Open Access

Fourth order Hamiltonian system with some singular nonlinear term and multiplicity result

Boundary Value Problems20162016:133

https://doi.org/10.1186/s13661-016-0638-z

Received: 12 April 2016

Accepted: 30 June 2016

Published: 19 July 2016

Abstract

We consider a fourth order Hamiltonian system with some singular nonlinear term and multiplicity result. We get two theorems which show the number of weak solutions of this problem. The first theorem is a result which shows that there exists a weak solution for this problem and the second one is an improved result of the first result, which shows that there exist infinitely many weak solutions for this problem. We get the first result by a variational method and critical point theory, and we get the second result by homology theory.

Keywords

fourth order Hamiltonian system singular nonlinear term variational method critical point theory minimax method homology theory \((P.S.)_{c}\) condition

MSC

35Q72

1 Introduction

Let \(\bar{N}_{\epsilon}(\theta)\) be a closure of an ϵ-neighborhood of \(\theta=(0,\ldots,0)\), \(\epsilon>0\) be a fixed small number, and D be an open subset in \(R^{n}\) with compact complement \(\bar{N}_{\epsilon}(\theta)=R^{n}\setminus D\), \(n\ge2\). Let \(c\in R\) and \(\vert \cdot \vert \) be a norm in \(R^{n}\). In this paper we consider the weak solutions \(z(t)=(z_{1}(t),\ldots,z_{n}(t))\in C^{4}([0,2\pi], D)\) of a fourth order Hamiltonian system with singular nonlinear term
$$\begin{aligned}& \ddddot{z}(t)+c\ddot{z}(t)+\operatorname{grad}_{z}\biggl(\frac {1}{\vert z(t)\vert ^{2p}} \biggr)=0,\quad p\ge1, \\& z(0)=z(2\pi),\qquad\ddot{z}(0)=\ddot{z}(2\pi). \end{aligned}$$
(1.1)
Our problems are characterized as a singular fourth order Hamiltonian system with singularity at \(\{z(t)=\theta\}\), \(\theta=(0,\ldots,0)\). The motivation of this paper is the fourth order elliptic problem with singular potential. We recommend the book [1] for the singular elliptic problems. Many authors considered the fourth order elliptic boundary value problem. In particular, Choi and Jung [2] showed that the problem
$$\begin{aligned}& \Delta^{2} u+c\Delta u=bu^{+}+s \quad\mbox{in }\Omega, \\& u=0,\qquad\Delta u=0\quad\mbox{on } \partial\Omega, \end{aligned}$$
(1.2)
has at least two nontrivial solutions when \(c<\lambda_{1}\), \(\lambda_{1}(\lambda_{1}-c)< b<\lambda_{2}(\lambda_{2}-c)\), and \(s<0\) or when \(\lambda_{1}< c<\lambda_{2}\), \(b<\lambda_{1}(\lambda_{1}-c)\), and \(s>0\). We obtained these results by using a variational reduction method. We [3] also proved that when \(c<\lambda_{1}\), \(\lambda_{1}(\lambda_{1}-c)< b<\lambda_{2}(\lambda_{2}-c)\), and \(s<0\), (1.2) has at least three nontrivial solutions by using degree theory. Tarantello [4] also studied
$$\begin{aligned}& \Delta^{2} u+c\Delta u=b\bigl((u+1)^{+}-1\bigr), \\& u=0,\qquad\Delta u=0\quad\mbox{on } \partial\Omega. \end{aligned}$$
(1.3)
She showed that if \(c<\lambda_{1}\) and \(b\ge\lambda_{1}(\lambda _{1}-c)\), then (1.3) has a negative solution. She obtained this result by degree theory. Micheletti and Pistoia [5] also proved that if \(c<\lambda_{1}\) and \(b\ge \lambda_{2}(\lambda_{2}-c)\) then (1.3) has at least three solutions by the variational linking theorem and Leray-Schauder degree theory.
The eigenvalue problem
$$\begin{aligned}& \ddot{u}+\lambda u=0\quad\mbox{in } (0,2\pi), \\& u(0)=u(2\pi)=0, \end{aligned}$$
has many eigenvalues \(\lambda_{j}\), \(j\ge1\), and corresponding eigenfunctions \(\phi_{j}\), \(j\ge 1\), suitably normalized with respect to \(L^{2}([0,2\pi])\) inner product and each eigenvalue \(\lambda_{j}\) is repeated as often as its multiplicity. The eigenvalue problem
$$\begin{aligned}& \ddddot{u}(t)+c\ddot{u}(t)=\mu u \quad\mbox{in } (0,2\pi), \\& u(0)=u(2\pi)=0,\qquad\ddot{u}(0)=\ddot{u}(2\pi)=0, \end{aligned}$$
has also infinitely many eigenvalues \(\mu_{j}=\lambda_{j}(\lambda_{j}-c)\), \(j\ge1\), and corresponding eigenfunctions \(\phi_{j}\), \(j\ge1\). We note that \(\mu_{1}<\mu_{2}\le\mu_{3},\ldots, \mu_{j} \to +\infty\).
In this paper we are trying to find the weak solutions \(z(t)\in C^{4}([0,2\pi],D)\cap\Lambda D\) of the system (1.1) satisfying
$$\int^{2\pi}_{0}\bigl[\ddot{z}(t) \cdot\ddot{\phi}(t)-c \dot{z}(t)\cdot \dot{\phi}(t)\bigr]\,dt+ \int_{\Omega}\operatorname{grad}_{u}\biggl(\frac {1}{\vert z(t)\vert ^{2p}} \biggr)\cdot\phi(t) \,dt=0 $$
for all \(\phi(t)\in C^{4}([0,2\pi],D)\cap\Lambda D\), where ΛD is introduced in Section 2.

