\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 14, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/14\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for Dirichlet problems involving the p(x)-Laplace operator} \author[M. Avci \hfil EJDE-2013/14\hfilneg] {Mustafa Avci} % in alphabetical order \address{Mustafa Avci \newline Department of Mathematics, Faculty of Science, Dicle University, 21280-Diyarbakir, Turkey} \email{mavci@dicle.edu.tr} \thanks{Submitted November 11, 2011. Published January 14, 2013.} \subjclass[2000]{35D05, 35J60, 35J70, 58E05} \keywords{$p(x)$-Laplace operator; variable exponent Lebesgue-Sobolev spaces; \hfill\break\indent variational approach; Fountain theorem} \begin{abstract} In this article, we study superlinear Dirichlet problems involving the $p(x)$-Laplace operator without using the Ambrosetti-Rabinowitz's superquadraticity condition. Using a variant Fountain theorem, but not including Palais-Smale type assumptions, we prove the existence and multiplicity of the solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} %\newtheorem{corollary}[theorem]{Corollary} %\newtheorem{definition}[theorem]{Definition} %\newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} We study the existence of infinitely many solutions for the Dirichlet boundary problems \begin{equation} \begin{gathered} -\Delta _{p(x)}u+| u|^{p(x) -2}u=f(x,u) \quad \text{in } \Omega ,\\ u=0\quad \text{on } \partial \Omega , \end{gathered}\label{P0} \end{equation} and \begin{equation} \begin{gathered} -\Delta _{p(x) }u=f(x,u) \quad \text{in } \Omega ,\\ u=0\quad \text{on } \partial \Omega , \end{gathered} \label{E0} \end{equation} where $\Omega $ is a bounded smooth domain of $\mathbb{R}^{N}$, $p\in C(\overline{\Omega }) $ such that $1
0$ and $\tau >p^{+}$ such that \[ 0<\tau F(x,s) \leq f(x,s) s,\quad |s| \geq M,\; x\in \Omega , \] where $f$ is the nonlinear term in the equation with $F(x,t)=\int_0^{t}f(x,s)ds$ for $x\in \Omega $ and $t\in\mathbb{R}$. \end{itemize} There are many articles dealing with superlinear Dirichlet problems involving $p(x)$-Laplacian, in which (AR) is the main assumption to get the existence and multiplicity of solutions \cite{Fan3,Fan4}. However, there are many functions which are superlinear but not satisfy (AR). It is well known that the main aim of using (AR) is to ensure the boundedness of the Palais-Smale type sequences of the corresponding functional. In the present paper we do not use (AR) and we know that without (AR) it becomes a very difficult task to get the boundedness. So, using a weaker assumption (G1) (see main results) instead of (AR), and some variant Fountain theorem, i.e., Theorem \ref{thm5}, we overcome these difficulties. \section{Abstract framework and preliminary results} We state some basic properties of the variable exponent Lebesgue-Sobolev spaces $L^{p(x) }(\Omega ) $ and $W^{1,p(x)}(\Omega ) $, where $\Omega \subset\mathbb{R}^{N}$ is a bounded domain (for more details, see \cite{Edmunds,Fan1,Fan2,Kovacik}). Set \[ C_{+}(\overline{\Omega }) =\{ p\in C(\overline{\Omega }): \inf p(x) >1,\forall x\in \overline{\Omega }\} . \] Let $p\in C_{+}(\overline{\Omega }) $ and denote \[ p^{-}:=\inf_{x\in \overline{\Omega }} p(x) \leq p(x) \leq p^{+}:=\sup_{x\in \overline{\Omega }} p(x) <\infty . \] For any $p\in C_{+}(\overline{\Omega }) $, we define the variable exponent Lebesgue space by \[ L^{p(x) }(\Omega ) =\{ u:\Omega \to\mathbb{R}\text{ is measurable, }\int_{\Omega }| u(x)| ^{p(x) }dx<\infty \} . \] Then $L^{p(x) }(\Omega ) $ endowed with the norm \[ | u| _{p(x) }=\inf \{ \mu>0:\int_{\Omega }| \frac{u(x) }{\mu } | ^{p(x) }dx\leq 1\} , \] becomes a Banach space. The modular of the $L^{p(x) }(\Omega ) $ space, which is the mapping $\rho :L^{p(x) }(\Omega ) \to \mathbb{R}$ defined by \[ \rho (u) =\int_{\Omega }| u(x)| ^{p(x) }dx,\quad \forall u\in L^{p(x) }(\Omega ) . \] \begin{proposition}[\cite{Fan1,Kovacik}] \label{prop1} If $u,u_n\in L^{p(x)}(\Omega )$ ($n=1,2,\dots$), then we have \begin{itemize} \item[(i)] $|u|_{p(x) }<1$ $(=1,>1)$ if and only if $\rho (u) <1$ $(=1,>1)$; \item[(ii)] $|u|_{p(x) }>1$ implies $|u|_{p(x)}^{p^{-}}\leq \rho (u) \leq |u|_{p(x) }^{p^{+}}$, $|u|_{p(x) }<1$ implies $|u|_{p(x) }^{p^{+}}\leq \rho (u) \leq |u|_{p(x) }^{p^{-}}$; \item[(iii)] $\lim_{n\to \infty }|u_n|_{p(x) }=0$ if and only if $\lim_{n\to \infty }\rho (u_n)=0$; $\lim_{n\to \infty }|u_n|_{p(x)}=\infty$ if and only if $\lim_{n\to \infty }\rho(u_n)=\infty$. \end{itemize} \end{proposition} \begin{proposition}[\cite{Fan1,Kovacik}] \label{prop2} If $u,u_n\in L^{p(x) }(\Omega ) $ ($n=1,2,\dots$), then the following statements are equivalent: \begin{itemize} \item[(i)] $\lim_{n\to \infty }|u_n-u|_{p(x) }=0$; \item[(ii)] $\lim_{n\to \infty }\rho (u_n-u)=0$; \item[(iii)] $u_n\to u$ in measure in $\Omega$ and $\lim_{n\to \infty}\rho (u_n)=\rho (u)$. \end{itemize} \end{proposition} The variable exponent Sobolev space $W^{1,p(x)}(\Omega) $ is defined by \[ W^{1,p(x)}(\Omega ) =\{u\in L^{p(x) }(\Omega) : |\nabla u|\in L^{p(x) }(\Omega ) \}, \] with the norm \[ \|u\|_{1,p(x) }=|u|_{p(x)}+|\nabla u|_{p(x)}, \] or equivalently \[ \|u\|_{1,p(x) }=\inf \big\{ \mu>0:\int_{\Omega }(| \frac{\nabla u(x) }{ \mu }| ^{p(x) }+| \frac{u(x) }{ \mu }| ^{p(x) }) dx\leq 1\big\} \] for all\ $u\in W^{1,p(x)}(\Omega ) $. The space $W_0^{1,p(x)}(\Omega ) $ is defined as the closure of $C_0^{\infty }(\Omega )$ in $W^{1,p(x)}(\Omega ) $ with respect to the norm $\|u\|_{1,p(x) }$. For $u\in W_0^{1,p(x)}(\Omega ) $, we define an equivalent norm \[ \|u\|=|\nabla u|_{p(x)}, \] since Poincar\'e inequality holds, i.e., there exists a positive constant $c$ such that \[ |u|_{p(x)}\leq c|\nabla u|_{p(x)}, \] for all $u\in W_0^{1,p(x)}(\Omega ) $, see \cite{Fan3}. \begin{proposition}[\cite{Fan1,Kovacik}] \label{prop3} If $1
r_k>0.
\end{gather*}
Let us consider the $C^{1}$-functional $I_{\lambda }:E\to\mathbb{R}$ defined by
\[
I_{\lambda }(u) :=A(u)-\lambda B(u),\quad \lambda \in [1,2] .
\]
Now we recall the following variant of the fountain theorem
\cite[Theorem 2.1]{Zou}, which is the main tool in the proof of the
main results of this article.
