# Global structure of radial sign-changing solutions for the prescribed mean
curvature problem in a ball

Ruyun Maa,∗ Hongliang Gaob
Department of Mathematics, Northwest Normal University, Lanzhou 730070, P R
China

000E-mail addresses<EMAIL_ADDRESS>(R.Ma<EMAIL_ADDRESS>(H.Gao)
000∗Supported by the NSFC (No.11361054), SRFDP(No.20126203110004) and Gansu
provincial National Science Foundation of China (No.1208RJZA258).

Abstract. In this paper, we are concerned with the global structure of radial
solutions, with prescribed nodal properties, to the boundary value problem

$\text{div}\big{(}\phi_{N}(\nabla v)\big{)}+\lambda
f(|x|,v)=0~{}~{}~{}\text{in}~{}~{}B(R),~{}~{}~{}v=0~{}~{}~{}\text{on}~{}~{}\partial
B(R),$

where $\phi_{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}},\;y\in\mathbb{R}^{N}$, $\lambda$
is a positive parameter, $B(R)=\\{x\in\mathbb{R}^{N}:|x|<R\\}$, and $|\cdot|$
denote the Euclidean norm in $\mathbb{R}^{N}$. All results, depending on the
behavior of nonlinear term $f$ near 0, are obtained by using global
bifurcation techniques.

Keywords. Mean curvature operator; Minkowski space; Radial solutions; Global
bifurcation.

MR(2000) 34B10, 34B18

##  1 Introduction

In this paper, we are concerned with the global structure of radial solutions,
with prescribed nodal properties, to the boundary value problem

$\text{div}\big{(}\phi_{N}(\nabla v)\big{)}+\lambda
f(|x|,v)=0~{}~{}~{}\text{in}~{}~{}B(R),~{}~{}~{}v=0~{}~{}~{}\text{on}~{}~{}\partial
B(R),$ $None$

where $\phi_{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}},\;y\in\mathbb{R}^{N}$, $\lambda$
is a positive parameter, $B(R)=\\{x\in\mathbb{R}^{N}:|x|<R\\}$, and $|\cdot|$
denote the Euclidean norm in $\mathbb{R}^{N}$, $f$ satisfies

$(H_{1})$ $f:[0,R]\times(-\alpha,\alpha)\to\mathbb{R}$ is a continuous
function, with $R<\alpha\leq\infty$ and such that $f(r,s)s>0$ for $r\in[0,R]$
and $s\in(-\alpha,0)\cup(0,\alpha)$.

Dirichlet problem (1.1) is associated to mean curvature operator in flat
Minkowski space

$\mathbb{L}^{N+1}:=\\{(x,t):x\in\mathbb{R}^{N},t\in\mathbb{R}\\}$

endowed with the Lorentzian metric

$\Sigma_{j=1}^{N}(dx_{j})^{2}-(dt)^{2},$

where $(x,t)$ are the canonical coordinates in $\mathbb{R}^{N+1}$.

It is known (see e.g. [1, 4, 12, 29, 35]) 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 type

$\mathcal{M}v=H(x,v)\ \ \ \text{in}\ \ \ \ \ \Omega,\ \ \ \ \ \ v=0\ \ \
\text{on}\ \partial\Omega,$ $None$

where

$\mathcal{M}v=\text{div}\Big{(}\frac{\nabla v}{\sqrt{1-|\nabla
v|^{2}}}\Big{)},$

$\Omega$ is a bounded domain in $\mathbb{R}^{N}$ and the nonlinearity
$H:\Omega\times\mathbb{R}\to\mathbb{R}$ is continuous.

The starting point of this type of problems is the seminal paper [12] which
deals with entire solutions of $\mathcal{M}v=0$. The equation
$\mathcal{M}v=\text{constant}$ is then analyzed in [35], while
$\mathcal{M}v=f(v)$ with a general nonlinearity $f$ is considered in [9]. More
general sign changing nonlinearities are studied in [5].

If $H$ is bounded, then it has been shown by Bartnik and Simon [4] that (1.2)
has at least one solution $u\in C^{1}(\Omega)\cap W^{2,2}(\Omega)$. Also, when
$\Omega$ is a ball or an annulus in $\mathbb{R}^{N}$ and the nonlinearity $H$
has a radial structure, then it has been proved in [6] that (1.2) has at least
one classical radial solution. This can be seen as a universal existence
result for the above problem in the radial case.

Very recently, Bereanu, Jebelean and Torres [7] use Leray-Schauder degree
arguments and critical point theory for convex, lower semicontinuous
perturbations of $C^{1}$-functionals, proved existence of classical positive
radial solutions for Dirichlet problems

$\mathcal{M}v+f(|x|,v)=0\ \
~{}~{}~{}\text{in}~{}~{}B(R),~{}~{}~{}v=0~{}~{}~{}\text{on}~{}~{}\partial
B(R),$

under the condition $(H_{1})$ and

$\lim_{s\to 0}\frac{f(r,s)}{\phi_{1}(s)}=\infty\ \ \ \ \text{uniformly for}\
r\in[0,R].$ $None$

Bereanu, Jebelean and Torres [8] used the upper and lower solutions and Leray-
Schauder degree type arguments to study the special case of

$\mathcal{M}v+\lambda\mu(|x|)v^{q}=0\ \
~{}~{}~{}\text{in}~{}~{}B(R),~{}~{}~{}v=0~{}~{}~{}\text{on}~{}~{}\partial
B(R),$ $None$

under the condition

($H_{2}$) $N\geq 2$ is an integer, $R>0,\;q>1$ and
$\mu:[0,\infty)\to\mathbb{R}$ is continuous, $\mu(r)>0$ for all $r>0$.

They proved that there exists $\Lambda>0$ such that problem (1.4) has zero, at
least one or at least two positive radial solutions according to
$\lambda\in(0,\Lambda)$, $\lambda=\Lambda$ or $\lambda>\Lambda$. Moreover,
$\Lambda$ is strictly decreasing with respect to $R$.

However, the existence of sign-changing radial solutions for (1.1) has been
scarcely explored in the related literature, see Capietto, Dambrosio and
Zanolin [10].

When dealing with radial solutions to (1.1) on a ball, one is led to study
(setting $|x|=r$) the BVP

$\left\\{\begin{aligned} &(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}+\lambda
r^{N-1}f(r,u)=0,\ \ \ r\in(0,R),\\\ &u^{\prime}(0)=0=u(R).\\\
\end{aligned}\right.$ $None$

Capietto, Dambrosio and Zanolin [10] used a degree approach combined with a
time-map technique (i.e. Abstract continuation theorem in [19]) to establish
the following

Theorem A. [10, Theorem 3.2] Assume that

$(H_{f})$ $f(r,0)\equiv 0$ and

$\lim_{s\to 0}\frac{f(r,s)}{\phi_{1}(s)}=\infty,\ \ \ \ \text{uniformly in}\
r\in[0,R];$

$(H_{F})$ $F(r,s)=\int^{s}_{0}f(r,x)dx$, and $F(r,s)$ is differentiable with
respect to $r\in[0,R]$ and there exists a continuous function
$\omega:[0,R]\to\mathbb{R}^{+}$ such that

$\Big{|}\frac{\partial F}{\partial r}(r,s)\Big{|}\leq\omega(r)F(r,s),\ \ \ \
r\in[0,R],\ s\in(-\alpha,\alpha).$

Then, there exists $n_{0}$ such that for every $n>n_{0}$, $(1.5)_{\lambda=1}$
has at least two solutions $u^{+}_{n}$ and $v^{-}_{n}$ with $u^{+}_{n}(0)>0$,
$v^{-}_{n}(0)<0$, all having exactly $n$ zeros in $[0,R)$. Moreover, we have

$\lim_{n\to\infty}\;|u^{+}_{n}(r)|+|(u_{n}^{+})^{\prime}(r)|=0=\lim_{n\to\infty}\;|v^{-}_{n}(r)|+|(v_{n}^{-})^{\prime}(r)|\
\ \ \ \text{uniformly in}\ r\in[0,R].$

Motivated above papers, in this paper, we investigate the global structure of
radial solutions, with prescribed nodal properties, to Dirichlet problem (1.1)
by the unilateral global bifurcation theory of Dancer [16-17] and López-Gómez
[24, Sections 6.4, 6.5] and some preliminary results on the superior limit of
a sequence of connected components due to Ma and An [26-27]. We shall use the
following assumptions

(A1) $R\in(0,\infty)$ and $\delta\in[0,R)$,
$f:[0,R]\times(-\alpha,\alpha)\to\mathbb{R}$ is a continuous function, with
$R<\alpha\leq\infty$ and such that $f(r,s)s>0$ for $r\in[0,R]$ and
$s\in(-\alpha,0)\cup(0,\alpha)$.

(A2) $\lim\limits_{s\to 0}\frac{f(r,s)}{s}=m(r)$ uniformly $r\in[\delta,R]$
for some $m\in C[\delta,R]$ and

$m(r)\geq 0,\ \ m(r)\not\equiv 0\ \ \text{on any subinterval of}\ [\delta,R];$

(A3) $f(r,0)\equiv 0$ and

$f_{0}:=\lim_{s\to 0}\frac{f(r,s)}{\phi_{1}(s)}=\infty,\ \ \ \ \text{uniformly
in}\ r\in[\delta,R];$

(A4) $F(r,s)$ is differentiable with respect to $r\in[\delta,R]$ and there
exists a continuous function $\omega:[\delta,R]\to\mathbb{R}^{+}$ such that

$\Big{|}\frac{\partial F}{\partial r}(r,s)\Big{|}\leq\omega(r)F(r,s),\ \ \ \
r\in[\delta,R],\ s\in(-\alpha,\alpha).$

To study the global structure of radial solutions of problem (1.1), we need to
study the family of auxiliary problems

$\left\\{\begin{aligned} &(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}+\lambda
r^{N-1}f(r,u)=0,\ \ \ \ r\in(\delta,R),\\\ &u^{\prime}(\delta)=0=u(R).\\\
\end{aligned}\right.$ $None$

For given $\delta\in[0,R)$. Let

$X_{\delta}=C[\delta,R],\ \ \ \ \ \ E_{\delta}=\\{u\in
C^{1}[\delta,R]:\,u^{\prime}(\delta)=u(R)=0\\}$

be the Banach spaces endowed with the normals

$||u||_{C[\delta,R]}=\sup_{r\in[\delta,R]}|u(r)|,\ \ \ \ \
||u||_{C^{1}[\delta,R]}=\sup_{r\in[\delta,R]}|u(r)|+\sup_{r\in[\delta,R]}|u^{\prime}(r)|,$

respectively. Denoted by $\Sigma_{\delta}$ be the closure of the set

$\\{(\lambda,u)\in[0,\infty)\times C^{1}[\delta,R]:u\ \text{satisfies}\
(1.6)_{\delta},\ \text{and}\ u\not\equiv 0\\}$

in $\mathbb{R}\times E_{\delta}$. Let $S^{+}_{k,\delta}$ denote the set of
function $u\in E_{\delta}$, which have exactly $k-1$ non-degenerate nodal
zeros in $(\delta,R)$ and there exists $\sigma_{1}>0$ such that $u$ is
positive in $(\delta,\delta+\sigma_{1})$, and set
$S_{k,\delta}^{-}=-S_{k,\delta}^{+}$, and $S_{k,\delta}=S^{+}_{k,\delta}\cup
S^{-}_{k,\delta}$. It is clear that $S^{+}_{k,\delta}$ and $S^{-}_{k,\delta}$
are disjoint and open in $E_{\delta}$. Finally, let
$\Phi^{\pm}_{k,\delta}=\mathbb{R}\times S^{\pm}_{k,\delta}$ and
$\Phi_{k,\delta}=\mathbb{R}\times S_{k,\delta}$ under the product topology.
Denoted by $\theta$ be the zero element in $E_{\delta}$.

