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\markboth{ On critical points for noncoercive functionals }
{ Klaus Schmitt \& Zhi-Qiang Wang }
\begin{document}
\setcounter{page}{237}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 237--245\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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On critical points for noncoercive functionals and subharmonic
solutions of some Hamiltonian systems
%
\thanks{ {\em Mathematics Subject Classifications:} 35J15, 37J45, 34C15.
\hfil\break\indent
{\em Key words:} Noncoercive functionals, Hamiltonian systems,
subharmonic solutions.
\hfil\break\indent
\copyright 2000 Southwest Texas State University.
\hfil\break\indent Published October 25, 2000. } }
\date{}
\author{ Klaus Schmitt \& Zhi-Qiang Wang }
\maketitle
\begin{abstract}
The paper is concerned with existence results about subharmonic
solutions of some Hamiltonian systems. The existence of such solutions
is established using a variational approach and results about minima
of noncoercive functionals.
\end{abstract}
\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\section{Introduction}
In this paper we use some critical point theorems for noncoercive
functionals (see \cite{le:mpn95}, \cite{le:mpn96}, \cite{le:mnf96})
(and also \cite{bennaoum:emr94})
to deduce the existence of periodic solutions of some
second order Hamiltonian systems and similar problems for semilinear
elliptic partial differential equations. Extensions of these results
to quasilinear differential equations are also indicated.
The results will be used to obtain existence results for subharmonic
solutions of such problems. The nonlinear terms involved have
superquadratic growth. Thus we obtain existence results for
subharmonic solutions complementing those in \cite{benci:btr88}
and \cite{felmer:sne93}, though the perturbation terms considered here
are different. For results concerning subharmonic solutions of
equations
with subquadratic perturbation terms, we refer to \cite{wang:sop96}.
\subsection*{The setting}
Let $E$ be a reflexive Banach space with norm norm $\| \cdot \|$, and
pairing $\langle \cdot, \cdot \rangle$. Let
$$
A : E \to E^*
$$
be a mapping such that the functional
$\varphi: E \to {\mathbb R}$
given by
$$
u \mapsto \langle Au, u \rangle
$$
is weakly lower semicontinuous.
Let us denote by
$$
\psi : E \to {\mathbb R}
$$
a weakly continuous mapping
whose level surfaces will be denoted by
$$
S = \{ u \in E : \psi (u) = \gamma \in {\mathbb R} \}.
$$
Suppose that $\varphi $ is nonnegative and
positive homogeneous of degree $p>1$.
Further assume that
$$
\mbox{\rm ker}\,\varphi=\{u:\varphi(u)=0\}
$$
is a finite dimensional subspace of $E$
such that
$$
\varphi(u+v)=\varphi(v),~\forall v\in E,~\forall
u\in\mathop{\rm ker}\varphi.
$$
We also assume that $E=\mbox{\rm ker}\,\varphi\oplus X$,
where $X$ is a closed subspace
of $E$ and $\varphi|_X$ is coercive in the sense that there exists $c>0$
such that
$$
\varphi(v)\geq c\|v\|^p,~\forall v\in X\,.
$$
Concerning the functional $\psi $ we also require that it be
positive homogeneous of degree
$\alpha > 1$.
We then have the following theorem. The theorem is established in
\cite{le:mpn95}, \cite{le:mpn96}. The first part follows from
properties of noncoercive functionals which are coercive on some
subsets,
satisfying suitable properties, and the second part from Liusternik's
theorem on Lagrange multipliers coupled with scaling arguments.
\begin{thm}
\label{critical}
(a)
Assume the above and also that
\begin{eqnarray*}
\varphi (v-u)&\leq& \varphi (v)\\
\psi (v-u)&=& \psi (v),~\forall v\in E,~\forall
u\in\mbox{\rm ker}\,\varphi\cap\mbox{\rm ker}\,\psi .
