# Sign Changing Critical Points for Locally Lipschitz Functionals ††thanks:
This paper is supported by NSFC 11871250; Qin Baoxia is the corresponding
author.

Xian Xu1, Baoxia Qin2

1Department of Mathematics, Jiangsu Normal University, Xuzhou,

Jiangsu, 221116, P. R. China

2School of Mathematics, Qilu Normal University, Jinan,

Shandong, 250013, P. R. China

Abstract In this paper, some existence results for sign-changing critical
points of locally Lipschitz functionals in real Banach space are obtained by
the method combining the invariant sets of descending flow method with a
quantitative deformation. First we assume the locally Lipschitz functionals to
be outwardly directed on the the boundary of some closed convex sets of the
real Banach space. By using the relation between the critical points on the
Banach space and those of the closed convex sets, we construct a quantitative
deformation lemma, and then we obtain some linking type of critical points
theorems. These theoretical results can be applied to the study of the
existence of sign-changing solutions for differential inclusion problems. In
contrast with the related results in the literatures, the main results of this
paper relax the requirement that the functional being of $C^{1}$ continuous to
locally Lipschitz.
Key words Locally Lipschitz Functionals, Sign-changing Critical Points,
Differential Inclusion Problems.
AMS Subject Classifications 35K57; 35K50; 45K05

##  1 Introduction

The main purpose of this paper is to obtain some existence results for sign-
changing critical points of Locally Lipschitz functionals in Banach spaces.
Chang [1], in ordered to study equations with discontinuities, developed an
extension of the classical smooth critical point theory, to non-smooth locally
Lipschitz functionals. The theory of Chang was based on the sub-differential
of locally Lipschitz functionals due to Clarke [2]. Using this sub-
differential, Chang proposed a generalization of the deformation lemma and
obtained a Mountain Pass Theorem for Locally Lipschitz functionals.
Subsequently, many other types critical points theorems for locally Lipschitz
functionals have been obtained and been applied to the studying of the
existence of solutions for various of elliptic equations boundary value with
discontinuities; see [3-15] and the references therein.

In the past 20 years or so, people have extensively studied the existence of
sign-changing solutions for elliptic boundary value problems; See the
literatures [19, 29, 30]. At the same time, some authors have also established
the theoretical results of the existence of sign-changing critical points for
smooth functionals in abstract space; see [20,25,26,27,32,33]. These
theoretical results can be applied to the study of sign-changing solutions for
elliptic boundary value problems. Let’s first recall some of these results. Li
and Wang [20] presented several variants of Ljusternik-Schnirelman type
theorems in partially ordered Hilbert spaces, which assert the locations of
the critical points constructed by the minimax method in terms of the order
structures. The method to show their main results is combining the invariant
sets of descending flow method with a quantitative deformation. Li and Wang’s
method have been applied to various of differential equations boundary value
problems to obtained the multiplicity of sign-changing solutions by many
authors; see [20, 25-28] and the references therein. T. Bartsch and T. Weth
[33] established an abstract critical theory in partially ordered Hilbert
spaces by virtue of critical groups and give some additional properties for
sign-changing critical points. By using the linking notation introduced by
Schechter and Tintarev and a type of Schauder invariance condition on a
neighborhood of the cone and negative cone in the Hilbert space, Schechter and
Zou [27] proved some critical points theorem of linking type ensuring the
existence for sign-changing critical points for a $C^{1}$ functional on the
Hilbert space. In order to prove that the neighborhoods of the cone and
negative cone are invariant set of descending flow, Schechter and Zou [27]
used the method of Sun and Liu [17] and Sun [16,18]. The abstract result was
used to solve semilinear eigenvalue problems and problems with jumping
nonlinearities resonant or not, and with the Fu$\check{c}i$k spectrum. Zou
[32] established some parameter-depending linking theorems, which allow to
produce a bounded and sign-changing Palais-Smale sequence. For even
functionals, a parameter-depending fountain theorem is also obtained in [32]
which provides infinitely many bounded and sign-changing Palais-Smale
sequences. The abstract results of [32] were applied to the
Schr$\ddot{o}$dinger equation with (or without) critical Sobolev exponents to
obtain the existence of sign-changing solutions. The positive and negative
solutions were also gained as by-products. For further theoretical results on
sign-changing critical points, readers can refer to Zou ’s monograph [25].

Recently, some authors have studied the existence of sign-changing solutions
for elliptic boundary value problems with discontinuous nonlinearities; See
[14,15] and the references therein. For instance, Iannizzotto A., M. Salvatore
A. and Motreanu D. [14] investigated the homogeneous Dirichlet problem for a
partial differential inclusion involving the $p$-Laplace operator and
depending on a parameter $\lambda>0$. The existence of three smooth solutions,
a smallest positive, a biggest negative, and a nodal one, is obtained for any
$\lambda$ sufficiently large by combining variational methods with truncation
techniques.

As far as we know, no one has theoretically studied the existence of the sign-
changing critical points for locally Lipschitz functionals. Now, a natural
question is whether we can generalize the theoretical existence results about
the sign-changing critical point of smooth functionals to locally Lipschitz
functionals. This paper will serve to fulfill this purpose. It is easy to see
that the main difficulty faced by this research is how to establish the
descending flow that ensuring cone and the negative cone being invariant. In
order to overcome this difficulty, we will introduce the conditions of
outwardly directed on convex closed sets and the conditions of the Schauder
invariance for set-valued mapping as in [11]. We will use the relation theorem
between the critical points of the whole space and the critical points of the
closed convex set in [11] when the above conditions hold for closed convex
sets. Also, we need to introduce a new class of Palais-Smale conditions on
closed convex sets. Based on these theorems and concepts, and based on the
method of constructing pseudo-gradient fields on closed convex sets in [6], we
establish a quantitative deformation lemma that asserts the neighbourhood of
the cone and the negative cone are invariant. On this basis, we obtain the
existence result for the sign-changing critical points in the presence of a
linking structure, and apply this abstract result to the study of the sign-
changing solution of differential inclusion problems.

The main results of this paper can be thought as a generalization of some main
results in Li and Wang [20], Zou [25-27] and Qian [28]. Comparing the related
results in Li and Wang [20], Zou [25-27] and Qian [28], the main results of
this paper relax the requirement that the functional being of $C^{1}$ to
locally Lipschitz. Different from the above literature, we use a different
method here. Obviously, the results here can also be applied to study the
existence of sign-changing critical points for smooth functionals.

##  2 Preliminaries

First let us recall some theories concerning the subdifferential theory of
locally Lipschitz functionals due to Clarke [2]. Let $X$ be a real Banach
space and $X^{*}$ its topological dual. A functional
$\varphi:X\rightarrow\mathbb{R}$ is said to be locally Lipschitz, if for
everey $x\in X$, there exists a neighbourhood $U$ of $x$ and a constant $k>0$
depending on $U$ such that $|\varphi(z)-\varphi(y)|\leqslant k\|z-y\|$ for all
$z,y\in U$. For such a functional we define generalized directional derivative
$\varphi^{0}(x;h)$ at $x\in X$ in the direction $h\in X$ by

$\varphi^{0}(x;h)=\lim\limits_{x^{\prime}\rightarrow
x}\sup\limits_{\lambda\downarrow 0^{+}}\frac{\varphi(x^{\prime}+\lambda
h)-\varphi(x^{\prime})}{\lambda}.$

The function $h\mapsto\varphi^{0}(x:h)$ is sublinear, continuous. So by the
Hahn-Banach theorem we know that $\varphi^{0}(x:\cdot)$ is the support
function of a nonempty, convex and $w^{*}$-compact set

$\partial\varphi(x)=\\{x^{*}\in X^{*}:\langle
x^{*},h\rangle\leqslant\varphi^{0}(x;h)\ \mbox{for all}\ h\in X\\}.$

The set $\partial\varphi(x)$ is called the generalized or Clarke
subdifferential of $\varphi$ at $x$.

If $\varphi,\psi:X\mapsto\mathbb{R}$ are locally Lipschitz functionals, then
$\partial(\varphi+\psi)(x)\subset\partial\varphi(x)+\partial\psi(x)$, while
for any $\lambda\in\mathbb{R}$ we have
$\partial(\lambda\varphi)(x)=\lambda\partial\varphi(x)$. Moreover, if
$\varphi:X\mapsto\mathbb{R}$ is also convex, then this sudifferential
coincides with the subdifferential in the sense of convex analysis. If
$\varphi$ is strictly differentiable, then
$\partial\varphi(x)=\\{\varphi^{\prime}(x)\\}$. A point $x\in X$ is a critical
point of $\varphi$ if $0\in\partial\varphi(x)$.

S.T. Kyritsi and N. S. Papageorgiou [6] developed a critical point theory for
non-smooth locally Lipschitz functionals defined on closed, convex set
extending this way the work of Struwe. Let $D$ be a nonempty, convex, closed
subset of $X$, and $\varphi:D\mapsto\mathbb{R}$ be a locally Lipschitz
continuous functional. For $x\in D$ we define

$m_{D}(x)=\inf\limits_{x^{*}}\sup\limits_{y}\big{\\{}\langle
x^{*},x-y\rangle:y\in D,\|x-y\|<1,x^{*}\in\partial\varphi(x)\big{\\}}.$

Evidently, $m_{D}(x)\geqslant 0$ for all $x\in D$. This quantity can be viewed
as a measure of the generalized slope of $\varphi$ at $x\in D$. If $\varphi$
admits an extension $\hat{\varphi}\in C^{1}(X)$, then
$\partial\varphi(x)=\\{\varphi^{\prime}(x)\\}$ and so we have

$m_{D}(x)=\sup\big{\\{}\langle\varphi^{\prime}(x),x-y\rangle:y\in
D,\|x-y\|<1\big{\\}},$

which is the quantity used by Stuwe [31 ,p.147]. Also if $D=X$, then we have

$m_{D}(x)=m(x)=\inf\\{\|x^{*}\|_{*}:x^{*}\in\partial\varphi(x)\\},$

which is the quantity used by Chang [1]. According to [6], a point $x\in D$ is
called a critical point of $\varphi$ on $D$, if $m_{D}(x)=0$. It follows from
[6] that $m_{D}:X\mapsto[0,+\infty)$ is a lsc (lower semi-continuous)
function.

Now we recall some theories concerning convex functionals. Let
$\psi:X\mapsto\mathbb{R}\cup\\{+\infty\\}$ be convex, proper and lsc. The
function $\psi$ turns out to be continuous on $D_{\psi}$, where, as usual

$D_{\psi}:=\\{x\in X:\psi(x)<+\infty\\}.$

If $\partial\psi(x)$ indicates the subdifferential of $\psi$ at the point
$x\in X$, namely,

$\partial\psi(x):=\\{x^{*}\in X^{*}:\psi(z)-\psi(x)\geqslant\langle
x^{*},z-x\rangle,\forall z\in X\\},$

and

$D_{\partial\psi}:=\\{x\in X:\partial\psi(x)\neq 0\\},$

then $D_{\partial\psi}\subset D_{\psi}$. Let
$\delta_{D}:X\rightarrow\mathbb{R}\cup\\{+\infty\\}$ be the indicator function
of $D$, namely

$\delta_{D}(x):=\left\\{\begin{array}[]{ll}0,&\mbox{if}\ x\in D,\\\
+\infty,&\mbox{otherwise}.\end{array}\right.$

Then we have

$\partial\delta_{D}(x)=\big{\\{}x^{*}\in X^{*}:\langle
x^{*},z-x\rangle\leqslant 0,\forall z\in D\big{\\}}.$

The set $\partial\delta_{D}(x)$ is usually called normal cone to $D$ at $x$.
If for $x\in\partial D$,

$(-\partial\delta_{D}(x))\cap\partial\varphi(x)\subset\\{0\\},$

then we say that $\partial\varphi$ turns out to be outwardly directed at
$x\in\partial D$. This clearly rewrites as

$\forall z^{*}\in\partial\varphi(x)\backslash\\{0\\}\ \mbox{there exists}\
z\in D\ \mbox{fulfilling}\ \langle z^{*},z-x\rangle<0.$ $None$

It follows from [11, Lemma 4.1 and 4.2] we have the following Lemma 2.1 and
2.2. Lemma 2.1. $m_{D}(x)=0$ if and only if
$0\in\partial\varphi(x)+\partial\delta_{D}(x)$, namely
$\partial\varphi(x)\cap(-\partial\delta_{D}(x))\neq\emptyset$.

Lemma 2.2. Let $D$ be a closed, convex set of $X$ and $\partial\varphi$ be
outwardly directed at all points of $\partial D$. If $x_{0}\in D$ and
$m_{D}(x_{0})=0$ then $x_{0}$ is a critical point of $\varphi$.

Now let us introduce the Schauder type condition for set value mappings in a
manner as [11]. Let $X$ be reflexive. As usual, we will identify $X^{**}$ with
$X$ while $\mathscr{F}:X^{*}\mapsto 2^{X}$ will denote the duality map, given
by

$\mathscr{F}(x^{*}):=\\{x\in X:\langle
x^{*},x\rangle=\|x^{*}\|^{2}_{*}=\|x\|^{2}\\},\forall x^{*}\in X^{*}.$

The set $\mathscr{F}(x^{*})$ turns out to be nonempty, convex, and closed;
see,e.g. [5, pp. 311-319]. Define

$\nabla\varphi(x):=\mathscr{F}(\partial\varphi(x)),\ \ x\in X.$ $None$

Clearly, $\nabla\varphi(x)$ depends on the choice of the duality pairing
between $X$ and $X^{*}$ whenever it is compatible with the topology of $X$. If
$X$ is a Hilbert space, the duality paring becomes the scalar product and
(2.2) gives the usual gradient. Write $I$ for the identity operator on $X$. It
follows from [11, Theorem 4.5] we have the following Lemma 2.3.