Let $\delta\in[0,R)$ be given and let $\lambda_{k}(m,\delta)$ be the $k$-th
eigenvalue of

$\left\\{\begin{aligned} &-(r^{N-1}u^{\prime})^{\prime}=\lambda r^{N-1}m(r)u,\
\ \ \ r\in(\delta,R),\\\ &u^{\prime}(\delta)=u(R)=0.\\\ \end{aligned}\right.$
$None$

The main results of the paper are as follows

Theorem 1.1. Assume that (A1) and (A2) hold. Then for $\nu\in\\{+,-\\}$ and
$k\in\mathbb{N}$, there exists a connected component
$\zeta^{\nu}\in\Sigma_{\delta}$, such that

(a)
$\big{(}\zeta^{\nu}\setminus\\{(\lambda_{k}(m,\delta),\theta)\\}\big{)}\subset\big{(}(0,\infty)\times\text{int}\,\Phi^{\nu}_{k,\delta}$);

(b) $\zeta^{\nu}$ joins $(\lambda_{k}(m,\delta),\theta)$ with infinity in
$\lambda$ direction;

(c)
$\text{Proj}_{\mathbb{R}}\,\zeta^{\nu}=[\lambda_{*},\infty)\subset(0,\infty)$
for some $\lambda_{*}>0$.

Theorem 1.2. Let $\delta\in[0,R)$ be given. Assume that (A1), (A3) and (A4)
hold. Then for $\nu\in\\{+,-\\}$ and $k\in\mathbb{N}$, there exists a
connected component $\zeta^{\nu}\in\Sigma_{\delta}$ such that

(a)
$\big{(}\zeta^{\nu}\setminus\\{(0,\theta)\\}\big{)}\subset\big{(}(0,\infty)\times\text{int}\,\Phi^{\nu}_{k,\delta}\big{)}$;

(b) $\zeta^{\nu}$ joins $(0,\theta)$ with infinity in $\lambda$ direction;

(c) $\text{Proj}_{\mathbb{R}}\,\zeta^{\nu}=[0,\infty)$.

Obviously, as the immediate consequences of Theorem 1.1-1.2, we have the
following

Corollary 1.1. Let $\delta\in[0,R)$ be given. Assume that (A1) and (A2) hold.
Then for $\nu\in\\{+,-\\}$ and $k\in\mathbb{N}$, there exists
$\lambda_{\ast}^{\nu}\in(0,\lambda_{k}(m,\delta)]$ such that, for all
$\lambda\in(0,\lambda_{\ast}^{\nu})$, the problem $(1.6)_{\delta}$ has no
solution in $S^{\nu}_{k,\delta}$ and, for all $\lambda>\lambda_{k}(m,\delta)$
has at least one solution in $S^{\nu}_{k,\delta}$.

Corollary 1.2. Let $\delta\in[0,R)$ be given. Assume that (A1), (A3) and (A4)
hold. Then for $\nu\in\\{+,-\\}$ and $n\in\mathbb{N}$, the problem
$(1.6)_{\delta}$ has at least one solution $u_{n}^{\nu}\in S^{\nu}_{n,\delta}$
for $\lambda=1$. Moreover,

$\lim_{n\to\infty}\;|u^{\nu}_{n}(r)|+|(u_{n}^{\nu})^{\prime}(r)|=0\ \
\text{uniformly in}\ r\in[0,R].$

Remark 1.1. Corollary 1.2 guarantees that for each $n\in\mathbb{N}$,
$(1.6)_{\delta}$ has two solutions $u^{+}_{n}$ and $v^{-}_{n}$ with
$u^{+}_{n}(0)>0$, $v^{-}_{n}(0)<0$, for all $\lambda>0$ and $\delta\in[0,R)$.
[11, Theorem 3.2] deal with the more general problem

$\left\\{\begin{aligned} &\text{div}\big{(}a(|\nabla u|)\nabla
u\big{)}+f(|x|,u)=0~{}~{}~{}\text{in}~{}~{}B(R),\\\
&u=0~{}~{}~{}\text{on}~{}~{}\partial B(R),\\\ \end{aligned}\right.$ $None$

where $a:[0,\epsilon_{1}]\to[0,+\infty),\;(\epsilon_{1}>0)$. However, they
only proved that there exists $n_{0}\in\mathbb{N}$ such that for $n\geq
n_{0}$, (1.8) has at least two radial solutions $u^{+}_{n}$ and $v^{-}_{n}$
with $u^{+}_{n}(0)>0$, $v^{-}_{n}(0)<0$, all having exactly $n$ zeros in
$[0,R)$.

Remark 1.2. Notice that $(1.6)_{0}$ a singularity appears for $r=0$. Beside
this intrinsic aspect of $(1.6)_{0}$, assumption (A3) means that
$\frac{\partial f}{\partial s}$ does not exist at $s=0$. Hence, in particular,
when one tries to develop some shooting argument, global existence and
uniqueness to initial value problems associated to the equation in
$(1.6)_{\delta}$ are not guaranteed. This is one of the reasons why few
results for the case $f_{0}=\infty$ is available in the literature: we refer
to the earlier works of Krasnosel’skii, Perov, Povolotskii and Zabreiko [22,
Section 15], Shekhter [34, Section 15] for a more classical approach in the
ODE’s case. For other results, in the PDE’s setting, we also refer to Omari
and Zanolin [31], Willem [37], mainly for the case $a(\cdot)\equiv 1$.

The rest of the paper is organized as follows. In Section 2 we state some
preliminary results on the superior limit of a sequence of connected
components due to Ma and An [26-27] and on the Unilateral global bifurcation
theorem of Dancer [16-17] and López-Gómez [24]. Section 3 is devoted to
establish the existence of connected component of radial solutions for the
prescribed mean curvature problem in an annular domain via global bifurcation
technique. Finally in Section 4, we shall use the components obtained in
Section 3 to construct the desired components of radial solutions for the
prescribed mean curvature problem in a ball and prove Theorem 1.1-1.2 and
their corollaries.

For other results concerning the problem associated to prescribed mean
curvature equations in Minkowski space we refer the reader to [5, 9, 23, 29].
The existence of radial solutions satisfying various boundary conditions has
been investigated by many authors, see Esteban [18], Castro and Kurepa [11],
Grillakis [20], Guo [21], Cheng [13], Ambrosetti, Garcia Azorero and Peral
[2], Njoku, Omari and Zanolin [30], Dai and Ma [15] and references therein.

##  2 Preliminary results

###  2.1 Unbounded connected component

Let $M$ be a metric space and $\\{C_{n}\,|\,n=1,2,\cdots\\}$ a family of
subsets of $M$. Then the superior limit $\mathscr{D}$ of $\\{C_{n}\\}$ is
defined by

$\mathscr{D}:=\limsup\limits_{n\to\infty}C_{n}=\\{x\in M\,|\,\exists\
\\{n_{k}\\}\subset\mathbb{N},\ x_{n_{k}}\in C_{n_{k}},\ \text{such that}\
x_{n_{k}}\to x\\}.$ $None$

Let $X$ be a Banach space with the norm $\|\cdot\|$. A component of a set
$M\subset X$ means a maximal connected subset of $M$, see [36] for the detail.

The following results are somewhat scattered in Ma and An [26-27], Ma and Gao
[28].

Lemma 2.1 ([26, Lemma 2.4; 27, Lemma 2.2]) Let $X$ be a Banach space. Let
$\\{C_{n}\\}$ be a family of closed connected subsets of $X$. Assume that

(i) there exist $z_{n}\in C_{n},\,n=1,2,\cdots$ and $z_{\ast}\in X$ such that
$z_{n}\to z_{\ast}$;

(ii)
$\lim\limits_{n\to\infty}r_{n}=\lim\limits_{n\to\infty}\sup\\{\|u\|\,|\,u\in
C_{n}\\}=\infty$;

(iii) for every $R>0$, $(\bigcup_{n=1}^{\infty}C_{n})\cap B_{R}$ is a
relatively compact of $X$.
Then there exists an unbounded component $\mathscr{C}$ in $\mathscr{D}$ and
$z_{\ast}\in\mathscr{C}$.

###  2.2 Global alternative of Rabinowitz

In order to formulate and prove main results of this section, it is convenient
to introduce Dancer [16-17] and López-Gómez’s notations [24]. Let
$\mathbb{X}=\mathbb{R}\times X$. Given any $\mu\in\mathbb{R}$ and
$0<s<+\infty$, we consider an open neighborhood of $(\mu,0)$ in $\mathbb{X}$
defined by

$\mathbb{B}(\mu,0):=\\{(\mu,u)\in\mathbb{X}:||u||+|\mu|<s\\}.$

A mapping $G:\mathbb{X}\to X$ is said to satisfy Assumption $\mathfrak{A}$ if

(1) $G(0,\lambda)=0$ for $\lambda\in\mathbb{R}$;

(2) $G$ is completely continuous, $G(x,\lambda)=\lambda Lx+H(\lambda,x)$,
where $L$ is a continuous linear operator on $X$;

(3) $||H(\lambda,x)||/||x||\to 0$ as $||x||\to 0$ uniformly on bounded subsets
of $\mathbb{R}$.

