Mountain pass lemma and new periodic solutions of the singular second order Hamiltonian system

Abstract

We generalize the classical Ambrosetti-Rabinowitz mountain pass lemma with the Palais-Smale condition for $C^1$ functional to some singular case with the Cerami-Palais-Smale condition and then we study the existence of new periodic solutions with a fixed period for the singular second-order Hamiltonian systems with a strong force potential.

MSC:34C15, 34C25, 58F.

1 Introduction

Many authors [1–19] studied the existence of periodic solutions $t\mapsto x(t)\in \mathrm{\Omega }$, with a prescribed period, of the following second-order differential equations:

$$\ddot{x}=-V^{\prime }(t,x),$$

(1.1)

where $\mathrm{\Omega }={\mathbb{R}}^N-\{0\}$ ($N\in \mathbb{N}$, $N\ge 2$) and $V\in C^1(\mathbb{R}\times \mathrm{\Omega },\mathbb{R})$; $V^{\prime }(t,\cdot )$ denotes the gradient of the function $V(t,\cdot )$ defined on Ω.

In 1975, Gordon [10] firstly used variational methods to study periodic solutions of planar 2-body type problems, he assumed the condition nowadays called Gordon’s strong force condition.

Condition (V₁): There exists a neighborhood of 0 and a function $U\in C^1(\mathrm{\Omega },\mathbb{R})$ such that:

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

• (ii) $-V(t,x)\ge {|U^{\prime }(x)|}^2$ for every $x\in \mathcal{N}-\{0\}$ and $t\in [0,T]$.

Moreover,

• (iii) ${lim}_{x\to 0}V(t,x)=-\mathrm{\infty }$.

In the 1980s and 1990s, Ambsosetti-Coti Zelati, Bahri-Rabinowitz, Greco etc. [1–9, 11–19] further studied 2-body type problems in $R^N$ ($N\ge 2$).

Suppose that $V(t,x)$ is T-periodic in t; as regards the behavior of $V(t,x)$ at infinity, they suppose that one of the following conditions holds.

Condition (V₂): ${lim}_{|x|\to \mathrm{\infty }}V(t,x)=0$, ${lim}_{|x|\to \mathrm{\infty }}{V}^{\prime }(t,x)=0$ (uniformly for t) and $V(t,x)<0$ for every $t\in [0,T]$, $x\in \mathrm{\Omega }$.

Condition (V₃): There exist $c_1,M_1,R_1,\nu >0$ such that, for every $t\in [0,T]$ and $x\in {\mathbb{R}}^N$ with $|x|\ge R_1$:

• (i) $|V^{\prime }(t,x)|\le M_1$;

• (ii) $V(t,x)\ge c_1{|x|}^{\nu }$.

Condition (V₄): There exist $c_1,R_1>0$, $\theta >\frac{1}{2}$, $\nu >1$ such that, for every $t\in [0,T]$, $|x|\ge R_1$:

• (i) $\theta V^{\prime }(t,x)x\le V(t,x)$;

• (ii) $V(t,x)\ge c_1{|x|}^{\nu }$.

Setting $K=\{x\in \mathrm{\Omega }|V^{\prime }(t,x)=0\text{ for every }t\in [0,T]\}$, they got the following results.

Theorem 1.1 (Greco [11])

If (V₁) and one of (V₂)-(V₄) hold, and moreover $K=\mathrm{\varnothing }$, then there is at least one non-constant T-periodic $C^2$ solution.

Theorem 1.2 (Bahri-Rabinowitz [3], Greco [11])

Suppose that $\partial V/\partial t\equiv 0$, so $V(t,x)\equiv V(x)$; moreover suppose we have the following condition.

Condition (V₅): K is compact (or empty).

Then, if (V₁) and one of (V₂)-(V₄) hold, there exist infinitely many non-constant T-periodic $C^2$ solutions.

In this paper, we prove the following new theorem.

Theorem 1.3 Suppose $V\in C^1(\mathbb{R}\times \mathrm{\Omega },\mathbb{R})$ satisfies the conditions:

• (V1) For the given $T>0$, $V(t+T,x)=V(t,x)$.