Theorem 1.1

Assume that \(\lambda _{j}< c<\lambda_{j+1}\), \(j\ge1\). Then the system (1.1) has at least one nontrivial weak solution.

Moreover, we improve Theorem 1.1 as follows.

Theorem 1.2

Assume that \(\lambda _{j}< c<\lambda_{j+1}\), \(j\ge1\). Then the system (1.1) has infinitely many nontrivial weak solutions.

For the proof of Theorem 1.1 we follow the approach of the variational method and use a minimax method in critical point theory on the loop space ΛD, and for the proof of Theorem 1.2 we follow homology theory. In Section 2, we introduce a loop subspace ΛD of the Banach space, and we prove that the associated functional J of (1.1) satisfies the \((P.S.)\) condition on the loop subspace ΛD. In Section 3, we use a minimax method and critical point theory for the existence of a nontrivial weak solution of (1.1) and prove Theorem 1.1. We also prove Theorem 1.2 by using critical point theory and homology theory to prove the existence of infinitely many nontrivial weak solutions.

2 Variational approach

Let \(L^{2}([0,2\pi],R)\) be a square integrable function space defined on \([0,2\pi]\). Any element x in \(L^{2}([0,2\pi],R)\) can be written as
$$x=\sum h_{k}\phi_{k}\quad\mbox{with }\sum h_{k}^{2}< \infty. $$
We shall denote the subset of \(L^{2}([0,2\pi],R)\) satisfying the 2π-periodic condition, by \(L^{2}(S^{1},R)\). Similar notations will be used for other 2π-periodic function spaces. We define a subspace W of \(L^{2}(S^{1},R)\) as follows:
$$W=\Bigl\{ x\in L^{2}\bigl(S^{1},R\bigr)\bigm| \sum \vert \mu_{k}\vert h_{k}^{2}< \infty\Bigr\} . $$
Then this is a complete normed space with a norm
$$\Vert x\Vert _{W}=\Bigl[\sum \vert \mu_{k} \vert h^{2}_{k}\Bigr]^{\frac{1}{2}}. $$
Let
$$\begin{aligned}& W^{+}=\{x\in W\mid h_{k}=0 \mbox{ if } \mu_{k}< 0\}, \\& W^{-}=\{x\in W\mid h_{k}=0 \mbox{ if } \mu_{k}>0\}. \end{aligned}$$
Then \(W=W^{-}\oplus W^{+}\), for \(x\in W\), \(x=x^{-}+x^{+}\in W^{-}\oplus W^{+}\). Let E be the n Cartesian product space of W, i.e.,
$$E=W\times W\times\cdots\times W. $$
Let \(E^{+}\) and \(E^{-}\) be the subspaces on which the functional
$$z\mapsto A(z)= \int^{2\pi}_{0}\bigl[\bigl\vert \ddot{z} (t)\bigr\vert ^{2}-c\bigl\vert \dot{z}(t)\bigr\vert ^{2}\bigr]\,dt $$
is positive definite and negative definite, respectively. Then
$$E=E^{+}\oplus E^{-}. $$
Let \(P^{+}\) be the projection from E onto \(E^{+}\) and \(P^{-}\) the projection from E onto \(E^{-}\). The norm in E is given by
$$\Vert z\Vert ^{2}_{E}=\bigl\Vert P^{+}z \bigr\Vert ^{2}_{E}+\bigl\Vert P^{-}z\bigr\Vert ^{2}_{E}, $$
where \(\Vert P^{+}z\Vert ^{2}_{E}=\sum^{n}_{i=1}\Vert P^{+}z_{i}\Vert ^{2}_{W}\), \(\Vert P^{-}z\Vert ^{2}_{E}=\sum^{n}_{i=1}\Vert P^{-}z_{i}\Vert ^{2}_{W}\), \(z=(z_{1},\ldots,z_{n})\). Let \(\nu^{1}_{\mu_{i}}, \nu^{2}_{\mu_{i}}, \ldots, \nu ^{n}_{\mu_{i}}\) be the eigenvalues of the matrix
$$\operatorname{det}(\mu _{i}I)=\begin{pmatrix} \mu_{i}&0&0&\ldots &0\\ 0&\mu_{i}&0&\ldots &0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots &\mu_{i} \end{pmatrix} \in M_{n\times n}(R),\quad i=1,\ldots,n, $$
that is,
$$\nu^{k}_{\mu_{i}}=\mu_{i},\quad i\ge1,\mbox{ for all }k=1,\ldots,n. $$
Let \((c^{1}_{1,\mu_{i}},\ldots,c^{1}_{n,\mu_{i}}), (c^{2}_{1,\mu _{i}},\ldots,c^{2}_{n,\mu_{i}}), \ldots, (c^{n}_{1,\mu _{i}},\ldots,c^{n}_{n,\mu_{i}})\) be the eigenvectors of the matrix
$$\operatorname{det}(\mu_{i} I)= \begin{pmatrix} \mu_{i}&0&0&\ldots &0\\ 0&\mu_{i}&0&\ldots &0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\ldots &\mu_{i} \end{pmatrix} \in M_{n\times n}(R) $$
corresponding to the eigenvalues \(\nu ^{1}_{\mu_{i}}\), \(\nu^{2}_{\mu_{i}}, \ldots, \nu^{n}_{\mu _{i}}\), respectively. Since \(\nu^{k}_{\mu_{i}}=\mu_{i}\) for all \(k=1, 2, \ldots, n\), \((c^{1}_{1,\mu_{i}},\ldots,c^{1}_{n,\mu _{i}})=\cdots=(c^{n}_{1,\mu_{i}},\ldots,c^{n}_{n,\mu_{i}})\). Let us set
$$(c_{1,\mu_{i}},\ldots,c_{n,\mu_{i}})=\bigl(c^{1}_{1,\mu_{i}}, \ldots,c^{1}_{n,\mu_{i}}\bigr)=\cdots=\bigl(c^{n}_{1,\mu_{i}}, \ldots,c^{n}_{n,\mu_{i}}\bigr). $$
Let us set
$$\begin{aligned}& W_{\mu_{i}} =\operatorname{span}\{\phi_{i}\mid \mu_{j}= \mu_{i}\}, \\& E_{\mu_{i}} = \bigl\{ (c_{1,\mu_{i}}\phi,\ldots,c_{n,\mu_{i}}\phi)\in E \mid (c_{1},\ldots,c_{n})\in R^{n}, \phi\in W_{\mu _{i}}\bigr\} , \\& E^{1}_{\mu_{i}} = \bigl\{ \bigl(c^{1}_{1,\mu_{i}} \phi,\ldots,c^{1}_{n,\mu _{i}}\phi\bigr)\in E \mid \phi\in W_{\mu_{i}}\bigr\} , \\& \phantom{E^{1}_{\mu_{i}} =}\vdots \\& E^{n}_{\mu_{i}} = \bigl\{ \bigl(c^{n}_{1,\mu_{i}} \phi,\ldots,c^{n}_{n,\mu _{i}}\phi\bigr)\in E \mid \phi\in W_{\mu_{i}}\bigr\} . \end{aligned}$$
We note that
$$E_{\mu_{i}}\equiv E^{1}_{\mu_{i}}\equiv\cdots\equiv E^{n}_{\mu_{i}} $$
and
$$E=\bigoplus_{i\ge1}E_{\mu_{i}}. $$
Let us introduce an open set of the Hilbert space E as follows:
$$\Lambda D=\bigl\{ z\in E\mid z(t)\in D=R^{n}\setminus\bar {N}_{\epsilon}(\theta), \epsilon>0 \mbox{ is a small number, } \forall t\in S^{1}\bigr\} . $$
Let us consider the functional on ΛD
$$ J(z)=\frac{1}{2} \int^{2\pi}_{0}\bigl[\bigl\vert \ddot{z} (t)\bigr\vert ^{2}-c\bigl\vert \dot{z}(t)\bigr\vert ^{2} \bigr]\,dt+ \int^{2\pi}_{0}\frac {1}{\vert z(t)\vert ^{2p}}\,dt,\quad p \ge1. $$
(2.1)
The Euler equation for J is (1.1).

