\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage[compress]{cite}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 125, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/125\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for sublinear
 ordinary differential equations at resonance}

\author[C. Li, F. Chen \hfil EJDE-2015/125\hfilneg]
{Chengyue Li, Fenfen Chen}

\address{Chengyue Li \newline
Department of Mathematics,
Minzu University of China, 
Beijing 100081, China}
\email{cunlcy@163.com}

\address{Fenfen Chen \newline
Department of Mathematics, 
Minzu University of China, 
Beijing 100081, China}
\email{chenfenfen359@163.com}

\thanks{Submitted February 5, 2015. Published May 6, 2015.}
\subjclass[2010]{58E05, 34C37, 70H05}
\keywords{Sublinear potential; $Z_2$ type index theorem;
 critical point;  resonance;  \hfill\break\indent Hamiltonian system}

\begin{abstract}
 Using a $Z_2$ type index theorem, we show the existence and multiplicity
 of solutions for the sublinear ordinary differential equation
 $$
 \mathcal{L} u(t)=\mu u(t)+W_u(t,u(t)),\quad 0\leq t\leq L
 $$
 with suitable periodic or boundary conditions.
 Here $\mathcal{L}$ is a linear positive selfadjoint operator,
 $\mu$ is a parameter between two egienvalues of this operator,
 and $W_u$ is the gradient of a potential function.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}

 In the study of physical, chemical and  biological systems, 
many ordinary differential equation models can be set in the form
\begin{equation}
\mathcal{L} u(t)=\mu u(t)+W_u(t,u(t)),\quad 0\leq t\leq L, \label{eP}
\end{equation}
(cf. \cite{c3,c4,g1,l1,l2,m1,r1,r2,s1,t1,t2} and their references)
where $\mathcal{L}$ is a linear positive selfadjoint operator on
 $L^2([0,L],\mathbb{R}^{n})$, $\mu$ is a real parameter,
the potential $W(t,u): \mathbb{R}\times \mathbb{R}^{n}
\to R $ is a $C^{1}$-function, and $W_u(t,u)=\partial W /
\partial u$ denotes the gradient of $W(t,u)$ with respect to the
variable $u$. We say that \eqref{eP} is sublinear if $W$ satisfies
$\lim_{| u | \to \infty}W(t,u)/| u |^2=0$.

Throughout this article, $\| \cdot \| _{L^{q}}$ denotes the norm of the usual space 
$L^{q} := L^{q}([0,L],\mathbb{R}^{n})$ with $1\leq q \leq\infty $, and we always assume that, for an appropriate Hilbert space $(X,\| \cdot \|)\subset L^2$ with the corresponding inner product $\langle \cdot ,\cdot \rangle$,
solutions of \eqref{eP} are exactly the critical points of the corresponding functional
\begin{equation} \label{e1.1}
\Phi (u)=I(u)-J(u), \quad u \in X
\end{equation}
where
\begin{equation} \label{e1.2}
I(u)=\frac{1}{2}(\| u \| ^2-\mu \| u \| ^2_{L^2}), \quad
J(u)=\int_{0}^{L} W(t,u(t))dt,
\end{equation}
 the problem
\begin{equation} \label{e1.3}
\mathcal{L} u(t)=\lambda u(t),
\end{equation}
has eigenvalues  $0<\lambda _{1} < \lambda _2 < \lambda
_{3} < \dots \to \infty$, the corresponding eigenspaces
$ \mathcal{N} _j=\{ v_j \}$ $(j \geq 1)$ have finite dimensions.
For simplicity, we first consider the case of dim
$\mathcal{N} _j=1$ for all $j \geq 1$, and more general case shall be discussed later.
Further, we assume that there exists some $k \in N$ such
that $\mu \in [\lambda_{k},\lambda_{k+1})$. One says \eqref{eP} is at
resonance if $\mu = \lambda _{k}$, and \eqref{eP} is sublinear.

Now we can state our main result as follows.

\begin{theorem} \label{thm1}
Suppose that $(X,\| \cdot \|)\subset L^2$ is a Hilbert space,
continuously embedded into $L^{q}$ for all $q \in [1,\infty ]$,  and 
$\{ v_j (t) \}$ is an orthogonal basis in $X$ and $L^2$ such that
\begin{equation} \label{e1.4}
\| v_j(t) \| ^2 =1=\lambda _j \| v_j(t) \| ^2
_{L^2}, \quad \forall j \geq 1.
\end{equation}
Furthermore, assume that the functional $J(u)\in C^{1}(X,R)$
satisfies $J(0)=0$, $J'(u)$ is a compact operator, and
\begin{itemize}
\item[(J1)] $J(u)=J(-u)$ for all $u \in X$,

\item[(J2)] there exists $K>0$ such that $| J'(u)w | \leq K \| w \| _{L^{1}}$ 
for all $u,w \in X$,