\end{eqnarray*}
Then the minimization problem
$$
\varphi (u)=\min _{v\in S}\varphi (v)
$$
has a solution $u\in S$.
(b) If $\varphi, ~\psi $ belong to class $C^1$,
$\alpha \ne p$, and if
\begin{description}
\item{(i)} $\psi (u) < 0$, for some $u \in E$
\item{(ii)} $\psi (u) \geq 0, \quad \forall u \in
\mbox{\rm ker}\,\varphi$
\item{(iii)} if $u \in \mbox{\rm ker}\,\varphi$
is such that $\psi (u) = 0$,
then
\begin{eqnarray*}
\varphi (v-u)&\leq&\varphi (v)\\
\psi (v-u)&=& \psi (v),~\forall v\in E \,.
\end{eqnarray*}
\end{description}
Then the functional $f=\varphi +\psi $
has a nontrivial critical point.
\end{thm}
\section{Periodic solutions of Hamiltonian systems} Let us consider the
following system of second order ordinary differential equations
defined by a function
$$
G:(0,T)\times {\mathbb R}^N\to {\mathbb R}^N
$$
\begin{equation}
\label{hsystem}
-u''(t)+\nabla _{u}G(t,u)=0,~02, ~u\in
{\mathbb R}^N \cr
G(t,\gamma)>0,~\forall \gamma \in {\mathbb R}^N,~\gamma \ne 0 \cr
\exists u\in E~\mbox{such that}~\psi (u)<0\,.
}$$
These requirements allow us to apply the critical point
theorem of the previous section and we conclude the existence
of a nontrivial solution $u$ of (\ref{hsystem}), (\ref{pbc}).
In particular one minimizes the functional $\varphi $ on sets
$S$ given by
$$
S=\{ u\in E: \int _0^TG(t,u)dt=c,\}
$$
where $c$ is a negative constant.
\subsection*{Subharmonic solutions}
We note that whenever the function
$G:{\mathbb R} \times {\mathbb R}^N\to {\mathbb R}$
is such that
$$
G(t+T, u)=G(t, u),~t\in {\mathbb R},~u\in {\mathbb R}^N
$$
then along with the problem (\ref{hsystem}), (\ref{pbc})
we may also consider, for any natural number $k\geq 1$,
the equations
\begin{equation}
\label{system}
-u''(t)+\nabla _{u}G(t,u)=0,
\end{equation}
subject to the periodic boundary conditions
\begin{equation}
\label{bc}
u(0)=u(kT),~u'(0)=u'(kT)
\end{equation}
whose solutions will be critical points of the functional
\begin{equation}
\label{fk}
f_k(u)=\frac{1}{2}\int_0^{kT}\left |u'\right |^2dt+\int _0^{kT}G(t,u)dt
\end{equation}
on the space
\begin{equation}
%\begin{array}{rcl}
E_k=W^{1,2}_{kT}\left ((0,kT),{\mathbb R}^N\right ).
\end{equation}
We recall how the existence of a critical point for the functional
$f_k$ was obtained. We write
$$
f_k=\varphi _k +\psi _k,
$$
where
the functionals $\varphi _k$ and $\psi _k$ are defined by
$$
\varphi _k(u)= \frac{1}{2}\int_0^{kT}\left |u'\right |^2dt
$$
and
$$
\psi _k(u)=\int _0^{kT}G(t,u)dt
$$
and have that the set
$$
S_k=\{ u:\psi _k(u)=-1\}\ne \emptyset
$$
with the Fr\'echet derivative $\psi' _k\ne 0$ on $S_k$.
(Note that the manifold $S_k$ could equally well have been chosen
as
$$
S_k=\{ u:\psi _k(u)=-c\}\ne \emptyset,
$$
where
$c$ is any positive number.