• (V2) $\mathrm{\forall }(t,x)\in \mathbb{R}\times \mathrm{\Omega }$, $V(t+\frac{T}{2},-x)=V(t,x)$.

• (V3) There is $a>0$, $\alpha \ge 2$ such that for any given $ϵ>0$ and

  $$\mathrm{\forall }t\in [0,T],|x|\le {\left(\frac{T}{12}\right)}^{\frac{1}{2}}[{(b\alpha )}^{\frac{1}{(\alpha +2)}}+ϵ],$$

  we have

  $$-V(t,x)\ge \frac{a}{{|x|}^{\alpha }},$$

  where

  $$b=a{(2\pi )}^{\alpha }{T}^{1-\frac{\alpha }{2}}.$$

• (V4) There exists $M>0$ such that $\mathrm{\forall }(t,x)\in \mathbb{R}\times \mathrm{\Omega }$,

  $$3V(t,x)-V^{\prime }(t,x)x\le M.$$

• (V5) $V(t,x)\to +\mathrm{\infty }$ as $|x|\to +\mathrm{\infty }$ uniformly for $0\le t\le T$.

  Then the system (1.1) has at least a non-constant T-periodic solution.

Corollary 1.1 Suppose $\alpha \ge 2$, $\beta \ge 3$, $a>0$, $a^{\prime }>0$, $V\in C^1(\mathrm{\Omega },\mathbb{R})$ and

$$\begin{array}{r}V(x)=-a{|x|}^{-\alpha },0<|x|\le r_1={\left(\frac{T}{12}\right)}^{\frac{1}{2}}[{(b\alpha )}^{\frac{1}{(\alpha +2)}}+ϵ];\\ V(x)=a^{\prime }{|x|}^{\beta },|x|\ge r_2>r_1;\end{array}$$

then $\mathrm{\forall }T>0$, (1.1) has at least a T-periodic solution.

2 A few lemmas

Lemma 2.1 (Sobolev-Rellich-Kondrachov [20])

We have

$$H^1=W^{1,2}(R/TZ,R^N)\subset C(R/TZ,R^N)$$

and the embedding is compact.

Lemma 2.2 (Eberlein-Shmulyan [20])

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

Lemma 2.3 ([21])

• (i) Let $q\in W^{1,2}(R/ZT,R^N)$ and ${\int }_0^{T}q(t)dt=0$, then we have Wirtinger’s inequality:

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

• (ii) Let $q\in W^{1,2}(R/ZT,R^N)$ and ${\int }_0^{T}q(t)dt=0$, then we have Sobolev’s inequality:

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

• (iii) Let ϕ be a convex function on the real line; $f:[a,b]\to R$ is a non-negative real-valued function which is Lebesgue-integrable, then

  $$\varphi ({\int }_a^{b}f(x)dx)\le \frac{1}{b-a}{\int }_a^b\varphi ((b-a)f(x))dx.$$

Lemma 2.4 (Ekeland [8])

Let X be a Banach space; suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous and bounded from below. Then there is a sequence $\{x_n\}$ such that

$$\begin{array}{c}\mathrm{\Phi }(x_n)\to inf\mathrm{\Phi },\hfill \\ (1+\parallel x_n\parallel )\parallel {\mathrm{\Phi }}^{\prime }(x_n)\parallel \to 0.\hfill \end{array}$$

Definition 2.1 (Palais and Smale [22])

Let X be a Banach space; $f\in C^1(X,R)$, if $\{x_n\}\subset X$ s.t.

$$f(x_n)\to c,f^{\prime }(x_n)\to 0,$$

and $\{x_n\}$ has a strongly convergent subsequence; then we say that f satisfies the ${(\mathit{PS})}_c$ condition.

Cerami [23] presented a weaker compact condition than the above classical ${(\mathit{PS})}_c$ condition.

Definition 2.2 ([8])

Let X be a Banach space, $\mathrm{\Lambda }\subset X$, and suppose that Φ is defined on Λ is Gateaux-differentiable, if the sequence $\{x_n\}$ is such that

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

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

Then we say that f satisfies the ${(\mathit{CPS})}_c$ condition.

