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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 180, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/180\hfil Multiple positive solutions]
{Multiple positive solutions for Dirichlet problem of prescribed
mean curvature equations in Minkowski spaces}

\author[R. Ma, T. Chen \hfil EJDE-2016/180\hfilneg]
{Ruyun Ma, Tianlan Chen}

\address{Ruyun Ma \newline
Department of Mathematics,
Northwest Normal University,
Lanzhou 730070, China}
\email{mary@nwnu.edu.cn}

\address{Tianlan Chen \newline
Northwest Normal University,
Lanzhou 730070, China}
\email{chentianlan511@126.com}

\thanks{Submitted June 3, 2016. Published July 7, 2016.}
\subjclass[2010]{35B15, 34K28, 34L30, 35J60, 35J65}
\keywords{Dirichlet problem; Minkowski-curvature;
 positive solutions; 
\hfill\break\indent variational methods}

\begin{abstract}
 In this article, we consider the Dirichlet problem for the prescribed
 mean curvature equation in the Minkowski space,
 \begin{gather*}
 -\operatorname{div}\Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)
 =\lambda f(u) \quad \text{in } B_R,\\
 u=0 \quad \text{on } \partial B_R,
 \end{gather*}
 where $B_R:=\{x\in \mathbb{R}^N: |x|< R\}$, $\lambda>0$
 is a parameter and $f:[0, \infty)\to\mathbb{R}$ is continuous. We apply
 some standard variational techniques to show how changes in the sign of
 $f$ lead to multiple positive solutions of the above problem for
 sufficiently large $\lambda$.
\end{abstract}

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

\section{Introduction}

In this article we show the existence of multiple positive solutions of
 Dirichlet problem in a ball, associated to the mean curvature operator
in the flat Minkowski space $\mathbb{L}^{N+1}$
with $(x_1,\dots,x_N,t)$ and metric $\sum^N_{i=1} (dx_i)^2-(dt)^2$.
These problems are of interest in differential geometry and
in general relativity.
 It is known  \cite{b1,g1} that the study of spacelike submanifolds of codimension
 one in $\mathbb{L}^{N+1}$ with prescribed mean extrinsic curvature leads
to Dirichlet problems of the form
\begin{equation}
-\operatorname{div}\Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)=f(x, u)
\quad \text{in } \Omega,
u=0 \quad\text{on } \partial \Omega,
\label{e1.1}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ and
$f:\Omega\times \mathbb{R}\to \mathbb{R}$ is continuous.

This topic has been largely discussed in the literature for \eqref{e1.1}
in the special cases that $N=1$ or $\Omega$ is a ball (or an annulus)
 in $\mathbb{R}^N$, see  \cite{b2,b3,b4,b5,c1,c2,m1,m2}
and the references contained therein.
Note that Coelho, Corsato, Obersnel and Omari \cite{c1} proved the
 existence of one or multiple positive solutions of \eqref{e1.1} with $N=1$ provided that $f$ is $L^p$-Caratheodory function,
but the positivity of $f$ is not required.
Moreover, Bereanu, Jebelean and Torres \cite{b3} applied Leray-Schauder degree
arguments and critical point theory to show existence of positive radial solutions
for \eqref{e1.1} when $\Omega=B_R:=\{x\in \mathbb{R}^N: |x|< R\}$ and
$f$ is positive on $[0, {R}]\times [0, \alpha)$ with $\alpha\geq R$. Of course,
 the natural question is what would happen if $f(|x|,s)\equiv f(s)$ and $f$
 changes its sign in $[0, \alpha)$.