Since this manifold is
weakly closed and the functionals $\varphi _k$ and $\psi _k$
satisfy the conditions of Theorem \ref{critical}
we have that
the minimization problem
\begin{equation}
\label{min}
\min _{v\in S_k} \varphi _k(v)=\varphi _k(u_k):=m_k
\end{equation}
has a solution $u_k$,
and hence there exists a Lagrange multiplier $\mu _k $
such that
\begin{equation}
\label{cpoint}
\varphi _k' (u_k)+\mu _k\psi _k'(u_k)=0,
\end{equation}
which by the homogeneity of the functionals implies that
$$
2\varphi_k (u_k)-\alpha \mu _k =0,
$$
i.e.
$$
\mu _k =\frac{2}{\alpha }m_k :=n_k,
$$
which implies that $\mu _k $ is positive and hence we may let
$$
v_k=n _k ^{1/(\alpha -2)}u_k,
$$
and obtain that $v_k$ is a nontrivial critical point of $f_k$.
The question now arises whether, for $k>1$, it is possible that
$v_k$ is a critical point of $f_p$, $p0,
$$ where $\mathop{\rm meas}\{A\}$ denotes the Lebesgue measure of a set $A$.
We hence have the problems
\begin{equation}
\label{g}
-u''(t)+g(t)|u|^{\alpha -2}u=0,
\end{equation}
subject to the same periodic boundary conditions.
Let us choose a nonzero function
$\eta \in C([0,T],{\mathbb R}^N)$ such that
$\eta (0)=0=\eta (T)$ whose support $\mbox{supp}\eta \subset A$
with
$A=\{t:g(t)<0\}$. It then follows that $\eta \in E_k$, for any $k\geq 1$
and
$\psi (\eta )<0$.
Hence
$$
\tilde \eta =\frac{\eta}{\left (-\frac{1}{\alpha}\int _0^{kT}g(t)|\eta|^{\alpha}\right
)^{\frac{1}{\alpha}}}\in S_k
$$
and therefore
$$
m_k\leq \frac{1}{2}\int _0^T|\tilde \eta '|^2 :=c.
$$
We therefore have the lemma.
\begin{lem} There exists a constant $c>0$, independent of $k$, such
that
$$
m_k\leq c,
$$
where $m_k$ is defined by (\ref{min}).
\end{lem}
We recall that
\begin{equation}
\label{1}
\frac{1}{\alpha}\int _0^{kT}g(t)|u_k|^{\alpha}=-1.
\end{equation}
Hence, since $u_k$ is not constant, if the minimal period of $u_k$ is
not
$kT$ it must equal $pT$, where $\theta =\frac{k}{p}$ is a positive
integer.
It therefore follows that
\begin{equation}
\label{2}
\frac{1}{\alpha}\theta \int _0^{pT}g(t)|u_k|^{\alpha}=-1
\end{equation}
and
\begin{equation}
\label{3}
\frac{1}{\alpha}\int _0^{pT}g(t)|\theta ^{\frac{1}{\alpha}}u_k|^{\alpha}=-1,
\end{equation}
i.e. $w_p:=\theta ^{\frac{1}{\alpha}}u_k\in S_p$.
We therefore obtain the following:
\begin{eqnarray}
\label{4}
m_p\leq \varphi _p(w_p)&=& \frac{1}{2}\int _0^{pT}|w_p'|^2 \nonumber\\
&=& \frac{1}{2}\theta ^{2/\alpha}\int _0^{pT}|u_k'|^2\\
&=& \frac{1}{2}\theta ^{2/\alpha}\theta ^{-1}\int _0^{kT}|u_k'|^2\nonumber\\
&=& \theta ^{(2-\alpha)/\alpha} m_k\,,\nonumber
\end{eqnarray}
or
$$
\left (\frac{k}{p}\right )^{\alpha/(\alpha -2)} m_p\leq m_k\leq c\,.
$$
The above formula lets us deduce the following theorem. The proof is a
simple indirect argument using this formula.
\begin{thm} Let $\{k_i\}$ be an unbounded increasing sequence of positive
integers. Then the minimal periods of $\{u_{k_i}\}$ tend to infinity.