We can give a weaker condition than the ${(\mathit{CPS})}_c$ condition.

Definition 2.3 Let X be a Banach space, $\mathrm{\Lambda }\subset X$, and suppose that Φ defined on Λ is Gateaux-differentiable; if the sequence $\{x_n\}$ is such that

$$\begin{array}{c}\mathrm{\Phi }(x_n)\to c,\hfill \\ (1+\parallel x_n\parallel )\parallel {\mathrm{\Phi }}^{\prime }(x_n)\parallel \to 0,\hfill \end{array}$$

and $\{x_n\}$ has a weakly convergent subsequence in Λ, then we say that f satisfies the ${(\mathit{WCPS})}_c$ condition.

Lemma 2.5 (Ambrosetti-Rabinowitz [24], mountain pass lemma)

Let X be a Banach space, $\mathrm{\Lambda }\subset X$, $f\in C^1(\mathrm{\Lambda },R)$. We have

$$\begin{array}{c}{B}_{\rho }=\{x\in \mathrm{\Lambda }|\parallel x\parallel \le \rho \},\hfill \\ S_{\rho }=\partial B_{\rho }\cap X,\rho >0.\hfill \end{array}$$

If there are two points $e_1\in B_{\rho }-S_{\rho }$, $e_2\in \mathrm{\Lambda }-B_{\rho }$ such that

$$f{|}_{S_{\rho }}\ge \alpha >0$$

and

$$f(e_1),f(e_2)\le 0,$$

then $C={inf}_{\varphi \in \mathrm{\Gamma }}{sup}_{t\in [0,1]}f(\varphi (t))\ge \alpha$, where $\mathrm{\Gamma }=\{h(t)\in C^1([0,1],\mathrm{\Lambda }),h(0)=e_1,h(1)=e_2\}$. If f satisfies the ${(\mathit{CPS})}_C$ condition on $\mathrm{\Lambda }\subset X$, furthermore, if $f(x_n)\to +\mathrm{\infty }$ as $x_n\to \partial \mathrm{\Lambda }$, then C is a critical value for f.

3 The proof of Theorem 1.3

Let

$$\begin{array}{c}{H}^1=\{q:R\to R^n|q\in L^2,\dot{q}\in L^2,q(t+T)=q(t)\},\hfill \\ \mathrm{\Lambda }=\{q\in H^1,q(t+\frac{T}{2})=-q(t),q(t)\ne 0,\mathrm{\forall }t\}.\hfill \end{array}$$

Lemma 3.1 ([2, 25])

If $V\in C^1(\mathbb{R}\times \mathrm{\Omega },\mathbb{R})$ satisfies the conditions (V1)-(V2), let

$$f(q)=\frac{1}{2}{\int }_0^T{|\dot{q}|}^{2}dt-{\int }_0^{T}V(t,q)dt,q\in \mathrm{\Lambda },$$

then the critical point of $f(q)$ on Λ is a T-periodic solution of (1.1).

Lemma 3.2 If V satisfies (V3), (V4) in Theorem  1.1, then f satisfies the Cerami-Palais-Smale condition for any $c>0$, that is, for any $\{x_n\}\subset \mathrm{\Lambda }$:

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

(3.1)

$\{x_n\}$ has a strongly convergent subsequence and the limit is in Λ.

Proof By the condition (V3), it is well known [10] that $f(x_n)\to +\mathrm{\infty }$ as $x_n\to \partial \mathrm{\Lambda }$. Since $f(x_n)\to c$, we know that for any given $ϵ>0$, there exists N such that when $n>N$, we have

$$\frac{1}{2}{\int }_0^T{|{\dot{x}}_n|}^{2}dt-{\int }_0^{T}V(x_n)dt\le c+ϵ.$$

(3.2)