Recently, Ma and Lu \cite{m2} used the quadrature arguments to study the existence
and multiplicity of positive solutions of the nonlinear eigenvalue problem
\begin{equation}
\Big(\frac{u'}{\sqrt{1-\kappa u'^2}}\Big)'+\lambda f(u)=0 \text{ in }(0, 1),\quad
u(0)=u(1)=0,\label{e1.2}
\end{equation}
where $\kappa> 0$ is a constant and $f$ satisfies
\begin{itemize}

\item[(A1)] $f\in C^1([0, \frac{1}{2\sqrt{\kappa}}));$

\item[(A2)] Either $f(0)>0$ or
$$
f(0)=0,\quad f_0=\lim_{s\to0+}\frac{f(s)}{\psi(s)}>0, \quad
\psi(s)=\frac{s}{\sqrt{1-\kappa s^2}};
$$

\item[(A3)] There exist $0<a_1<b_1<a_2<b_2<\dots<b_{m-1}<a_m
<\frac{1}{2\sqrt{\kappa}}$ such that
$f(a_i)\leq0,\ f(b_i)>0$ and $F(b_i)>F(u)$ for all
$0\leq u\leq b_i$, $i=1, 2, \dots, m-1$.
\end{itemize}
They showed the existence of at least $2m-1$ positive solutions provided
$\lambda$ is large enough.
 Their result is an analogous of the well-known result due to Brown
and Budin \cite{b6}, who established the result of \eqref{e1.2} with $\kappa=0$
by using a generalization of a quadrature technique of Laetsch \cite{l1}.

Motivated by above papers, this article is devoted to studying how
changes in the sign of $f$ lead to multiple positive solutions
for the Dirichlet problem
\begin{equation}
\begin{gathered}
-\operatorname{div}\Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)
=\lambda f(u) \quad \text{in } B_R,\\
u=0 \quad \text{on } \partial B_R
\end{gathered}\label{e1.3}
\end{equation}
for $\lambda>0$ sufficiently large.
Assume throughout that $f:[0, \infty)\to\mathbb{R}$ is continuous and
satisfies:
\begin{itemize}
\item[(A4)] $f(0) \geq 0$ and there exist
$0 < a_1 < b_1 < a_2 < b_2 < \dots < b_{m-1} <
a_m<R$ such that $f(s)\leq 0$ if $s\in(a_k, b_k)$ and $f(s)\geq 0$
 if $s\in(b_k, a_{k+1})$ for all $k = 1, \dots, m-1$;

\item[(A5)] $\int_{a_k}^{a_{k+1}}f(s)ds>0$ for all $k\in\{1, \dots, m-1\}$.
\end{itemize}

Our main result is the following theorem.


\begin{theorem} \label{thm1.1}
Assume {\rm (A4), (A5)}. Then there exists a number
$\bar\lambda > 0$ such that for all $\lambda>\bar \lambda$,
problem \eqref{e1.3} has at least $m-1$ positive solutions
$u_1, u_2,\dots, u_{m-1} \in H^1_0(B_R)\cap L^\infty(B_R)$ and
$\|u_k\|_\infty\in (a_k, a_{k+1}]$ for all $ k=1,\dots, m-1$.
\end{theorem}


 \begin{remark} \label{rmk1.1} \rm
It would be interesting to investigate a similar version of
 Theorem \ref{thm1.1} for Dirichlet problem \eqref{e1.1} with
 $\Omega\subset\mathbb{R}^N$ bounded, sufficiently smooth.
\end{remark}

The proof of our main result will be given in the next section and
follows ideas used in \cite{c1,c2,l2},
suitably modified and expanded for the case being considered.
For the earlier results on the semilinear problem, see \cite{d1,f1}.

Now we list a few notation that will be used in this paper.
Let $E=H^1_0(B_R)$
with the usual norm
$\|u\|=\Big(\int_{B_R} |\nabla u|^2dx\Big)^{1/2}$.
The norm $\|\cdot\|_\infty$ is considered on $L^\infty(B_R)$.
We also define $\phi:(-1, 1)\to\mathbb{R}$ by $\phi(s)=\frac{s}{\sqrt{1-s^2}}$
and $\phi_N(y)=\frac{y}{\sqrt{1-|y|^2}},\ y\in\mathbb{R}^N$ with
$|\cdot|$ stands for the Euclidean norm in $\mathbb{R}^N$.