In particular for all primes $k$, sufficiently large, $u_k$ has
minimal period $kT$.
\end{thm}
\section{Periodic solutions of elliptic problems}
In this section we briefly point out how the results
on Hamiltonian systems may be extended to elliptic problems. We shall
discuss this in the case of dimension $2$, it will be clear
that similar results may be obtained (with suitable restrictions on
the power $\alpha $) for systems in more independent variables.
\subsection*{Periodic and subharmonic solutions}
Let
$g:{\mathbb R}^2 \to {\mathbb R}$
be such that there exists $X=(X_1,X_2),~X_i>0,~i=1,2$ such that
\begin{equation}
\label{Xper}
g(x+X)=g(x),~x\in {\mathbb R}^2.
\end{equation}
(We say $g$ is periodic with period $X.$) Further assume that $g$ is
essentially bounded and $X_1,X_2$ are the least positive numbers such
that
(\ref{Xper}) holds. We let
$$
{\mathbb P}=\{x=(x_1,x_2): 0\leq x_1\leq X_1,0\leq x_2\leq X_2\}
$$ and assume
$g$ has the property that
$$
\mathop{\rm meas}\{x\in {\mathbb P}:g(x)<0\}\ne 0\,,
$$
and $\int _{{\mathbb P}}g>0$.
We now consider the problem (again $\alpha >2$)
\begin{equation}
\label{perpde}
-\Delta u+g(x)|u|^{\alpha -2}u=0,~x\in {\mathbb R}^2
\end{equation}
subject to the periodic boundary condition
\begin{equation}
\label{per}
u(x+X)=u(x),~x\in {\mathbb R}^2.
\end{equation}
We set
$$
X_{1,1}=X,~X_{k,l}=(kX_1,lX_2),
$$
where $k,l$ are positive integers.
We, of course, then have
$$
g(x+X_{k,l})=g(x),~x\in {\mathbb R}^2
$$
and hence, in analogy with the previous section, we also consider
equation (\ref{perpde}) subject to the (subharmonic)
constraints
\begin{equation}
\label{subper}
u(x+X_{k,l})=u(x),~x\in {\mathbb R}^2.
\end{equation}
The appropriate Sobolev spaces and functionals will be
\begin{equation}
%\begin{array}{space}
E_{k,l}=\{u\in W^{1,2}_{loc}\left ({\mathbb R}^2,{\mathbb R}^N\right ):u(x+X_{k,l})=u(x),~x\in {\mathbb R}^2\},
\end{equation}
$$
f_{k,l}=\varphi _{k,l} +\psi _{k,l},
$$
where
the functionals $\varphi _{k,l}$ and $\psi _{k,l}$ are defined by
$$
\varphi _{k,l}(u)= \frac{1}{2}\int_{{\mathbb P}_{k,l}} |\nabla u|^2\,dx
$$
and
$$
\psi _{k,l}(u)=\int _{{\mathbb P}_{k,l}}g(x)|u|^{\alpha}\,dx
$$
and have that the set
$$
S_{k,l}=\{ u:\psi _{k,l}(u)=-1\}\ne \emptyset,
$$
with
$$
{\mathbb P}_{k,l}=\{x: 0\leq x\leq X_{k,l}\},
$$
employing the usual partial ordering of ${\mathbb R}^2$.
We proceed as in the previous section and find that the functional
$\varphi _{k,l}$ will assume its minimum on the manifold $S_{k,l}$
and then one finds a Lagrange multiplier and after rescaling a critical
point for the functional $f_{k,l}$. Such critical points, on the other
hand will be solutions $u_{k,l}$
of (\ref{perpde}) subject to the constraint (\ref{subper}).
Following the calculations of the previous section one may
now prove the following theorem.
\begin{thm} Let $\{(k,l)\}$ be an unbounded increasing sequence of
tuples
of positive
integers. Then the minimal periods of $\{u_{k,l}\}$ tend to infinity.