\item[(J3)] there exist $p \in N,M>0,\rho >0$ such that 
$M>\lambda_{k+p} -\lambda_{k}$ , and
\[
J(u)\geq \frac{1}{2} M \| u \| ^2_{L^2} \quad  \text{for } \| u \| _{L^{\infty}} \leq \rho,
\]

\item[(J4)] $J(u) \to \pm \infty$ if $u \in \mathcal{N} _j$ for all $j \geq 1$,
 and $\| u \| \to \infty$.
\end{itemize}
Then, there exist at least $p$ distinct pairs $(u,-u)$ of critical
points of $\Phi(u)$. 
If $\mu \in (\lambda _{k},\lambda_{k+1})$, then
{\rm (J4)} can be ommitted.
\end{theorem}

The above theorem will be proved using the following $Z_2$ type index theorem.

\begin{theorem}[\cite{c1}] \label{thm2}. 
Let $Y$ be a Banach space, and $f \in C^{1}(Y,R)$ be even satisfying 
the Palais-Smale condition. 
Suppose that: (i) there exist a subspace $V$ of $Y$ with
$\operatorname{dim}V=r$ and $\delta >0$ such that 
$\sup_{w \in V,\| w \| =\delta} f(w)<f(0)$; 
(ii) there exists a closed subspace $W$ of $Y$
with $\operatorname{Codim}W=s<r $ such that $\inf_{w \in W} f(w)>- \infty $. 
Then $f$ possesses at least $r-s$ distinct pairs $(u,-u)$ of critical
points.
\end{theorem}

For the convenience of the reader, let us recall that the functional
$f$ is said to satisfy the Palais-Smale condition: if any sequence
$\{ u_j \} $ in $Y$ be such that $f(u_j)$ is bounded
and $f'(u_j)\to 0$, possesses a convergent subsequence.

This article is organized as follows. 
In Section 2, we  prove some lemmas for the functional $\Phi (u)$ defined by
\eqref{e1.1}. In section 3, the proof of Theorem \ref{thm1} and its some extensions 
shall be given. Section 4 is devoted to apply Theorem \ref{thm1} to sublinear 
Hamiltonian systems as well as Extended Fisher-Kolmogorov type equations, 
and the existence and multiplicity results of their solutions shall be obtained.

\section{Preliminaries}

  In this section, we shall study the properties of the functionals 
$\Phi (u), I(u), J(u)$ defined in  \eqref{e1.1}-\eqref{e1.2}.

  With the hypotheses of Theorem \ref{thm1}, for all $u \in X$,we can write 
$u=\sum_{j=1}^{\infty} \alpha_j v_j$, thus 
$\| u \|^2=\sum_{j=1}^{\infty} \alpha_j^2$, and
  \begin{equation} \label{e2.1}
I(u)=\frac{1}{2}\sum_{j=1}^{\infty} \alpha_j^2[1-\mu \int_{0}^{L}| v_j |^2dt]
=\frac{1}{2}\sum_{j=1}^{\infty}(1-\frac{\mu}{\lambda_j})\alpha_j^2.
\end{equation}
\smallskip

\noindent\textbf{Case (i).} If $\mu =\lambda _{k}$, then we set
\begin{gather} \label{e2.2}
u^{+}=\sum_{j=k+1}^{\infty}\alpha_j v_j,\quad 
u^{0}=\alpha_{k} v_{k},\quad 
u^{-}=\sum_{j=1}^{k-1} \alpha_j v_j,\\
\label{e2.3}
\begin{gathered}
 X^{+}=\operatorname{span}\{v_j: j\geq k+1 \},\quad
 X^{-}=\operatorname{span}\{v_j: 1 \leq j \leq k-1 \},\\
  X^{0}=\mathcal{N}  _{k}=\operatorname{span}\{v_{k}\}.
\end{gathered}
\end{gather}
Thus, we have $u= u^{+}+u^{0}+u^{-},X= X^{-} \oplus X^{0} \oplus X^{+}$.
\smallskip

\noindent\textbf{Case (ii).} 
If $\lambda _{k} < \mu < \lambda_{k+1}$, then we let
\begin{gather} \label{e2.4}
u^{+}=\sum_{j=k+1}^{\infty}\alpha_j v_j,\quad 
u^{-}=\sum_{j=1}^{k} \alpha_j v_j, \\
\label{e2.5}
 X^{+}=\operatorname{span}\{v_j: j\geq k+1 \},\quad
 X^{-} =\operatorname{span}\{v_j: 1 \leq j \leq k \},
\end{gather}
so we have $u= u^{+}+u^{-}$, $X= X^{+} \oplus X^{-}$.