By Lemma 2.1, \(J\in C^{1}(\Lambda D,R)\), and so the weak solutions of system (1.1) coincide with the critical points of the associated functional \(J(z)\).

Lemma 2.1

\(J(z)\) is continuous and Fréchet differentiable in ΛD with Fréchet derivative
$$ DJ(z)v= \int^{2\pi}_{0}\biggl[\ddot{z}(t)\cdot\ddot{w}(t)-c\dot {z}(t)\cdot\dot{w}(t)+\operatorname{grad}_{z}\frac{1}{\vert z(t)\vert ^{2p}}\cdot w(t) \biggr]\,dx\quad\forall w\in\Lambda D. $$
(2.2)
Moreover, \(DJ\in C\). That is, \(J\in C^{1}\).

Proof

First we prove that \(J(z)\) is continuous. For \(z, w\in\Lambda D\),
$$\begin{aligned}& \bigl\vert J(z+w)-J(z)\bigr\vert \\& \quad = \biggl\vert \frac{1}{2} \int^{2\pi}_{0}(\ddddot{z+w}+c\ddot {z+w})\cdot(z+w)\,dt \\& \qquad{} + \int^{2\pi}_{0}\frac{1}{\vert z(t)+w(t)\vert ^{2p}}\,dt \\& \qquad{} -\frac{1}{2} \int^{2\pi}_{0}(\ddddot{z}+c\ddot{z}) \cdot z \,dt- \int^{2\pi}_{0}\frac{1}{\vert z(t)\vert ^{2p}}\,dt\biggr\vert \\& \quad =\biggl\vert \frac{1}{2} \int^{2\pi}_{0}\bigl[(\ddddot{z}+c\ddot{z}) \cdot w+( \ddddot{w}+c\ddot{w}) \cdot z+(\ddddot{w}+c\ddot{w}) \cdot w\bigr]\,dt \\& \qquad{} + \int^{2\pi}_{0}\biggl(\frac{1}{\vert z(t)+w(t)\vert ^{2p}}- \frac {1}{\vert z(t)\vert ^{2p}}\biggr)\,dt\biggr\vert . \end{aligned}$$
We have
$$\begin{aligned}& \biggl\vert \int^{2\pi}_{0}\biggl[\frac{1}{\vert z(t)+w(t)\vert ^{2p}}- \frac{1}{\vert z(t)\vert ^{2p}}\biggr]\,dt\biggr\vert \\& \quad \le\biggl\vert \int^{2\pi}_{0}\biggl[\operatorname{grad}_{z} \frac{1}{\vert z(t)\vert ^{2p}}\cdot w+O\bigl(\Vert w\Vert _{E}\bigr)\biggr]\,dt \biggr\vert =O\bigl(\Vert w\Vert _{E}\bigr). \end{aligned}$$
(2.3)
Thus we have
$$\bigl\vert J(z+w)-J(z)\bigr\vert = O\bigl(\Vert w\Vert _{E} \bigr). $$
Next we shall prove that \(J(z)\) is Fréchet differentiable in ΛD. For \(z, w\in\Lambda D\),
$$\begin{aligned}& \bigl\vert J(z+w)-J(z)-DJ(z)w\bigr\vert \\& \quad =\biggl\vert \frac{1}{2} \int^{2\pi}_{0}(\ddddot{z+w}+c\ddot {z+w})\cdot(z+w)\,dt+ \int^{2\pi}_{0}\frac{1}{\vert z(t)+w(t)\vert ^{2p}}\,dt \\& \qquad{} -\frac{1}{2} \int^{2\pi}_{0}(\ddddot{z}+c\ddot{z}) \cdot z\,dt- \int^{2\pi}_{0}\frac{1}{\vert z(t)\vert ^{2p}}\,dt \\& \qquad{} - \int^{2\pi}_{0}\biggl(\ddddot{z}+c\ddot{z}+\operatorname{grad}_{z}\frac{1}{\vert z(t)\vert ^{2p}}\biggr)\cdot w \,dt\biggr\vert \\& \quad =\biggl\vert \frac{1}{2} \int^{2\pi}_{0}\bigl[(\ddddot{w}+c\ddot{w})\cdot z + ( \ddddot{w}+c\ddot{w} )\cdot w\bigr]\,dt \\& \qquad{} + \int^{2\pi}_{0}\biggl(\frac{1}{\vert z(t)+w(t)\vert ^{2p}}- \frac {1}{\vert z(t)\vert ^{2p}}\biggr)\,dt- \int^{2\pi}_{0}\operatorname{grad}_{z} \frac {1}{\vert z(t)\vert ^{2p}}\cdot w\,dt\biggr\vert . \end{aligned}$$