Lemma 2.3 (Schauder invariance condition). Suppose $X$ is reflexive, $D$ is a
convex and closed set of $X$, $x_{0}\in\partial D$ and,

$\big{(}I-\nabla\varphi\big{)}(x_{0})\subset D.$ $None$

Then $\partial\varphi$ is outwardly directed at $x_{0}$.

Remark 2.1. The well known Schauder invariance condition for a
$C^{1}$-functional $\varphi$ on a Hilbert space $X$ reds as
$(I-\varphi^{\prime})(C)\subset C$; see [16-18,21,22]. It has been extend to
Banach spaces in [22]. The notation of Schauder invariance condition was
firstly put forward by Sun Jingxian in [17]. As pointed out in [11], the
geometric condition that $\partial\varphi$ is outwardly directed at
$x_{0}\in\partial D$ is more general than (2.3).

Corollary 2.1. Suppose $X$ is reflexive, $D$ is a convex and closed set of
$X$, $\partial D=\Gamma_{1}\cup\Gamma_{2}$. Moreover,

$\big{(}I-\nabla\varphi\big{)}(x)\subset D,\ \ \ \ \forall x\in\Gamma_{1},$

and

$\partial\varphi(x)\cap\big{(}-\partial\delta_{D}(x)\big{)}\subset\\{0\\},\ \
\ \ \forall x\in\Gamma_{2}.$

Then $m_{D}(x_{0})=0$ if only if $m(x_{0})=0$ for any $x_{0}\in D$.

Lemma 2.4([34], Von Neumann). Let $X,Y$ be two Hausdorff topological linear
spaces, $C\subset X$, $D\subset Y$ be two convex and compact sets. Let
$\psi:X\times Y\mapsto\mathbb{R}$ satisfy: 1) $x\mapsto\psi(x,y)$ is upper
semi-continuous (usc.) and concave; 2) $y\mapsto\psi(x,y)$ is lower semi-
continuous (lsc.) and convex. Then $\psi$ has at least one saddle point
$(\bar{x},\bar{y})\in C\times D$, that is

$\psi(x,\bar{y})\leqslant\psi(\bar{x},\bar{y})\leqslant\psi(\bar{x},y)\
\mbox{for}\ (x,y)\in C\times D.$

Definition 2.1. Let $D$ be a nonempty closed subset of $X$, $D_{1}\subset D$,
$r\in\mathbb{R}$. We say that $\varphi$ satisfies the non-smooth $C$-condition
on $D_{1}$ with respect to $D$ at level $r$, denoted by $(PS)_{D,D_{1}}^{r}$,
if every sequence $\\{x_{n}\\}\subset D_{1}$ such that
$\varphi(x_{n})\rightarrow r$ and $(1+\|x_{n}\|)m_{D}(x_{n})\rightarrow 0$ as
$n\rightarrow\infty$, has a convergent subsequence. If $\varphi$ satisfies the
non-smooth $C$-condition on $D_{1}$ with respect to $D$ at every level
$r\in\mathbb{R}$, we say $\varphi$ satisfies the non-smooth $C$-condition on
$D_{1}$ with respect to $D$, denoted by $(PS)_{D,D_{1}}$. We say $\varphi$
satisfies the non-smooth $C$-condition on $D_{1}$ if $D=X$, denoted by
$(PS)_{D_{1}}$.

##  3 Main Results

Let $X$ be a real reflexive Banach space and $J:X\mapsto\mathbb{R}$ be locally
Lipschitz. Let $P$ be a cone of $X$, that is, $P$ is closed convex set in $X$,
$\lambda x\in P$ for all $x\in P$ and $\lambda\geqslant 0$, and
$P\cap(-P)=\\{0\\}$. For each $D\subset X$, denoted by $\mbox{int}D$ the
interior of $D$. Let $x_{0}$ be a critical point of $J$. Then, we call $x_{0}$
a positive, or negative, or sign-changing critical point of $J$, if $x_{0}\in
P$, or $x_{0}\in(-P)$, or $x_{0}\in X\backslash\big{(}P\cup(-P)\big{)}$,
respectively. For each $\mu>0$, let

$D^{\pm}(\mu)=\\{x\in X:\mbox{dist}\ (x,\pm P)\leqslant\mu\\}.$

Obviously, $D^{\pm}(\mu)$ are closed convex set in $X$, whose interiors are
non-empty. Let us denote by $\pi_{P}:X\mapsto P$ the orthogonal projection on
the cone $P$ so that $T_{P}(u)=u-\pi(u)$ realizes the distance of $u$ from
$P$, that is $\mbox{dist}(u,P)=\|T_{P}(u)\|$. Since $X$ is reflexive, the the
distance of $u$ from $P$ is achievable.

We will employ the following conditions in this paper.

(H${}_{1})$ $(I-\nabla J)\big{(}\partial D^{\pm}(\mu_{0})\big{)}\subset
D^{\pm}(\frac{1}{2}\mu_{0})$ for some $1>\mu_{0}>0$;

(H${}_{2})$ $J$ satisfies the conditions $(PS)_{X}$,
$(PS)_{D^{+}(\mu_{0}),D^{+}(\frac{3}{4}\mu_{0})}$,
$(PS)_{D^{-}(\mu_{0}),D^{-}(\frac{3}{4}\mu_{0})}$, and
$(PS)_{D^{+}(\mu_{0})\cap D^{-}(\mu_{0})}$.

Remark 3.1. Obviously, the Schauder invariance condition holds for
$D^{\pm}(\mu_{0})$ if the condition (H${}_{1})$ holds. The condition
(H${}_{1})$ is a variant of the so-called $K$-invariant condition for the
functional being locally Lipschitz. The $K$-invariant condition for the the
functional beng of $C^{1}$ class was put forward by Monica Conti, et.al [24].
Some authors have employed this condition to show the multiplicity results for
solutions of various elliptic differential equations in the presence of lower
and upper solutions, and some authors have employed this condition to show the
existence of sign-changing solutions; see [25-27].

Lemma 3.1. Suppose that (H${}_{1})$ holds. Then the following conclusions
hold:

(1) If $m_{D^{+}(\mu_{0})}(x_{0})=0$ for some $x_{0}\in
D^{+}(\mu_{0})\backslash D^{-}(\mu_{0})$, then $m(x_{0})=0$;

(2) If $m_{D^{-}(\mu_{0})}(x_{0})=0$ for some $x_{0}\in
D^{-}(\mu_{0})\backslash D^{+}(\mu_{0})$, then $m(x_{0})=0$;

(3) If $m_{D^{+}(\mu_{0})\cap D^{-}(\mu_{0})}(x_{0})=0$ for some $x_{0}\in
D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$, then $m(x_{0})=0$.

Proof. The the conclusions (1) and (2) can be easily proved by Lemma 2.2. We
need only to show the conclusion (3).

Now we show that $\partial J$ is outwardly directed on
$\partial(D^{+}(\mu_{0})\cap D^{-}(\mu_{0}))$, that is

$\partial J(x)\cap\big{(}-\delta_{D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})}(x)\big{)}\subset\\{0\\},\ \ \forall
x\in\partial\big{(}D^{+}(\mu_{0})\cap D^{-}(\mu_{0})\big{)}.$ $None$

Obviously, we have

$\partial\big{(}D^{+}(\mu_{0})\cap D^{-}(\mu_{0})\big{)}=\big{(}\partial
D^{+}(\mu_{0})\cap D^{-}(\mu_{0})\big{)}\cup\big{(}\partial D^{-}(\mu_{0})\cap
D^{+}(\mu_{0})\big{)}.$

Let $x_{0}\in\partial\big{(}D^{+}(\mu_{0})\cap D^{-}(\mu_{0})\big{)}$. Assume
that $x_{0}\in\partial D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$. Then we have
following two cases:

(a) $x_{0}\in\partial D^{+}(\mu_{0})\cap\mbox{int}(D^{-}(\mu_{0}))$. We claim
that

$\partial\delta_{D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})}(x_{0})\subset\partial\delta_{D^{+}(\mu_{0})}(x_{0}).$

In fact, for any $x^{*}\in\partial\delta_{D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})}(x_{0})$, by the definition we have

$\langle x^{*},z-x_{0}\rangle\leqslant 0,\ \ \forall z\in D^{+}(\mu_{0})\cap
D^{-}(\mu_{0}).$

Since $x_{0}\in\partial D^{+}(\mu_{0})\cap\mbox{int}(D^{-}(\mu_{0}))$, there
exists $r_{0}>0$ such that

$B(x_{0},r_{0})\cap D^{+}(\mu_{0})\subset D^{+}(\mu_{0})\cap D^{-}(\mu_{0}).$

For each $z\in D^{+}(\mu_{0})\backslash B(x_{0},r_{0})$, we have

$z_{1}:=x_{0}+r_{0}\frac{z-x_{0}}{\|z-x_{0}\|}=\Big{(}1-\frac{r_{0}}{\|z-x_{0}\|}\Big{)}x_{0}+\frac{r_{0}}{\|z-x_{0}\|}z\in
D^{+}(\mu_{0}),$

and $\|z_{1}-x_{0}\|=r_{0}$. Thus,

$z_{1}\in B(x_{0},r_{0})\cap D^{+}(\mu_{0})\subset D^{+}(\mu_{0})\cap
D^{-}(\mu_{0}).$

So, we have

$0\geqslant\langle x^{*},z_{1}-x_{0}\rangle=\frac{r_{0}}{\|z-x_{0}\|}\langle
x^{*},z-x_{0}\rangle.$

Consequently, we have $\langle x^{*},z-x_{0}\rangle\leqslant 0$ for all $z\in
D^{+}(\mu_{0})$. This implies that
$x^{*}\in\partial\delta_{D^{+}(\mu_{0})}(x_{0})$. Hence,

$\partial\delta_{D^{+}(\mu_{0})}(x_{0})\supset\partial\delta_{D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})}(x_{0}).$ $None$

By (H${}_{1})$, for any $z^{*}\in\partial J(x_{0})\backslash\\{0\\}$ and
$y\in\mathscr{F}(z^{*})$, there exists $z\in D^{+}(\mu_{0})$ such that
$x_{0}-y=z$. So,

$\langle z^{*},z-x_{0}\rangle=-\langle z^{*},y\rangle=-\|z^{*}\|^{2}_{*}<0.$

Hence,

$\partial J(x_{0})\cap\partial\delta_{D^{+}(\mu_{0})}(x_{0})\subset\\{0\\}.$
$None$

It follows from (3.2) and (3.3) that (3.1) holds.

(b) $x_{0}\in\partial D^{+}(\mu_{0})\cap\partial D^{-}(\mu_{0})$. By
(H${}_{1})$, we have

$x_{0}-\nabla J(x_{0})\subset D^{+}(\mu_{0})\cap D^{-}(\mu_{0}).$

So, for all $z^{*}\in\partial J(x_{0})\backslash\\{0\\}$ and
$y\in\mathscr{F}(z^{*})$, there exists $z\in D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})$ such that $x_{0}-y=z$. Then we have

$\langle z^{*},z-x_{0}\rangle=-\langle z^{*},y\rangle=-\|z^{*}\|^{2}_{*}<0.$

This implies (3.1) holds.

It follows from Lemma 2.2 that $m(x_{0})=0$ if $m_{D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})}(x_{0})=0$. The proof is complete.

Let $r,r_{1},r_{2}\in\mathbb{R}$ with $r_{1}\leqslant r_{2}$. In what follows
we use the following notations:

$\mathbb{K}=\\{x\in X:m(x)=0\\},\mathbb{K}_{r}=\\{x\in X:m(x)=0\ \mbox{and}\
J(x)=r\\},$ $(\mathbb{K}_{r})_{\delta}=\\{x\in X:\mbox{dist}\
(x,\mathbb{K}_{r})<\delta\\},J^{r}=\\{x\in X:J(x)\leqslant r\\},$
$J_{r}=\\{x\in X:J(x)\geqslant r\\},J_{r_{1}}^{r_{2}}=\\{x\in X:r_{1}\leqslant
J(x)\leqslant r_{2}\\}.$

Lemma 3.2. Suppose that (H${}_{1})$ and (H${}_{2})$ hold. Then, for any
$\delta>0$, there exist $b,\bar{\varepsilon}>0$ such that

$(1+\|x\|)m(x)\geqslant b,\forall x\in
B(r,\bar{\varepsilon},\delta)\backslash(D^{+}(\mu_{0})\cup D^{-}(\mu_{0})),$
$None$ $(1+\|x\|)m_{D^{+}(\mu_{0})}(x)\geqslant b,\forall x\in
B(r,\bar{\varepsilon},\delta)\cap(D^{+}(\mu_{0})\backslash D^{-}(\mu_{0})),$
$None$ $(1+\|x\|)m_{D^{-}(\mu_{0})}(x)\geqslant b,\forall x\in
B(r,\bar{\varepsilon},\delta)\cap(D^{-}(\mu_{0})\backslash D^{+}(\mu_{0})),$
$None$ $(1+\|x\|)m_{D^{+}(\mu_{0})\cap D^{-}(\mu_{0})}(x)\geqslant b,\forall
x\in B(r,\bar{\varepsilon},\delta)\cap D^{+}(\mu_{0})\cap D^{-}(\mu_{0}),$
$None$

where
$B(r,\bar{\varepsilon},\delta)=J^{r+\bar{\varepsilon}}_{r-\bar{\varepsilon}}\backslash(\mathbb{K}_{r})_{\delta}$.