\begin{lemma} \label{lem1}
 Under the assumptions of Theorem \ref{thm1}, there exists a norm
 $\| \cdot \|_{*}$ of $X$, equivalent with $\| \cdot \|$, such that
\[
I(u)=\frac{1}{2}(\| u^{+} \| ^2_{*}-\| u^{-} \| ^2_{*}).
\]
\end{lemma}

\begin{proof} 
Without loss of generality, we only consider  the case 
$\mu= \lambda _{k}$ in the following. Thus
\begin{gather}
\begin{aligned}
\Big(1-\frac{\lambda_{k}}{\lambda_{k+1}}\Big)\| u^{+}\|^2
&=\Big(1-\frac{\lambda_{k}}{\lambda_{k+1}}\Big)
\sum_{j=k+1}^{\infty}\alpha_j^2\\
&\leq \sum_{j=k+1}^{\infty}\Big(1-\frac{\lambda_{k}}{\lambda_j}\Big)
\alpha_j^2\\
&\leq \sum_{j=k+1}^{\infty} \alpha_j^2=\| u^{+} \|^2,
\end{aligned} \label{e2.6}\\
\begin{aligned}
\Big(\frac{\lambda_{k}}{\lambda_{k-1}}-1\Big)\| u^{-}\|^2
&=\Big(\frac{\lambda_{k}}{\lambda_{k-1}}-1\Big)\sum_{j=1}^{k-1}\alpha_j^2\\
&\leq \sum_{j=1}^{k-1}\Big(\frac{\lambda_{k}}{\lambda_j}-1\Big)\alpha_j^2\\
&\leq \frac{\lambda_{k}}{\lambda_{1}}\sum_{j=1}^{k-1}
\alpha_j^2=\frac{\lambda_{k}}{\lambda_{1}}\| u^{-}\|^2.
\end{aligned} \label{e2.7}
\end{gather}
Let
\begin{equation} \label{e2.8}
\| u\|_{*}^2=\sum_{j=1}^{k-1}\Big(\frac{\lambda_{k}}{\lambda_j}-1\Big)
\alpha_j^2+\sum_{j=k+1}^{\infty}\Big(1-\frac{\lambda_{k}}{\lambda_j}\Big)\alpha_j^2+
\alpha_{k}^2 \,.
\end{equation}
Clearly, $\| \cdot \| _{*}$ is a norm on $X$, and is equivalent with the norm 
$\| \cdot \|$. The corresponding inner product is
\begin{equation} \label{e2.9}
\langle u,w\rangle_{*}=\sum_{j=1}^{k-1}
\Big(\frac{\lambda_{k}}{\lambda_j}-1\Big)\alpha_j \beta_j +
\sum_{j=k+1}^{\infty}\Big(1-\frac{\lambda_{k}}{\lambda_j}\Big)\alpha_j
\beta_j +\alpha_{k} \beta_{k},
\end{equation}
where $u=\sum_{j=1}^{\infty} \alpha_j v_j$,
$w=\sum_{j=1}^{\infty} \beta_j v_j \in X$. Consequently, according 
to \eqref{e2.1} and \eqref{e2.8}, one obtains
\begin{equation} \label{e2.10}
I(u)=\frac{1}{2}(\| u^{+} \| ^2_{*}-\| u^{-} \| ^2_{*})\,.
\end{equation}
Then 
\begin{equation} \label{e2.11}
\Phi'(u)w=\langle u^{+},w\rangle_{*}-\langle u^{-},w\rangle_{*}-\int_{0}^{L}
W_u(t,u)w\,\mathrm{d}t,\quad \forall u,w \in X.
\end{equation}

Finally, we point out that, in the nonresonant case of 
$\lambda _{k} < \mu < \lambda _{k+1}$, \eqref{e2.8} and \eqref{e2.9} 
should be replaced by 
\begin{gather}
 \| u\|_{*}^2=\sum_{j=1}^{k}\Big(\frac{\mu}{\lambda_j}-1\Big)
\alpha_j^2+\sum_{j=k+1}^{\infty}\Big(1-\frac{\mu}{\lambda_j}\Big)\alpha_j^2 ,
\label{e2.8'}\\
\langle u,w\rangle_{*}=\sum_{j=1}^{k}\Big(\frac{\mu}{\lambda_j}-1\Big)\alpha_j
\beta_j +
\sum_{j=k+1}^{\infty}\Big(1-\frac{\mu}{\lambda_j}\Big)\alpha_j
\beta_j , \label{e2.9'}
\end{gather}
respectively. The proof is complete.
\end{proof}

\begin{lemma} \label{lem2} 
Under the assumptions of Theorem \ref{thm1}, the functional
$\Phi (u)$ satisfies the Palais-Smale condition on $X$.
\end{lemma}

\begin{proof}
 We shall use the idea given by Rabinowitz \cite[Theorem 4.12]{r3} 
and Costa \cite[Proposition 3.2]{c2} for a PDE existence problem. 
Let $\{ u_j \} \subset X$ be such that $\Phi (u_j)$ is bounded, and 
$\Phi'(u_j)\to 0$. We shall prove $\{ u_j \}$ has a convergent subsequence.