Thus by (2.3), we have
$$ \bigl\vert J(z+w)-J(z)-DJ(z)w\bigr\vert = O\bigl(\Vert w\Vert _{E}\bigr). $$
(2.4)
Similarly, it is easily checked that \(J\in C^{1}\). □

Lemma 2.2

Assume that \(\lambda _{j}< c<\lambda_{j+1}\), \(j\ge1\). Let \(\{z_{k}\}\subset\Lambda D\), \(z_{k}(t)\in Z\), and \(z_{k}\rightharpoonup z\) weakly in ΛD with \(z\in\partial \Lambda D\). Then \(J(z_{k})\to\infty\), where Z is a neighborhood of \(\theta=(0,\ldots,0)\).

Proof

Since \(\frac{1}{z(t)^{2p}}\) has a singular point \(\theta =(0,\ldots,0)\) in \(R^{n}\), the conclusion follows. □

Now, we shall prove that \(J(z)\) satisfies \((P.S.)_{\gamma}\) condition for any \(\gamma\in R\).

Lemma 2.3

Assume that \(\lambda _{j}< c<\lambda_{j+1}\), \(j\ge1\). Then if \(\Vert z_{k}\Vert _{E}\to\infty\), then there exist \((z_{h_{k}})_{k}\) and z in ΛD such that
$$\operatorname{grad}_{z}\frac{1}{\vert z_{h_{k}}(t)\vert ^{2p}}\to z\in\Lambda D, \qquad \frac{z_{h_{k}}}{\Vert z_{h_{k}}\Vert _{E}}\to\theta, \quad \theta=(0,\ldots,0). $$

Proof

Let \(\Vert z_{k}\Vert _{E}\to\infty\). Then \(\frac{1}{| z_{k}(t)|^{2p}}\) is bounded, it follows that
$$ \int^{2\pi}_{0}\frac{\frac{1}{\vert z_{k}(t)\vert ^{2p}}}{\Vert z_{k}\Vert _{E}}\,dt\to 0. $$
(2.5)
Since
$$\int^{2\pi}_{0}\frac{[\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}} \cdot z_{k}(t)-2\frac{1}{\vert z_{k}(t)\vert ^{2p}}]}{\Vert z_{k}\Vert _{E}}\,dt\longrightarrow0, $$
by (2.5), we have
$$ \int^{2\pi}_{0}\frac{\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}} \cdot z_{k}(t)}{\Vert z_{k}\Vert _{E}} \,dt \longrightarrow0. $$
(2.6)
Thus the sequence \((\int^{2\pi}_{0}\frac{\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}} \cdot z_{k}(t)}{\Vert z_{k}\Vert _{E}} \,dt)_{k}\) is bounded. It follows from (2.6) that there exists a subsequence \((z_{h_{k}})_{k}\) such that
$$ \lim_{k\to\infty}\frac{\int^{2\pi}_{0}[\operatorname{grad}_{z}\frac {1}{\vert z_{h_{k}}(t)\vert ^{2p}} \cdot z_{h_{k}}(t)]\,dt}{\Vert z_{h_{k}}\Vert _{E}} =\lim_{k\to \infty} \int^{2\pi}_{0}\operatorname{grad}_{z} \frac{1}{\vert z_{h_{k}}(t)\vert ^{2p}} \cdot \frac{z_{h_{k}}(t)}{\Vert z_{h_{k}}\Vert _{E}}\,dt=0. $$
(2.7)
Since \(\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}}\) is bounded when \(\Vert z_{k}\Vert _{E}\to\infty\), it follows from (2.7) that there exists z in ΛD such that
$$\operatorname{grad}_{z}\frac{1}{\vert z_{h_{k}}(t)\vert ^{2p}}\to z\in\Lambda D, \qquad \frac{z_{h_{k}}}{\Vert z_{h_{k}}\Vert _{E}}\to\theta. $$
Thus the lemma is proved. □