\section{Proof of the main result}

The following Lemma is a consequence of the weak maximum principle
for the $\phi$-Laplace operator.


\begin{lemma} \label{lem2.1}
 Let $g: \mathbb{R}\to \mathbb{R}$ be a continuous function and there
exists $a_0\in \big(0, R)$ such that $g(s)\geq 0$ if
$s\in (-\infty, 0)$ and $g(s)\leq 0$ if $s\geq a_0$.
If $u$ is a non-trivial solution of
\begin{equation}
-\operatorname{div} \Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)=g(u)
\text{ in } B_R,\quad u=0 \text{ on } \partial B_R,
\label{e2.1}
\end{equation}
then $u$ is positive a.e. and belongs to $L^\infty(B_R)$. Moreover,
$\|u\|_\infty\leq a_0$.
\end{lemma}

\begin{proof}
Let $v=u^-=\max\{-u, 0\}\in E$, then
\begin{equation}
\nabla v=\begin{cases}
-\nabla u, & u<0,\\
0, & u\geq 0.
\end{cases}\label{e2.2}
\end{equation}
Multiplying the equation in \eqref{e2.1} by $v$ and integrating by parts, we have
$$
0\geq-\int_{B_R} \frac {|\nabla v|^2}{\sqrt{1-|\nabla v|^2}} dx
=\int_{B_R} \frac {\nabla u\cdot\nabla v}{\sqrt{1-|\nabla u|^2}} dx
=\int_{B_R} g(u)v dx\geq0.
$$
Hence $\nabla v=0$ a.e. in ${B_R}$ and we conclude that $u\geq 0$ in ${B_R}$.

Next, choosing the test function $w = (u-a_0)^+ = \max\{u-a_0, 0\}\in E$
in the equation
$$
\int_{B_R} \frac {\nabla u\cdot\nabla w}{\sqrt{1-|\nabla u|^2}} dx
=\int_{B_R} g(u)w dx,
$$
we have $\nabla w=0$ a.e. in ${B_R}$ and therefore $u\leq a_0$, i.e.,
 $\|u\|_\infty\leq a_0$.
\end{proof}

Observe that there exists a constant $M\in (0, \infty)$ such that
\begin{equation}
|f(s)|\leq M, \quad s\in \big[0, R].\label{e2.3}
\end{equation}
With the aim of finding positive solutions of \eqref{e1.3},
 we introduce an equivalent formulation
of the problem aforementioned.
For $k = 2,\dots,m$, let us define $f_k:\mathbb{R}\to\mathbb{R}$, by
\begin{equation}
f_k(s) =\begin{cases}
f(0), &s\leq 0,\\
f(s), &s\in (0,a_k),\\
0, &s\geq a_k.
\end{cases} \label{e2.4}
\end{equation}
We notice that the function $f_k$ shares the assumed properties of $f$.
Moreover, if $u$ is a non-trivial solution of
\begin{equation}
-\operatorname{div}\Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)
=\lambda f_k(u) \text{ in } B_R,\quad
u=0 \text{ on } \partial B_R,
\label{e2.5}
\end{equation}
by Lemma \ref{lem2.1}, $u$ is positive and $\|u\|_\infty\leq a_k$.
Thus, $u$ is also a positive solution of \eqref{e1.3}
and belongs to $L^\infty(B_R)$ with $\|u\|_\infty\leq a_k$.