Setting $u_j= u^{+}_j+u^{0}_j+u^{-}_j$ with 
$ u^{+}_j \in  X^{+}$, $u^{0}_j \in  X^{0}$,
$u^{-}_j \in  X^{-}$ for all $j\geq 1$. For $j$ sufficiently large, we have
\begin{equation}
\| u^{\pm}_j \|_{*} \geq \Phi'(u_j)u^{\pm}_j
=\langle u^{+}_j,u^{\pm}_j\rangle_{*}-<u^{-}_j,u^{\pm}_j\rangle_{*}
-J'(u_j)u^{\pm}_j.   \label{e2.12pm}
\end{equation}
From (J2),  it follows that
\begin{equation}
| J'(u_j)u^{\pm}_j | \leq K \| u^{\pm}_j \|_{L^{1}}
\leq K_{1} \| u^{\pm}_j \|_{*}     \label{e2.13pm}
\end{equation}
with $K_{1}>0$ coming from the continuous embedding
$L^{1}\to (X,\| \cdot \|)\to (X,\| \cdot \|_{*})$.
Combining \eqref{e2.12pm} with $+$ in the exponents,
and \eqref{e2.13pm} with $+$ in the exponents, we obtain
\begin{equation} \label{e2.14}
\| u^{+}_j \|_{*} \geq \| u^{+}_j \|_{*}^2-K_{1}\| u^{+}_j \|_{*} ,
\end{equation}
thus, $\{ u_j^{+} \}$ is bounded on $X$. Similarly, we also deduce
that $\{ u_j^{-} \}$ is  bounded.
Therefore, there exists $d>0$ such that
\begin{gather} \label{e2.15}
\| u_j-u_j^{0} \|_{*} = \| u_j^{+}+u_j^{-} \|_{*} \leq d, \\
\begin{aligned}
\Big| J(u_j)-J(u^{0}_j) \Big|
&= \Big| \int_{0}^{1} \frac{d}{dt}J((1-t)u^{0}_j+tu_j)dt \Big|\\
&= \Big| \int_{0}^{1} J'((1-t)u^{0}_j+tu_j)(u_j-u^{0}_j)dt \Big|  \\
&\leq  K\| u_j-u^{0}_j \| _{L^{1}} \leq K_{1}\| u_j-u^{0}_j \| _{*} \\
&\leq K_{1}d,
\end{aligned}  \label{e2.16}
\end{gather}
which together with
\[
J(u^{0}_j)=\frac{1}{2}(\| u^{+}_j \| ^2_{*}-\| u^{-}_j \| ^2_{*})
-\Phi(u_j)-[J(u_j)-J(u_{0})]
\]
yields $J(u^{0}_j)$ is bounded. By (J4), we get $\{ u^{0}_j \}$ is bounded.
 Thus $\{ u_j \}$ is bounded on $X$.

It should be noted that the gradient of 
$\Phi(u)$, $\nabla \Phi(u): X \to X$ satisifes
\begin{equation} \label{e2.17}
\nabla \Phi(u)=u- G(u)
\end{equation}
with $G(u):X \to X$ being a compact operator defined by 
\[
\langle G(u),z\rangle =\mu \int_{0}^{L}u(t)z(t)dt+J'(u)z, \quad u,z \in X.
\]
From the boundedness of $\{ u_j \} $ and \eqref{e2.17}, we
infer that $\{ u_j \}$ has at least one convergent
subsequence on $X$. So the Palais-Smale condition holds.
\end{proof}

\begin{lemma} \label{lem3} 
Under the hypotheses of Theorem \ref{thm1}, the functional
$\Phi(u)$ is bounded from below on $X^{+}$.
\end{lemma}

\begin{proof}
 From (J2), we have the  estimate
\begin{equation} \label{e2.18}
J(u)=\int_{0}^{1}\frac{d}{dt}J(tu)dt
=\int_{0}^{1}J'(tu)u\,dt 
\leq K \| u \| _{L^{1}} \leq K_{1} \| u \| _{*}, \quad \forall u \in X.
\end{equation}
Then for $u\in X^{+}$, we infer that
\begin{equation} \label{e2.19}
\Phi (u)=\frac{1}{2} \| u \| ^2_{*}-J(u) 
\geq \frac{1}{2} \| u \| ^2_{*} - K_{1} \| u \|_{*} \to \infty 
\quad (\| u \|_{*} \to \infty).
\end{equation}
Namely, $\Phi (u)$ is coercive and bounded from below on $X^{+}$.
\end{proof}

\begin{lemma} \label{lem4} 
Under the assumptions of Theorem \ref{thm1}, there exists a
subspace $V$ of $X$ with $\dim V=k+p$ and $\widetilde{\rho} >0$ such
that $\sup_{u \in V,\| u \| =\widetilde{\rho}} \Phi(u) <0$.
\end{lemma}

\begin{proof} Put
\begin{gather} 
V = \Big\{ u=\sum _{j=1}^{k+p} \alpha_j v_j: \alpha_j \in \mathbb{R} \;
 (1 \leq j \leq k+p) \Big\}, \label{e2.20} \\
Z = \Big\{ u \in V: \sum _{j=1}^{k+p}
\alpha_j^2=\widetilde{\rho}^2 \Big\}, \label{e2.21}
\end{gather}
where $\widetilde{\rho}=\rho/ (c_{\infty} \sqrt{k+p})$,
 $c_{\infty}$ satisfies $\| z \| _{L^{\infty}} \leq c_{\infty} \| z \|$ for all
$ z \in X$.