Lemma 2.4

Assume that \(\lambda _{j}< c<\lambda_{j+1}\), \(j\ge1\). Then \(J(z)\) satisfies the \((P.S.)_{\gamma}\) condition for any \(\gamma\in R\).

Proof

Let \(\gamma\in R\) and \((z_{k})_{k}\subset\Lambda D\) be a sequence such that \(J(z_{k})\to\gamma\) and
$$DJ(z_{k})=\ddddot{z_{k}}(t)+c\ddot{z_{k}}(t)+ \operatorname{grad}_{z}\biggl(\frac {1}{\vert z_{k}(t)\vert ^{2p}}\biggr)\longrightarrow\theta, \quad \theta=(0,\ldots,0) \mbox{ in }\Lambda D $$
or equivalently
$$ \bigl\Vert P_{+}z_{k}(t)\bigr\Vert -\bigl\Vert P_{-}z_{k}(t)\bigr\Vert +(D_{tttt}+cD_{tt})^{-1} \biggl(\operatorname{grad}_{z}\biggl(\frac{1}{\vert z_{k}(t)\vert ^{2p}}\biggr)\biggr) \longrightarrow\theta, $$
(2.8)
where \(D_{tttt}z_{k}(t)=\ddddot{z_{k}}(t)\), \((D_{tttt}+cD_{tt})^{-1}\) is a compact operator. We shall show that \((z_{k})_{k}\) has a convergent subsequence. We claim that \(\{z_{k}\}\) is bounded in ΛD. By contradiction, we suppose that \(\Vert z_{k}\Vert _{E}\to\infty\) and set \(w_{k}=\frac{z_{k}}{\Vert z_{k}\Vert _{E}}\). Since \((w_{k})_{k}\) is bounded, up to a subsequence, \((w_{k})_{k}\) converges weakly to some \(w_{0}\) in ΛD. Since \(J(z_{k})\to\gamma\) and \(DJ(z_{k})\to0\), we have
$$\frac{DJ(z_{k})\cdot(z_{k})}{\Vert z_{k}\Vert _{E}}=\frac{2J(z_{k})}{\Vert z_{k}\Vert _{E}} +\frac{\int^{2\pi}_{0} [\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}} \cdot z_{k}(t) -2\frac{1}{\vert z_{k}(t)\vert ^{2p}}]\,dt}{\Vert z_{k}\Vert _{E}}\longrightarrow0. $$
Thus we have
$$\frac{\int^{2\pi}_{0}[\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}} \cdot z_{k}(t) -2\frac{1}{\vert z_{k}(t)\vert ^{2p}}]\,dt}{\Vert z_{k}\Vert _{E}}\longrightarrow0. $$
By Lemma 2.3 and (2.8), there exist \((z_{h_{k}})_{k}\) and z in ΛD such that
$$\operatorname{grad}_{u}\frac{1}{\vert z_{h_{k}}(t)\vert ^{2p}}\to z\in\Lambda D, \quad\mbox{and}\quad\frac{z_{h_{k}}}{\Vert z_{h_{k}}\Vert _{E}}\to \theta. $$
Thus we have \(w_{0}=0\), which is absurd because \(\Vert w_{0}\Vert _{E}=1\). Thus \(\{z_{k}\}\) is bounded in ΛD. Thus \((z_{k})_{k}\) has a convergent subsequence converging weakly to some z in ΛD. We claim that this subsequence of \((z_{k})_{k}\) converges strongly to z. By \(DJ(z_{k})\to\theta\), we have
$$DJ(z_{k})=\ddddot{z_{k}}+c\ddot{z_{k}}+ \operatorname{grad}_{z}\frac {1}{\vert z_{k}(t)\vert ^{2p}}\longrightarrow\theta. $$
We claim that the mapping \(z_{k}\to\mapsto (\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}})_{k}\) is compact. Since the embedding \(\Lambda D\hookrightarrow C^{2}([0,2\pi]\times \Lambda D,R^{n})\) is compact, the sequence \((\int^{2\pi}_{0}[\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}}\cdot z_{k}(t)\,dt)_{n}\) has a convergent subsequence which converges to \(\int^{2\pi}_{0} [\operatorname{grad}_{z}\frac{1}{\vert z(t)\vert ^{2p}} \cdot z(t)\,dt\). Because \(\{z_{k}\}\) is bounded and the subsequence of \((z_{k})_{k}\) converges weakly to some z in ΛD, \((\operatorname{grad}_{z}\frac{1}{\vert z_{k}(t)\vert ^{2p}})_{k}\) is bounded. Since \((D_{tttt}+cD_{tt})^{-1}\) is compact, by (2.8), \((P_{+}z_{k})_{k}\) and \((P_{-}z_{k})_{k}\) have subsequences converging strongly. Thus \((z_{k})_{k}\) has a subsequence converging strongly. Thus the lemma is proved. □

3 Proofs of Theorems 1.1 and 1.2

Lemma 3.1

There exists a sequence of integers
$$b_{1}< b_{2}< \cdots< b_{i}< \cdots,\quad b_{i}\to\infty, $$
such that \(H_{b_{i}}(\Lambda D)\neq0\).