Proof. First we prove that there exist $\bar{\varepsilon}_{1},b_{1}>0$ such
that

$(1+\|x\|)m(x)\geqslant b_{1},\forall x\in
B(r,\bar{\varepsilon}_{1},\delta)\backslash\big{(}D^{+}(\frac{3}{4}\mu_{0})\cup
D^{-}(\frac{3}{4}\mu_{0})\big{)}.$ $None$

Arguing by make contradiction that this is not the case. Then there exist
positive numbers sequences $\\{\bar{b}_{n}\\}$, $\\{\bar{\varepsilon}_{n}\\}$
and $\\{x_{n}\\}\subset
B(r,\varepsilon_{n},\delta)\backslash(D^{+}(\frac{3}{4}\mu_{0})\cup
D^{-}(\frac{3}{4}\mu_{0}))$ such that $\bar{b}_{n}\rightarrow 0$,
$\bar{\varepsilon}_{n}\rightarrow 0$ as $n\rightarrow\infty$, and

$(1+\|x_{n}\|)m(x_{n})<\bar{b}_{n}.$

By using the condition (H${}_{2})$, there is a subsequence of $\\{x_{n}\\}$
and $x_{0}\in\overline{X\backslash(D^{+}(\frac{3}{4}\mu_{0})\cup
D^{-}(\frac{3}{4}\mu_{0})\cup(\mathbb{K}_{r})_{\delta})}$, up to a
subsequence, we assume that $x_{n}\rightarrow x_{0}$ as $n\rightarrow\infty$.
Obviously, we have $J(x_{0})=r$ and $m(x_{0})=0$, that is
$x_{0}\in\mathbb{K}_{r}\subset(\mathbb{K}_{r})_{\delta}$. Since
$\\{x_{n}\\}\subset
B(r,\varepsilon_{n},\delta)\backslash(D^{+}(\frac{3}{4}\mu_{0})\cup
D^{-}(\frac{3}{4}\mu_{0}))$ for $n=1,2,\cdots$, we have
$x_{0}\not\in(\mathbb{K}_{r})_{\delta}$, which is a contradiction. Hence,
(3.8) holds.

It follows from (H${}_{1})$ that

$\mathbb{K}_{r}\cap\big{(}D^{+}(\mu_{0})\backslash(D^{-}(\mu_{0})\cup
D^{+}(\frac{3}{4}\mu_{0}))\big{)}=\emptyset.$

Thus, by (3.8) we have

$(1+\|x\|)m(x)\geqslant b_{1},\ \forall x\in
B(r,\bar{\varepsilon}_{1},\delta)\cap\big{(}D^{+}(\mu_{0})\backslash(D^{-}(\mu_{0})\cup
D^{+}(\frac{3}{4}\mu_{0}))\big{)}.$

For each $x_{0}\in
B(r,\bar{\varepsilon}_{1},\delta)\cap\big{(}D^{+}(\mu_{0})\backslash(D^{-}(\mu_{0})\cup
D^{+}(\frac{3}{4}\mu_{0}))\big{)}$ and $x^{*}\in\partial J(x_{0})$, by
(H${}_{1})$ we may assume that $\mathscr{F}x^{*}=x_{0}-w$ for some $w\in
D^{+}(\frac{1}{2}\mu_{0})$. Let
$\bar{x}=x_{0}+\frac{\mu_{0}}{8\|w-x_{0}\|}(w-x_{0})$. Then we have
$\|w-x_{0}\|\geqslant\frac{1}{4}\mu_{0}$. In fact, if
$\|w-x_{0}\|<\frac{1}{4}\mu_{0}$, then we have

$\begin{array}[]{ll}\frac{3\mu_{0}}{4}&\leqslant\mbox{dist}\
(x_{0},P)=\|T_{P}x_{0}\|\leqslant\|x_{0}-\pi_{P}(w)\|\\\
&\leqslant\|x_{0}-w\|+\|w-\pi_{P}(w)\|=\|x_{0}-w\|+\|T_{P}(w)\|\\\
&<\frac{\mu_{0}}{4}+\frac{\mu_{0}}{2}<\frac{3\mu_{0}}{4},\end{array}$

which is a contradiction. This implies that
$\|w-x_{0}\|\geqslant\frac{1}{4}\mu_{0}$. Consequently, we have

$\bar{x}=\Big{(}1-\frac{\mu_{0}}{8\|w-x_{0}\|}\Big{)}x_{0}+\frac{\mu_{0}}{8\|w-x_{0}\|}w\in
D^{+}(\mu_{0})$

and $\|x_{0}-\bar{x}\|=\frac{1}{8}\mu_{0}<1$. So,

$\sup\limits_{v\in D^{+}(\mu_{0}),\|x_{0}-v\|<1}\langle
x^{*},x_{0}-v\rangle\geqslant\langle
x^{*},x_{0}-\bar{x}\rangle=\frac{\mu_{0}}{8\|w-x_{0}\|}\langle
x^{*},x_{0}-w\rangle=\frac{1}{8}\mu_{0}\|x^{*}\|_{*},$

and

$m_{D^{+}(\mu_{0})}(x_{0})\geqslant\frac{1}{8}\mu_{0}\inf\limits_{x^{*}\in\partial
J(x_{0})}\|x^{*}\|_{*}=\frac{1}{8}\mu_{0}m(x_{0}).$

Hence,

$(1+\|x\|)m_{D^{+}(\mu_{0})}(x)\geqslant\frac{1}{8}\mu_{0}b_{1},\ x\in
B(r,\bar{\varepsilon}^{+},\delta)\cap\big{(}D^{+}(\mu_{0})\backslash(D^{-}(\mu_{0})\cup
D^{+}(\frac{3}{4}\mu_{0}))\big{)}.$

Now we show that there exist $\widetilde{\varepsilon}_{2},\widetilde{b}_{2}>0$
such that

$(1+\|x\|)m_{D^{+}(\mu_{0})}(x)\geqslant\widetilde{b}_{2},\forall x\in
B(r,\widetilde{\varepsilon}_{2},\delta)\cap(D^{+}(\frac{3}{4}\mu_{0})\backslash
D^{-}(\mu_{0})).$

Arguing by make contradiction that this is not the case. Then there exist
positive numbers sequences $\\{\bar{b}_{n}\\}$, $\\{\bar{\varepsilon}_{n}\\}$
and sequence $\\{x_{n}\\}\subset
B(r,\bar{\varepsilon}_{n},\delta)\cap(D^{+}(\frac{3}{4}\mu_{0})\backslash
D^{-}(\mu_{0}))$ such that $\bar{b}_{n}\rightarrow 0$,
$\bar{\varepsilon}_{n}\rightarrow 0$ as $n\rightarrow\infty$, and

$(1+\|x_{n}\|)m_{D^{+}(\mu_{0})}(x_{n})<\bar{b}_{n}.$

Since $J$ satisfies the condition
$(PS)_{D^{+}(\mu_{0}),D^{+}(\frac{3}{4}\mu_{0})}$, then, up to a subsequence
if necessary, we assume that $x_{n}\rightarrow x_{0}$ as $n\rightarrow\infty$,
where $m_{D^{+}(\mu_{0})}(x_{0})=0$ and $J(x_{0})=r$. It follows from Lemma
3.1 that $m(x_{0})=m_{D^{+}(\mu_{0})}(x_{0})=0$, so $x_{0}\in\mathbb{K}_{r}$.
Since $\\{x_{n}\\}\subset
B(r,\widetilde{\varepsilon}_{n},\delta)\cap(D^{+}(\frac{3}{4}\mu_{0})\backslash
D^{-}(\mu_{0}))$ for each $n=1,2,\cdots$, we have
$x_{0}\not\in(\mathbb{K}_{r})_{\delta}$, which is a contradiction. Let
$b_{2}=\min\\{\widetilde{b}_{2},\frac{1}{8}\mu_{0}b_{1}\\}$ and
$\bar{\varepsilon}_{2}=\min\\{\bar{\varepsilon}_{1},\widetilde{\varepsilon}_{2}\\}$.
Then we have

$(1+\|x\|)m_{D^{+}(\mu_{0})}(x)\geqslant b_{2},\forall x\in
B(r,\bar{\varepsilon}_{2},\delta)\cap\big{(}D^{+}(\frac{3}{4}\mu_{0})\backslash
D^{-}(\mu_{0})\big{)},$ $None$

In the same way as above we can prove that, there exist
$b_{3},b_{4},\bar{\varepsilon}_{3},\bar{\varepsilon}_{4}>0$ such that

$(1+\|x\|)m_{D^{-}(\mu_{0})}(x)\geqslant b_{3},\forall x\in
B(r,\bar{\varepsilon}_{3},\delta)\cap(D^{-}(\mu_{0})\backslash
D^{+}(\mu_{0})),$ $None$

and

$(1+\|x\|)m_{D^{+}(\mu_{0})\cap D^{-}(\mu_{0})}(x)\geqslant b_{4},\forall x\in
B(r,\bar{\varepsilon}_{4},\delta)\cap D^{+}(\mu_{0})\cap D^{-}(\mu_{0}).$
$None$

Now let $b=\min\\{b_{1},b_{2},b_{3},b_{4}\\}$ and
$\bar{\varepsilon}=\min\\{\bar{\varepsilon}_{1},\bar{\varepsilon}_{2},\bar{\varepsilon}_{3},\bar{\varepsilon}_{4}\\}$.
It follows from (3.8)$\sim$(3.11) that (3.4)$\sim$(3.7) hold. The proof is
complete.

In what follows of this section, for brevity, let
$S=B(r,\bar{\varepsilon},\delta)\backslash(D^{+}(\mu_{0})\cup
D^{-}(\mu_{0}))$.

Lemma 3.3. Suppose that (H${}_{1})$ and (H${}_{2})$ hold. Then there exists
$v:B(r,\bar{\varepsilon},\delta)\rightarrow X$ such that $v$ is locally
Lipschitz continuous, $\|v(x)\|\leqslant 2(1+\|x\|)$ for all $x\in
B(r,\bar{\varepsilon},\delta)$ and

$x-\frac{1}{1+\|x\|}v(x)\in D^{+}(\mu_{0}),\forall x\in
B(r,\bar{\varepsilon},\delta)\cap D^{+}(\mu_{0}),$ $x-\frac{1}{1+\|x\|}v(x)\in
D^{-}(\mu_{0}),\forall x\in B(r,\bar{\varepsilon},\delta)\cap D^{-}(\mu_{0}),$
$\langle x^{*},v(x)\rangle\geqslant\frac{b}{8},\ \forall x\in
B(r,\bar{\varepsilon},\delta),x^{*}\in\partial J(x).$

Moreover, $v$ is odd if $J$ is even.

Proof. For each $x_{0}\in S$, take $w_{0}^{*}\in\partial J(x_{0})$ such that
$m(x_{0})=\|w_{0}^{*}\|_{*}>0$. Then we have
$B(0,\|w_{0}^{*}\|_{*})\cap\partial J(x_{0})=\emptyset$, where
$B(0,\|w_{0}^{*}\|)=\\{z^{*}\in X^{*}:\|z^{*}\|_{*}<\|w_{0}^{*}\|_{*}\\}$. So,
by Hahn-Banach separation theorem, we can find $u_{1}(x_{0})\in X$ with
$\|u_{1}(x_{0})\|=1$ such that for all $z^{*}\in B(0,\|w_{0}^{*}\|_{*})$ and
$y^{*}\in\partial J(x_{0})$,

$\langle z^{*},u_{1}(x_{0})\rangle\leqslant\langle
w_{0}^{*},u_{1}(x_{0})\rangle\leqslant\langle y^{*},u_{1}(x_{0})\rangle.$

Recall that

$\|w_{0}^{*}\|_{*}=\sup\big{\\{}\langle z^{*},u_{1}(x_{0})\rangle:z^{*}\in
B(0,\|w_{0}^{*}\|_{*})\big{\\}},$

then we have by Lemma 3.2,

$\langle
y^{*},u_{1}(x_{0})\rangle\geqslant\|w_{0}^{*}\|_{*}>\frac{b}{2(1+\|x_{0}\|)}.$

Since the map $x\mapsto\partial J(x)$ is usc from $X$ into $X^{*}_{w}$, we may
take an open neighborhood $B_{1}(x_{0},r_{1}(x_{0}))$ of $x_{0}$, such that

$\langle y^{*},u_{1}(x_{0})\rangle>\frac{b}{4(1+\|y\|)},\ \ \forall
y^{*}\in\partial J(y),y\in U_{1}(x_{0}),$ $None$

where $U_{1}(x_{0})=B_{1}(x_{0},r_{1}(x_{0}))\cap
B(r,\bar{\varepsilon},\delta)$. Since $x_{0}\in S\subset
B(r,\bar{\varepsilon},\delta)\backslash(D^{+}(\mu_{0})\cup D^{-}(\mu_{0}))$,
we may take $r_{1}(x_{0})>0$ small enough such that $U_{1}(x_{0})\subset S$.