Define $\Phi:\mathbb{X}\to X$ by

$\Phi(\lambda,x)=x-G(\lambda,x)$

and

$\mathcal{S}:=\overline{\\{(\mu,u)\in\mathbb{X}:\ \Phi(\mu,u)=0,\ u\neq
0\\}}^{\mathbb{X}}.$

Assume that $\mu\in r(L)$ such that $\mu$ has algebraic multiplicity $1$.
Suppose that $\varphi\in X\setminus\\{0\\}$ such that

$\varphi=\mu L\varphi.$

Let $X_{0}$ be a closed subspace of $X$ such that

$X=\text{span}\\{\varphi\\}\oplus X_{0}.$

Let $C_{\mu}$ to be the component of $\mathcal{S}$ containing $(\mu,0)$.

Theorem B. Global bifurcation of Rabinowitz, see López-Gómez [24, Corollary
6.3.1]
Assume that $\mu\in r(L)$ has algebraic multiplicity $1$. Then, one of the
following non-excluding options occurs. Either
1\. $C_{\mu}$ is unbounded in $\mathbb{R}\times X$.
2\. There exists $\mu_{1}\in r(L)\setminus\\{\mu\\}$ such that $(\mu_{1},0)\in
C_{\mu}$. $\Box$

###  2.3 Unilateral global bifurcation theorem

In this subsection, we shall introduce the unilateral global bifurcation
theorem, see Dancer [16-17] and López-Gómez [24].

According to the Hahn-Banach theorem, there exists a linear functional $l\in
X^{*}$, here $X^{*}$ denotes the dual space of $X$, such that

$l(\varphi)=1,\ \ \ X_{0}=\\{u\in X:\,l(u)=0\\}.$

Finally, for any $0<\eta<1$, we define

$K_{\eta}:=\\{(\mu,u)\in\mathbb{X}:\,|l(u)|>\eta||u||\\}.$

Since

$u\mapsto|l(u)|-\eta||u||$

is continuous, $K_{\eta}$ is an open subset of $\mathbb{X}$ consisting of two
disjoint components $K^{+}_{\eta}$ and $K^{-}_{\eta}$, where

$K^{+}_{\eta}:=\\{(\mu,u)\in\mathbb{X}:\,l(u)>\eta||u||\\}.\ \ $
$K^{-}_{\eta}:=\\{(\mu,u)\in\mathbb{X}:\,l(u)<-\eta||u||\\}.$

In particular, both $K^{+}_{\eta}$ and $K^{-}_{\eta}$ are convex cones,
$K^{+}_{\eta}=-K^{-}_{\eta}$, and $\nu t\varphi\in K^{\nu}_{\eta}$ for every
$t>0$, where $\nu\in\\{+,-\\}$. Applying the similar method to prove [24,
Lemma 6.4.1] with obvious changes, we may obtain the following result.

Lemma 2.2. For every $\eta\in(0,1)$, there exists a number $\delta_{0}>0$ such
that for each $0<\delta<\delta_{0}$,

$((S\setminus\\{(\mu,0)\\})\cap\bar{B}_{\delta}(\mu,0))\subset K_{\eta}.$

Moreover, for each

$(\mu,u)\in(S\setminus\\{(\mu,0)\\})\cap\bar{B}_{\delta}(\mu,0)),$

there are $s\in\mathbb{R}$ and a unique $y\in X_{0}$ such that

$u=s\varphi+y\ \ \ \text{and}\ \ |s|>\eta||u||.$

Furthermore, for these solutions $(\lambda,u)$

$\lambda=\mu+\circ(1)\ \ \ \text{and}\ \ y=\circ(s)$

as $s\to 0$.

Let $\delta>0$ be the constant from Lemma 2.2. For $0<\epsilon<\delta$ we
define $\mathcal{D}^{\nu}_{\mu,\epsilon}$ to be the component of
$\\{(\mu,\theta)\\}\cup(\mathcal{S}\cap\bar{B}_{\epsilon}\cap K^{\nu}_{\eta})$
containing $(\mu,\theta)$, $\mathcal{C}^{\nu}_{\mu,\epsilon}$ to be the
component of $C_{\mu}\setminus\mathcal{D}^{-\nu}_{\mu,\epsilon}$ containing
$(\mu,\theta)$, and $C^{\nu}_{\mu}$ to be the closure of
$\cup_{0<\epsilon\leq\delta}C^{\nu}_{\mu,\epsilon}$. Clearly, $C^{\nu}_{\mu}$
is connected. Thanks to Lemma 2.2, the definition of $C^{\nu}_{\mu}$ is
independent from the choice of $\eta$ and

$C_{\mu}=C^{+}_{\mu}\cup C^{-}_{\mu}.$

Theorem C. Unilateral global bifurcation, see Dancer [16-17]
Either $C^{+}_{\mu}$ and $C^{-}_{\mu}$ are both unbounded or $C^{+}_{\mu}\cap
C^{-}_{\mu}\neq\\{(\mu,\theta)\\}$. $\Box$

###  2.4 Uniqueness of solutions of Cauchy problem

Lemma 2.3. Let $\tau\in(\delta,R)$ be given. Let
$f:[\delta,R]\times\mathbb{R}\to\mathbb{R}$ be continuous and let $f(r,u)$ be
Lipschitz continuous in $u$ on bounded sets, and $f(r,0)\equiv 0$ for
$r\in[\delta,R]$. If $u$ is a solution of

$\left\\{\begin{aligned} &(r^{N-1}\phi(u^{\prime}))^{\prime}+\lambda
r^{N-1}f(r,u)=0,\\\ &u^{\prime}(\tau)=0=u(\tau),\ \ \ \\\
\end{aligned}\right.$ $None$

then $u\equiv 0$ in $[\delta,R]$.

Proof. (2.2) is equivalent to

$\left\\{\begin{aligned} &-(r^{N-1}u^{\prime})^{\prime}=r^{N-1}[\lambda
f(r,u)h(u^{\prime})-\frac{N-1}{r}u^{\prime 3}],\ \ \ \ r\in(\delta,R),\\\
&u^{\prime}(\tau)=0=u(\tau).\\\ \end{aligned}\right.$

Since the function

$g(r,u,p):=r^{N-1}[\lambda f(r,u)(1-p^{2})^{3/2}-\frac{N-1}{r}p^{3}]$

is continuous in $[\delta,R]\times\mathbb{R}\times[-1,1]$ and is Lipschitz in
$(u,p)$ on any bounded subset of $\mathbb{R}\times[-1,1]$, it deduce that
$u\equiv 0$ in $[\delta,R]$. $\Box$

Finally, we consider the case that $f$ does not meet the Lipschitz condition
at $u=0$, i.e. we allow $f_{0}=\infty$.

Lemma 2.4. [10, Lemma 2.3] Let $(A4)$ hold. Let $\epsilon_{0}=0.9$. Then, for
every $\epsilon\leq\epsilon_{0}$ there exists $d_{\epsilon}\in(0,\epsilon]$
such that if $u$ is a (local) solution of

$\left\\{\begin{aligned} &(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}+\lambda
r^{N-1}f(r,u)=0,\\\ &u(\delta)=d,\ \ \ u^{\prime}(\delta)=0\\\
\end{aligned}\right.$

with $|d|\leq d_{\epsilon}$, then $u$ can be defined on $[\delta,R]$ and
$||u||_{C^{1}[\delta,R]}\leq\epsilon$.

Lemma 2.5. [10, Lemma 2.5] Let $(A4)$ hold. Let $u$ be a solution of

$\left\\{\begin{aligned} &(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}+\lambda
r^{N-1}f(r,u)=0,\\\ &u(r_{0})=u^{\prime}(r_{0})=0,\ \ \ \
r_{0}\in(\delta,R].\\\ \end{aligned}\right.$

Then $u\equiv 0$ in $[\delta,R]$. $\Box$

##  3 Radial solutions for the prescribed mean curvature problem in an annular
domain

Let $\delta\in(0,R)$ be a given constant in this section.

Let us consider the following boundary value problem

$\left\\{\begin{aligned} &\text{div}\big{(}\phi_{N}(\nabla v)\big{)}+\lambda
f(|x|,v)=0~{}~{}~{}\text{in}~{}~{}\mathcal{A},\\\ &\frac{\partial
v}{\partial\nu}=0~{}~{}~{}\text{on}~{}~{}\Gamma_{1},\ \ \ \ \ \ \ \ \ \
v=0~{}~{}~{}\text{on}~{}~{}\Gamma_{2},\\\ \end{aligned}\right.$ $None$

where

$\mathcal{A}=\\{x\in\mathbb{R}^{N}:\delta<|x|<R\\},$
$\Gamma_{1}=\\{x\in\mathbb{R}^{N}:|x|=\delta\\},\ \ \ \ \
~{}\Gamma_{2}=\\{x\in\mathbb{R}^{N}:|x|=R\\},$

$\frac{\partial v}{\partial\nu}$ and $|\cdot|$ denote the outward normal
derivative of $v$ and the Euclidean norm in $\mathbb{R}^{N}$, respectively.

Setting, as usual, $|x|=r$ and $v(x)=u(r)$, the above problem (3.1) reduces to

$\left\\{\begin{aligned} &-(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}=\lambda
r^{N-1}f(r,u),\\\ &u^{\prime}(\delta)=0=u(R).\\\ \end{aligned}\right.$ $None$

To find a radial solution of (3.1), it is enough to find a solution of
$(3.2)_{\delta}$.

Remark 3.1. It is worth remarking that $(3.2)_{\delta}$ is equivalent to

$\left\\{\begin{array}[]{ll}-(r^{N-1}u^{\prime})^{\prime}=r^{N-1}[\lambda
f(r,u)h(u^{\prime})-\frac{N-1}{r}u^{\prime 3}],\ \ \ \ r\in(\delta,R),\\\
u^{\prime}(\delta)=0=u(R).\\\ \end{array}\right.$ $None$

Since the nonlinearity $F(r,u,p):=\lambda f(r,u)h(p)-\frac{N-1}{r}p^{3}$ is
singular at $r=0$ if $\delta=0$, we cannot deal with $(3.3)_{0}$ via the
spectrum of $(1.4)_{0}$ directly. However, $F(r,u,p)$ is regular at $r=\delta$
if $\delta>0$, in this case, $(3.3)_{\delta}$ with $\delta>0$ can be treated
via the spectrum of $(1.4)_{\delta}$ and the standard bifurcation technique.
This is why we firstly study the prescribed mean curvature problem in an
annular domain.