Proof

Let \(\epsilon>0\) be a fixed small number such that \(\bar {N}_{\epsilon}(\theta)\) contains θ, and choose \(R>0\) such that \(\bar{N}_{\epsilon}(\theta)\subset\operatorname{int}(B_{R})\). Then we have
$$R^{n}\setminus B_{R}\subset D \subset R^{n} \setminus\{\theta\}. $$
Since \(R^{n}\setminus B_{R}\) is a deformation retract of \(R^{n}\setminus\{\theta\}\), \(\Lambda(R^{n}\setminus B_{R})\) is a deformation retract of \(\Lambda(R^{n}\setminus\{\theta\})\), so \(\Lambda(R^{n}\setminus B_{R})\) is a deformation retract of λD. Then we have
$$\begin{aligned} H_{*}(\Lambda D) \cong& H_{*}\bigl(\Lambda \bigl(R^{n}\setminus B_{R}\bigr)\bigr)\oplus H_{*}\bigl(\Lambda D,\Lambda\bigl(R^{n}\setminus B_{R}\bigr)\bigr) \\ \cong& H_{*}\bigl(\Lambda\bigl(S^{n-1}\bigr)\bigr)\oplus H_{*}\bigl(\Lambda D,\Lambda \bigl(S^{n-1}\bigr)\bigr). \end{aligned}$$
By [6], the Poincaré series of \(\Lambda(S^{n-1})\) is written as
$$P_{t}\bigl(\Lambda\bigl(S^{n-1}\bigr)\bigr)= \bigl(1+t^{n}\bigr)+\frac {t^{n-1}}{1-t^{2(n-1)}}\bigl(1+t^{n}\bigr) \bigl(1+t^{n-1}\bigr) $$
with \(Z_{2}\) coefficients. Thus the lemma is proved. □
Let us set a level set
$$J_{\gamma}=\bigl\{ z\in\Lambda D\mid J(z)\le\gamma\bigr\} $$
and
$$\beta=\bigl\{ [z]\subset\Lambda D\mid z\in\Lambda D, z(t) \mbox{ is a loop on }D, \forall t\in S^{1}\bigr\} . $$

Lemma 3.2

Assume that \(\lambda _{j}< c<\lambda_{j+1}\), \(j\ge1\). For each \(\gamma>0\), there exists a finite dimensional singular complex \(\Omega=\Omega_{\gamma}\) such that the level set \(J_{\gamma}\) is deformed into Ω.