For each $u(t)=\sum _{j=1}^{k+p} \alpha_j v_j(t)\in Z$,
 we have by Cauchy-Schwarz inequality
\begin{equation} \label{e2.22}
| u(t) |^2 
\leq \Big(\sum_{j=1}^{k+p} | v_j(t) |^2\Big)
\Big(\sum _{j=1}^{k+p} \alpha_j^2\Big) 
\leq (k+p)c^2_{\infty} \widetilde{\rho}^2=\rho^2,
\end{equation}
using $\| v_j \| _{L^{\infty}} \leq c_{\infty}\| v_j \|=c_{\infty}$ for all 
$j \geq 1$. Hence
\begin{align*}
\Phi(u)&= \frac{1}{2} \| u \| ^2 -\frac{\mu}{2}\| u \| ^2_{L^2}-J(u) \\
&\leq \frac{1}{2} \| u \| ^2 -\frac{\mu +M}{2}\| u \| ^2_{L^2}        \\
&= \frac{1}{2}\sum _{j=1}^{k+p} \alpha_j^2-\frac{\mu +M}{2}\sum _{j=1}^{k+p} 
 \frac{1}{\lambda_j}\alpha_j^2\\
&=\frac{1}{2}\sum _{j=1}^{k+p}\frac{\lambda_j-\mu -M}{\lambda_j}\alpha_j^2     \\
& \leq   \frac{1}{2}(\lambda_{k+p}-\lambda_{k} -M)
 \sum _{j=1}^{k+p}\frac{\alpha _j^2}{\lambda_j}<0,
\end{align*}
which implies $\sup \{ \Phi(u): u \in Z \} <0$.   
\end{proof}

\section{Proof and extension of Theorem \ref{thm1}}

\subsection*{Proof of Theorem \ref{thm1}}
With the aid of Lemmas \ref{lem2}-\ref{lem4} , by
Theorem \ref{thm2}, we conclude that $\Phi(u)$ \eqref{e1.1} possesses at least $p$
distinct pairs $(u_{i},-u_{i})$ of critical points.

\begin{corollary} \label{coro1}
 Under the assumptions of Theorem \ref{thm1}, if condition {\rm (J3)} is replaced by
\begin{itemize}
\item[(J3')] $ \lim_{| u | \to 0} \frac{W(t,u)}{| u | ^2} = \infty$  
uniformly in $t \in [0,L]$,
\end{itemize}
then the functional $\Phi(u)$ defined in \eqref{e1.1} has infinitely many
distinct pairs $(u,-u)$ of critical points.
\end{corollary}

\begin{proof} 
For any fixed $p \in N$, we may take $M$ large enough such that 
$M>\lambda_{k+p} -\lambda_{k}$. By (J3'), there exists $\rho$ sufficiently
 small satisfying 
\begin{equation} \label{e3.1}
W(t,w)\geq \frac{1}{2} M | w |^2,\quad \forall w \in
\mathbb{R}^{n},\; | w | \leq \rho
\end{equation}
uniformly in $t\in [0,L]$. Thus, if $u=u(t) \in X $  with 
$ \| u \|_{L^{\infty}} \leq \rho $, then 
\begin{equation} \label{e3.2}
W(t,u(t))\geq \frac{1}{2} M | u(t) |^2
\end{equation}
uniformly in $t\in [0,L]$, and one obtains
\begin{equation} \label{e3.3}
J(u)\geq \frac{1}{2} M \| u \|^2_{L^2}.
\end{equation}
Therefore, in view of Theorem \ref{thm1}, the functional $\Phi(u)$ has at least 
$p$ distinct pairs $(u_{i},-u_{i})$ of critical points $(1\leq i \leq p)$.
 Since $p$ is arbitrary, there exist infinitely many distinct pairs 
$(u_{i},-u_{i})$ of critical points of $\Phi(u)$ $(i=1,2,3,\dots)$.
\end{proof}

\begin{remark} \label{rmk1} \rm
For all $\beta \in (0,1/2)$, $\gamma \in (0,1)$, we can take a function
 $H(s) \in C^{1}([0,\infty),R)$ such that 
\begin{gather}
s^{1+2\beta} \leq  H(s) \leq s^{1+\beta},\quad \forall s\in [0,1], \label{e3.4}\\
-\frac{1}{8} s^{\gamma -1} \leq  H'(s) \leq \frac{1}{8} s^{\gamma -1} 
\ quad \forall s\in [2,\infty),  \label{e3.5}\\
 H(s)\to \pm \infty \quad \text{as }s\to \infty. \label{e3.6}
\end{gather}
Define $W(t,u)=H(| u |)((\sin t)^{2m}+2),m \geq 1$.
 A straightforward computation shows that \eqref{e3.4} and \eqref{e3.5} imply
(J1)--(J3). In addition, (J4) can be easily deduced
by \eqref{e3.6}, see \cite[Lemma 4.21]{r3}.
\end{remark}