Lemma 3.1. Let $u\in S^{+}_{k,\delta}$ be a solution of

$(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}+r^{N-1}f(r,u)=0,\ \ \ r\in(\delta,R)$
$None$

Assume that (A1) holds. Then $u(\delta)>0$.

Proof. Denote by $\tau_{1}$ the first positive zero of $u$. Let
$u(\delta):=d$. Then $u$ satisfies

$\left\\{\begin{aligned}
&(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}+r^{N-1}f(r,u)=0,\ \ \
r\in(\delta,\tau_{1}),\\\ &u^{\prime}(\delta)=0,\ \ \ \ \ u(\tau_{1})=0.\\\
\end{aligned}\right.$ $None$

It is easy to check that (3.5) is equivalent to

$r^{N-1}\phi_{1}(u^{\prime}(r))=-\int^{r}_{\delta}t^{N-1}f(t,u(t))dt,\ \ \ \
r\in[\delta,\tau_{1}]$

it follows $u^{\prime}\leq 0$ because $f(r,s)\geq 0$ for all $r\in[\delta,R]$
and $s\in[0,\alpha)$, so $u$ is decreasing. Since $u(\tau_{1})=0$, we have
$u\geq 0$ on $[\delta,\tau_{1}]$. As $u$ is not identically zero, one has
$u(\delta)>0$, and subsequently, $u^{\prime}<0$ on $(\delta,\tau_{1}]$, which
ensures that actually $u$ is strictly decreasing in $[\delta,\tau_{1}]$ and
$u>0$ on $[\delta,\tau_{1})$. $\Box$

Definition 3.1. For $y\in S_{k}$, denote the zeros of $y$ by

$(\delta<)\,\tau_{1}<\cdots<\tau_{k}\,(=R).$

If $y^{\prime}(\delta)=0$ and, for each $j\in\\{1,\cdots,k-1\\}$, there exists
exactly one $\xi_{j}\in(\tau_{j},\tau_{j+1})$, such that
$y^{\prime}(\xi_{j})=0$, then we call that $y$ possesses the Property
$[P(k)]$.

Let

$\Pi_{k}=\\{y\in C^{1}[\delta,R]:\,y^{\prime}(\delta)=y(R)=0,\,y\
\text{possesses the Property $[P(k)]$}\\}.$

###  3.1 Eigenvalue problem in an annular domain

Let $\delta\in(0,R)$ be given. Let us recall the weighted eigenvalue problem

$\left\\{\begin{array}[]{ll}-(r^{N-1}u^{\prime})^{\prime}=\lambda
r^{N-1}m(r)u,\ \ \ \ r\in(\delta,R),\\\ u^{\prime}(\delta)=0=u(R),\\\
\end{array}\right.$ $None$

where

(A5) $m\in C[\delta,R]$ and $m(r)\geq 0,m(r)\not\equiv 0$ on any subinterval
of $[\delta,R]$.

The following result is a special case of [32, Theorem 1.5.3] when $p=2$.

Lemma 3.2. Let (A5) hold. Then the eigenvalue problem $(3.7)_{\delta}$ has
infinitely many simple real eigenvalues

$0<\lambda_{1}(m,\delta)<\lambda_{2}(m,\delta)<\cdot\cdot\cdot<\lambda_{k}(m,\delta)<\cdots\to+\infty\
\ \text{as}\ \ k\to+\infty$

and no other eigenvalues. Moreover, the algebraic multiplicity of
$\lambda_{k}(m,\delta)$ is $1$, and the eigenfunction $\varphi_{k}$
corresponding to $\lambda_{k}(m,\delta)$ has exactly $k-1$ simple zeros in
$(\delta,R)$.

Let us consider the following auxiliary problem

$\left\\{\begin{array}[]{ll}-(r^{N-1}u^{\prime})^{\prime}=r^{N-1}h(r),\ \ \ \
\ r\in(\delta,R)\ \text{with}\ \delta>0,\\\ u^{\prime}(\delta)=0=u(R)\\\
\end{array}\right.$

for a given $h\in X_{\delta}$. Its Green function for $N\geq 3$ is explicitly
given by

$K_{\delta}(t,s)=\left\\{\begin{aligned} &\frac{1}{2-N}[R^{2-N}-t^{2-N}],\ \ \
\ \ \delta\leq s\leq t\leq R,\\\ &\frac{1}{2-N}[R^{2-N}-s^{2-N}],\ \ \ \ \
\delta\leq t\leq s\leq R.\\\ \end{aligned}\right.$

and its Green function for $N=2$ is explicitly given by

$K_{\delta}(t,s)=\left\\{\begin{aligned} &\ln\frac{R}{t},\ \ \ \ \qquad\
\delta\leq s\leq t\leq R,\\\ &\ln\frac{R}{s},\ \ \ \ \qquad\ \delta\leq t\leq
s\leq R.\\\ \end{aligned}\right.$

It is well-known that for every $h\in X_{\delta}$, the above auxiliary problem
has a unique solution

$u=\int^{R}_{\delta}K_{\delta}(t,s)s^{N-1}h(s)ds=:\mathcal{G}_{\delta}(h)$

It is easy to check that $\mathcal{G}_{\delta}:X_{\delta}\to E_{\delta}$ is
continuous and compact (see [3]).

Define a linear operator $\mathcal{L}_{\delta}:X_{\delta}\to E_{\delta}\
(\hookrightarrow X_{\delta})$.

$\mathcal{L}_{\delta}(u)(r):=\mathcal{G}_{\delta}(mu)(r).$

Then $\mathcal{L}_{\delta}$ is compact and $(3.7)_{\delta}$ is equivalent to

$u=\lambda\mathcal{L}_{\delta}(u).$

Moreover, $\mathcal{L}_{\delta}|_{E_{\delta}}:\ E_{\delta}\to E_{\delta}$ is
compact.

###  3.2 An equivalent formulation

We will use some idea in Coelho et. al. [14]. Let us define a function
$\tilde{f}:[\delta,R]\times\mathbb{R}\to\mathbb{R}$ by setting, for
$r\in[\delta,R]$,

$\tilde{f}(r,s)=\left\\{\begin{aligned} &f(r,s),\ \ \ \ \ \ \ \ \text{if}\ \
0\leq|s|\leq R-\delta,\\\ &0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ \
|s|\geq(R-\delta)+1,\\\ &\text{linear},\ \ \ \ \ \ \ \ \ \text{if}\ \
R-\delta<|s|<(R-\delta)+1.\\\ \end{aligned}\right.$

Since any solution $u$ of $(3.2)_{\delta}$ satisfies

$||u^{\prime}||_{C[\delta,R]}<1,\ \ \ \ ||u||_{C[\delta,R]}<R-\delta,$

it follows that $(3.2)_{\delta}$ is equivalent to the same problem with $f$
replaced by $\tilde{f}$. Clearly, $\tilde{f}$ satisfies all the properties
assumed in the statement of the theorem. In the sequel, we shall replace $f$
with $\tilde{f}$; however, for the sake of simplicity, the modified function
$\tilde{f}$ will still be denoted by $f$. Next, let us define
$h:\mathbb{R}\to\mathbb{R}$ by setting

$h(y)=\left\\{\begin{aligned} &(1-y^{2})^{\frac{3}{2}},\ \ \ \ \text{if}\ \
|y|\leq 1,\\\ &0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ \ |y|>1.\\\
\end{aligned}\right.$

Claim. A function $u\in C^{1}[\delta,R]$ is a solution of $(3.2)_{\delta}$ if
and only if it is a solution of the problem

$\left\\{\begin{array}[]{ll}-(r^{N-1}u^{\prime})^{\prime}=\lambda
r^{N-1}f(r,u)h(u^{\prime})-(N-1)r^{N-2}u^{\prime 3},\ \ \ \ r\in(\delta,R),\\\
u^{\prime}(\delta)=0=u(R).\\\ \end{array}\right.$ $None$

It is clear that a solution $u\in C^{1}[\delta,R]$ of $(3.2)_{\delta}$ is a
solution of $(3.8)_{\delta}$ as well. Conversely, suppose that $u\in
C^{1}[\delta,R]$ is a solution of $(3.8)_{\delta}$ . We aim to show that

$\|u^{\prime}\|_{C[\delta,R]}<1.$

Assume by contradiction that this is not the case. Then we can easily find an
interval $[a,b]\subseteq[\delta,R]$ such that, either $u^{\prime}(a)=0,\
0<|u^{\prime}(r)|<1$ in $(a,b)$ and $|u^{\prime}(b)|=1$, or
$|u^{\prime}(a)|=1,\ 0<|u^{\prime}(r)|<1$ in $(a,b)$ and $u^{\prime}(b)=0$.
Suppose the former case occurs (in the latter one the argument would be
similar). The function $u$ satisfies the equation

$-\big{(}r^{N-1}\frac{u^{\prime}}{\sqrt{1-u^{\prime
2}}}\big{)}^{\prime}=\lambda r^{N-1}f(r,u)$

in $[a,b)$. For each $r\in(a,b)$, integrating over the interval $[a,r]$ and
using (A1), we obtain

$|\phi_{1}(u^{\prime}(r))|=\Big{|}\frac{1}{r^{N-1}}\int_{a}^{r}\lambda
t^{N-1}f(t,u)dt\Big{|}\leq M$

and hence

$|u^{\prime}(r)|\leq\phi_{1}^{-1}(M)$

for every $r\in[a,b)$. Since $\phi_{1}^{-1}(M)<1$, taking the limit as $r\to
b^{-}$ we get $|u^{\prime}(b)|<1$. This is a contradiction. Therefore
$\|u^{\prime}\|_{C[\delta,R]}<1$ and, as a consequence, $u$ is a solution of
$(3.2)_{\delta}$.

###  3.3 Proof of Theorem 1.1-1.2 with $\delta\in(0,R)$

In this subsection, we shall prove Theorem 1.1-1.2 in the case $\delta>0$.