Pick $x_{0}\in
B(r,\bar{\varepsilon},\delta)\cap\big{(}D^{+}(\mu_{0})\backslash
D^{-}(\mu_{0})\big{)}$, we have by Lemma 3.2,

$m_{D^{+}(\mu_{0})}(x_{0})>\frac{b}{1+\|x_{0}\|}.$

Let $C=(\\{x_{0}\\}-D^{+}(\mu_{0}))\cap\bar{B}(0,1)$ and
$D=\partial\varphi(x_{0})$, where $\bar{B}(0,1)=\\{x\in X:\|x\|\leqslant
1\\}$. Let $X_{w}$ and $X^{*}_{w}$ be the spaces $X$ and $X^{*}$ furnished
their weak topology respectively. Since $X$ is reflexive, $D$ is compact in
$X^{*}_{w}$ and $C$ is compact in $X_{w}$. Obviously, both $C$ and $D$ are
convex. Let $\psi:C\times D\mapsto\mathbb{R}$ be defined by
$\psi(x,y^{*})=\langle y^{*},x\rangle$ for any $(x,y^{*})\in C\times D$. It
follows from Lemma 2.4 that $\psi$ has at least one saddle point
$(u_{+}(x_{0}),x_{0}^{*})\in C\times D$, that is

$\min\limits_{x^{*}\in D}\max\limits_{x\in C}\langle x^{*},x\rangle=\langle
x_{0}^{*},u_{+}(x_{0})\rangle=\max\limits_{x\in C}\min\limits_{x^{*}\in
D}\langle x^{*},x\rangle,\ \forall
x^{*}\in\partial\varphi(x_{0}),x\in(\\{x_{0}\\}-D^{+}(\mu_{0}))\cap\bar{B}(0,1).$

Thus, we have

$\langle x_{0}^{*},x\rangle\leqslant\langle
x_{0}^{*},u_{+}(x_{0})\rangle\leqslant\langle x^{*},u_{+}(x_{0})\rangle,\
\forall
x^{*}\in\partial\varphi(x_{0}),x\in(\\{x_{0}\\}-D^{+}(\mu_{0}))\cap\bar{B}(0,1).$

Then, for any $x^{*}\in\partial\varphi(x_{0})$, we have

$\langle x^{*},u_{+}(x_{0})\rangle\geqslant\langle
x_{0}^{*},u_{+}(x_{0})\rangle=m_{D^{+}(\mu_{0})}(x_{0})>\frac{b}{2(1+\|x_{0}\|)}.$

Again by using the fact that $x\mapsto\partial J(x)$ is usc, we know that
there exists an open neighborhood $B_{+}(x_{0},r_{+}(x_{0}))$ of $x_{0}$ such
that for any $y\in U_{+}(x_{0})$, $y^{*}\in\partial J(y)$,

$\langle y^{*},u_{+}(x_{0})\rangle>\frac{b}{4(1+\|y\|)}.$ $None$

where $U_{+}(x_{0})=B_{+}(x_{0},r_{+}(x_{0}))\cap
B(r,\bar{\varepsilon},\delta)$. Since $B(r,\bar{\varepsilon},\delta)\backslash
D^{-}(\mu_{0})$ is open in $B(r,\bar{\varepsilon},\delta)$ and $x_{0}\in
B(r,\bar{\varepsilon},\delta)\backslash D^{-}(\mu_{0})$, we may assume that
$U_{+}(x_{0})\subset B(r,\bar{\varepsilon},\delta)\backslash D^{-}(\mu_{0})$.

By using the same method as above, for each $x_{0}\in D^{-}(\mu_{0})\backslash
D^{+}(\mu_{0})$, there exist an open neighborhood of $x_{0}$ in
$B(r,\bar{\varepsilon},\delta)$, denoted by $U_{-}(x_{0})$, and $u_{-}(x_{0})$
such that $\|u_{-}(x_{0})\|\leqslant 1$, $x_{0}-u_{-}(x_{0})\in
D^{-}(\mu_{0})$, $U_{-}(x_{0})\subset B(r,\bar{\varepsilon},\delta)\backslash
D^{+}(\mu_{0})$ and

$\langle y^{*},u_{-}(x_{0})\rangle>\frac{b}{4(1+\|y\|)}$ $None$

for any $y\in U_{-}(x_{0})$, $y^{*}\in\partial J(y)$.

Also, for each $x_{0}\in D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$, there exist
$u_{2}(x_{0})\in X$ and an open neighborhood $U_{2}(x_{0})$ of $x_{0}$ in
$B(r,\bar{\varepsilon},\delta)$ with $\|u_{2}(x_{0})\|\leqslant 1$, such that
$x_{0}-u_{2}(x_{0})\in D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$, and

$\langle y^{*},u_{-}(x_{0})\rangle>\frac{b}{4(1+\|y\|)}$ $None$

for any $y\in U_{2}(x_{0})$, $y^{*}\in\partial J(y)$.

Set

$\mathscr{A}_{1}=\\{U_{1}(x_{0}):x_{0}\in
B(r,\bar{\varepsilon},\delta)\backslash(D^{+}(\mu_{0})\cup
D^{-}(\mu_{0}))\\},$ $\mathscr{A}_{2}=\\{U_{+}(x_{0}):x_{0}\in
B(r,\bar{\varepsilon},\delta)\cap(D^{+}(\mu_{0})\backslash
D^{-}(\mu_{0}))\\},$ $\mathscr{A}_{3}=\\{U_{-}(x_{0}):x_{0}\in
B(r,\bar{\varepsilon},\delta)\cap(D^{-}(\mu_{0})\backslash
D^{+}(\mu_{0}))\\},$ $\mathscr{A}_{4}=\\{U_{2}(x_{0}):x_{0}\in
B(r,\bar{\varepsilon},\delta)\cap D^{-}(\mu_{0})\cap D^{+}(\mu_{0})\\},$

and
$\mathscr{A}=\mathscr{A}_{1}\cup\mathscr{A}_{2}\cup\mathscr{A}_{3}\cup\mathscr{A}_{4}$.
Obviously, $\mathscr{A}$ is an open cover of $B(r,\bar{\varepsilon},\delta)$.
By paracompactness we can find a locally finite refinement
$\mathscr{D}=\\{V_{\alpha}:\alpha\in\Lambda\\}$ and a locally Lipschitz
partition of unit $\\{\gamma_{\alpha}:\alpha\in\Lambda\\}$ sub-ordinate to it.
For each $\alpha\in\Lambda$ we can find $x_{\alpha}\in
B(r,\bar{\varepsilon},\delta)$ such that $V_{\alpha}\subset
U_{i(\alpha)}(x_{\alpha})$ for some $i(\alpha)\in\\{+,-,1,2\\}$ and
$U_{i(\alpha)}(x_{\alpha})\in\mathscr{A}$. To this $x_{\alpha}\in
B(r,\bar{\varepsilon},\delta)$ corresponds the element $w_{\alpha}$ such that
$\|w_{\alpha}\|\leqslant 1$, and

$w_{\alpha}=\left\\{\begin{array}[]{ll}u_{1}(x_{\alpha}),&\mbox{if}\
V_{\alpha}\subset U_{1}(x_{\alpha});\\\ u_{+}(x_{\alpha}),&\mbox{if}\
V_{\alpha}\subset U_{+}(x_{\alpha});\\\ u_{-}(x_{\alpha}),&\mbox{if}\
V_{\alpha}\subset U_{-}(x_{\alpha});\\\ u_{2}(x_{\alpha}),&\mbox{if}\
V_{\alpha}\subset U_{2}(x_{\alpha}).\end{array}\right.$

Now, let $v:B(r,\bar{\varepsilon},\delta)\rightarrow X$ be defined by

$v(x)=(1+\|x\|)\sum\limits_{\alpha\in\Lambda}\gamma_{\alpha}(x)(w_{\alpha}-x_{\alpha}+x).$
$None$

For each $x\in U_{i}(x_{\alpha})$ with $i\in\\{+,-,1,2\\}$ and
$x^{*}\in\partial J(x)$, there exists $L_{i,\alpha}>0$ such that
$\|x^{*}\|_{*}\leqslant L_{i,\alpha}$. We may also assume that
$B_{i}(x_{\alpha},r_{i}(x_{\alpha}))$ has a small radium $r_{i}(x_{\alpha})>0$
such that $(1+\|x\|)(1+\|x_{\alpha}\|)^{-1}\leqslant 2$ for each $x\in
U_{i}(x_{\alpha})$, and

$0<r_{i}(x_{\alpha})\leqslant\min\Big{\\{}1,\frac{b}{16(1+\|x_{\alpha}\|)L_{i,\alpha}}\Big{\\}}.$

Then, we have

$\|v(x)\|\leqslant(1+\|x\|)\sum\limits_{\alpha\in\Lambda}\gamma_{\alpha}(x)(\|w_{\alpha}\|+\|x_{\alpha}-x\|)\leqslant
2(1+\|x\|),$

and

$\begin{array}[]{ll}(1+\|x\|)&\big{|}\sum\limits_{\alpha\in\Lambda}\gamma_{\alpha}(x)\langle
x^{*},x-x_{\alpha}\rangle\big{|}\\\
&\leqslant\sum\limits_{\alpha\in\Lambda}\frac{1+\|x\|}{1+\|x_{\alpha}\|}(1+\|x_{\alpha}\|)\gamma_{\alpha}(x)\|x^{*}\|_{*}\|x-x_{\alpha}\|\\\
&\leqslant
2\sum\limits_{\alpha\in\Lambda}L_{i,\alpha}\|x-x_{\alpha}\|(1+\|x_{\alpha}\|)\gamma_{\alpha}(x)\\\
&<\frac{b}{8}\end{array}$

for each $x\in B(r,\bar{\varepsilon},\delta)$. On the other hand, by
(3.12)$\sim$(3.16) we have

$\sum\limits_{\alpha\in\Lambda}\gamma_{\alpha}(x)(1+\|x\|)\langle
x^{*},w_{\alpha}\rangle\geqslant\frac{b}{4}.$

So, we have

$\begin{array}[]{ll}\langle
x^{*},v(x)\rangle&=\sum\limits_{\alpha\in\Lambda}\gamma_{\alpha}(x)(1+\|x\|)\langle
x^{*},w_{\alpha}\rangle\\\
&+\sum\limits_{\alpha\in\Lambda}\gamma_{\alpha}(x)(1+\|x\|)\langle
x^{*},x-x_{\alpha}\rangle\geqslant\frac{b}{8}.\end{array}$

Next we show that, $x-\frac{1}{1+\|x\|}v(x)\in D^{+}(\mu_{0})$ for any $x\in
D^{+}(\mu_{0})\backslash D^{-}(\mu_{0})$. In fact,

$x-\frac{1}{1+\|x\|}v(x)=\sum\limits_{\alpha\in\Lambda}\gamma_{\alpha}(x)(x_{\alpha}-w_{\alpha}).$

If $\gamma_{\alpha}(x)\neq 0$ for some $\alpha\in\Lambda$, then $x\in
V_{\alpha}\subset U_{i(\alpha)}(x_{\alpha})$ for some
$i(\alpha)\in\\{+,-,1,2\\}$. As $x\in D^{+}(\mu_{0})\backslash
D^{-}(\mu_{0})$, by the definition of
$\mathscr{A}_{1},\mathscr{A}_{2},\mathscr{A}_{3},\mathscr{A}_{4}$, we must
have $i(\alpha)\in\\{+,2\\}$. So, $x_{\alpha}-w_{\alpha}\in D^{+}(\mu_{0})$.
By (3.16) and using the convexity of $D^{+}(\mu_{0})$, we have

$x-\frac{1}{1+\|x\|}v(x)\in D^{+}(\mu_{0}).$

Similarly we can show that $x-\frac{1}{1+\|x\|}v(x)\in D^{-}(\mu_{0})$ for
each $x\in D^{-}(\mu_{0})\backslash D^{+}(\mu_{0})$, and
$x-\frac{1}{1+\|x\|}v(x)\in D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$ for each $x\in
D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$.

At last we show that $v$ is odd whenever $J$ is even. Obviously,
$B(r,\bar{\varepsilon},\delta)$ is a symmetric closed set with respect to 0.
Assume without loss of generality that $B(r,\bar{\varepsilon},\delta)\cap
D^{+}(\mu_{0})\neq\emptyset$. Then we have $B(r,\bar{\varepsilon},\delta)\cap
D^{-}(\mu_{0})\neq\emptyset$. Let $v:B(r,\bar{\varepsilon},\delta)\rightarrow
X$ be defined by (3.16) and let

$v_{0}(x)=\frac{1}{2}\big{[}v(x)-v(-x)\big{]},\ \forall x\in
B(r,\bar{\varepsilon},\delta).$

Obviously, $v_{0}:B(r,\bar{\varepsilon},\delta)\mapsto X$ is locally
Lipschitz, and odd map, such that $\|v_{0}(x)\|\leqslant 2(1+\|x\|)$. For each
$x\in B(r,\bar{\varepsilon},\delta)$ and $x^{*}\in\partial J(x)$, we have

$\langle x^{*},v_{0}(x)\rangle=\frac{1}{2}\big{(}\langle
x^{*},v(x)\rangle+\langle-x^{*},v(-x)\rangle\big{)}\geqslant\frac{b}{8}.$

This implies that $v_{0}$ is a generalized normalized pseudo-gradient vector
field of the locally Lipschitz functional $J$.