We will use the following assumptions:
\begin{itemize}
\item[(F1)] $I_{\lambda }$ maps bounded sets to bounded
sets uniformly for $\lambda \in [1,2] $. Moreover,
$I_{\lambda }(-u)=I_{\lambda }(u)$ for all
$(\lambda ,u)\in [1,2] \times E$;
\item[(F2)] $B(u)\geq 0$ for all $u\in E$, and $A(u)\to \infty $
or $B(u)\to \infty $ as $\|u\|\to \infty, $
\item[(F3)] $B(u)\leq 0$ for all $u\in E$; $B(u)\to -\infty$ as
$\|u\|\to \infty$.
\end{itemize}
\begin{theorem} \label{thm5}
Assume the functional $I_{\lambda }$ satisfies {\rm (F1)}, and either
{\rm(F2)} or {\rm (F3)}.
For $k\geq 2$, let
\begin{gather*}
\Gamma _k:=\{ \psi \in C(B_k,E):\psi \text{ is odd, }
\psi \big| _{\partial B_k} ={\rm id}\}, \\
c_k(\lambda ):=\inf_{\psi \in \Gamma _k}\max_{u\in
B_k}I_{\lambda }(\gamma (u)),\\
b_k(\lambda ):=\inf_{u\in Z_k,\|u\|=r_k}I_{\lambda }(u),\\
a_k(\lambda ):=\max_{u\in Y_k,\|u\|=\rho _k}I_{\lambda}(u).
\end{gather*}
If $b_k(\lambda )>a_k(\lambda )$ for all
$\lambda \in [1,2] $, then
$c_k(\lambda )\geq b_k(\lambda )$ for all $\lambda \in [1,2] $.
Moreover, for a.e $\lambda \in [1,2] $, there exists a sequence
$\{ u_n^{k}(\lambda )\} _{n=1}^{\infty }$ such that
$\sup_n\|u_n^{k}(\lambda )\|<\infty$,
$ I_{\lambda }'(u_n^{k}(\lambda ))\to 0$ and
$I_{\lambda }(u_n^{k}(\lambda ))\to c_k(\lambda )$ as
$n\to \infty$.
\end{theorem}
\section{Main results}
First, we study the Dirichlet boundary-value problem
\begin{equation} \label{P}
\begin{gathered}
-\Delta _{p(x) }u+| u|^{p(x) -2}u=f(x,u) \quad \text{in } \Omega ,\\
u=0\quad \text{on } \partial \Omega ,
\end{gathered}
\end{equation}
where $\Omega $ is a bounded smooth domain of $\mathbb{R}^{N}$.
We assume the following conditions:
\begin{itemize}
\item[(S1)] $f:\overline{\Omega }\times\mathbb{R}\to\mathbb{R}$
is a Carath\'eodory function and
$| f(x,t)| \leq c(1+| t| ^{q(x) -1}) $ for a.e. $x\in \overline{\Omega }$
and all $t\in\mathbb{R}$, $f(x,t)t\geq 0$ for all $t>0$,
where $p,q\in C_{+}(\overline{\Omega }) $ such that
$p(x) 0$ uniformly for $x\in \overline{\Omega }$, where $p^{+}<\theta \leq q^{-}$;
\item[(S3)] $\lim_{t\to 0}\frac{f(x,t)}{t^{p^{-}-1}}=0$ uniformly for
$x\in \overline{\Omega }$, $\frac{f(x,u)}{u^{p^{-}-1}}$ is an increasing
function of $t\in\mathbb{R}$ for all $x\in \overline{\Omega }$.
\item[(S4)] $f(x,-t)=-f(x,t)$ for all $x\in \overline{\Omega }$,
$t\in\mathbb{R}$.
\item[(G1)] There exists a constant $\xi \geq 1$, such that
for any $s\in [0,1] $, $t\in\mathbb{R}$, and for each
$G_{\gamma }\in \mathcal{F} $, and all
$\eta \in [p^{-},p^{+}] $, the inequality
$\xi G_{\gamma }(x,t)\geq G_{\eta }(x,st) $ hold for a.e.
$x\in \overline{\Omega }$,
where
\[
\mathcal{F} =\{ G_{\gamma }:G_{\gamma }(x,t) =f(x,t)
t-\gamma F(x,t),\gamma \in [p^{-},p^{+}] \}.