By $(1+\parallel x_n\parallel )f^{\prime }(x_n)\to 0$, we have

$$f^{\prime }(x_n)x_n\to 0,$$

(3.3)

$$f^{\prime }(x_n)x_n=2f(x_n)+{\int }_0^T[2V(t,x_n)-V^{\prime }(t,x_n)x_n]dt\to 0.$$

(3.4)

So by (V4) and (3.2) and (3.4), we have $d>0$ such that when n large enough, we have

$${\int }_0^T{|{\dot{x}}_n|}^{2}dt\le d.$$

(3.5)

So ${\int }_0^T{|{\dot{x}}_n|}^{2}dt$ is bounded. Then $\{x_n\}$ has a weakly convergence subsequence, and it is standard to further prove that this subsequence is strongly convergent in Λ.

Now we can prove our theorem.

In order to apply for Ambrosetti-Rabinowitz’s mountain pass lemma, we notice that

$$\mathrm{\forall }x\in \mathrm{\Lambda },{\int }_0^{T}x(t)dt=0,$$

so by (V3) and Wirtinger’s inequality we have

$$f(x)=\frac{1}{2}{\int }_0^T{|\dot{x}|}^{2}dt-{\int }_0^{T}V(t,x)dt$$

(3.6)

$$\ge \frac{1}{2}{\int }_0^T{|\dot{x}|}^{2}dt+a{\int }_0^T{|x|}^{-\alpha }dt$$

(3.7)

$$\ge \frac{1}{2}{\int }_0^T{|\dot{x}|}^2+a{T}^{1+\frac{\alpha }{2}}{\left({\int }_0^T{|x|}^{2}dt\right)}^{\frac{-\alpha }{2}}$$

(3.8)

$$\ge \frac{1}{2}{\int }_0^T{|\dot{x}|}^2+a{T}^{1+\frac{\alpha }{2}}{\left(\frac{T}{2\pi }\right)}^{-\alpha }{\left({\int }_0^T{|\dot{x}|}^{2}dt\right)}^{\frac{-\alpha }{2}}$$

(3.9)

$$=\frac{1}{2}{s}^2+b{s}^{-\alpha }=\varphi (s),$$

(3.10)

where

$$s={\left({\int }_0^T{|\dot{x}|}^{2}dt\right)}^{1/2},b=a{T}^{1+\frac{\alpha }{2}}{\left(\frac{T}{2\pi }\right)}^{-\alpha }=a{(2\pi )}^{\alpha }{T}^{1-\frac{\alpha }{2}}.$$

(3.11)

It is easy to see that if $s_0={(b\alpha )}^{\frac{1}{\alpha +2}}$, ϕ attains its infimum which is a positive number. $\mathrm{\forall }ϵ>0$, we can take $\rho =s_0+ϵ$, take $e_1(t)\ne 0$, $\parallel \dot{e_1}\parallel =s_0<\rho$. By Sobolev’s inequality, we know that $({\int }_0^T{|\dot{x}|}^{2}dt)\ge \frac{12}{T}{\parallel x\parallel }_{\mathrm{\infty }}^2$, so if ${\parallel x(t)\parallel }_{\mathrm{\infty }}\le {(\frac{T}{12})}^{\frac{1}{2}}[{(b\alpha )}^{\frac{1}{(\alpha +2)}}+ϵ]$, then the above proofs hold.

Let us choose $e_2=$ constant value vector in $R^n$, $\dot{e_2}=0$. Then by (V1) and (V5), we have

$$f(e_2)=-{\int }_0^{T}V(t,e_2)\le -T\underset{0\le t\le T}{min}|V(t,e_2)|\to -\mathrm{\infty }\text{as }|e_2|=R\to +\mathrm{\infty }.$$

(3.12)

So if $|e_2|=R$ is large enough, we have

$$f{|}_{e_2}\le 0.$$

By Lemmas 2.5 and 3.2, f has a critical value $C>0$, and the corresponding critical point is a T-periodic solution of the system (1.1). Furthermore, we claim that the critical point is non-constant; in fact, if otherwise, by the anti-$T/2$ periodic property, we know that the critical point must be constant zero, which is impossible since $f(0)=+\mathrm{\infty }$. □