By using the symmetric property of $D^{+}(\mu_{0})=-D^{-}(\mu_{0})$, we have
$-x\in D^{-}(\mu_{0})\backslash D^{+}(\mu_{0})$ for each $x\in
D^{+}(\mu_{0})\backslash D^{-}(\mu_{0})$. So, we have

$x-\frac{1}{1+\|x\|}v(x)\in D^{+}(\mu_{0})\ \mbox{and}\
-x-\frac{1}{1+\|x\|}v(-x)\in D^{-}(\mu_{0})$

for $x\in D^{+}(\mu_{0})\backslash D^{-}(\mu_{0})$. Therefore, for $x\in
D^{+}(\mu_{0})\backslash D^{-}(\mu_{0})$, we have

$-\Big{(}-x-\frac{1}{1+\|x\|}v(-x)\Big{)}\in D^{+}(\mu_{0}),$

and so

$x-\frac{1}{2(1+\|x\|)}\big{(}v(x)-v(-x)\big{)}=\frac{1}{2}\Big{(}x-\frac{1}{1+\|x\|}\Big{)}+\frac{1}{2}\Big{(}-\big{(}-x-\frac{1}{1+\|x\|}v(-x)\big{)}\Big{)}\in
D^{+}(\mu_{0}).$

Similarly, we have

$x-\frac{1}{2(1+\|x\|)}\big{(}v(x)-v(-x)\big{)}\in D^{-}(\mu_{0})$

for each $x\in D^{-}(\mu_{0})\backslash D^{+}(\mu_{0})$, and

$x-\frac{1}{2(1+\|x\|)}\big{(}v(x)-v(-x)\big{)}\in D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})$

for each $x\in D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$. The proof is complete.

The next theorem (deformation theorem) is the key tool for our main results.

Obviously, if the condition (H${}_{1})$ holds then
$\mathbb{K}_{r}\cap\partial\big{(}D^{+}(\mu_{0})\cup
D^{-}(\mu_{0})\big{)}=\emptyset$ for any $r\in\mathbb{R}$. Let
$\mathbb{K}_{r}^{1}=\mathbb{K}_{r}\cap\big{(}D^{+}(\mu_{0})\cup
D^{-}(\mu_{0})\big{)}$ and $\mathbb{K}_{r}^{2}=S\cap\mathbb{K}_{r}$. Assume
(H${}_{2})$ holds. Then, for each $r\in\mathbb{R}$, $\mathbb{K}_{r}^{1}$ and
$\mathbb{K}_{r}^{2}$ are compact sets, and so we may take $\delta>0$ small
such that
$(\mathbb{K}_{r}^{1})_{3\delta}\cap(\mathbb{K}_{r}^{2})_{3\delta}=\emptyset$.

Lemma 3.4. Suppose that (H${}_{1})$ and (H${}_{2})$ hold. Then for each
$\varepsilon_{0}>0$, there exists $0<\bar{\varepsilon}<\varepsilon_{0}$, such
that for each $0<\varepsilon<\bar{\varepsilon}$, compact set $A\subset
J^{r+\varepsilon}$, there exists $\eta\in C([0,1]\times X,X)$ with the
following properties:

1) $\eta(t,u)=u$ for $t=0$ or $u\not\in
J^{-1}(r-\varepsilon_{0},r+\varepsilon_{0}])\backslash(\mathbb{K}_{r}^{2})_{\delta}$;

2) $\eta(1,A\backslash(\mathbb{K}_{r}^{2})_{3\delta})\subset
J^{r-\varepsilon}\cup\mbox{int}(D^{+}(\mu_{0}))\cup\mbox{int}(D^{-}(\mu_{0}))$;
and $\eta(1,A)\subset
J^{r-\varepsilon}\cup\mbox{int}(D^{+}(\mu_{0}))\cup\mbox{int}(D^{-}(\mu_{0}))$
if $\mathbb{K}_{r}^{2}=\emptyset$;

3) $\eta(t,\cdot)$ is a homeomorphism of $X$ for $t\in[0,1]$;

4) $J(\eta(\cdot,u))$ is non-increasing for any $u\in X$;

5) $\eta(t,D^{+}(\mu_{0}))\subset D^{+}(\mu_{0})$,
$\eta(t,D^{-}(\mu_{0}))\subset D^{-}(\mu_{0})$ and $\eta(t,D^{+}(\mu_{0})\cap
D^{-}(\mu_{0}))\subset D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$ for $t\in[0,1]$; and

6) if $J$ is even, $\eta$ is odd in $u\in X$.

Proof. Since $\mathbb{K}_{r}$ is a compact set, and so
$(\mathbb{K}_{r})_{3\delta}$ is a bounded set. Assume that $\|u\|\leqslant
M_{0}$ for all $u\in(\mathbb{K}_{r})_{3\delta}$ and some $M_{0}>0$. It follows
from Lemma 3.2 that there exists $\bar{\varepsilon},b>0$ such that
(3.4)$\sim$(3.7) hold. By Lemma 3.3, there exists a locally Lipschitz mapping
$v:B(r,\bar{\varepsilon},\delta)\rightarrow X$ such that $\|v(x)\|\leqslant
2(1+\|x\|)$, and

$\langle x^{*},v(x)\rangle\geqslant\frac{b}{8},\ \ \forall x\in
B(r,\bar{\varepsilon},\delta),x^{*}\in\partial J(x),$ $None$
$x-\frac{1}{1+\|x\|}v(x)\in D^{+}(\mu_{0}),\ \ \forall x\in D^{+}(\mu_{0}),$
$None$ $x-\frac{1}{1+\|x\|}v(x)\in D^{-}(\mu_{0}),\ \ \forall x\in
D^{-}(\mu_{0}),$ $None$ $x-\frac{1}{1+\|x\|}v(x)\in D^{+}(\mu_{0})\cap
D^{-}(\mu_{0}),\ \ \forall x\in D^{+}(\mu_{0})\cap D^{-}(\mu_{0}).$ $None$

Without loss of generality, we may assume that

$0<\bar{\varepsilon}<\min\big{\\{}\varepsilon_{0},\frac{b\delta}{32(1+M_{0})}\big{\\}}.$

For each $0<\varepsilon<\bar{\varepsilon}$, we set

$D_{1}=\\{x\in X:r-\varepsilon\leqslant J(x)\leqslant r+\varepsilon\\},$
$D_{2}=\\{x\in X:r-\bar{\varepsilon}\leqslant J(x)\leqslant
r+\bar{\varepsilon}\\},$

and

$\varphi(x)=\frac{\mbox{dist}\ (x,X\backslash D_{2})}{\mbox{dist}\
(x,X\backslash D_{2})+\mbox{dist}\ (x,D_{1})}.$

Obviously, $\varphi:X\mapsto\mathbb{R}$ is locally Lipschitz continuous on
$X$. Let

$\psi(x)=\frac{\mbox{dist}\ (x,(\mathbb{K}_{r})_{\delta})}{\mbox{dist}\
(x,(\mathbb{K}_{r})_{\delta}))+\mbox{dist}\
(x,X\backslash(\mathbb{K}_{r})_{2\delta})}.$

Also, $\psi:X\mapsto\mathbb{R}$ is locally Lipschitz continuous on $X$.

Consider the following initial value problem (IVP) in $X$

$\left\\{\begin{array}[]{l}\frac{du}{dt}=-V(u),\\\
u(0)=u_{0},\end{array}\right.$ $None$

where $V(u)=\varphi(u)\psi(u)v(u)$ for $u\in X$. Since $V:X\mapsto X$ is
locally Lipschitz continuous on $X$, by the theory of ordinary differential
equations in Banach spaces, (3.21) has a unique solution in $X$, denoted by
$\sigma(t,u_{0})$, with right maximal interval of existence $[0,T(u_{0}))$.

Now we show that $T(u_{0})=+\infty$ for each $u_{0}\in X$. Arguing by make
contradiction that $T(u_{0})<+\infty$. By (3.21) we have

$\|\sigma(t,u_{0})-u_{0}\|\leqslant\int^{t}_{0}\|V(\sigma(s,u_{0}))\|ds\leqslant
2\int^{t}_{0}\big{(}1+\|\sigma(s,u_{0})\|\big{)}ds.$

So, we have

$\begin{array}[]{ll}\frac{1}{2}\|\sigma(t,u_{0})-u_{0}\|&\leqslant\displaystyle\int^{t}_{0}(1+\|\sigma(s,u_{0})\|)ds\\\
&\leqslant\displaystyle\int^{t}_{0}\|\sigma(s,u_{0})-u_{0}\|ds+(1+\|u_{0}\|)t,\end{array}$

By the well known Gronwall’s inequality, we have

$\begin{array}[]{ll}\frac{1}{2}\|\sigma(t,u_{0})-u_{0}\|&\leqslant\displaystyle\int^{t}_{0}(1+\|u_{0}\|)e^{t-s}ds+(1+\|u_{0}\|)t\\\
&\leqslant(1+\|u_{0}\|)(e^{t}-1)+(1+\|u_{0}\|)t\\\
&\leqslant(1+\|u_{0}\|)(t+e^{t}-1)\\\
&\leqslant(1+\|u_{0}\|)(T(u_{0})+e^{T(u_{0})}).\end{array}$

So, we have

$\|\sigma(t,u_{0})\|\leqslant
2(1+\|u_{0}\|)(T(u_{0})+e^{T(u_{0})})+2\|u_{0}\|=:M_{1}.$

Take $\\{t_{n}\\}\subset[0,T(u_{0}))$ such that $t_{n}\rightarrow
T^{-}(u_{0})$ and for $n=1,2,\cdots$,

$|t_{n}-t_{n-1}|<\frac{1}{2\cdot 2^{n}(1+M_{1})}.$

Then we have

$\begin{array}[]{ll}\|\sigma(t_{n},u_{0})-\sigma(t_{n-1},u_{0})\|&\leqslant\displaystyle\int_{t_{n-1}}^{t_{n}}\|V(\sigma(s,u_{0})\|ds\\\
&\leqslant
2\displaystyle\int_{t_{n-1}}^{t_{n}}\big{(}1+\|\sigma(s,u_{0})\|\big{)}ds\\\
&\leqslant 2(1+M_{1})(t_{n}-t_{n-1})<\frac{1}{2^{n}}.\end{array}$

This implies that $\\{\sigma(t_{n},u_{0})\\}$ is a Cauchy sequence. Thus,
there exists $\bar{u}\in X$ such that $\sigma(t_{n},u_{0})\rightarrow\bar{u}$
as $t\rightarrow T^{-}(u_{0})$.

Now we consider the initial value problem

$\left\\{\begin{array}[]{l}\frac{du}{dt}=-V(u),\\\
u(0)=\sigma(T(u_{0}),u_{0}),\end{array}\right.$ $None$

Then (3.22) has a unique solution on $[0,\bar{\delta})$ for some
$\bar{\delta}>0$ since $V$ is locally Lipschitz on $X$, and so (3.21) has a
unique solution on $[0,T(u_{0})+\bar{\delta})$, which is a contradiction.
Thus, we have $T(u_{0})=+\infty$.

For each $u_{0}\in X$, denote by $o(u_{0})$ the orbit of (3.21) starting from
$u_{0}$, that is

$o(u_{0})=\\{\sigma(t,u_{0}):t\in[0,+\infty)\\}.$

Now we show that $D^{+}(\mu_{0})$ and $D^{-}(\mu_{0})$ are flow invariant sets
of (3.21). We only show that $D^{+}(\mu_{0})$ is a flow invariant set of
(3.21). For a given $u_{0}\in\partial D^{+}(\mu_{0})$, we have the following
three cases:

(1) $u_{0}\in J^{-1}([r-\varepsilon,r+\varepsilon])\cap\partial
D^{+}(\mu_{0})$.

In this case, we have $\varphi(u_{0})=\psi(u_{0})=1$. Thus, for $\lambda>0$
small enough, by (3.18) and (3.20) we have

$\begin{array}[]{ll}u_{0}+\lambda(-V(u_{0}))&=\lambda(1+\|u_{0}\|)\big{(}u_{0}-\frac{1}{1+\|u_{0}\|}v(u_{0})\big{)}\\\
&+(1-\lambda(1+\|u_{0}\|))u_{0}\in D^{+}(\mu_{0}).\end{array}$

(2) $u_{0}\not\in
J^{-1}([r-\bar{\varepsilon},r+\bar{\varepsilon}])\cap\partial D^{+}(\mu_{0})$.

In this case, $V(u_{0})=0$. Obviously, for $\lambda>0$ small enough, by (3.18)
and (3.20) we have $u_{0}+\lambda(-V(u_{0}))=u_{0}\in D^{+}(\mu_{0})$.

(3)
$u_{0}\in\big{(}J^{-1}([r-\bar{\varepsilon},r+\bar{\varepsilon}])\backslash
J^{-1}([r-\varepsilon,r+\varepsilon])\big{)}\cap\partial D^{+}(\mu_{0})$.