For every $\lambda>0$, set
$\beta:=\phi'\Big(\phi^{-1}\big(\frac{\lambda MR}{N}\big)\Big)$ and define
 $\chi_{\lambda}:\mathbb{ R}\to \mathbb{R}$ such that
\begin{equation}
\chi_{\lambda}(s)=\begin{cases}
\beta \Big(s-\phi^{-1}\big(\frac{\lambda MR}{N}\big)\Big)
+\frac{\lambda MR}{N}, & \text{if } s>\phi^{-1}\big(\frac{\lambda MR}{N}\big),\\
\phi(s), & \text{if } |s|\leq \phi^{-1}\big(\frac{\lambda MR}{N}\big),\\
\beta \Big(s+\phi^{-1}\big(\frac{\lambda MR}{N}\big)\Big)-\frac{\lambda MR}{N},
 & \text{if } s<-\phi^{-1}\big(\frac{\lambda MR}{N}\big).
\end{cases}\label{e2.6}
\end{equation}
Let $\Pi_\lambda: \mathbb{R}\to \mathbb{R}$ be given by
$$
\Pi_\lambda(y)=\int_0^y \chi_{\lambda}(\zeta)d\zeta.
$$
Then
\begin{equation}
\frac 12 y^2\leq \Pi_\lambda(y)\leq \frac 12 \beta y^2, \quad y\in \mathbb{R}.
\label{e2.7}
\end{equation}
Let the functional $\mathcal{I}_k(\lambda, \cdot):E\to \mathbb{R}$
be defined by
$$
\mathcal{I}_k(\lambda, u)=\int_{B_R} \Pi_\lambda(|\nabla u|)dx
-\lambda \int_{B_R} F_k(u)dx,
$$
where
$F_k(s)=\int_0^s f_k(\sigma)d \sigma.
$
We denote by $K_k(\lambda)$ the set of critical points of $\mathcal{I}_k$.

\begin{lemma} \label{lem2.2}
If $u$ is in $K_k(\lambda)$, then $u$ is a weak solution of
\begin{equation}
-\operatorname{div}(\psi_N(\nabla u))=\lambda f_k(u) \text{ in }B_R,\quad
 u=0 \text{ on } \partial B_R, \label{e2.8}
\end{equation}
where
\begin{equation}
\psi_N(\nabla u)=\frac {\chi_{\lambda} \big(| \nabla u|\big)}{| \nabla u|}\nabla u.
\label{e2.9}
\end{equation}
\end{lemma}

\begin{proof}
Let $u\in K_k(\lambda)$. For any $\varphi\in C_0^{\infty}(B_R)$ and
$\epsilon\in\mathbb{R}$, then
$u+\epsilon\varphi\in E$. Since
\begin{align*}
& \mathcal{I}_k(\lambda, u+\epsilon\varphi)-\mathcal{I}_k(\lambda, u)\\
&=\int_{B_R} \big[\Pi_\lambda(|\nabla u+\epsilon\nabla \varphi|)
 -\Pi_\lambda(|\nabla u|)\big]dx
 -\lambda \int_{B_R}\big[F_k(u+\epsilon\varphi)-F_k(u)\big]dx\\
&=\int_{B_R} \chi_{\lambda}\big[|\nabla u|
 +\theta_1(|\nabla u+\epsilon\nabla \varphi|
 -|\nabla u|)\big]\big(|\nabla u+\epsilon\nabla \varphi|-|\nabla u|\big)dx \\
&\quad  -\lambda \epsilon\int_{B_R} f_k(u+\theta_2\epsilon\varphi)\varphi dx\\
&=\int_{B_R} \chi_{\lambda}\big[|\nabla u|
 +\theta_1(|\nabla u+\epsilon\nabla \varphi|-|\nabla u|)\big]
\frac{2\epsilon\nabla u\cdot\nabla \varphi
 +\epsilon^2|\nabla \varphi|^2}{|\nabla u+\epsilon\nabla \varphi|
 +|\nabla u|}dx \\
&\quad -\lambda \epsilon\int_{B_R} f_k(u+\theta_2\epsilon\varphi)\varphi dx,
\end{align*}
for some constants $\theta_1, \theta_2\in(0, 1)$, it follows that
\begin{align*}
0&=\lim_{\epsilon\to0}\frac{\mathcal{I}_k(\lambda, u+\epsilon\varphi)
 -\mathcal{I}_k(\lambda, u)}{\epsilon}\\
&= \int_{B_R} \chi_{\lambda}(|\nabla u|)\frac{\nabla u\cdot\nabla \varphi}
 {|\nabla u|} dx-\lambda \int_{B_R} f_k(u)\varphi dx\\
&= \int_{B_R} \psi_N(\nabla u)\cdot\nabla \varphi dx
 -\lambda \int_{B_R} f_k(u)\varphi dx \\
&= \int_{B_R} \big[-\operatorname{div}\,\big(\psi_N(\nabla u)\big)
 -\lambda f_k(u)\big] \varphi dx.
\end{align*}
Thus, for any $\varphi\in C_0^{\infty}(B_R)$, $u$ is a weak solution
of \eqref{e2.8}.
\end{proof}