Proof of Theorem 1.1 with $\delta\in(0,R)$. By (A1) and (A2) we can write, for
any $r\in[\delta,R]$ and every $s\in\mathbb{R}$,

$f(r,s)=(m(r)+l(r,s))s,$

where $l:[\delta,R]\times\mathbb{R}\to\mathbb{R}$ is a continuous function and

$\lim_{s\to 0}l(r,s)=0$ $None$

uniformly in $[\delta,R]$. Let us set, for convenience, $k(y)=h(y)-1$ for
$y\in\mathbb{R}$. We have

$\lim_{y\to 0}\frac{k(y)}{y}=0.$ $None$

Define the operator $\mathcal{H}_{\delta}:\mathbb{R}\times E_{\delta}\to
E_{\delta}$ by

$\mathcal{H}_{\delta}(\lambda,u)(\cdot)=\mathcal{G}_{\delta}\big{(}\lambda[l(\cdot,u)+(m(\cdot)+l(\cdot,u))k(u^{\prime})]u-\gamma(\cdot)u^{\prime
3}\big{)}$

where $\gamma(r)=\frac{N-1}{r}$. Clearly, $\mathcal{H}_{\delta}$ is completely
continuous and, by (3.9) and (3.10),

$\lim_{\|u\|_{C^{1}[\delta,R]}\to
0}\frac{\|\mathcal{H}_{\delta}(\lambda,u)\|_{C^{1}[\delta,R]}}{\|u\|_{C^{1}[\delta,R]}}=0,$
$None$

uniformly with respect to $\lambda$ varying in bounded intervals. Observe
that, for any $\lambda$, the couple $(\lambda,u)\in\mathbb{R}\times
E_{\delta}$ is a solution of the equation

$u=\lambda\mathcal{L}_{\delta}(u)+\mathcal{H}_{\delta}(\lambda,u)$ $None$

if and only if $u$ is a solution of $(3.2)_{\delta}$.

Recall that $\Sigma_{\delta}\subset\mathbb{R}\times E_{\delta}$ be the closure
of the set of all nontrivial solutions $(\lambda,u)$ of (3.12) with
$\lambda>0$. Note that the set $\\{u\in
E_{\delta}\,|\,(\lambda,u)\in\Sigma_{\delta}\\}$ is bounded in $E_{\delta}$.

As the algebraic multiplicity of $\lambda_{k}(m,\delta)$ equals 1 [25], the
local index of $0$ as a fixed point of $\lambda\mathcal{L}_{\delta}$ changes
sign as $\lambda$ crosses $\lambda_{k}(m,\delta)$. Therefore, according to a
revised version of [24, Theorem 6.2.1], there exists a component, denoted by
$\mathfrak{C}_{k}\subset\Sigma_{\delta}$, emanating from
$(\lambda_{k}(m,\delta),\theta)$.

Now, we use some notations and preliminary results on Unilateral global
bifurcation.

We shall show that both $\mathfrak{C}^{+}_{k}$ and $\mathfrak{C}_{k}^{-}$ are
unbounded , and $\mathfrak{C}_{k}^{\nu}\subset\Phi^{\nu}_{k}$ for
$\nu\in\\{+,-\\}$.

It is easy to check that (3.12) enjoys the structural requirements for
applying the unilateral global bifurcation theory of Dancer and López-Gómez
(see Theorem C (by a counterexample of Dancer [16], the global unilateral
theorem of Rabinowitz [33] is false as stated. So, it cannot be used).
$\mathfrak{C}_{k}=\mathfrak{C}^{+}_{k}\cup\mathfrak{C}_{k}^{-}$ and either
both $\mathfrak{C}^{+}_{k}$ and $\mathfrak{C}_{k}^{-}$ are unbounded, or

$\mathfrak{C}^{+}_{k}\cap\mathfrak{C}_{k}^{-}\neq\\{(\lambda_{k}(m,\delta),\theta)\\}.$

We claim that the second case can never occur.

In fact, the uniqueness of IVP guarantees that

$\mathfrak{C}^{+}_{k}\subset\Phi^{+}_{k},\ \ \ \
\mathfrak{C}^{-}_{k}\subset\Phi^{-}_{k}.$ $None$

Suppose on the contrary that
$(\eta,z)\in(\mathfrak{C}^{+}_{k}\cap\mathfrak{C}_{k}^{-})$ for some
$(\eta,z)\neq(\lambda_{k}(m,\delta),\theta)$. Then it follows from (3.13) that
$z=\theta$. In this case, $\eta=\lambda_{j}(m,\delta)$ for some $k\neq j$.
Suppose $(\lambda_{m},u_{m})\to(\lambda_{j}(m,\delta),\theta)$ when
$m\to+\infty$ with $(\lambda_{m},u_{m})\in\mathfrak{C}_{k}$. Let
$v_{m}=\frac{u_{m}}{\|u_{m}\|_{C^{1}[\delta,R]}}$, then $v_{m}$ should be a
solution of the problem

$v_{m}=\lambda_{m}\mathcal{L}_{\delta}(v_{m})+\frac{\mathcal{H}_{\delta}(\lambda,u_{m})}{\|u_{m}\|_{C^{1}[\delta,R]}}$
$None$

This together with the compactness of $\mathcal{L}_{\delta}$ and
$\mathcal{H}_{\delta}$ imply that for some convenient subsequence $v_{m}\to
v_{0}\neq 0$ as $m\to+\infty$. Now $v_{0}$ verifies the equation

$-(r^{N-1}v_{0}^{\prime})^{\prime}=\lambda_{j}r^{N-1}m(r)v_{0}$

and $\|v_{0}\|=1$. Hence $v_{0}\in S_{j,\delta}$. Since $S_{j,\delta}$ is an
open in $E_{\delta}$, and as a consequence for some $m$ large enough,
$v_{m}\in S_{j,\delta}$, and this is a contradiction.

Therefore, both $\mathfrak{C}^{+}_{k}$ and $\mathfrak{C}_{k}^{-}$ are
unbounded.

Take

$\zeta^{+}:=\mathfrak{C}^{+},\ \ \ \ \ \ \zeta^{-}:=\mathfrak{C}^{-}.$

Obviously, (a) is true.

(b) can be deduced from the fact that

$\sup\\{||u^{\prime}||_{C[\delta,R]}:\,(\lambda,u)\in\zeta^{\nu}\\}\leq 1,\ \
\ \ \sup\\{||u||_{C[\delta,R]}:\,(\lambda,u)\in\zeta^{\nu}\\}\leq R-\delta.$

(c) Let

$\lambda_{*}^{\nu}:=\inf\\{\lambda:\,(\lambda,u)\in\zeta^{\nu}\\}.$

We claim that $\lambda^{\nu}_{*}\in(0,\infty)$.

Suppose on the contrary that $\lambda^{\nu}_{*}=0$. Then there exists a
sequence $\\{(\mu_{n},u_{n})\\}\subset\zeta^{\nu}$ satisfying $u_{n}>0$, and

$\lim_{n\to\infty}(\mu_{n},u_{n})=(0,u^{*})\ \ \ \ \ \ \ \text{in}\
\mathbb{R}\times X_{\delta}$

for some $u^{*}\geq 0$. Then it follows from

$-(r^{N-1}\phi_{1}(u_{n}^{\prime}))^{\prime}=\mu_{n}r^{N-1}f(r,u_{n}),\ \ \ \
\ \ u_{n}^{\prime}(\delta)=0=u_{n}(R)$

that, after taking a subsequence and relabeling, if necessary,

$u_{n}\to 0.$

On the other hand,

$\left\\{\begin{array}[]{ll}-(r^{N-1}u_{n}^{\prime})^{\prime}=\mu_{n}r^{N-1}f(r,u_{n})h(u_{n}^{\prime})-(N-1)r^{N-2}u_{n}^{\prime
3},\ \ \ \ \ r\in(\delta,R),\\\ u_{n}^{\prime}(\delta)=0=u_{n}(R).\\\
\end{array}\right.$

Setting, for all $n$, $v_{n}=u_{n}/||u_{n}||_{C[\delta,R]}$, we have that

$\left\\{\begin{array}[]{ll}-(r^{N-1}v_{n}^{\prime})^{\prime}=\mu_{n}r^{N-1}\frac{f(r,u_{n})}{u_{n}}h(u_{n}^{\prime})v_{n}-(N-1)r^{N-2}u_{n}^{\prime
2}\,v_{n}^{\prime},\ \ \ \ r\in(\delta,R),\\\
v_{n}^{\prime}(\delta)=0=v_{n}(R).\\\ \end{array}\right.$ $None$

Notice that

$r^{N-1}\phi_{1}(u_{n}^{\prime}(r))=-\mu_{n}\int^{r}_{\delta}\tau^{N-1}f(\tau,u_{n}(\tau))d\tau,\
\ \ \ \ r\in[\delta,R].$

This together with $f(r,0)=0$ for $r\in[\delta,R]$ imply that

$\lim_{n\to\infty}\,||u^{\prime}_{n}||_{C[\delta,R]}=0.$

Combining this with (3.15) and the facts $f_{0}=m(r),\ u_{n}\to 0$ and
$\lim_{n\to\infty}h(u_{n}^{\prime})=1$, it concludes that
$\mu_{n}\to\lambda_{k}(m,\delta)$. This is a contradiction. $\Box$

In the following, we will deal with the case that $f_{0}=\infty$.