Proof

Let us choose \(z\in J_{\gamma}\). Then \(z\in\Lambda D\) and we have
$$J(z)=\frac{1}{2} \int^{2\pi}_{0}\bigl[\bigl\vert \ddot{z}(t)\bigr\vert ^{2}-c\bigl\vert \dot {z}(t)\bigr\vert ^{2} \bigr]\,dt+ \int^{2\pi}_{0}\frac{1}{\vert z(t)\vert ^{2p}}\,dt\le\gamma. $$
We note that there exists a constant \(R_{0}>0\) such that
$$\begin{aligned}& \mbox{if }\bigl(t,z(t)\bigr)\in[0,2\pi]\times R^{n}\setminus B_{R_{0}}, \\ & \quad\mbox{then }\frac{1}{\vert z(t)\vert ^{2p}}< +\infty\quad\mbox{and}\quad \biggl\vert \operatorname{grad}_{z}\frac {1}{\vert z(t)\vert ^{2p}}\biggr\vert < + \infty. \end{aligned}$$
(3.1)
We also note that there exists a neighborhood Z of \(\bar {N}_{\epsilon}(\theta)\) such that
$$ \frac{1}{\vert z(t)\vert ^{2p}}\ge\frac{C}{d^{2}(z,Z)}\quad\mbox{for }(t,z)\in[0,2\pi] \times Z. $$
(3.2)
It follows that there exists a constant \(\gamma_{0}>0\) such that
$$\int^{2\pi}_{0}\bigl[\bigl\vert \ddot{z}(t)\bigr\vert ^{2}-c\bigl\vert \dot{z}(t)\bigr\vert ^{2}\bigr]\,dt \le\gamma_{0}, $$
i.e., we have
$$\bigl\Vert \dot{z}(t)\bigr\Vert ^{2}_{E}-c \int^{2\pi}_{0}\bigl\vert \dot{z}(t)\bigr\vert ^{2}\,dt\le\gamma_{0}. $$
Since the number of elements of the set \(\{\lambda_{i}-c\mid \lambda_{i}-c<0\}\) is finite and \(\lambda_{i}-c\to\infty\) as \(i\in \infty\), there exists a constant \(\gamma_{1}>0\) such that
$$ \int^{2\pi}_{0}\bigl\vert \dot{z}(t)\bigr\vert ^{2}\,dt\le\gamma_{1}. $$
(3.3)
By Lemma 2.2, there exists \(\epsilon_{0}=\epsilon(\gamma,\gamma _{1})\) such that
$$d\bigl(z,\bar{N}_{\epsilon}(\theta)\bigr)\ge\epsilon_{0}\quad \forall z\in J_{\gamma}, \forall t\in S^{1}. $$
Let us choose an integer \(M=M_{\gamma}>2\pi\frac{\gamma_{1}^{\frac {1}{2}}}{\epsilon_{0}}\) and let
$$t_{i}=\frac{2\pi i}{M},\quad i=1,2,\ldots,M. $$
Let us define a broken line
$$\bar{z}(t)=\biggl(1-\frac{1}{2\pi}M(t-t_{i-1})\biggr)z(t_{i-1})+ \frac{1}{2\pi }M(t-t_{i-1})z(t_{i}), $$
\(\forall t\in[t_{i-1},t_{i}]\), \(i=0, 1, 2,\ldots, M\), \(\forall x\in J_{\gamma}\). Let
$$\Omega=\bigl\{ \bar{z}(t)\mid z\in J_{\gamma}\bigr\} . $$
The corresponding \(\bar{z}\mapsto(z(t_{1}),z(t_{2}),\ldots,z(t_{M}))\) define a homeomorphism between Ω and a certain open subset of the M-fold product \(D\times D\times\cdots\times D\). We first claim that \(\Omega\subset\Lambda D\). In fact, \(\forall z\in J_{\gamma}\), for \(t_{2}>t_{1}\), by (3.3), we have
$$\begin{aligned} \bigl\Vert z(t_{2})-z(t_{1})\bigr\Vert _{R^{n}} \le& \int^{t_{2}}_{t_{1}}\bigl\vert \dot {z}(t)\bigr\vert \,dt \\ \le& \biggl( \int^{2\pi}_{0}\bigl\vert \dot{z}(t)\bigr\vert ^{2}\,dt \biggr)^{\frac {1}{2}}\vert t_{2}-t_{1} \vert ^{\frac{1}{2}} \\ \le&\gamma_{1}^{\frac{1}{2}} \vert t_{2}-t_{1} \vert . \end{aligned}$$
Therefore
$$\begin{aligned} d\bigl(\bar{z}(t),\bar{N}_{\epsilon}(\theta)\bigr) \ge& d \bigl(z(t_{i}),\bar {N}_{\epsilon}(\theta)\bigr)- \biggl(1- \frac{1}{2\pi}M(t-t_{i-1}) \biggr)\bigl\Vert z(t_{i})-z(t_{i-1}) \bigr\Vert _{R^{n}} \\ \ge& \epsilon_{0}-2\pi{M}^{-1}\gamma_{1}^{\frac{1}{2}}>0 \end{aligned}$$
\(\forall s\in[t_{i-1},t_{i}]\), \(i=0, 1, 2,\ldots,M\). We next claim that there exists \(\nu\in C([0,1]\times J_{\gamma },\Lambda D)\) such that \(\nu(0,\cdot)=\operatorname{id}\), and \(\nu (1,J_{\gamma})=\Omega\). In fact, let us choose \(z(t)\in\Lambda D\) and let us define ν as follows:
$$\nu(s,z) (t)= \textstyle\begin{cases} z(t) &\mbox{for } t\ge2\pi s,\\ (1-\frac{t-t_{i-1}}{2\pi s-t_{i-1}})z(t_{i-1})+\frac{t-t_{i-1}}{2\pi s-t_{i-1}}z(2\pi s) &\mbox{for } t_{i-1}< t< 2\pi s,\\ \bar{z}(t) & \mbox{for }t\le t_{i-1}\le2\pi s\le t_{i}. \end{cases} $$
Then \(\nu(0,\cdot)=\operatorname{id}\), and \(\nu(1,J_{\gamma})=\Omega\). Thus we prove that \(J_{\gamma}\) is deformed into Ω in the loop space ΛD. Thus the lemma is proved. □

Proof of Theorem 1.1 (Existence of a weak solution)

We shall show that the functional \(J(z)\) has a critical value by the generalized mountain pass theorem. Thus we first shall show that \(J(z)\) satisfies the geometric assumptions of the generalized mountain pass theorem.