Consequently, from Lemma \ref{lem2.2}, if $u$ is in $K_k(\lambda)$, then $u$ is a
weak solution of \eqref{e2.8}.
By a similar argument of Lemma \ref{lem2.1} with $\psi_N(\nabla u)$ instead of
$\phi_N(\nabla u)$, we can deduce that
$u$ is nonnegative and belongs to $L^\infty(B_R)$ with $\|u\|_\infty\leq a_k$.

 We next claim that $K_k(\lambda)$ is not empty. Since $f_k$ is bounded and
vanishes on $(a_k,\infty)$, $\mathcal{I}_k(\lambda,\cdot)$ is coercive
and bounded from below. Further, it is weakly lower semi-continuous.
Therefore there exists $u_k(\lambda)$ such that
$$
\mathcal{I}_k\big(\lambda, u_k(\lambda)\big)
= \inf\{\mathcal{I}_k(\lambda, v):\, v \in E\}.
$$
The following Lemma shows that for $k = 2,\dots,m$,
$a_{k-1} < \|u_k\|_\infty\leq a_k$ and therefore,
\eqref{e2.8} has at least $m-1$ solutions when $\lambda > 0$ sufficiently
large.

\begin{lemma} \label{lem2.3}
For $k = 2,\dots,m$, there exists $\lambda_k>0$ such that for all
$\lambda>\lambda_k$, $u_k\not\in K_{k-1}(\lambda)$.
\end{lemma}

\begin{proof}
 We shall show that there exist $\lambda_k > 0$ and $\varphi\in E$,
$\varphi\geq 0$ and $\|\varphi\|_\infty\leq a_k$, such that
$$
\mathcal{I}_k(\lambda, \varphi)<\mathcal{I}_{k-1}(\lambda, u), \quad
\lambda>\lambda_k
$$
for all $u\in E$ satisfying $0\leq u\leq a_{k-1}$.

From (A5), $\alpha:=F(a_k)-\max\{F(s):0\leq s<a_{k-1}\}>0$.
Then, for all $u\in E$ satisfying
$0 \leq u \leq a_{k-1}$,
\begin{equation}
\int_{B_R} F(u)dx\leq \int_{B_R} F(a_k)dx-\alpha w_NR^N,\label{e2.10}
\end{equation}
where $w_N$ is the measure of the unit ball in $\mathbb{R}^N$.
For $\delta > 0$, let $\Omega_\delta:= \{x \in B_R:
\operatorname{dist}(x, \partial {B_R}) < \delta\}$.
By Lebesgue's Theorem, $|\Omega_\delta|\to 0$ as $\delta\to 0$.
Moreover, for each $\delta > 0$, there
exists $\varphi_\delta\in C^\infty_0({B_R})$ with $0\leq \varphi_\delta\leq a_k$,
$\varphi_\delta(x) = a_k$, for all $x\in {B_R}\backslash\Omega_\delta$. Thus
\begin{equation}
\begin{aligned}
\int_{B_R} F(\varphi_\delta)dx
&=\int_{{B_R}\setminus\Omega_\delta} F(a_k)dx
 +\int_{\Omega_\delta} F(\varphi_\delta)dx\\
&=\int_{B_R} F(a_k)dx-\int_{\Omega_\delta} \big(F(a_k)-F(\varphi_\delta)\big)dx\\
&\geq \int_{B_R}F(a_k)dx-2C|\Omega_\delta|,\\
\end{aligned}\label{e2.11}
\end{equation}
where $C = \max\{|F(s)| : 0 \leq s \leq a_k\}$.