Define $f^{[n]}:[\delta,R]\times\mathbb{R}\to\mathbb{R}$ as follows

$f^{[n]}(r,s)=\left\\{\begin{array}[]{lll}nf(r,\frac{1}{n})s,&|s|\in[0,\frac{1}{n}],\\\
f(r,s),&|s|\in(\frac{1}{n},\infty).\\\ \end{array}\right.$

Then $f^{[n]}$ is continuous and satisfies (A1) and

$(f^{[n]})_{0}=nf(r,\frac{1}{n})=f(r,\frac{1}{n})/(1/n)=:m^{[n]}(r)\ \ \
\text{uniformly for}\ r\in[\delta,R].$

Now, let us consider the auxiliary family of the problems

$\left\\{\begin{array}[]{ll}-(r^{N-1}u^{\prime})^{\prime}=\lambda
r^{N-1}f^{[n]}(r,u)h(u^{\prime})-(N-1)r^{N-2}u^{\prime 3},\ \ \ \
r\in(\delta,R),\\\ u^{\prime}(\delta)=0=u(R).\\\ \end{array}\right.$ $None$

From the definition of $f^{[n]}$, it follows that for $r\in[\delta,R]$ and
every $u\in\mathbb{R}$,

$f^{[n]}(r,s)=(m^{[n]}(r)+\xi^{[n]}(r,s))s,$

where $\xi^{[n]}:[\delta,R]\times\mathbb{R}\to\mathbb{R}$ is continuous and

$\lim\limits_{s\to 0}\xi^{[n]}(r,s)=0\ \ \quad\text{uniformly for}\
r\in[\delta,R].$

Let us set, for convenience, $k(v)=h(v)-1$ for $v\in\mathbb{R}$. We have

$\lim\limits_{v\to 0}\frac{k(v)}{v}=0.$ $None$

Define the operator $\mathcal{H}_{\delta}^{[n]}:\mathbb{R}\times E_{\delta}\to
E_{\delta}$ by

$\mathcal{H}_{\delta}^{[n]}(\lambda,u)=\mathcal{G}_{\delta}\Big{(}\lambda\big{(}\xi^{[n]}(\cdot,u)+[m^{[n]}+\xi^{[n]}(\cdot,u)]k(u^{\prime})\big{)}u-\gamma(\cdot)u^{\prime
3}\Big{)}.$

Clearly, $\mathcal{H}_{\delta}^{[n]}$ is completely continuous and by (3.16)
and (3.17), it follows that

$\lim\limits_{\|u\|_{C^{1}[\delta,R]}\to
0}\frac{\|\mathcal{H}_{\delta}^{[n]}(\lambda,u)\|_{C^{1}[\delta,R]}}{\|u\|_{C^{1}[\delta,R]}}=0,$

uniformly with respect to $\lambda$ varying in bounded intervals. Observe
that, for any $\lambda$, the couple $(\lambda,u)\in\mathbb{R}\times
E_{\delta}$ is a solution of the equation

$u=\lambda\mathcal{L}_{\delta}^{[n]}(u)+\mathcal{H}_{\delta}^{[n]}(\lambda,u)$
$None$

if and only if $u$ is a solution of (3.16), here
$\mathcal{L}_{\delta}^{[n]}:X_{\delta}\to E_{\delta}$ be defined by
$\mathcal{L}_{\delta}^{[n]}(u)=\mathcal{G}_{\delta}(m^{[n]}u)$.

Let $\Sigma_{\delta}^{[n]}\subset\mathbb{R}\times E_{\delta}$ be the closure
of the set of all nontrivial solutions $(\lambda,u)$ of (3.18) with
$\lambda>0$. Note that the set $\\{u\in
E_{\delta}|(\lambda,u)\in\Sigma_{\delta}^{[n]}\\}$ is bounded in $E_{\delta}$.

Remark 3.2. Note that from the compactness of the embedding
$E_{\delta}\hookrightarrow X_{\delta}$, it concluded that
$\mathfrak{C}_{+}^{[n]}$ is also an unbounded connected component in
$[0,\infty)\times X_{\delta}$.

Proof of Theorem 1.2 with $\delta\in(0,R)$. Similar to the proof of Theorem
1.1 with $\delta\in(0,R)$, for each fixed $n$ and $\nu\in\\{+,-\\}$, there
exists an unbounded component
$\mathfrak{C}_{\nu,k}^{[n]}\subset\Sigma_{\delta}^{[n]}$ of solutions of
(3.18) joining $(\lambda_{k}(m^{[n]},\delta),0)\in\mathfrak{C}_{\nu,k}^{[n]}$
to infinity in $[0,\infty)\times\mathcal{S}_{k,\delta}$. Moreover,
$(\lambda_{k}(m^{[n]},\delta),0)\in\mathfrak{C}_{\nu,k}^{[n]}$ is the a
bifurcation point of (3.18) lying on a trivial solution line $u\equiv 0$ and
the component $\mathfrak{C}_{\nu,k}^{[n]}\subset\Phi^{\nu}_{k}$ joins the
infinity in the direction of $\lambda$ since $u$ is bounded.

It is not difficult to verify that $\mathfrak{C}^{[n]}_{\nu,k}$ satisfies all
conditions in Lemma 2.1 and consequently
$\limsup\limits_{n\to\infty}\mathfrak{C}^{[n]}_{\nu,k}$ contains a component
$\mathfrak{C}_{\nu,k}$ which is unbounded.

From (A3), it follows that for $r\in[\delta,R]$,

$\lim\limits_{n\to\infty}\frac{f^{[n]}(r,u)}{u}=\lim\limits_{n\to\infty}\frac{f(r,\frac{1}{n})}{1/n}=\infty,$

and consequently,

$\lim\limits_{n\to\infty}\lambda_{k}(m^{[n]},\delta)=0.$ $None$

Thus, from (3.19), we have that the component $\mathfrak{C}_{\nu,k}$ joins
$(0,\theta)$ with infinity in the direction of $\lambda$ in $[0,\infty)\times
S_{k}^{\nu}$.

We claim that

$(\mathfrak{C}_{\nu,k}\setminus\\{(0,\theta)\\})\subset(0,\infty)\times
S^{\nu}_{k}.$ $None$

Suppose on the contrary that there exists a sequence
$\\{(\mu_{n},u_{n})\\}\subset\mathfrak{C}_{\nu,k}$ satisfying

$\lim_{n\to\infty}(\mu_{n},u_{n})=(\mu*,\theta)\ \ \ \ \ \ \ \text{in}\
\mathbb{R}\times X_{\delta}$

for some $\mu*>0$. Then

$\left\\{\begin{array}[]{ll}-(r^{N-1}u_{n}^{\prime})^{\prime}=\mu_{n}r^{N-1}f^{[n]}(r,u_{n})h(u_{n}^{\prime})-(N-1)r^{N-2}u_{n}^{\prime
3},\ \ \ \ \ r\in(\delta,R),\\\ u_{n}^{\prime}(\delta)=0=u_{n}(R).\\\
\end{array}\right.$

Setting, for all $n$, $v_{n}=u_{n}/||u_{n}||_{C[\delta,R]}$, we have that

$\left\\{\begin{array}[]{ll}-(r^{N-1}v_{n}^{\prime})^{\prime}=\mu_{n}r^{N-1}\frac{f^{[n]}(r,u_{n})}{u_{n}}h(u_{n}^{\prime})v_{n}-(N-1)r^{N-2}u_{n}^{\prime
2}\,v_{n}^{\prime},\ \ \ \ r\in(\delta,R),\\\
v_{n}^{\prime}(\delta)=0=v_{n}(R).\\\ \end{array}\right.$ $None$

Notice that

$r^{N-1}\phi_{1}(u_{n}^{\prime}(r))=-\mu_{n}\int^{r}_{0}\tau^{N-1}f^{[n]}(\tau,u_{n}(\tau))d\tau,\
\ \ \ \ r\in[0,R].$ $None$

This together with $f^{[n]}(r,0)=0$ for $r\in[\delta,R]$ imply that

$\lim_{n\to\infty}\,||u^{\prime}_{n}||_{C[0,R]}=0.$ $None$

Combining this with (3.21) and the facts $f_{0}=\infty$ and
$\lim_{n\to\infty}h(u_{n}^{\prime})=1$, it concludes that $\mu^{*}=0$. This is
a contradiction.

Therefore, (3.20) holds. $\Box$

##  4 Radial solutions for the prescribed mean curvature problem in a ball

In this section, we shall deal with $(1.6)_{\delta}$ with $\delta=0$.

Let

$g_{n}(r,s)=\left\\{\begin{aligned} 0,\qquad\ \ \ \ \ \ \ \ \ \ \ \
&(r,s)\in(0,\frac{1}{n}]\times(-\alpha,\alpha),\\\ f(r-\frac{1}{n},s),\ \ \ \
\ \ &(r,s)\in(\frac{1}{n},R)\times(-\alpha,\alpha).\\\ \end{aligned}\right.$
$None$

In the following, we shall use the solutions of the family of problems

$\left\\{\begin{aligned} &-(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}=\lambda
r^{N-1}g_{n}(r,u),\ \ \ \ \ r\in(\frac{1}{n},R),\\\ &\
u^{\prime}(1/n)=0=u(R),\\\ \end{aligned}\right.$ $None$

to construct the radial solutions of the prescribed mean curvature problem in
a ball

$\mathcal{M}v+\lambda f(|x|,v)=0\ \
~{}~{}~{}\text{in}~{}~{}\mathcal{B}(R),~{}~{}~{}v=0~{}~{}~{}\text{on}~{}~{}\partial\mathcal{B}(R).$
$None$

To find a radial solution of (4.3), it is enough to find a solution of the
problem

$\left\\{\begin{aligned} &-(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}=\lambda
r^{N-1}f(r,u),\\\ &\ u^{\prime}(0)=0=u(R).\\\ \end{aligned}\right.$ $None$

For given $n\in\mathbb{N}$ , let $(\lambda,u)$ be a solution of $(4.2)_{n}$.
For each $n$, define a function $y_{n}:[0,R]\to[0,\infty)$ by

$y_{n}(r)=\left\\{\begin{aligned} u(r),\ \ \ \ \ \ \ \ \ \ \ &\frac{1}{n}\leq
r\leq R,\\\ u(\frac{1}{n}),\ \ \ \ \ \ \ \ \ \ &0\leq r\leq\frac{1}{n}.\\\
\end{aligned}\right.$ $None$

Then

$y_{n}\in\\{w\in C^{2}[0,R]:\,w^{\prime}(0)=w(R)=0\\}.$

Moreover, $y_{n}$ is a solution of the problem

$\left\\{\begin{aligned} &-(r^{N-1}\phi_{1}(u^{\prime}))^{\prime}=\lambda
r^{N-1}g_{n}(r,u),\ \ \ \ \ r\in(0,R)\\\ &\ u^{\prime}(0)=0=u(R),\\\
\end{aligned}\right.$ $None$

i.e. $y_{n}$ is a solution of the problem

$\left\\{\begin{aligned}
&-(r^{N-1}u^{\prime}(r))^{\prime}+(N-1)r^{N-2}[u^{\prime}(r)]^{3}=\lambda
r^{N-1}g_{n}(r,u(r))h(u^{\prime}(r)),\ \ \ \ r\in(0,R),\\\
&u^{\prime}(0)=u(R)=0.\end{aligned}\right.$ $None$

On the other hand, if $(\lambda,y)$ is a solution of $(4.7)_{n}$, then
$(\lambda,y|_{[\frac{1}{n},R]})$ is a solution of $(4.2)_{n}$.

Lemma 4.1. Let (A1) and (A2) hold. Let
$\hat{\lambda}:\,\hat{\lambda}\neq\lambda_{k}(m,0)$ be given. Then there
exists $\hat{b}>0$, such that

$||u||_{C[0,R]}\geq\hat{b}$

for any solution $(\hat{\lambda},u)\in\Phi_{k,0}^{\nu}$ of $(4.7)_{n}$. Here
$b$ is independent of $n$ and $u$.