Let
$$\Lambda D^{+}=\Lambda D\cap E^{+},\qquad\Lambda D^{-}=\Lambda D\cap E^{-}. $$
Then
$$\Lambda D=\Lambda D^{+}\oplus\Lambda D^{-}. $$
Let \(z\in\Lambda D^{+}\). Then we have
$$\begin{aligned} J(z) =&\frac{1}{2}\bigl\Vert P^{+}z(t)\bigr\Vert ^{2}_{E}-\frac{1}{2}\bigl\Vert P^{-}z(t) \bigr\Vert ^{2}_{E}+ \int^{2\pi}_{0}\frac{1}{\vert z(t)\vert ^{2p}}\,dt \\ =&\frac{1}{2}\bigl\Vert P^{+}z(t)\bigr\Vert ^{2}_{E}+ \int^{2\pi}_{0}\frac {1}{\vert z(t)\vert ^{2p}}\,dt. \end{aligned}$$
Since \(\frac{1}{\vert z(t)\vert ^{2p}}\) is positive and bounded, if \(z\in \Lambda D^{+}\), then there exists a number \(r>0\) such that if \(z\in \partial B_{r}\cap\Lambda D^{+}\), then \(J(z)>0\). Thus \(\inf_{z\in \partial B_{r}\cap\Lambda D^{+}}J(z)>0\). We note that by (3.1), there exists \(R>R_{0}\) such that
$$\mbox{if}\quad \bigl(t,z(t)\bigr)\in[0,2\pi]\times R^{n}\setminus B_{R},\quad\mbox{then }\frac{1}{\vert z(t)\vert ^{2p}}< +\infty \quad\mbox{and}\quad \biggl\vert \operatorname{grad}_{z}\frac {1}{\vert z(t)\vert ^{2p}}\biggr\vert < +\infty, $$
and by (3.2), there exists a neighborhood Z of \(\bar{N}_{\epsilon }(\theta)\) such that
$$\frac{1}{\vert z(t)\vert ^{2p}}\ge\frac{C}{d^{2}(z,Z)}\quad\mbox{for }(t,z)\in[0,2\pi] \times Z. $$
Let us choose \(e\in B_{1}\cap\Lambda D^{+}\). Let \(z\in\Lambda D^{-}\oplus\{\rho e\mid \rho>0\}\). Then \(z=x+y\), \(x\in\Lambda D^{-}\), \(y=\rho e\). Then we have
$$\begin{aligned} J(z) =&\frac{1}{2}\bigl\Vert P^{+}z(t)\bigr\Vert ^{2}_{E}-\frac{1}{2}\bigl\Vert P^{-}z(t) \bigr\Vert ^{2}_{E}+ \int^{2\pi}_{0}\frac{1}{\vert z(t)\vert ^{2p}}\,dt \\ =&\frac{1}{2}\rho^{2}-\frac{1}{2}\bigl\Vert P^{-}x\bigr\Vert ^{2}+ \int _{\Omega}\frac{1}{\vert x+\rho e\vert ^{2p}}\,dt. \end{aligned}$$
By (3.1), there exists constant \(R_{0}>0\) such that if \((t,z(t))\in [0,2\pi]\times R^{n}\setminus B_{R_{0}}\), then \(\vert \frac{1}{\vert z(t)\vert ^{2p}}\vert <+\infty\) and \(\vert \operatorname{grad}_{z}\frac{1}{\vert z(t)\vert ^{2p}}\vert <+\infty\). Thus there exist a large number \(R>R_{0}\) and a small number \(\rho>0\) such that if \(z=x+\rho e\in\partial Q=\partial (((\bar{B}_{R}\cap\Lambda D^{-})\oplus\{re\mid e\in B_{1}\cap\Lambda D^{+}, 0< r< R\})\setminus B_{R_{0}})\), then \(J(z)<0\). Thus we have \(\sup_{z\in\partial Q}J(z)<0\). By Lemma 2.1, \(J(z)\) is continuous and Fréchet differentiable in ΛD and, moreover, \(DJ\in C\). By Lemma 2.4, \(J(z)\) satisfies the \((P.S.)\) condition. Thus by the generalized mountain pass theorem [7], \(J(z)\) possesses a critical value \(c>0\), which is characterized as
$$c=\inf_{h\in\Gamma}\sup_{z\in Q}J\bigl(h(z)\bigr), $$
where
$$\Gamma=\bigl\{ h\in C(\bar{Q},\Lambda D)\mid h=\operatorname{id} \mbox{ on }\partial Q\bigr\} . $$
Thus (1.1) has at least one nontrivial weak solution. Thus we prove Theorem 1.1. □

Proof of Theorem 1.2 (Existence of infinitely many weak nontrivial solutions)

By contradiction, we assume that \(J(z)\) has only finitely many critical points \(z_{1}, z_{2}, \ldots, z_{l}\) such that by the process of the proof of Theorem 1.1, we can obtain \(J(z_{j})>0\), \(1\le j\le l\). Let us set
$$K=\{z_{1},z_{2},\ldots,z_{l}\}. $$
We note that \(\dim\operatorname{ker}(D^{2}J(z_{j}))\le2n\), for all j. Letting
$$b^{*}>\max\bigl\{ n M_{0},\operatorname{ind}(J,z_{j})+\operatorname{dim}\operatorname{ker}\bigl(D^{2}J(z_{j})\bigr)\mid 1\le j\le l\bigr\} , $$
where \(M_{\gamma}\) is defined in the proof of Lemma 3.2, and
$$\tau>\max\bigl\{ 0,J(z_{j})\mid 1\le j\le l\bigr\} , $$
we have
$$C_{b}(J,z_{j})=0\quad\forall b\ge b^{*}, j=1, 2,\ldots, $$
and
$$H_{*}(\Lambda D,J_{0})=H_{*}(J_{\tau},J_{0}). $$
It follows that
$$H_{b}(\Lambda D,J_{0})=0\quad\forall b>b^{*}. $$
Since
$$\begin{aligned}& i_{*}:H_{b}(\Lambda D)\longrightarrow H_{b}( \Lambda D,J_{0})\quad\mbox{is injective for }b\ge b^{*}, \\& H_{b}(\Lambda D)=0\quad\mbox{for } b\ge b^{*}, \end{aligned}$$
which is a contradiction to Lemma 3.1. Thus \(J(z)\) has infinitely many critical points \(z_{j}\), \(j=1, 2,\ldots\) , in ΛD. □

Declarations

Acknowledgements

This work was supported by Inha University Research Grant.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Kunsan National University
(2)
Department of Mathematics Education, Inha University

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© Jung and Choi 2016