In particular for all tuples of primes ${k,l}$ with $k+l$,
sufficiently large,
$u_{k,l}$ has
minimal period $(kX_1,lX_2)$.
\end{thm}
\section{Quasilinear problems}
Results similar to the above may be obtained for the quasilinear problem
(now $\alpha >p>1$)
\begin{equation}
\label{plap}
-\mathop{\rm div}\left (|\nabla u|^{p-2}\nabla u\right )+g(x)|u|^{\alpha
-2}u=0,~x\in {\mathbb R}^n,~n\geq 1,
\end{equation}
subject to the periodic boundary condition
$$
u(x+X)=u(x),~x\in {\mathbb R}^n,~X\in {\mathbb R}^n.
$$
The appropriate Sobolev spaces and functionals will be
$$ \displaylines{
E=\{u\in W^{1,p}_{loc}\left ({\mathbb R}^n,{\mathbb R}^N\right ):
u(x+X)=u(x),~x\in {\mathbb R}^n\}, \cr
f=\varphi +\psi \,,
}$$
where the functionals $\varphi $ are defined by
$$
\varphi (u)= \frac{1}{p}\int_{{\mathbb P}} |\nabla u|^p\,dx\,,
$$
and $\psi $ as above.
\begin{thebibliography}{0}
\bibitem{benci:btr88}
{\sc V.~Benci and D.~Fortunato,}
{\em A Birkhoff-Lewis type result for non autonomous differential
equations.} pp. 85--96, in Partial Differential Equations, Cardoso,
Defigueiredo,
I\'orio, Lopes (editors), Springer Lecture Notes in Mathh., vol. 1324,
Berlin, New York, 1988.
\bibitem{bennaoum:emr94}
{\sc A.~BenNaoum, C.~Troestler, and M.~Willem,}
{\em Existence and multiplicity results for homogeneous second order
differential equations.} J. Differential Equations, {\bf 112}(1994), 239--249.
\bibitem{felmer:sne93}
{\sc P.~Felmer and E.~de B. e Silva,}
{\em Subharmonics near an equilibrium for some second order
Hamiltonian systems.}
Proc. Roy. Soc. Edinburgh, {\bf 123A}(1993), 819--834.
\bibitem{le:mpn95}
{\sc V.~Le and K.~Schmitt,}
{\em Minimization problems for noncoercive functionals
subject to constraints.}
Trans. Amer. Math. Soc., {\bf 347}(1995), 4485--4513.
\bibitem{le:mpn96}
{\sc V.~Le and K.~Schmitt,}
{\em Minimization problems for noncoercive functionals
subject to constraints II.}
Advances Differential Equations, {\bf 1}(1996), 453--498.
\bibitem{le:mnf96}
{\sc V.~Le and K.~Schmitt,}
{\em On minimizing noncoercive functionals on weakly closed sets.}
Topology in Nonlinear Analysis, Banach Center Publications,
{\bf 35}(1996), 51--72.
\bibitem{mawhin:cpt89}
{\sc J.~Mawhin and M.~Willem,}
{\em Critical Point Theory and Hamiltonian Systems.}
Springer Verlag, New York, 1989.
\bibitem{wang:sop96}
{\sc Q.~Wang, Z.~Wang, and J.~Shi.}
{\em Subharmonic oscillations with prescribed minimal period for a
class of Hamiltonian systems.}
Nonl. Anal., TMA, {\bf 28}(1996), 1273--1282.
\end{thebibliography}
\noindent{\sc Klaus Schmitt }\\
Department of Mathematics, University of Utah\\
155 South 1400 East\\
Salt Lake City, UT 84112, USA\\
e-mail: schmitt@math.utah.edu \medskip
\noindent{\sc Zhi-Qiang Wang }\\
Department of Mathematics and Statistics \\
Utah State University \\
Logan, UT 84322, USA \\
e-mail: wang@math.usu.edu
\end{document}