\]
Note that when $p(x) \equiv p$ a constant,
$\mathcal{F} =\{ f(x,t) t-pF(x,t)\} $ is consist of only one element.
\end{itemize}
\begin{remark} \label{rmk6} \rm
It is not difficult to show that if $f(x,t) $ is increasing in
$ t $, then (AR) implies (G1) when $t$ is large enough.
However, in general, (AR) does not imply (G1);
see \cite[Remark 3.3]{Zang}.
\end{remark}
\begin{theorem} \label{thm7}
Assume that {\rm (S1)--(S4), (G1)} hold.
Then problem \eqref{P} has infinitely many solutions
$\{ u_k\} $ satisfying
\[
J(u_k) =\int_{\Omega }\frac{1}{p(x) }(| \nabla u_k| ^{p(x) }+|
u_k| ^{p(x) })dx-\int_{\Omega}F(x,u_k)dx\to \infty \quad
\text{as } k\to \infty ,
\]
where $J:W^{1,p(x)}(\Omega ) \to\mathbb{R}$
is the functional corresponding to problem \eqref{P}
and $F(x,t)=\int_0^{t}f(x,s)ds$.
\end{theorem}
\begin{remark} \label{rmk8} \rm
Condition (S1) implies that the functional $J$ is
well defined and of class $C^{1}$. It is well known that the critical points
of $J$ are weak solutions of \eqref{P}. Moreover, the derivative of $J$
is given by
\[
\langle J'(u) ,\upsilon \rangle =\int_{\Omega
}(| \nabla u| ^{p(x) -2}\nabla u\nabla
\upsilon +| u| ^{p(x) -2}u\upsilon
)dx-\int_{\Omega }f(x,u)\upsilon dx,
\]
for any $u,\upsilon \in W^{1,p(x)}(\Omega ) $.
\end{remark}
Second, we consider the Dirichlet boundary problem
\begin{equation}
\begin{gathered}
-\Delta _{p(x) }u=f(x,u) \quad \text{in }\Omega ,\\
u=0\quad \text{on } \partial \Omega ,
\end{gathered} \label{E}
\end{equation}
where $\Omega $ is a bounded smooth domain of $\mathbb{R}^{N}$.
We will use the following assumptions:
\begin{itemize}
\item[(E1)] $f:\overline{\Omega }\times\mathbb{R}\to\mathbb{R}$
is a Carath\'eodory function and
$| f(x,t)| \leq c(1+| t| ^{q(x) -1})$ a.e. $x\in \overline{\Omega }$
and all $t\in\mathbb{R}$, where $p,q\in C_{+}(\overline{\Omega }) $
such that $p(x)
\overline{b}_k>0$, and
$\{ z_n\} _{n=1}^{\infty }\subset W^{1,p(x)}(\Omega ) $,
such that
\[
J_{\lambda }'(z_n)=0,\quad J_{\lambda }(z_n)\in [\overline{b}_k,\overline{c}_k] .
\]
\end{lemma}
\begin{proof}
It is easy to prove that, for some $\rho _k>0$ large enough, we have
$a_k(\lambda ):=\max_{u\in Y_{k,}\|u\|=p_k}J_{\lambda }(u) \leq 0$
uniformly for $\lambda \in [1,2] $. Indeed, by the conditions
(S1)--(S3), for any $\varepsilon >0$ there exists
$C_{\varepsilon }>0$ such that
$f(x,u)u\geq C_{\varepsilon}| u| ^{\theta }-\varepsilon | u|^{p^{-}}$.
Further, on the finite dimensional subspace $Y_k$, we can find
some constants $c>0$ such that
\[
| u| _{\theta }\geq c\|u\|_{1,p(x) }, \quad
| u| _{p^{-}}\leq c\| u\|_{1,p(x) },\quad \forall u\in Y_k.