From a carefully analyzing the constructions of $V$ and $Z$ in 
\eqref{e2.20}-\eqref{e2.21}, 
we have the following result which is more general than Theorem \ref{thm1}.

\begin{theorem} \label{thm3} 
Suppose that $(X,\| \cdot \|)\subset L^2$ is a Hilbert space, continuously
 embedded in $L^{q}, \forall q \in [1,\infty]$. 
Let $n_j=$ dim $\mathcal{N} _j$ and ${\{ v_{j1},v_{j2},\dots,v_{jn_j} \}}$ 
be an orthogonal basis of $\mathcal{N} _j(\forall j \geq 1)$ such that 
${\{ v_{ji}(t):j\geq 1,1\leq i\leq n_j \}}$ is an orthogonal basis in $X$ 
and $L^2$ with
\[
\| v_{ji}(x) \| ^2=1=\lambda _j \| v_{ji}(x) \| ^2_{L^2}, \quad
\forall j \geq 1,1\leq i \leq n_j.
\]
Furthermore, assume that the functional $J(u) \in C^{1}(X,R)$ satisfies 
$J(0)=0$, $J'(u)$ is a compact operator, and 
{\rm (J1)-(J4)} hold. Then, there exist at least
 $\sum_{j=k+1}^{k+p}n_j$ distinct pairs $(u,-u)$ of critical points 
of $\Phi (u)$ 
(If $\mu \in (\lambda_{k},\lambda_{k+1})$, then {\rm (J4)} can be omitted).
\end{theorem}

To prove this theorem, we need only changes in Lemmas \ref{lem1} and \ref{lem4}. 
Especially, $V,Z$ in \eqref{e2.20}-\eqref{e2.21} shall be replaced by 
\begin{gather}
\widetilde V = \big\{ u=\sum_{j=1}^{k+p}\sum_{i=1}^{n_j}\alpha_{ji}v_{ji}: 
\alpha_{ji} \in \mathbb{R}\; (1 \leq j \leq k+p,1 \leq i \leq n_j) \big\},  \\
\widetilde Z = \big\{ u \in \widetilde V :
\sum_{j=1}^{k+p}\sum_{i=1}^{n_j}\alpha_{ji}^2 = \widetilde
\rho^2  \big\},
\end{gather}
respectively.

\section{Applications}

\noindent\textbf{Application i.}  
Given $T>0$,  we discuss the existence of  $T$-periodic solutions to the
second-order Hamiltonian system 
\begin{equation}
\ddot u(t) + \mu u(t)+ W_u(t,u(t)) =0,   \quad   t \in \mathbb{R},
\label{HS}
\end{equation}
where $W(t,u) \in C^{1}(R \times \mathbb{R}^{n},R)$ is a $T$-periodic
function in the variable $t$ and $W(t,0)\equiv 0$.

Since 1973, many authors studied periodic solutions for Hamiltonian
systems via critical point theory. Clarke and Ekeland \cite{c3} studied
a family of convex sublinear Hamiltonian systems where $W(t,u)=W(u)$
satisfies $\lim_{| u | \to 0} \frac{W(t,u)}{| u
| ^2}=\infty$, and they used the dual variational method to
obtain the first variational result on periodic solutions having a
prescribed minimal period. Later, Mawhin and Willem \cite{m1} made a
good improvement. Rabinowitz \cite{r1,r2},
Tang \cite{t1} and others proved
the existence under the sublinear condition $uW_u(t,u) \leq \alpha
W(t,u) (0<\alpha <2)$, which plays an important role. 
Schechter \cite{s1} assumed that $W(t,u)$ is sublinear, and
$2W(t,u)-uW_u(t,u) \to - \infty (| u | \to
\infty )$ or $2W(t,u)-uW_u(t,u) \leq W_{0}(t)$, then he proved
that \eqref{HS} has one non-constant periodic solution. 
Long \cite{l2} also studied this problem for bi-even sublinear potentials, and got the
existence of one odd periodic solution.
Li-Wang-Xiao \cite{l1} considered the existence and multiplicity of odd periodic solution
for bi-even sublinear \eqref{HS} in the case of $\mu < \lambda_{1}$.