In this case, $\psi(u_{0})=1$ and $0\leqslant\varphi(u_{0})\leqslant 1$. Then,
for $\lambda>0$ small enough, by (3.18) and (3.20) we have

$\begin{array}[]{ll}u_{0}+\lambda(-V(u_{0}))&=(1-\lambda\varphi(u_{0})(1+\|u_{0}\|)u_{0}\\\
&+\lambda(1+\|u_{0}\|)\varphi(u_{0}))\Big{(}u_{0}-\frac{1}{1+\|u_{0}\|}v(u_{0})\Big{)}\in
D^{+}(\mu_{0}).\end{array}$

By the arguments above and a well known theorem due to Brezis-Martin on flow
invariant set, we see that $D^{+}(\mu_{0})$ is a flow invariant set of (3.21).

Now we show the following claim: for each $u_{0}\in A\cap
J^{r+\varepsilon}\backslash\mathbb{K}_{r}^{2})_{3\delta}$, there exist an open
neighborhood $U(x_{0})$ of $x_{0}$ and $\tau(u_{0})>0$, such that

$\sigma(t,u)\in\mbox{int}(D^{+}(\mu_{0}))\cup\mbox{int}(D^{-}(\mu_{0}))\cup
J^{r-\varepsilon}$

for $t\geqslant t(u_{0})$ and $u\in A\cap U(x_{0})\cap J^{r+\varepsilon}$.

To prove the claim, we need to consider the following three cases:

(1) $o(u_{0})\cap\mbox{int}(D^{+}(\mu_{0}))\neq\emptyset$.

Assume without loss of generality that
$u_{1}:=\sigma(t_{1},u_{0})\in\mbox{int}(D^{+}(\mu_{0}))$ for some
$t_{1}\geqslant 0$. Take an open neighborhood $U(u_{1})$ of $u_{1}$ such that
$U(u_{1})\subset\mbox{int}(D^{+}(\mu_{0}))$. By the continuous dependence of
ordinary differential equations on initial data, there exists an open
neighborhood $U(u_{0})$ of $u_{0}$, and $\tau(u_{0})>0$, such that
$\sigma(t,u)\in U(u_{1})\subset\mbox{int}(D^{+}(\mu_{0}))$ for all
$u\in\big{(}A\cap
J^{r+\varepsilon}\backslash(\mathbb{K}_{r}^{2})_{3\delta}\big{)}\cap U(u_{0})$
and $t\geqslant\tau(u_{0})$.

(2)  $o(u_{0})\cap\mbox{int}(D^{-}(\mu_{0}))\neq\emptyset$.

By the same way as the case (1) we can prove that, there exists an open
neighborhood $U(u_{0})$ of $u_{0}$, and $\tau(u_{0})>0$, such that
$\sigma(t,u_{0})\in\mbox{int}(D^{-}(\mu_{0}))$ for all $u\in\big{(}A\cap
J^{r+\varepsilon}\backslash(\mathbb{K}_{r}^{2})_{3\delta}\big{)}\cap U(u_{0})$
and $t\geqslant\tau(u_{0})$.

(3)
$o(u_{0})\cap\big{(}\mbox{int}(D^{+}(\mu_{0}))\cup\mbox{int}(D^{+}(\mu_{0}))\big{)}=\emptyset$,
$o(u_{0})\cap(\mathbb{K}^{2}_{r})_{3\delta}=\emptyset$.

Let $h(t,u_{0})=J(\sigma(t,u_{0}))$ for all $t\in[0,\infty)$. It is easy to
see that $h(t,u_{0})$ is locally Lipschitz continuous in $t\in[0,+\infty)$.
According to Leburng’s Mean Theorem we have

$\begin{array}[]{ll}\frac{\partial}{\partial
s}h(s,u_{0})&\leqslant\max\\{\langle w^{*},\frac{\partial}{\partial
s}\sigma(s,u_{0})\rangle:w^{*}\in\partial J(\sigma(s,u_{0}))\\}\
\mbox{a.e.}\\\ &=-\min\\{\langle
w^{*},V(\sigma(s,u_{0}))\rangle:w^{*}\in\partial J(\sigma(s,u_{0}))\\}\ \mbox{
a.e.}\\\ &\leqslant\left\\{\begin{array}[]{ll}-\frac{b}{8},&\mbox{if}\
\sigma(s,u_{0})\in B(r,\bar{\varepsilon},2\delta);\\\
0,&\mbox{otherwise}.\end{array}\right.\end{array}$ $None$

Consequently, we have for any $0\leqslant t_{1}<t_{2}<\infty$,

$J(\sigma(t_{1},u_{0}))-J(\sigma(t_{2},u_{0}))=-\int^{t_{2}}_{t_{1}}\frac{\partial}{\partial
s}h(s,u_{0})ds\geqslant\frac{b}{8}(t_{2}-t_{1})\ \mbox{if}\ \sigma(s,u_{0})\in
B(r,\varepsilon,2\delta)\ \mbox{and}\ s\in[t_{1},t_{2}).$ $None$

So, if we let $\tau(u_{0})>\frac{16\varepsilon}{b}$, we have by (3.23),

$J(\sigma(t,u_{0}))\leqslant J(u_{0})-\frac{bt}{8}\leqslant
r+\varepsilon-\frac{b}{8}<r-\varepsilon$

for $t\geqslant\tau(u_{0})$. Take an open neighborhood
$U(\sigma(\tau(u_{0}),u_{0}))$ of $\sigma(\tau(u_{0}),u_{0})$ such that
$U(\sigma(\tau(u_{0}),u_{0}))\subset J^{r-\varepsilon}$. By the continuous
dependence of ordinary differential equations on initial data, there exists an
open neighborhood $U(u_{0})$ of $u_{0}$, such that $\sigma(\tau(u_{0}),u)\in
U(\sigma(\tau(u_{0}),u_{0}))$ for all $u\in\big{(}A\cap
J^{r+\varepsilon}\backslash(\mathbb{K}_{r}^{2})_{3\delta}\big{)}\cap
U(u_{0})$. Since $h(t,u_{0})$ is non-increasing in $t\in[0,+\infty)$, we have
$\sigma(t,u)\in J^{r-\varepsilon}$ for all $u\in\big{(}A\cap
J^{r+\varepsilon}\backslash(\mathbb{K}_{r}^{2})_{3\delta}\big{)}\cap U(u_{0})$
and $t\geqslant\tau(u_{0})$.

(4)
$o(u_{0})\cap\big{(}\mbox{int}(D^{+}(\mu_{0}))\cup\mbox{int}(D^{+}(\mu_{0}))\big{)}=\emptyset$,
$o(u_{0})\cap(\mathbb{K}^{2}_{r})_{3\delta}\neq\emptyset$.

In this case we may take $[t_{1},t_{2}]\subset[0,+\infty)$ such that
$\sigma(t_{1},u_{0})\in\partial(K_{r})_{3\delta}$,
$\sigma(t_{2},u_{0})\in\partial(K_{r})_{2\delta}$, and
$\sigma(t,u_{0})\in(K_{r})_{3\delta}\backslash{(K_{r})}_{2\delta}$ for
$t\in[t_{1},t_{2}]$. By (3.21) we have

$\begin{array}[]{ll}\|\sigma(t_{2},u_{0})-\sigma(t_{1},u_{0})\|&\leqslant\displaystyle\int^{t_{2}}_{t_{1}}\|V(\sigma(s,u_{0})\|ds\\\
&\leqslant
2\displaystyle\int^{t_{2}}_{t_{1}}\big{(}1+\|\sigma(s,u_{0})\|\big{)}ds\\\
&\leqslant 2\big{(}1+M_{0}\big{)}(t_{2}-t_{1}).\end{array}$ $None$

So, we have

$t_{2}-t_{1}\geqslant\frac{\|\sigma(t_{2},u_{0})-\sigma(t_{1},u_{0})\|}{2\big{(}1+M_{0}\big{)}}\geqslant\frac{\delta}{2\big{(}1+M_{0}\big{)}}.$

Now we show that there must exist $t_{0}\in[t_{1},t_{2}]$ such that
$\sigma(t_{0},u_{0})\in\\{x\in X:J(x)<r-\varepsilon\\}$. Assume by make
contradiction that this is not the case, then

$\\{\sigma(t,u_{0}):t\in[t_{1},t_{2}]\\}\cap\\{x\in
X:J(x)<r-\varepsilon\\}=\emptyset.$

Then, by (3.24) and (3.25) we have

$\begin{array}[]{ll}J(\sigma(t_{2},u_{0}))&\leqslant
J(\sigma(t_{1},u_{0}))-\frac{\gamma}{4}(t_{2}-t_{1})\\\ &\leqslant
J(u_{0})-\frac{\gamma}{4}(t_{2}-t_{1})\\\ &\leqslant
r-\frac{\gamma\delta}{8\big{(}1+M_{0}\big{)}}\\\
&<r-\bar{\varepsilon}_{0}<r-\varepsilon,\end{array}$

which is a contradiction. Thus, there must exist $t_{0}\in[t_{1},t_{2}]$ such
that $\sigma(t_{0},u_{0})\in\\{x\in X:J(x)<r-\varepsilon\\}$. Then, as the
arguments in the case (3), we can show that there exists a open neighborhood
of $U(u_{0})$ of $u_{0}$, and $\tau(u_{0})\geqslant 0$, such that
$\sigma(t,u)\subset\\{x\in X:J(x)<r-\varepsilon\\}$ for any $u\in U(u_{0})$
and $t\geqslant\tau(u_{0})$.

From the arguments above we see the claim holds. Obviously,
$\mathscr{U}=\\{U(u_{0}):u_{0}\in A\cap
J^{r+\varepsilon}\backslash\mathbb{K}_{r}^{2})_{3\delta}\\}$ is an open cover
of $A$. Since $A\cap J^{r+\varepsilon}\backslash\mathbb{K}_{r}^{2})_{3\delta}$
is compact, there exist finite sets in $\mathscr{U}$, say
$U(u_{1}),U(u_{2}),\cdots,U(u_{m})$ such that $\big{(}A\cap
J^{r+\varepsilon}\backslash\mathbb{K}_{r}^{2})_{3\delta}\big{)}\subset\bigcup\limits_{i=1}^{m}U(u_{i})$.
Let

$T=\max\\{\tau(u_{1}),\tau(u_{2}),\cdots,\tau(u_{m})\\}.$

Then for each $u\in A\cap
J^{r+\varepsilon}\backslash\mathbb{K}_{r}^{2})_{3\delta}$, $\sigma(T,u)\in
J^{-1}((-\infty,r-\varepsilon))\cup D^{+}(\mu_{0})D^{-}(\mu_{0})$. If we let
$\eta(t,x)=\sigma(Tt,x)$ for $t\in[0,1]$ and $x\in X$. Then we can easily see
the conclusions (1)$\sim$(5) hold.

Since $D^{+}(\mu_{0})\cup D^{-}(\mu_{0})$ is a flow invariant set of (3.21),
we have $\eta(t,D^{+}(\mu_{0})\cup D^{-}(\mu_{0}))\subset D^{+}(\mu_{0})\cup
D^{-}(\mu_{0})$. This means that the conclusion (5) holds. The proof is
complete.

Definition 3.1. Let $Q,T$ be two closed subset of $X$, satisfies that (1)
$\partial Q\cap T=\emptyset$; (2) for each $\gamma\in C(Q,X)$ with
$\gamma|_{\partial Q}=I$, we have $\partial Q\cap T\neq\emptyset$. Then we
call $(Q,T)$ is linking.

There have been many papers studied the critical points by using the linking
structure. For Example, the paper [20] study the case where the functional $J$
being of $C^{2}$; while [13] studied the case where the functional $J$ is
locally Lipschitz continuous. Now we will use the linking structure and the
deformation theorem to show the existence of sign-changing solutions.

Theorem 3.1. Suppose that (H${}_{1})$ and (H${}_{2})$ hold. Let
$W=D^{+}(\mu_{0})\cup D^{-}(\mu_{0})$. Let $(Q,T)$ is a linking. $T\subset
X\backslash W$ and $Q\subset X$ is compact, and for each $\gamma\in\Gamma$,
$\gamma(Q)\cap S\neq\emptyset$, where $S=X\backslash W$ and

$\Gamma=\\{\gamma\in C(Q,X):\gamma|_{\partial Q\cap S}=I,\gamma(\partial Q\cap
W)\subset W\\}.$

Assume that there exist $\beta>\alpha$ such that

$\sup\limits_{\partial Q\cap
S}J(x)\leqslant\alpha<\beta\leqslant\inf\limits_{T}J(x)$

and $\sup\limits_{Q}J(x)<\infty$. Let
$r=\inf\limits_{\gamma\in\Gamma}\sup\limits_{\gamma(Q)\cap S}J(x)$. Then
$r\geqslant\beta$ and $\mathbb{K}_{r}\cap S\neq\emptyset$.