\]
By Propositions \ref{prop1} and \ref{prop4}, we have
\begin{align*}
J_{\lambda }(u)
&\leq \frac{1}{p^{-}}\int_{\Omega}(| \nabla u| ^{p(x) }+|
u| ^{p(x) }) dx-\frac{\lambda C_{\varepsilon }}{
\theta }\int_{\Omega }| u| ^{\theta }dx+\frac{
\lambda \varepsilon }{p^{-}}\int_{\Omega }| u|^{p^{-}}dx \\
&\leq \frac{1}{p^{-}}\|u\|_{1,p(x) }^{p^{+}}-
\frac{\lambda C_{\varepsilon }c^{\theta }}{\theta }\|u\|
_{1,p(x) }^{\theta }+\frac{\lambda c^{p^{-}}}{p^{-}}\|
u\|_{1,p(x) }^{p^{-}}\text{\ }.
\end{align*}
Since $\theta >p^{+}$, it follows that
\[
a_k(\lambda ):=\max_{u\in Y_{k,}\|u\|_{1,p(x) }
=\rho _k}J_{\lambda }(u) \to -\infty \quad \text{as }\|u\|_{1,p(x) }\to +\infty
\]
uniformly for $\lambda \in [1,2] $ and for all $u\in Y_k$.
On the other hand, by conditions
(S1) and (S3), for any $\varepsilon >0$ there exists
$C_{\varepsilon }>0$ such that
$| f(x,u)| \leq \varepsilon| u| ^{p^{-}-1}+C_{\varepsilon }| u|^{q(x) -1}$.
Let
\[
\beta _k:=\sup_{u\in Z_k,\|u\|_{1,p(x) }=1}| u| _{q(x) },\quad
\vartheta _k:=\sup_{u\in Z_k,\|u\|_{1,p(x) }=1}| u| _{p^{-}}.
\]
Then $\beta _k\to 0$ and $\vartheta _k\to 0$
as $k\to \infty $ (see \cite{Fan4}). Therefore, when
$u\in Z_k $ and $\|u\|_{1,p(x) }>1$, we have
\begin{align*}
J_{\lambda }(u)
&\geq \frac{1}{p^{+}}\int_{\Omega}(| \nabla u| ^{p(x) }+|u| ^{p(x) }) dx
-\lambda \varepsilon \int_{\Omega }| u| ^{p^{-}}dx
-\lambda C_{\varepsilon }\int_{\Omega }| u| ^{q(x)}dx \\
&\geq \frac{1}{p^{+}}\|u\|_{1,p(x)}^{p^{-}}-c| u| _{p^{-}}^{p^{-}}-c| u|
_{q(x) }^{q^{+}} \\
&\geq \frac{1}{p^{+}}\|u\|_{1,p(x)}^{p^{-}}
-c\vartheta _k^{p^{-}}\|u\|_{1,p(x) }^{p^{-}}
-c\beta _k^{q^{+}}\|u\|_{1,p(x) }^{q^{+}},
\end{align*}
where $c=\max \{ 2\varepsilon ,2C_{\varepsilon }\} $. For
sufficiently large $k$, we have $c\vartheta _k^{p^{-}}<\frac{1}{2p^{+}}$.
Then, it follows
\[
J_{\lambda }(u) \geq \frac{1}{2p^{+}}\|u\|
_{1,p(x) }^{p^{-}}-c\beta _k^{q^{+}}\|u\|_{1,p(x) }^{q^{+}}.
\]
If we choose $r_k:=(2cq^{+}\beta _k^{q^{+}}) ^{\frac{1}{
p^{-}-q^{+}}}$, then for $u\in Z_k$ with $\|u\|_{1,p(x) }=r_k$, we obtain
\begin{align*}
J_{\lambda }(u)
&\geq \frac{1}{2p^{+}}(2cq^{+}\beta_k^{q^{+}}) ^{\frac{p^{-}}{p^{-}-q^{+}}}
-c\beta _k^{q^{+}}(2cq^{+}\beta _k^{q^{+}}) ^{\frac{q^{+}}{p^{-}-q^{+}}} \\
&\geq \frac{q^{+}-p^{+}}{2p^{+}q^{+}}(2cq^{+}\beta
_k^{q^{+}}) ^{\frac{p^{-}}{p^{-}-q^{+}}}:=\overline{b}_k,
\end{align*}
which implies
\[
b_k(\lambda ):=\inf_{u\in Z_k,\|u\|_{1,p(x) }=r_k}
J_{\lambda }(u) \to \infty \quad \text{as }k\to \infty
\]
uniformly for $\lambda $. So, by Theorem \ref{thm5}, for a.e.
