt{\int }_0^1(h-V(u))dt$$

satisfies the ${(CPS)}^{+}$ condition on ${\mathrm{\Lambda }}_1$, that is, if $\{u_n\}\subset {\mathrm{\Lambda }}_1$ satisfies

$$f(u_n)\to c>0,(1+\parallel u_n\parallel )f^{\prime }(u_n)\to O,$$

(4.1)

then $\{u_n\}$ has a strongly convergent subsequence in ${\mathrm{\Lambda }}_1$.

Proof Since $f^{\prime }(u_n)$ makes sense, we know

$$\{u_n\}\subset {\mathrm{\Lambda }}_1.$$

We claim ${\int }_0^1{|{\dot{u}}_n|}^{2}dt$ is bounded. In fact, by $f(u_n)\to c$, we have

$$-\frac{1}{2}{\parallel {\dot{u}}_n\parallel }_{L^2}^2\cdot {\int }_0^{1}V(u_n)dt\to c-\frac{h}{2}{\parallel {\dot{u}}_n\parallel }_{L^2}^2.$$

(4.2)

By (A3)′ we have

$$\begin{array}{rcl}〈f^{\prime }(u_n),u_n〉& =& {\parallel {\dot{u}}_n\parallel }_{L^2}^2\cdot {\int }_0^1(h-V(u_n)-\frac{1}{2}〈V^{\prime }(u_n),u_n〉)dt\\ \ge & {\parallel {\dot{u}}_n\parallel }_{L^2}^2{\int }_0^1[h-\frac{{\mu }_2}{2}-(1-\frac{\alpha }{2})V(u_n)]dt.\end{array}$$

(4.3)

By (4.2) and (4.3) we have

$$\begin{array}{rcl}〈f^{\prime }(u_n),u_n〉& \ge & (h-\frac{{\mu }_2}{2}){\parallel {\dot{u}}_n\parallel }_{L^2}^2+(1-\frac{\alpha }{2})(2c-h{\parallel {\dot{u}}_n\parallel }_{L^2}^2)\\ =& (\frac{\alpha }{2}h-\frac{{\mu }_2}{2}){\parallel {\dot{u}}_n\parallel }_{L^2}^2+C_1,\end{array}$$

(4.4)

where $C_1=2(1-\frac{\alpha }{2})c$, $\alpha >2$, $h>\frac{{\mu }_2}{\alpha }$. So ${\parallel {\dot{u}}_n\parallel }_2\le C_2$.

Then we claim $|u_n(0)|$ is bounded.

We notice that

$$\begin{array}{r}{f}^{\prime }(u_n)\cdot (u_n-u_n(0))\\ ={\int }_0^1{|{\dot{u}}_n|}^{2}dt{\int }_0^1(h-V(u_n))dt\\ -\frac{1}{2}{\int }_0^1{|{\dot{u}}_n|}^{2}dt{\int }_0^1〈V^{\prime }(u_n),u_n-u_n(0)〉dt\\ =2f(u_n)-\frac{1}{2}{\int }_0^1{|{\dot{u}}_n|}^2{\int }_0^1〈V^{\prime }(u_n),u_n-u_n(0)〉dt.\end{array}$$

(4.5)

If $|u_n(0)|$ is unbounded, then there is a subsequence, still denoted by $u_n$ s.t. $|u_n(0)|\to +\mathrm{\infty }$. Since

$$\parallel {\dot{u}}_n\parallel \le M_1,$$

we have

$$\underset{0\le t\le 1}{min}|u_n(t)|\ge |u_n(0)|-{\parallel {\dot{u}}_n\parallel }_2\to +\mathrm{\infty },\text{as }n\to +\mathrm{\infty }.$$

(4.6)

By Friedrics-Poincaré’s inequality and the condition (P1), we have

$${\int }_0^1{|{\dot{u}}_n(t)|}^{2}dt\ge {\pi }^2{\int }_0^1{|u_n(t)-u_n(0)|}^{2}dt,$$

(4.7)

$${\int }_0^1{V}^{\prime }(u_n)(u_n-u_n(0))dt\to 0,$$

(4.8)

$$f^{\prime }(u_n)\cdot (u_n-u_n(0))\to 0.$$

(4.9)

So $f(u_n)\to 0$, which contradicts $f(u_n)\to c>0$, hence $u_n(0)$ is bounded, and $\parallel u_n\parallel ={\parallel {\dot{u}}_n\parallel }_{L^2}+|u_n(0)|$ is bounded. Furthermore, similar to the proof of Ambrosetti and Coti Zelati [15], $u_n$ strongly converges to $u\in \mathrm{\Lambda }$. □

It is easy to prove the following.

Lemma 4.3 Under the assumption (B1)′, $f(u)\ge 0$ on Λ, that is, f has a lower bound.

Lemma 4.4 Under the assumptions of Theorem  1.8, $f(u)$ is weakly lower semi-continuous on the closure $\overline{\mathrm{\Lambda }}$ of Λ.

Now we can prove our Theorem 1.8, in fact, by Lemma 4.1, we know that the infimum of f on ${\mathrm{\Lambda }}_1$ is equal to the infimum of f on the closure of ${\mathrm{\Lambda }}_1$. Furthermore, we can prove the infimum of f on ${\mathrm{\Lambda }}_1$ is greater than zero, otherwise if it is zero, the corresponding minimizer must be constant, then the winding number is zero, which is a contradiction. Now by the above lemmas, especially Lemma 2.11, we know that f attains the positive infimum on ${\mathrm{\Lambda }}_1$ and the corresponding minimizer must be nonconstant.