Proof. Suppose on the contrary that $(4.7)_{n},\ n\in\mathbb{N}$, has a
sequence of solution $(\hat{\lambda},y_{j})\in\Phi_{k,0}^{\nu}$ with

$\lim_{j\to\infty}\,||y_{j}||_{C[0,R]}=0.$ $None$

Then

$\left\\{\begin{aligned}
&(r^{N-1}\phi_{1}(y_{j}^{\prime}(r)))^{\prime}+\hat{\lambda}r^{N-1}g_{n}(r,y_{j}(r))=0,\
\ \ \ r\in(0,R),\\\ &y_{j}^{\prime}(0)=y_{j}(R)=0,\\\ \end{aligned}\right.$
$None$

and consequently,

$r^{N-1}\phi_{1}(y_{j}^{\prime}(r))=-\hat{\lambda}\int^{r}_{0}\tau^{N-1}g_{n}(\tau,y_{j}(\tau))d\tau,\
\ \ \ r\in[0,R].$

This together with (4.8) and the fact that $g_{n}(r,0)=0$ for $r\in[0,R]$
imply that

$\lim_{j\to\infty}\,||y^{\prime}_{j}||_{C[0,R]}=0.$ $None$

Recall that (4.9) can be rewritten as

$\left\\{\begin{aligned}
&-(r^{N-1}y_{j}^{\prime}(r))^{\prime}+(N-1)r^{N-2}[y_{j}^{\prime}(r)]^{3}=\hat{\lambda}r^{N-1}g_{n}(r,y_{j}(r))h(y_{j}^{\prime}(r)),\\\
&y_{j}^{\prime}(0)=y_{j}(R)=0.\end{aligned}\right.$ $None$

Setting, for all $j$, $v_{j}=y_{j}/||y_{j}||_{C[0,R]}$ , we have that

$\left\\{\begin{aligned}
&-(r^{N-1}v_{j}^{\prime}(r))^{\prime}+(N-1)r^{N-2}[y_{j}^{\prime}(r)]^{2}v_{j}^{\prime}(r)=\hat{\lambda}r^{N-1}\frac{g_{n}(r,y_{j}(r))}{y_{j}(r)}v_{j}(r)h(y_{j}^{\prime}(r)),\\\
&v_{j}^{\prime}(0)=v_{j}(R)=0.\end{aligned}\right.$ $None$

Letting $j\to\infty$, it follows from (4.8), (4.10) and $(4.12)_{n}$ that
there exists $w\in C^{2}[0,R]$ with $||w||_{C[0,R]}=1$, such that

$\left\\{\begin{aligned}
&-(r^{N-1}w^{\prime}(r))^{\prime}=\hat{\lambda}r^{N-1}m(r)w(r),\\\
&w^{\prime}(0)=w(R)=0,\end{aligned}\right.$ $None$

which implies that $\hat{\lambda}=\lambda_{k}(m,0)$. However, this contradicts
the assumption $\hat{\lambda}\neq\lambda_{k}(m,0)$. $\Box$

Using the same argument with obvious changes, we may prove the following Lemma
4.2. Let (A1) and (A3) hold. Let $\hat{\lambda}\in(0,\infty)$ be given. Then
there exists $\hat{b}>0$, such that

$||u||_{C[0,R]}\geq\hat{b}$

for any solution $(\hat{\lambda},u)\in\Phi^{\nu}_{k,0}$ of $(4.7)_{n}$. $\Box$

Now, we are in the position to prove Theorem 1.1-1.2 with $\delta=0$.

Proof of Theorem 1.1 with $\delta=0$. For given $n$, let $\xi_{n}$ be the
component obtained by Theorem 1.1 with $\delta\in(0,R)$ for $(4.2)_{n}$. Let

$\zeta_{n}:=\\{(\lambda,y_{n}):\,\ y_{n}\ \text{is determined by}\ u\
\text{via}\ (4.5)_{n}\ \text{for}\ (\lambda,u)\in\xi_{n}\\}.$

Then $\zeta_{n}$ is a component in $[0,\infty)\times C^{1}[0,R]$ which joins
$(\lambda_{k}(m^{[n]},\frac{1}{n}),\theta)$ with infinity in the direction of
$\lambda$ and

$\sup\\{||y||_{C^{1}[0,R]}:(\lambda,y)\in\xi_{n}\\}<M$ $None$

for some constant $M>0$, independent of $y$ and $n$. Here

$m^{[n]}(r):=m(r-\frac{1}{n}),\ \ \ \ \ \ \ \ \ \ \ \frac{1}{n}\leq r\leq R,$

and $\lambda_{k}(m^{[n]},\frac{1}{n})$ is the principal eigenvalue of the
linear problem

$\left\\{\begin{aligned} &-(r^{N-1}u^{\prime}(r))^{\prime}=\lambda
r^{N-1}m^{[n]}(r)u(r),\ \ \ \ r\in(\frac{1}{n},R),\\\
&u^{\prime}(\frac{1}{n})=u(R)=0.\end{aligned}\right.$ $None$

Since $\lim_{n\to\infty}\,\lambda_{k}(m^{[n]},\frac{1}{n})=\lambda_{k}(m,0)$,
it follows from Lemma 2.1 that there exists a component $\zeta$ in
$\underset{n\to\infty}{\limsup}\;\zeta_{n}$ which joins
$(\lambda_{k}(m,0),\theta)$ with infinity in the direction of $\lambda$ and

$\sup\\{||y||_{C^{1}[0,R]}:(\lambda,y)\in\zeta\\}\leq M.$ $None$

Now, Lemma 4.1 ensures that

$\zeta\cap\big{(}[0,\infty)\times\\{\theta\\}\big{)}=\\{(\lambda_{k}(m,0),\theta)\\}.$

$\Box$

Proof of Theorem 1.2 with $\delta=0$. It is an immediate consequence of
Theorem 1.2 with $\delta>0$ and Lemma 4.2. $\Box$

The proof of Corollary 1.1 is a direct consequence of Theorem 1.1.

Proof of Corollary 1.2. From Theorem 1.2, for each $\nu\in\\{+,-\\}$ and
$n\in\mathbb{N}$, $(1.6)_{\delta}$ has a solution $u_{n}^{\nu}\in
S^{\nu}_{n,\delta}$. We only need to show that

$\lim_{n\to\infty}\;|u^{\nu}_{n}(r)|+|(u_{n}^{\nu})^{\prime}(r)|=0\ \
\text{uniformly in}\ r\in[0,R].$

We firstly deal with the case that $\delta>0$. In the following, we shall
replace $u_{n}^{\nu}$ with $u_{n}$ for fixed $\nu\in\\{+,-\\}$.

Step 1 We show that

$\lim_{n\to\infty}||u_{n}||_{C[\delta,R]}=0.$ $None$

Let $\tau_{1}(n),\cdots,\tau_{n}(n)$ be zeros of $u_{n}$ in $(0,R]$:

$\big{(}\delta=\tau_{0}(n)<\big{)}\,\tau_{1}(n)<\cdots<\tau_{n-1}(n)<\tau_{n}(n)\;\big{(}=R\big{)}.$
$None$

Since $||u_{n}^{\prime}||_{C[\delta,R]}<1$, it follows that

$||u_{n}||_{C[\delta,R]}\leq\sup\\{|\tau_{j+1}(n)-\tau_{j}(n)|:\;j=0,\cdots,n-1\\}.$
$None$

Suppose on the contrary that (4.17) is not true. Then there exist a positive
constant $\sigma_{0}$ and a subsequence of solution of $(1.6)_{\delta}$,
$\\{(1,u_{n_{j}})\\}\subseteq\\{(1,u_{n})\\}$, such that

$||u_{n_{j}}||_{C[\delta,R]}\geq\sigma_{0}.$ $None$

It follows from (4.20) and (4.19) that for $n_{j}\geq 6$, there exist
$t_{1}(n_{j}),t_{2}(n_{j})\in\\{\tau_{0}(n_{j}),\tau_{1}(n_{j}),\cdots,$
$\tau_{n_{j}}(n_{j})\\}$ satisfying

$t_{2}(n_{j})-t_{1}(n_{j})\geq\sigma_{0},$ $None$
$||u_{n_{j}}||_{C[\delta,R]}=\sup\\{|u_{n_{j}}(r)|:\,r\in[t_{1}(n_{j}),t_{2}(n_{j})]\\},$
$None$ $u_{n_{j}}(r)\neq 0,\ \ \ \ r\in(t_{1}(n_{j}),t_{2}(n_{j})).$ $None$

Without loss of generality, we may assume that there exists a closed
subinterval $I_{1}$ with $\text{meas}\,I_{1}\geq\frac{\sigma_{0}}{2}$, such
that (after taking a subsequence and relabeling, if necessary,)

$u_{n_{j}}(r)\neq 0,\ \ \ \ \ r\in I_{1},\ j\geq j_{0}\ \text{for some}\
j_{0}\in\mathbb{N}.$ $None$

Now, applying the facts

$u_{n_{j}}(r)=\int_{r}^{R}\phi_{1}^{-1}\big{(}\frac{1}{s^{N-1}}\int^{s}_{\delta}t^{N-1}f(t,u_{n_{j}}(t)dt)\big{)}ds$
$None$

and

$||u^{\prime}_{n_{j}}||_{C[\delta,R]}<1,\ \ \ \
||u_{n_{j}}||_{C[\delta,R]}<R,$

and the standard argument, we deduce that after taking a subsequence and
relabeling, if necessary,

$u_{n_{j}}\to u_{\diamond}\ \ \ \ \text{in}\ C^{1}[\delta,R],$

for some $u_{\diamond}\in C^{1}[\delta,R]$. Obviously,

$u_{\diamond}(r)\geq\frac{\sigma_{0}}{2},\ \ \ \ \ r\in I_{1},$ $None$

and it is easy to check that $u_{\diamond}$ satisfies

$\left\\{\begin{aligned}
&-(r^{N-1}u_{\diamond}^{\prime}(r))^{\prime}=r^{N-1}f(r,u_{\diamond}),\ \ \ \
r\in(\delta,R),\\\
&u_{\diamond}^{\prime}(\delta)=u_{\diamond}(R)=0.\end{aligned}\right.$