Proof. Since $Q\subset X$ is compact, $J$ is continuous on $Q$, then $J$ is
bounded on $Q$. Since $I\in\Gamma$, we have

$r\leqslant\sup\limits_{Q\cap S}J(x)\leqslant\sup\limits_{Q}J(x)<+\infty.$

Now we show $r\geqslant\beta$. Since for any $\gamma\in\Gamma$, $\gamma(Q)\cap
T\neq\emptyset$ and $T\subset S$, then $\gamma(Q)\cap S\cap T\neq\emptyset$.
Thus, we have

$\sup\limits_{\gamma(Q)\cap S}J(x)\geqslant\sup\limits_{\gamma(Q)\cap T\cap
S}J(x)=\sup\limits_{\gamma(Q)\cap T}J(x)\geqslant\inf\limits_{\gamma(Q)\cap
T}J(x)\geqslant\inf\limits_{T}J(x)\geqslant\beta.$

Assume that $\mathbb{K}_{r}\cap\partial W=\emptyset$. Otherwise, the
conclusion would hold. We claim that $\mathbb{K}_{r}\cap S\neq\emptyset$.
Arguing by make contradiction that $\mathbb{K}_{r}\cap S=\emptyset$. Take
$\varepsilon_{0}>0$ such that $r-2\varepsilon_{0}>\alpha$. Let
$0<\varepsilon<\varepsilon_{0}$. Take $\gamma\in\Gamma$ such that
$\sup\limits_{\gamma(Q)\cap S}J(x)\leqslant r+\varepsilon$. Let $A=\gamma(Q)$.
By using the deformation theorem (Lemma 3.4), there exists $\eta\in
C([0,1]\times X,X)$ such that the conclusions (1)$\sim$ (5) hold. In
particular, we have $\eta(1,\gamma(Q))\subset J^{r-\varepsilon}\cup W$.

Set $\gamma_{1}=\eta(1,\gamma(\cdot))$. Then we have

$\gamma_{1}(\partial Q\cap W)=\eta(1,\gamma(\partial Q\cap
W))\subset\eta(1,W)\subset W.$

For any $x\in\partial Q\cap S$ with $J(x)\leqslant\alpha<r-2\varepsilon$, we
have

$\eta(1,\gamma(x))=\eta(1,x)=x,$

which implies that $\gamma_{1}\in\Gamma$.

On the other hand, we have

$\sup\limits_{\gamma_{1}(Q)\cap
S}J(x)\leqslant\sup\limits_{(J^{r-\varepsilon}\cup W)\cap
S}J(x)\leqslant\sup\limits_{J^{r-\varepsilon}}J(x)\leqslant r-\varepsilon.$

which is a contradiction. The proof is complete.

Now we apply Theorem 3.1 to a special linking structure.

Suppose that $X=X_{1}\oplus X_{1}^{\perp}$, where $\infty>\mbox{dim}\
X_{1}\geqslant 1$. Take $y_{1}\in X_{1}^{\perp}$ with $\|y_{1}\|=1$. Let $R>0$
and

$Q=\\{e=e_{1}+ty_{1}:e_{1}\in X_{1},\|e_{1}\|\leqslant R,0\leqslant t\leqslant
R,\|e\|\leqslant R\\}.$

For any $0<r<R$, let $T=\\{e\in X_{1}^{\perp}:\|e\|=r\\}$. Then $(Q,T)$ is
linking.

Corollary 3.1. Suppose that $J$ satisfies all conditions on Theorem 3.1,
$(Q,T)$ is defined as above, $W\subset X_{1}^{\perp}\backslash\\{0\\}$, and
for $\alpha<\beta$,

$\max\limits_{\partial Q\cap
S}J(x)\leqslant\alpha<\beta\leqslant\inf\limits_{T}J(x).$ $None$

Let $\Gamma,r$ be defined as Theorem 2.1. Then

$\mathbb{K}_{r}\cap(S\cup\partial W)\neq\emptyset.$

Remark 3.2. Obviously, we can obtain the existence results of infinitely many
critical points of Locally Lipschitz and even functionals by using the
deformation lemma (Lemma 3.4). Here, for simplicity, we will not give the
related results. We think that readers of this paper can easily obtain the
related results using Lemma 3.4 by themselves.

##  4 Sign Changing Solutions for Differential Inclusion Problems

Consider the following differential inclusions problems

$\left\\{\begin{array}[]{ll}-\Delta u\in\lambda\partial j(x,u)&\ \mbox{in}\
\Omega,\\\ u=0&\ \mbox{on}\ \partial\Omega,\end{array}\right.$ $None$

where $\Omega$ is a bounded open domain in $\mathbb{R}^{N}$ with a $C^{1}$
boundary, $\lambda>0$ is a parameter, the reaction term $\partial j(x,s)$ is
the generalized gradient of a non-smooth potential $s\mapsto j(x,s)$, which is
subject to the following conditions.

(H${}_{j})$ $j:\Omega\times\mathbb{R}\mapsto\mathbb{R}$ is a Carathéodory
function and there exist constants $a_{1}>0$, $2<q<2^{*}$ such that

(i) $j(x,\cdot)$ is locally Lipschitz for almost every $x\in\Omega$ and
$j(\cdot,0)=0$;

(ii) $|\xi|\leqslant a_{1}(1+|s|^{q-1})$ a.e. in $\Omega$ and for all
$s\in\mathbb{R}$, $\xi\in\partial j(x,s)$;

(iii) For some $\mu>N(\frac{q}{2}-1)$ one has

$\lim\inf\limits_{|x|\rightarrow\infty}\frac{u\inf\partial
j(x,z)-2j(x,z)}{|z|^{\mu}}>0$

uniformly for almost $x\in\Omega$;

(iv)

$\lim\sup_{z\rightarrow 0}\frac{2j(x,z)}{z^{2}}=0$

uniformly with respect to $x\in\Omega$;

(v) $zw\geqslant 0$ for each $w\in\partial j(x,z)$ a.e. $\Omega$ and
$z\in\mathbb{R}$.

Theorem 4.1. Assume (H${}_{j})$ holds. Then (4.$1_{\lambda}$) has at least one
sign-changing solution $\bar{u}$ for $\lambda>0$.

Let

$\|u\|=\Big{(}\int_{\Omega}|\nabla u|^{2}dx\Big{)}^{1\over
2},|u|_{p}=\Big{(}\int_{\Omega}|u|^{p}dx\Big{)}^{1\over p}\ \mbox{and}\
|u|_{\infty}=\sup\limits_{x\in\Omega}|u(x)|$

be the standard norms of $H_{0}^{1}(\Omega)$, $L^{p}(\Omega)$, respectively
$C(\Omega)$. Let $X=H_{0}^{1}(\Omega)$. We introduce the energy functional
$J_{\lambda}:X\mapsto\mathbb{R}$ for $\lambda>0$ by

$J_{\lambda}(u)=\frac{1}{2}\|u\|^{2}-\lambda\varphi(u),$

where the functional $\varphi:X\mapsto\mathbb{R}$ be defined by

$\varphi(u)=\int_{\Omega}j(x,u(x))dx.$

Let $P=\\{u\in X:u(x)\geqslant 0\ \mbox{a.e.}\ \Omega\\}$. Let
$\lambda_{1},\lambda_{2}$ be the first and second eigenvalues of $-\Delta$
under the Dirichlet condition, and $u_{1},u_{2}$ be the eigenfunctions
corresponding to $\lambda_{1}$ and $\lambda_{2}$ respectively.

Lemma 4.1. Assume that (H${}_{j})$ holds. Then there exists $\mu_{0}>0$ such
that $(I-\partial J_{\lambda})\big{(}{D}^{\nu}(\mu_{0})\big{)}\subset
D^{\nu}(\frac{1}{2}\mu_{0})$ for $\nu\in\\{+,-\\}$.

Proof. For each $u\in X$, denote $u^{\pm}=\max\\{\pm u,0\\}$. Then, for each
$u\in X$ we have

$\begin{array}[]{ll}|u^{+}|_{2}&=\min\limits_{w\in(-P)}|u-w|_{2}\\\
&\leqslant\frac{1}{\sqrt{\lambda_{1}}}\min\limits_{w\in(-P)}\|u-w\|\\\
&=\frac{1}{\sqrt{\lambda_{1}}}\mbox{dist}\ (u,-P).\end{array}$

Similarly, we have for some constant $C_{\mu}>0$,

$|u^{+}|_{\mu}\leqslant C_{\mu}\mbox{dist}\ (u,-P).$

Obviously, $u\mapsto\frac{1}{2}\|u\|^{2}$ is a $C^{1}$-functional whose
derivative is the identity operator $I$. Aubin-Clarke’s Theorem [4, Theorem
1.3.10] ensures that the functional $\varphi:X\rightarrow\mathbb{R}$ is
Lipschitz continuous on any bounded subset of $L^{2}(\Omega)$ and its gradient
is included in the set

$N(u)=\\{w\in L^{2}(\Omega):w(x)\in\partial j(x,u(x))\ \mbox{for almost
every}\ x\in\Omega\\}.$

Since $X$ continuously embedded in $L^{2}(\Omega)$, the function $J_{\lambda}$
turns out to be locally Lipschitz on $X$. So, we have

$\partial J_{\lambda}(u)\subset\\{u\\}-\lambda N(u).$ $None$

Now, if $u\in X$ complies with $0\in\partial J_{\lambda}(u)$ then

$u=\lambda w\ \mbox{in}\ X^{*}$

for some $w\in N(u)$.

By (H${}_{j})$ (i) and (iv), for any $u\in X$, $z(x)\in\partial j(x,u(x))$, we
have

$(z(x),u(x))\leqslant\lambda\varepsilon|u(x)|^{2}+\lambda
C_{\varepsilon}|u(x)|^{q}.$

Now we check the Schauder invariance condition. Take
$\mathscr{F}:X^{*}\rightarrow X$ as the identity operator $I:X\rightarrow X$.
By (4.2) we have

$(I-\partial J_{\lambda})(u)\subset\lambda N(u).$

Let $w_{\lambda}\in\lambda N(u)$ with $w_{\lambda}=\lambda w$ for some $w\in
N(u)$. So, if $\varepsilon=\varepsilon(\lambda)>0$ small enough,

$\begin{array}[]{ll}\mbox{dist}(w_{\lambda},-P)\|w_{\lambda}^{+}\|&\leqslant\|w_{\lambda}^{+}\|^{2}=(w_{\lambda},w_{\lambda}^{+})\\\
&\leqslant\displaystyle\int_{\Omega}\big{(}\lambda\varepsilon|u^{+}|+\lambda
C_{\varepsilon}|u^{+}|^{q-1}\big{)}|w^{+}_{\lambda}|dx\\\
&\leqslant\Big{(}\frac{1}{3}\mbox{dist}\ (u,-P)+C\big{(}\mbox{dist}\
(u,-P)\big{)}^{q-1}\Big{)}\|w^{+}_{\lambda}\|,\end{array}$

where $C>0$ is a constant, here we have employed some method of [25,p.71].
Thus, we have

$\mbox{dist}\ \big{(}(I-\partial
J_{\lambda})(u),-P\big{)}\leqslant\frac{1}{3}\mbox{dist}\
(u,-P)+C\big{(}\mbox{dist}\ (u,-P)\big{)}^{q-1}.$

Hence, there exists $\mu_{1}>0$ such that for all $0<\bar{\mu}<\mu_{1}$

$(I-\partial J_{\lambda})\big{(}{D}^{-}(\bar{\mu})\big{)}\subset
D^{-}(\frac{1}{2}\bar{\mu}).$

Similarly, we can show that there exists a $\mu_{2}>0$ such that for all
$0<\bar{\mu}<\mu_{2}$,

$(I-\partial J_{\lambda})\big{(}{D}^{+}(\bar{\mu})\big{)}\subset
D^{+}(\frac{1}{2}\bar{\mu}).$

Let $\mu_{0}=\min\\{\mu_{1},\mu_{2}\\}$. Then we have

$(I-\partial J_{\lambda})\big{(}{D}^{\nu}(\mu_{0})\big{)}\subset
D^{\nu}(\frac{1}{2}\mu_{0})$

for $\nu\in\\{+,-\\}$. So, the condition (H${}_{1})$ holds. The proof is
complete.

Now we make a decomposition for the real Banach space $X:X=\mbox{span}\
\\{u_{1}\\}\oplus V$, where $V=\\{v\in X:\int_{\Omega}vu_{1}=0\\}$.

Lemma 4.2. Assume (H${}_{j})$ holds. Then $J_{\lambda}$ satisfies the
condition (H${}_{2})$.

Proof.  The proof is similar to Lemma 5.1 in [11] and Proposition 7 in [6].
For reader’s convenience we give the details of the process. We divide the
proof into three steps.

Step 1. $J_{\lambda}$ satisfies the condition $(PS)_{X}$.