$\lambda \in [1,2] $, there exists a sequence
$\{ u_n^{k}(\lambda )\} _{n=1}^{\infty }$ such that
\begin{gather*}
\sup_n \|u_n^{k}(\lambda )\|_{1,p(x) }<\infty ,\quad
J_{\lambda }'(u_n^{k}(\lambda ))\to 0,\\
J_{\lambda }(u_n^{k}(\lambda )) \to c_k(\lambda
)\geq b_k(\lambda )\geq \overline{b}_k\quad \text{as }n\to \infty .
\end{gather*}
Moreover, since $c_k(\lambda )\leq \sup_{u\in B_k}J_{\lambda
}(u):=\overline{c}_k$ and $W^{1,p(x)}(\Omega ) $ is embedded
compactly to $L^{q(x) }(\Omega ) $, and thanks to
the standard arguments,
$\{ u_n^{k}(\lambda )\} _{n=1}^{\infty} $ has a convergent subsequence.
Hence, there exists $z^{k}(\lambda )$ such that
$J_{\lambda }'(z^{k}(\lambda )) =0$ and
$J_{\lambda }(z^{k}(\lambda )) \in [\overline{b}_k,\overline{c}_k] $.
Consequently, we can find $\lambda _n\to 1$ and $\{ z_n\} $ desired
as the claim.
\end{proof}
\begin{lemma} \label{lem11}
$\{ z_n\} _{n=1}^{\infty }$ is bounded in $W^{1,p(x)}(\Omega ) $.
\end{lemma}
\begin{proof}
We argue by contradiction. Passing to a subsequence if necessary, still
denoted by $\{ z_n\} $, we may assume that
$\| z_n\|_{1,p(x) }\to \infty $ as $n\to \infty $.
Let $\{ \omega _n\} \subset W^{1,p(x)}(\Omega) $ and put
$\omega _n:=\frac{z_n}{\|z_n\|_{1,p(x) }}$.
Since $\|\omega _n\|_{1,p(x) }=1$, up to subsequences, we obtain
\begin{gather*}
\omega _n \rightharpoonup \omega \quad \text{in }W^{1,p(x)}(\Omega) , \\
\omega _n \to \omega \quad \text{in }L^{\gamma (x)}(\Omega ) , \\
\omega _n(x) \to \omega (x) \quad \text{a.e. }x\in \Omega .
\end{gather*}
Then, the main concern is whether
$\{ \omega _n\} \subset W^{1,p(x)}(\Omega ) $ vanish or not.
We shall prove that none of
these alternatives can occur and this contradiction will prove that
$\{\omega _n\} \subset W^{1,p(x)}(\Omega ) $ is bounded.
If $\omega =0$, we can define a sequence
$\{ t_n\} \subset\mathbb{R}$, as argued in \cite{Zang}, such that
\begin{equation}
J_{\lambda _n}(t_nz_{n)}:=\max_{t\in [0,1]}J_{\lambda _n}(tz_n) . \label{3.1}
\end{equation}
Let $\overline{\omega }_n:=(2p^{+}c)^{\frac{1}{p^{-}}}\omega _n$ with
$c>0$. Then for $n$ is large enough, we have
\begin{equation}
\begin{split}
J_{\lambda _n}(t_nz_n)
&\geq J_{\lambda _n}(\overline{\omega }_n)
\geq A((2p^{+}c)^{\frac{1}{p^{-}}}\omega_n) -\lambda _nB(\overline{\omega }_n)
\\
&\geq \frac{1}{p^{+}}(2p^{+}c)A(\omega _n) -\lambda
_nB(\overline{\omega }_n) \geq 2c-\lambda _nB(
\overline{\omega }_n) \geq c,
\end{split} \label{e3.2}
\end{equation}
which implies that $\lim_{n\to \infty } J_{\lambda_n}(t_nz_n) =\infty $
by the fact $c>0$ can be large arbitrarily.
Noting that $J_{\lambda _n}(0) =0$ and $J_{\lambda _n}(z_n) \to c$,
then $0