On the other hand, we may take
$s_{1}(n_{j}),s_{2}(n_{j})\in\\{\tau_{1}(n_{j}),\cdots,\tau_{n_{j}}(n_{j})\\}$
with

$s_{2}(n_{j})>s_{1}(n_{j}),$ $None$
$s_{2}(n_{j}),s_{1}(n_{j})\not\in[t_{1}(n_{j}),t_{2}(n_{j})],$ $None$
$u_{n_{j}}(r)\neq 0,\ \ \ \ r\in(s_{1}(n_{j}),s_{2}(n_{j})),$ $None$
$s_{2}(n_{j})-s_{1}(n_{j})\to 0,\ \ \ \ \ j\to\infty.$ $None$

For each $n_{j}\geq 6$, $u_{n_{j}}(s_{1}(n_{j}))=u_{n_{j}}(s_{2}(n_{j}))=0$
yields

$u^{\prime}_{n_{j}}(x(n_{j}))=0,\ \ \ \ \text{for some}\
x(n_{j})\in((s_{1}(n_{j}),\;s_{2}(n_{j})).$ $None$

After taking a subsequence and relabeling, if necessary, we may assume

$x(n_{j})\to x_{0},\ \ \ \ \ j\to\infty.$

Combining this with (4.30), (4.29) and using (4.24), it follows that

$u_{\diamond}(x_{0})=u^{\prime}_{\diamond}(x_{0})=0\ \ \ \ \text{for some}\
x_{0}\in[\delta,R]$ $None$

which means that $u_{\diamond}\equiv 0$ in $[\delta,R]$ (see Lemma 2.5 where
$x_{0}>0$ is needed). However, this contradicts (4.25).

Therefore, (4.17) is true.

Step 2 We show that

$\lim_{n\to\infty}||u^{\prime}_{n}||_{C[\delta,R]}=0.$ $None$

In fact, it is an immediate consequence of (4.17) and the relation

$u_{n}^{\prime}(r)=-\phi_{1}^{-1}\big{(}\frac{1}{r^{N-1}}\int^{r}_{0}t^{N-1}f(t,u_{n}(t)dt)\big{)}.$

Next, we consider the case $\delta=0$.

Let

$g_{m}(r,s)=\left\\{\begin{aligned} 0,\qquad\ \ \ \ \ \ \ \ \ \ \ \
&(r,s)\in(0,\frac{1}{m}]\times(-\alpha,\alpha),\\\ f(r-\frac{1}{m},s),\ \ \ \
\ \ &(r,s)\in(\frac{1}{m},R)\times(-\alpha,\alpha).\\\ \end{aligned}\right.$

Let $u^{\nu}_{m}\in S^{\nu}_{n,\frac{1}{m}}$. Then

$\left\\{\begin{aligned}
&-(r^{N-1}\phi_{1}((u^{\nu}_{m})^{\prime}))^{\prime}=r^{N-1}g_{m}(r,u^{\nu}_{m}),\
\ \ \ \ r\in(\frac{1}{m},R),\\\ &\
(u^{\nu}_{m})^{\prime}(1/m)=0=u^{\nu}_{m}(R).\\\ \end{aligned}\right.$

For each $n$ and $m$, define a function $y^{\nu}_{m}\in S^{\nu}_{n,0}$ by

$y^{\nu}_{m}(r)=\left\\{\begin{aligned} u_{m}(r),\ \ \ \ \ \ \ \ \ \ \
&\frac{1}{m}\leq r\leq R,\\\ u_{m}(\frac{1}{m}),\ \ \ \ \ \ \ \ \ \ &0\leq
r\leq\frac{1}{m}.\\\ \end{aligned}\right.$

By the same argument used in proof of Theorem 1.1, with obvious changes, we
may use $y^{\nu}_{m}$ to construct two solutions, $z_{n}^{\nu}\in
S^{\nu}_{n,0},\ \nu\in\\{+,-\\}$, for $(1.5)_{\lambda=1}$. Obviously, (4.17)
and (4.32) imply that

$\lim_{n\to\infty}(||z^{\nu}_{n}||_{C[0,R]}+||(z^{\nu})^{\prime}_{n}||_{C[0,R]})=0.$

$\Box$

REFERENCES

[1] L. J. Aías, B. Palmer, On the Gaussian curvature of maximal surfaces and
the Calabi-Bernstein theorem, Bull. London Math. Soc. 33 (2001) 454-458.

[2] A. Ambrosetti, J. Garcia-Azorero and I. Pekal, Quasilinear equations with
a multiple bifurcation, Differential Integral Equations 10 (1997) 37-50.

[3] V. Anuradha, D. D. Hai, R. Shivaji, Existence results for superlinear
semipositone BVP’s, Proc. Amer. Math. Soc. 124(3) (1996) 757-763.

[4] R. Bartnik, L. Simon, Spacelike hypersurfaces with prescribed boundary
values and mean curvature, Comm. Math. Phys. 87 (1982-1983) 131-152.

[5] C. Bereanu, P. Jebelean, J. Mawhin, Radial solutions for Neumann problems
involving mean curvature operators in Euclidean and Minkowski spaces, Math.
Nachr. 283 (2010) 379-391.

[6] C. Bereanu, P. Jebelean, J. Mawhin, Radial solutions for some nonlinear
problems involving mean curvature operators in Euclidean and Minkowski spaces,
Proc. Amer. Math. Soc. 137 (2009) 171-178.

[7] C. Bereanu, P. Jebelean, P. J. Torres, Positive radial solutions for
Dirichlet problems with mean curvature operators in Minkowski space, J. Funct.
Anal. 264 (2013) 270-287.

[8] C. Bereanu, P. Jebelean, P. J. Torres, Multiple positive radial solutions
for a Dirichlet problem involving the mean curvature operator in Minkowski
space, J. Funct. Anal. 265(4) (2013) 644-659.

[9] M. F. Bidaut-Véron, A. Ratto, Spacelike graphs with prescribed mean
curvature, Differential Integral Equations 10 (1997) 1003-1017.

[10] A. Capietto, W. Dambrosio, F. Zanolin, Infinitely many radial solutions
to a boundary value problem in a ball, Ann. Mat. Pura Appl. (4) 179 (2001)
159-188.

[11] A. Castro, A. Kurepa, Infinitely many radially symmetric solutions to a
superlinear Dirichlet problem in a ball, Proc. Amer. Math. Soc. 101 (1987)
57-64.

[12] S.-Y. Cheng, S.-T. Yau, Maximal spacelike hypersurfaces in the Lorentz-
Minkowski spaces, Ann. of Math. 104 (1976) 407-419.

[13] Y. Cheng, On the existence of radial solutions of a nonlinear elliptic
equation on the unit ball, Nonlinear Anal. 24 (1995) 287-307.

[14] I. Coelho, C. Corsato, F. Obersnel, P. Omari, Positive solutions of the
Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv.
Nonlinear Stud. 12(3) (2012) 621-638.

[15] G. Dai, R. Ma, Unilateral global bifurcation phenomena and nodal
solutions for $p$-Laplacian, J. Differential Equations 252(3) (2012)
2448-2468.

[16] E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of
geometric multiplicity one, Bull. Lond. Math. Soc. 34 (2002) 533-538.

[17] E. N. Dancer On the structure of solutions of non-linear eigenvalue
problems, Indiana Univ. Math. J. 23 (1973/74) 1067-1076.

[18] M. J. Esteban, Multiple solutions of semilinear elliptic problems in a
ball, J. Differential Equations 57 (1985) 112-137.

[19] M. García-Huidobro, R. Manásevich, F. Zanolin, Strongly nonlinear second-
order ODEs with rapidly growing terms, J. Math. Anal. Appl. 202(1) (1996)
1-26.

[20] M. Grillakis, Existence of nodal solutions of semilinear equations in
$\mathbb{R}^{N}$, J. Differential Equations 85 (1990) 367-400.

[21] Z. Guo, Boundary value problems for a class of quasilinear ordinary
differential equations, Differential Integral Equations 6 (1993) 705-719.

[22] M. A. Krasnosel’skii, A. I. Perov, A. I. Povolotskiy, P. P. Zabreiko,
Plane vector fields, Translated by Scripta Technica, Ltd. Academic Press, New
York-London 1966.

[23] R. López, Stationary surfaces in Lorentz-Minkowski space, Proc. Roy. Soc.
Edinburgh Sect. A 138 (2008) 1067-1096.

[24] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, in:
Research Notes in Mathematics, vol. 426, Chapman & Hall/CRC, Boca Raton,
Florida, 2001.

[25] J. López-Gómez, C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of
Linear Operators, Oper. Theory Adv. Appl. vol. 177, Birkhäuser/Springer,
Basel, Boston/Berlin, 2007.

[26] R. Ma, Y. An, Global structure of positive solutions for superlinear
second order $m$-point boundary value problems, Topol. Methods Nonlinear Anal.
34(2) (2009) 279-290.

[27] R. Ma, Y. An, Global structure of positive solutions for nonlocal
boundary value problems involving integral conditions, Nonlinear Anal. 71(10)
(2009) 4364-4376.

[28] R. Ma, C. Gao, Bifurcation of positive solutions of a nonlinear discrete
fourth-order boundary value problem, Z. Angew. Math. Phys. 64 (2013) 493-506.

[29] J. Mawhin, Radial solution of Neumann problem for periodic perturbations
of the mean extrinsic curvature operator, Milan J. Math. 79 (2011) 95-112.

[30] F. I. Njoku, P. Omari, F. Zanolin, Multiplicity of positive radial
solutions of a quasilinear elliptic problem in a ball, Adv. Differential
Equations, 5 (2000) 1545-1570.

[31] P. Omari, F. Zanolin, Infinitely many solutions of a quasilinear elliptic
problem with an oscillatory potential, Comm. Partial Differential Equations 21
(1996) 721-733.

[32] I. Peral, Multiplicity of solutions for the $p$-Laplacian, ICTP SMR
990/1, 1997.

[33] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,
J. Funct. Anal. 7 (1971) 487-513.

[34] B. L. Shekhter, On existence and zeros of solutions of a nonlinear two-
point boundary value problem, J. Math. Anal. Appl. 97 (1983) 1-20.

[35] A. E. Treibergs, Entire spacelike hypersurfaces of constant mean
curvature in Minkowski space, Invent. Math. 66 (1982) 39-56.

[36] G. T. Whyburn, Topological Analysis, Princeton University Press,
Princeton, 1958.

[37] M. Willem, Minimax theorems, Birkhäuser, Boston, 1996.