Let $\\{x_{n}\\}\subset X$ be such that $|J_{\lambda}(x_{n})|\leqslant M_{1}$
for some $M_{1}>0$ and $(1+\|x_{n}\|)m(x_{n})\rightarrow 0$ as
$n\rightarrow\infty$. Take $x_{n}^{*}\in\partial J_{\lambda}(x_{n})$ such that
$m(x_{n})=\|x_{n}^{*}\|_{*}$ for $n\geqslant 1$. Then we have
$x_{n}^{*}=A(x_{n})-u_{n}$ with $A:X\rightarrow X^{*}$ being defined by

$(A(u),y)=\int_{\Omega}\nabla u\cdot\nabla ydx\ \ \mbox{for}\ u,y\in X,$

and $u_{n}\in L^{2}(\Omega)$, $u_{n}(x)\in\partial\varphi(x,x_{n}(x))$. Since
$(1+\|x_{n}\|)m(x_{n})\rightarrow 0$ as $n\rightarrow\infty$, we can say that
$|(x_{n}^{*},x_{n})|\leqslant\frac{1}{n}$. So,

$-\|x_{n}\|^{2}+\lambda(u_{n},x_{n})\leqslant\frac{1}{n}.$ $None$

Similarly, since $|2J_{\lambda}(x_{n})|\leqslant 2M_{1}$ for all $n\geqslant
1$, we have

$\|x_{n}\|^{2}-2\lambda\int_{\Omega}j(x,x_{n}(x))dx\leqslant 2M_{1}.$ $None$

It follows from (4.3) and (4.4) that

$\int_{\Omega}\big{(}\lambda u_{n}(x)\cdot
x_{n}(x)-2j(x,x_{n}(x))\big{)}dx\leqslant\frac{1}{n}+2M_{1}.$ $None$

By the condition (H${}_{j})$ (iii), there exists $M_{2}>0$ such that for a.e.
$x\in\Omega$ and $|z|\leqslant M_{2}$, and all $u\in\partial j(x,z)$, we have

$\frac{\beta}{2}|z|^{\mu}<uz-2j(x,z).$

By using (H${}_{j})$ (ii), for a.e. $x\in\Omega$ and $|z|\leqslant M_{2}$, and
all $u\in\partial j(x,z)$, we have

$\frac{\beta}{2}|z|^{\mu}-a_{2}(x)<uz-2j(x,z),a_{2}\in
L^{\infty}(\Omega),a_{2}(x)\geqslant 0\ \mbox{a.e. on }\ \partial\Omega.$
$None$

By (4.4) and (4.5) we have

$\frac{\beta}{2}|x_{n}|^{\mu}_{\mu}\leqslant\int_{\Omega}(u_{n}(x)x_{n}(x)-2j(x,x_{n}(x)))dx+|a_{2}(x)|_{1}<\frac{1}{n}+2M_{1}+|a_{2}|_{\infty}.$

This implies that $\\{|x_{n}|_{\mu}\\}$ is bounded.

Take $s>1$ such that $2<s<\min\big{\\{}2^{*},\frac{2(N+\mu)}{N}\big{\\}}$, and
for a.e. $x\in\Omega$, $z\in\mathbb{R}$,

$j(x,z)\leqslant c_{2}+c_{3}|z|^{s},c_{2},c_{3}>0.$

Take

$\theta=\left\\{\begin{array}[]{ll}\frac{2^{*}(s-\mu)}{s(2^{*}-\mu)},&{N>2},\\\
1-\frac{\mu}{s},&N=2.\end{array}\right.$

Then we have $0<\theta<1$, and

$\frac{1}{s}=\frac{1-\theta}{\mu}+\frac{\theta}{2^{*}}.$

Thus, by using the interpolation inequality and the Sobolev embedding Theorem,
we obtain

$|x_{n}|_{s}\leqslant|x_{n}|_{\mu}^{1-\theta}|x_{n}|_{2^{*}}^{\theta}\leqslant
c_{4}\|x_{n}\|^{\theta}\ \mbox{for some }\ c_{4}>0.$

Since $J_{\lambda}(x_{n})\leqslant M_{1}$ and $L^{s}(\Omega)\hookrightarrow
L^{2}(\Omega)$, by using the well known Young’s inequality we have

$\begin{array}[]{ll}\frac{1}{2}\|x_{n}\|^{2}&\leqslant
c_{2}|\Omega|+c_{3}|x_{n}|_{s}^{s}+M_{1}\\\ &\leqslant
c_{5}+c_{6}|x_{n}|_{s}^{s}\leqslant c_{7}+c_{8}\|x_{n}\|^{\theta
s},\end{array}$

where $c_{5},\cdots,c_{8}$ are positive constants.

If $N>2$, then $Ns<2(N+\mu)$, and so we have

$\theta
s=\frac{2^{*}(s-\mu)}{2^{*}-\mu}=\frac{2N}{N-2}\cdot\frac{(s-\mu)(N-2)}{2N-N\mu+2\mu}<\frac{2N}{N-2}\cdot\frac{(s-\mu)(N-2)}{Ns-N\mu}=2.$

If $N=2$, then $s<\min\\{2^{*},\frac{2(N+\mu)}{N}\\}=2+\mu$. So,

$\theta s=\big{(}1-\frac{\mu}{s}\big{)}s=s-\mu<2.$

Thus, we have $\theta s<2$ for $N\geqslant 2$. Thus, $\\{\|x_{n}\|\\}$ is
bounded.

Assume that $x_{n}\rightharpoonup x$ in $X$ and $x_{n}\rightarrow x$ in
$L^{2}(\Omega)$. By usual method we can show that $x_{n}\rightarrow x$ in $X$.
So, $J_{\lambda}$ satisfies the condition on $(PS)_{X}$.

Step 2. $J_{\lambda}$ satisfies the condition
$(PS)_{D^{+}(\mu_{0}),D^{+}(\frac{3}{4}\mu_{0})}$.

Let $\\{x_{n}\\}\subset D^{+}(\frac{3}{4}\mu_{0})$ be such that
$|J_{\lambda}(x_{n})|\leqslant M_{2}$ for some $M_{2}>0$, and

$(1+\|x_{n}\|)m_{D^{+}(\mu_{0})}(x_{n})\rightarrow 0\ \mbox{as}\
n\rightarrow\infty.$

Let $h_{D^{+}(\mu_{0})}:X^{*}_{w}\times X\mapsto\mathbb{R}$ be defined in the
manner in [6] by

$h_{D^{+}(\mu_{0})}(x^{*},x)=\sup\big{\\{}\langle x^{*},x-y\rangle:y\in
D^{+}(\mu_{0}),\|y-x\|<1\big{\\}}.$

Then, we can find $x_{n}^{*}\in\partial J_{\lambda}(x_{n})$ such that
$m_{D^{+}(\mu_{0})}(x_{n})=h_{D^{+}(\mu_{0})}(x_{n}^{*},C_{1}(x_{n}))$, where
$C_{1}(x_{n})=(x_{n}-D^{+}(\mu_{0}))\cap B(0,1)$. That is

$(x_{n}^{*},x_{n}-y)\leqslant h_{D^{+}(\mu_{0})}(x_{n}^{*},C_{1}(x_{n}))\
\mbox{for all}\ y\in D^{+}(\mu_{0}),\|x_{n}-y\|<1.$ $None$

Obviously, $x_{n}^{*}=A(x_{n})-\lambda u_{n}$ with $A:X\rightarrow X^{*}$
being defined as Step 1. Let us denote by $\pi_{P}:X\mapsto P$ the orthogonal
projection on the cone $P$ so that $T_{P}(u)=u-\pi(u)$ realizes the distance
of $u$ from $P$. Let $y_{n}=x_{n}+\frac{\mu_{0}}{8\|x_{n}\|}x_{n}$ for
$n\geqslant 1$. Since $\pi(x_{n})\in P$, we have

$\begin{array}[]{ll}\mbox{dist}\
(y_{n},P)&=\|T_{P}y_{n}\|\leqslant\|y_{n}-\pi(x_{n})\|\\\
&\leqslant\|y_{n}-x_{n}\|+\|x_{n}-\pi(x_{n})\|=\|y_{n}-x_{n}\|+\|T_{p}x_{n}\|\\\
&=\|y_{n}-x_{n}\|+\mbox{dist}\
(x_{n},P)\leqslant\frac{\mu_{0}}{8}+\frac{3\mu_{0}}{4}\leqslant\mu_{0}.\end{array}$

Thus, $y_{n}\in D^{+}(\mu_{0})$, $\|x_{n}-y_{n}\|<1$. And so, by (4.7) we have

$\begin{array}[]{ll}(1+\|x_{n}\|)(x_{n}^{*},x_{n}-y_{n})&=-\frac{\mu_{0}(1+\|x_{n}\|)}{8\|x_{n}\|}(x_{n}^{*},x_{n})\\\
&=-\frac{\mu_{0}(1+\|x_{n}\|)}{8\|x_{n}\|}\Big{(}(A(x_{n}),x_{n})-\lambda\displaystyle\int_{\Omega}u_{n}x_{n}dx\Big{)}\\\
&\leqslant(1+\|x_{n}\|)h_{D^{+}(\mu_{0})}(x_{n}^{*},C_{1}(x_{n}))=:\varepsilon_{n}\
\mbox{with}\ \varepsilon_{n}\downarrow 0.\end{array}$

So we obtain

$-\|x_{n}\|^{2}+\lambda\int_{\Omega}u_{n}x_{n}\leqslant\frac{8\|x_{n}\|}{\mu_{0}(1+\|x_{n}\|)}\varepsilon_{n}=\varepsilon^{\prime}_{n},\varepsilon_{n}^{\prime}\downarrow
0.$

Then, by using the arguments from (4.3) to the end of the step 1 we can show
that $\\{\|x_{n}\|\\}$ is bounded, and $\\{x_{n}\\}$ has a convergent
subsequence.

Step 3. $J_{\lambda}$ satisfies the condition
$(PS)_{D^{-}(\mu_{0}),D^{-}(\frac{3}{4}\mu_{0})}$.

In a similar way as Step 2, we can show the claim above hold.

Step 4. $J_{\lambda}$ satisfies the condition $(PS)_{D^{+}(\mu_{0})\cap
D^{-}(\mu_{0})}$.

Obviously, $D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$ is bounded in $X$. So, by using
the Sobolev inequality, we see that $D^{+}(\mu_{0})\cap D^{-}(\mu_{0})$ is
bounded in $L^{\mu}(\Omega)$ for $\mu\in(1,2^{*})$. Then, by using the
arguments above we can show that $J_{\lambda}$ satisfies the condition
$(PS)_{D^{+}(\mu_{0})\cap D^{-}(\mu_{0})}$.

The proof is complete.

Lemma 4.3. Assume that (H${}_{j})$ holds. Then there exists $\delta>0$ such
that $J_{\lambda}(v)>0$ for all $v\in V$ with $0<\|v\|\leqslant\delta$.

Proof. For each $\varepsilon>0$, there exists $C_{1}(\varepsilon)>0$ such that

$j(x,z)\leqslant\frac{\varepsilon}{2}|z|^{2}+C_{1}|z|^{q}.$

So, for any $v\in V$ we have

$\begin{array}[]{ll}J_{\lambda}(v)&=\frac{1}{2}\|v\|^{2}-\lambda\displaystyle\int_{\Omega}j(x,v(x))dx\\\
&\geqslant\frac{1}{2}\|v\|^{2}-\lambda\displaystyle\int_{\Omega}\big{(}\frac{\varepsilon}{2}|v(x)|^{2}+C_{1}|v(x)|^{q}\big{)}dx\\\
&\geqslant\frac{1}{2}\big{(}1-\frac{\lambda}{\lambda_{V}}\varepsilon)\|v\|^{2}-C_{2}\|v(x)\|^{q},\end{array}$

where $C_{2}>0$ is a constant. Take $\varepsilon>0$ small such that
$\frac{\lambda}{\lambda_{V}}\varepsilon<1$. Then, we have $J_{\lambda}(v)>0$
for all $v\in V$ with $0<\|v\|\leqslant\delta$. The proof is complete.

Lemma 4.4. Assume that (H${}_{j})$ holds. Then there exists $R>0$ such that
$J_{\lambda}(u)<0$ for all $u\in E_{2}$ with $\|u\|\geqslant R$, where
$E_{2}=\mbox{span}\ \\{\varphi_{1},\varphi_{2}\\}$.

Proof. For each $u\in E_{2}$, we have $\|u\|^{2}<\lambda_{2}|u|_{2}^{2}$. By
the condition (H${}_{j})(iii)$ we have

$j(x,u)\geqslant a_{0}|u|^{\mu}-C_{3}$

for some $a_{0},C_{3}>0$. So, we have

$\begin{array}[]{ll}J_{\lambda}(u)&=\frac{1}{2}\|u\|^{2}-\lambda\displaystyle\int_{\Omega}j(x,u)dx\\\
&\leqslant\frac{1}{2}\|u\|^{2}-\lambda a_{0}|u|^{\mu}_{\mu}+C_{4}\\\
&\leqslant\frac{1}{2}\|u\|^{2}-\lambda C_{\mu}\|u\|^{\mu}+C_{4},\end{array}$

where $C_{4}>0$ is a constant. Then we have

$\lim\limits_{u\in E_{2},\|u\|\rightarrow\infty}J_{\lambda}(u)=-\infty,$

and so the conclusion holds. The proof is complete.

The Proof of Theorem 4.1. We will apply Corollary 3.1 to show Theorem 4.1. It
follows from Lemma 4.1 and 4.2 that the conditions (H${}_{1})$ and (H${}_{2})$
hold. Take $R>0$ large enough such as that in Lemma 4.4. Let

$Q=\\{e=\varphi_{1}+t\varphi_{2}\in E_{2}:e_{1}\in X_{1},\|e_{1}\|\leqslant
R,0\leqslant t\leqslant R,\|e\|\leqslant R\\},$

and $T=\\{x\in V:\|x\|=\delta\\}$, where $\delta>0$ be defined as that in
Lemma 4.3. It follows from Lemma 4.3 and 4.4 that (3.26) hold. Hence, by
Corollary 3.1, we see that Theorem 4.1 hold. The proof is complete.

Remark 4.1. There have been some papers studied the existence for sign-
changing solutions of differential inclusion problems; see [14,15] and the
references therein. For example, by combing variational methods with
truncation techniques the paper [14] obtained the existence of positive,
negative and nodal solutions to differential inclusion problems with a
parameter. Here, our method is different to that in [14,15].

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