Motivated by the above papers, using Theorem \ref{thm3}, we shall give a
multiplicity result for \eqref{HS} with sublinear potentials in the case
of $\lambda_{k} \leq \mu < \lambda_{k+1}$.

\begin{theorem} \label{thm4}
 Assume that $L=T/2$, and there exists some $k \in N$ such that 
$(\frac{k \pi }{L})^2 \leq \mu <(\frac{(k+1) \pi }{L})^2$. 
Let $W(t,u)\in C^{1}(R \times \mathbb{R}^{n},R)$ be $T$-periodic in $t$,
 and bi-even, namely 
\[
W_u(t,u)=-W_u(-t,-u),\quad  \forall t \in \mathbb{R},\; u \in \mathbb{R}^{n}.
\]
Suppose that
\begin{itemize}
\item[(W11)] $W(t,u)=W(t,-u)$ for all $t \in \mathbb{R}$, $u \in \mathbb{R}^{n}$;

\item[(W12)] there exists $K>0$ such that
$| W_u(t,u) | \leq K$ for all $t \in \mathbb{R}$, $u \in \mathbb{R}^{n}$;

\item[(W13)] there exist $p \in N$, $M>0$, $\rho >0$ such that if
 $M>\frac{p(p+2k)}{L^2} \pi^2$ then
\[
W(t,u)\geq \frac{1}{2} M | u | ^2 \quad \forall t \in \mathbb{R}, | u | \leq \rho ;
\]

\item[(W14)] for $u=c\sin\frac{j \pi t}{L} \theta_{i}$ with 
$\theta_{i}=(0,0,\dots,0,1,0,\dots,0)\in \mathbb{R}^{n}$ 
(the $i-$th element is 1, $1 \leq i \leq n)$, for all 
$j \geq 1$,  $\int_{0}^{L}W(t,u(t))dt \to \pm \infty $ as $| c | \to \infty $.
\end{itemize}
Then, \eqref{HS} has $np-$ distinct pairs $(u(t),-u(t))$ of odd 
$T$-periodic solutions. 
If $(\frac{k \pi }{L})^2 < \mu <(\frac{(k+1) \pi }{L})^2$, then
{\rm (W14)} can be omitted.
\end{theorem}

\begin{remark} \label{rmk3}  \rm
If $W(t,u)$ satisfies
\[
W(t,u)=W(t,-u)=W(-t,-u),
\]
then $W(t,u)$ is bi-even, and $(W_{11})$ holds. For this, a typical
example is, $W(t,u)=b(t)\widetilde W (u)$, where $b(t)$ and
$\widetilde W (u)$ are even in the variable $t,u$, respectively.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm4}]
 Firstly, consider the boundary value problem 
\begin{equation} \label{e4.1}
 \begin{gathered}
-\ddot{u}(t)=\mu u(t)+W_u(t,u(t)), \quad 0<t<L,\\
u(0)=u(L)=0.\\
\end{gathered}
\end{equation}
If $u=u(t)$ is a solution of \eqref{e4.1}, then we define 
\begin{equation} \label{e4.2}
\overline{u}=\overline{u}(t)=\begin{cases}
u(t), &0 \leq t \leq L,\\
-u(-t), &-L \leq t \leq 0.
\end{cases}
\end{equation}
By the bi-even condition, $\overline{u}=\overline{u}(t)$ is a solution
of \eqref{HS} restricted on $[-L,L]$, so its odd extension in $(-\infty ,
\infty)$ is an odd $T$-periodic solution of \eqref{HS}.

Secondly, let $X=H^{1}_{0}([0,L],\mathbb{R}^{n})$ be the usual Hilbert
 space with the inner product $(x,y)=\int_{0}^{L}\dot{x}(t)\cdot \dot{y}(t)dt$ 
and the norm $\| x \|=(\int_{0}^{L}| \dot{x}(t) |^2dt)^{1/2}$.  
Set 
\begin{equation} \label{e4.3}
\Phi(u)=\frac{1}{2} \int_{0}^{L}[| \dot{u}(t) |^2-\mu| u(t) |^2]
\,\mathrm{d}t-\int_{0}^{L} W(t,u(t))\,\mathrm{d}t,
\end{equation}
then $\Phi(u) \in C^{1}(X,R)$, and its critical points are the classical 
solutions of \eqref{e4.1}.

By direct computations, we know that the problem 
\[
-\ddot{u}(t)=\lambda u(t), \quad  u(0)=u(L)=0
\]
possesses eigenvalues $\lambda_j=({\frac{j \pi}{L}})^2,j \geq 1$, 
and the corresponding eigenfunctions are 
$u_{ji}=c \theta_{i}\sin {\frac{j \pi t}{L}}, 1\leq i \leq n, c \in \mathbb{R}$. 
Furthermore,
\begin{equation} \label{e4.4}
 \big\{ \theta_{i} \sin \frac{ \pi t}{L},\theta_{i} \sin 
\frac{ 2\pi t}{L},\theta_{i} \sin \frac{ 3\pi t}{L},\dots , 1 \leq i \leq n \big\}
\end{equation}
is an orthogonal basis on both $X$ and $L^2$ . Since 
\begin{equation} \label{e4.5}
\int_{0}^{L}| \dot u_{ji}(t) |^2\,\mathrm{d}t=\lambda _j
\frac{L}{2}=\lambda _j\int_{0}^{L}| u_{ji}(t) |^2\,\mathrm{d}t,
\end{equation}
writing $v_{ji}=\sqrt{\frac{2}{L\lambda_j}} u_{ji}$, 
we have $\| v_{ji} \|^2=\int_{0}^{L}| \dot{v} _{ji}
|^2\,\mathrm{d}t=1=\lambda_j \int_{0}^{L}| {v}_{ji} |^2\,\mathrm{d}t$.