By \eqref{e2.10} and \eqref{e2.11} we can choose and fix $\delta$ sufficiently
small so that there exists
$\eta:= \alpha|\Omega|- 2C|\Omega_\delta| > 0$ such that
$\varphi:= \varphi_\delta$ satisfies
$$
\int_\Omega F(\varphi)dx\geq\int_\Omega F(u)+\eta
$$
for all $u\in E$ with $0\leq u \leq a_{k-1}$. Therefore for all such $u$,
\begin{align*}
\mathcal{I}_k(\lambda, \varphi)-\mathcal{I}_{k-1}(\lambda, u)
 &=\int_{B_R} \big[\Pi_\lambda(|\nabla \varphi|)-\Pi_\lambda(|\nabla u|)\big]dx
 -\lambda \int_{B_R} \big[F(\varphi)-F(u)\big]dx\\
 &\leq\int_{B_R} \Pi_\lambda(|\nabla \varphi|)dx-\lambda \eta<0,
\end{align*}
provided $\lambda > 0$ is chosen sufficiently large. Hence for such
$\lambda$ the global minimum of $\mathcal{I}_k$
 cannot be obtained at any $u\in E$ such that $0\leq u \leq a_{k-1}$,
i.e. $u_k\not\in K_{k-1}(\lambda)$.
\end{proof}

\begin{lemma} \label{lem2.4}
A function $u\in E$ is a positive solution of \eqref{e2.5}
 if and only if it is a positive solution of \eqref{e2.8}.
\end{lemma}

\begin{proof}
Suppose that $u$ is a positive solution of \eqref{e2.5}. Hence, for fixed
$r\in(0, R]$,  from
\begin{equation}
\begin{aligned}
\int_{B_r} \operatorname{div}\Big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)dx
&=\int_{B_r} \operatorname{div}\Big(\frac{\phi(|\nabla u|)}{|\nabla u|}
 \nabla u\Big)dx \\
&=\int_{\partial{B_r}} \frac{\phi(|\nabla u|)}{|\nabla u|}\nabla u \cdot \mathbf{n} dS
\end{aligned} \label{e2.12}
\end{equation}
it follows that
\begin{equation}
-\int_{\partial{B_r}} \frac{\phi(|\nabla u|)}{|\nabla u|}\nabla u \cdot \mathbf{n} dS
 =\lambda \int_{B_r} f_k(u)dx, \label{e2.13}
\end{equation}
where $\mathbf{n}$ denotes the unit outward normal to $B_R$.

Since $\nabla u \cdot \mathbf{n}=|\nabla u|$ on $\partial B_r$,
we have
 $$
-\int_{\partial{B_r}} \phi(|\nabla u|)dS =\lambda \int_{B_r} f_k(u)dx.
$$
By radial symmetry, this can be rewritten as
\begin{equation}
|\nabla u(r)|\leq \phi^{-1}\Big(\frac{\lambda M r}{N}\big)\quad
\text{for all } r\in(0, R],
\label{e2.14}
\end{equation}
i.e. $\|\nabla u\|_\infty\leq \phi^{-1}\Big(\frac{\lambda M R}{N}\big)$.
Therefore, $\phi_N(\nabla u)=\psi_N(\nabla u)$ and we conclude that $u$
is a positive solution of \eqref{e2.8}.

Suppose now that $u$ is a positive solution of \eqref{e2.8}.
Arguing as above we see that
\begin{equation}
\|\nabla u\|_\infty\leq \chi^{-1}\Big(\frac{\lambda M R}{N}\big).
\label{e2.15}
\end{equation}
Therefore, $\psi_N(\nabla u)=\phi_N(\nabla u)$. In particular,
$\|\nabla u\|_\infty<1$ and we conclude that $u$ is a positive solution
of \eqref{e2.5}.
\end{proof}

Note that by Lemmas \ref{lem2.3} and \ref{lem2.4}, for all $\lambda$ large enough, there are
$m-1$ positive solutions $u_2(\lambda),\dots, u_m(\lambda)$ as asserted
by Theorem \ref{thm1.1}.

\subsection*{Acknowledgments}
Ruyun Ma was supported by the
NSFC (No. 11361054), and SRFDP (No. 20126203110004).


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\end{document}