Noticing that
\[
\frac{p(p+2k)}{L^2} \pi^2 = \lambda_{k+p} - \lambda _{k}, 
\]
the functional \eqref{e4.3} satisfies all hypotheses of Theorem \ref{thm3},
hence it has at least $np$ distinct pairs $(u_{i},-u_{i})$ of
critical points $(1 \leq i \leq np)$ . Consequently, in the way of
\eqref{e4.2}, the extensions of $\pm \overline{u}_{i}(t)(1 \leq i \leq np)$
are $np$ distinct pairs of odd $T$-periodic solutions of \eqref{HS}.
\end{proof}

\noindent\textbf{Application ii.} 
We are concerned with a class of Extended Fisher-Kolmogorov type equations
(see \cite{c4,g1,t2} and their references) 
\begin{equation} 
u^{(4)}(t) = \mu u(t)+ W_u(t,u(t))    \quad   0 \leq t \leq L  \label{EFK}
\end{equation}
with the boundary condition 
\[
u(0)=u(L)=u''(0)=u''(L)=0,
\]
which appears in the formation of spatial patterns in bistable systems.

\begin{theorem} \label{thm5}
Assume that there exists some $k \in N$ such that 
$(\frac{k \pi }{L})^{4} \leq \mu <(\frac{(k+1) \pi }{L})^{4}$.  
Let $W(t,u)\in C^{1}([0,L] \times R,R)$ satisfy the following conditions:
\begin{itemize}

\item[(W21)] $ W(t,u)=W(t,-u)$ for all $t \in [0,L]$, $u \in \mathbb{R}$;

\item[(W22)] there exists $K>0$ such that 
$| W_u(t,u) | \leq K$ for all $t \in [0,L], u \in \mathbb{R}$;

\item[(W23)] there exist $p \in N, M>0, \rho >0$ such that if
 $M>\frac{(p+k)^{4}-k^{4}}{L^{4}} \pi^{4}$  then
\[
W(t,u)\geq \frac{1}{2} M | u | ^2 \quad \forall t \in [0,L], | u | \leq \rho ;
\]

\item[(W24)] for $u=c\sin\frac{j \pi t}{L} $ , for all $j \geq 1$,
$ c \in \mathbb{R}$, $\int_{0}^{L}W(t,u(t))dt \to \pm \infty $ as
 $| c | \to \infty $.
\end{itemize}
Then, \eqref{EFK} has $p$ distinct pairs $(u(t),-u(t))$ of classical
solutions. If $(\frac{k \pi }{L})^{4} < \mu <(\frac{(k+1) \pi
}{L})^{4}$, then {\rm (W24)} can be omitted.
\end{theorem}

\begin{proof} 
Similarly to the proof of Theorem \ref{thm4}, we sketch it. Set 
\begin{equation} \label{e4.6}
X=H^2(0,L) \cap H^{1}_{0}(0,L),
\end{equation}
by \cite[Lemma 2.1]{g1},
 $\| u \|=(\int_{0}^{T}| \ddot{u}(t) |^2dt)^{1/2}$ is a norm of $X$, and
\begin{equation} \label{e4.7}
v_j(t)=\sin \frac{j \pi t}{L} \big( \sqrt{\frac{L}{2}} (\frac{j \pi }{L})^2\Big)^{-1}
\end{equation}
is an orthogonal basis on $X$ and $L^2$ such that 
\begin{equation} \label{e4.8}
\| v_j(t) \| ^2 = 1 = (\frac{j \pi }{L})^{4} \|
v_j(t) \| ^2_{L^2},\quad j \geq 1.
\end{equation}
In addition, the problem 
\[
u^{(4)}(t)=\lambda u(t)
\]
has eigenvalues $\lambda_j=(\frac{j \pi }{L})^{4}$,  $j \geq 1$, 
and the corresponding eigenfunctions are exactly $v_j(t)$ in \eqref{e4.7}. 
Define the functional 
\begin{equation} \label{e4.9}
\Phi(u)=\frac{1}{2} \int_{0}^{L}| \ddot{u}(t)
|^2dt-\frac{1}{2} \mu  \int_{0}^{L} | u(t) |^2\,\mathrm{d}t
-\int_{0}^{L} W(t,u(t))\,\mathrm{d}t, \quad u \in X,
\end{equation}
then the critical points of $\Phi(u)$ in \eqref{e4.9} are the classical 
solutions of the problem \eqref{EFK}. 
Therefore, by Theorem \ref{thm1}, we have the statement in Theorem \ref{thm5}.
\end{proof}

\subsection*{Acknowledgments}
 The authors would like to thank the anonymous referees for their valuable 
suggestions.

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\end{document}