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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 13, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/13\hfil Bifurcation from infinity]
{Bifurcation from infinity and nodal solutions of quasilinear 
elliptic differential equations}

\author[B.-X. Yang \hfil EJDE-2014/13\hfilneg]
{Bian-Xia Yang}  % in alphabetical order

\address{Bian-Xia Yang \newline
School of Mathematics and Statistics, Lanzhou University,
Lanzhou, Gansu 730000, China}
\email{yanglina7765309@163.com}

\thanks{Submitted November 29, 2013. Published January 8, 2014.}
\subjclass[2000]{35P30，35B32}
\keywords{$p$-Laplacian; bifurcation; nodal solutions}

\begin{abstract}
 In this article, we establish a unilateral global bifurcation theorem
 from infinity for a class of $N$-dimensional p-Laplacian problems.
 As an application, we  study the global behavior
 of the components of nodal solutions of the  problem
 \begin{gather*}
 \operatorname{div}(\varphi_p(\nabla u))+\lambda a(x)f(u)=0,\quad x\in B,\\
 u=0,\quad x\in\partial B,
 \end{gather*}
 where $1<p<\infty$, $\varphi_p(s)=|s|^{p-2}s$,
 $B=\{x\in \mathbb{R}^N: |x|<1\}$, and $a\in C(\bar{B}, [0,\infty))$
 is radially symmetric with $a\not\equiv 0$ on any subset of
 $\bar{B}$, $f\in C(\mathbb{R}, \mathbb{R})$ and there exist two constants
 $s_2<0<s_1$, such that $f(s_2)=f(s_1)=0$, and $f(s)s>0$ for
 $s\in \mathbb{R}\setminus\{s_2, 0,s_1\}$.
 Moreover, we give intervals for the parameter $\lambda$, where 
 the problem has multiple nodal solutions if
 $\lim_{s\to 0}f(s)/\varphi_p(s)=f_0>0$ and
 $\lim_{s\to \infty}f(s)/\varphi_p(s)=f_\infty>0$.
 We use topological methods and nonlinear
 analysis techniques to prove our main results.
\end{abstract}

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

\section{Introduction}

In natural sciences, there are various concrete problems involving 
bifurcation phenomena, for example, Taylor vortices \cite{MSB}, 
catastrophic shifts in ecosystems \cite{MS} and shimmy
oscillations of an aircraft nose landing gear \cite{PBM}.
The existence of bifurcation phenomena have called the attention 
of several mathematicians.  Dai et al \cite{DaiML} established a unilateral 
global bifurcation theorem from infinity for one-dimensional p-Laplacian 
problem, and studied the global behavior of the components of nodal solutions of
nonlinear one-dimensional p-Laplacian eigenvalue problem.

Dai and Ma \cite{Dai} established a result from trivial
solutions line about the continua of radial solutions for the
$N$-dimensional p-Laplacian problem  on the unit ball of
$\mathbb{R}^N$ with $N\geq 1$ and $1<p<\infty$. 
Ambrosetti and Hess  \cite{AP} studied the global behavior of the
components of positive solutions of quasilinear elliptic
differential equation under the asymptotically linear growth
condition.  Ambrosetti et al  \cite{AJI} studied the existence of
branches of positive solutions for quasilinear elliptic differential
equation under the equidiffusive growth condition, which extend the
main result in \cite{AP}. However, these references  gave no information
about the sign-changing solution.

Motivated by the above articles, it is our main purpose 
to use the results in \cite{Dai} and in line with the global
bifurcation results from infinity by Rabinowitz \cite{Rabinowitz}.
We shall establish the unilateral global bifurcation result from
infinity for the following $N$-dimensional $p$-Laplacian problem
\begin{equation}\label{e1.1}
\begin{gathered}
-\operatorname{div}(\varphi_p(\nabla u))=\lambda a(x)\varphi_p(u)+g(x,u;\lambda),
\quad x\in B,\\
u=0,\quad x\in\partial B,
\end{gathered}
\end{equation}
where $1<p<\infty$, $\varphi_p(s)=|s|^{p-2}s, B$ is the unit ball of 
$\mathbb{R}^N$, $a\in M(B)$ is a non-negative function with
$$
M(B)=\{a\in C(\bar{B})\ \text{is radially symmetric with }
 a(\cdot)\not\equiv0 \text{ on any subset of } \bar{B} \},
$$
the function $g: B\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}$ satisfies the 
Carath\'{e}odory condition in the first two variables and is radially 
symmetric with respect $x$.

It is clear that the radial solutions of \eqref{e1.1} are 
the solutions of 
\begin{equation}\label{e1.6}
\begin{gathered}
-(r^{N-1}\varphi_p(u'))'=\lambda r^{N-1}a(r)\varphi_p(u)+r^{N-1}g(r,u;\lambda),
\quad\text{a.e. }r\in (0,1),\\
u'(0)=u(1)=0,
\end{gathered}
\end{equation}
where $r=|x|$ with $x\in B$, $a\in M(I)$ is a non-negative function with 
$I=(0,1)$ and
$$
M(I)=\{a\in C(\bar{I}) \text{ is radially symmetric with }
 a(\cdot)\not\equiv0 \text{ on any subset of } \bar{I} \}.
$$
We also assume the perturbation function $g$ satisfies the assumption
\begin{equation}\label{e1.2}
\lim_{|s|\to\infty}\frac{g(r,s;\lambda)}{|s|^{p-1}}=0
\end{equation}
uniformly for a.e. $r\in I$ and $\lambda$ on bounded sets.

Based on the unilateral global bifurcation results from zero by
\cite{Dai}, and the global bifurcation results from infinity,
Theorem \ref{thm2.2}, we shall study the existence of radial  nodal solutions for
the  nonlinear eigenvalue problem
\begin{equation}\label{e1.3}
\begin{gathered}
\operatorname{div}(\varphi_p(\nabla u))+\lambda a(x)f(u)=0,\quad x\in B,\\
u=0,\quad x\in\partial B,
\end{gathered}
\end{equation}
where $a$ and $f$ satisfy the following assumptions:
\begin{itemize}
\item[(H1)] $a\in C(\bar{B}, [0,\infty))$ with $a\not\equiv 0$ 
 on any subset of $\bar{B}$;

\item[(H2)] there exist $f_0, f_\infty\in (0,\infty)$ such that
$$
f_0=\lim_{s\to0}\frac{f(s)}{|s|^{p-2}s}\quad \text{and}\quad
f_\infty=\lim_{s\to\infty}\frac{f(s)}{|s|^{p-2}s};
$$

\item[(H3)] $f\in C(\mathbb{R},\mathbb{R})$, there exist two constants
$s_2<0<s_1$, such that $f(s_2)=f(s_1)=f(0)=0$, and $f(s)s>0$ for
$s\in \mathbb{R}\setminus\{s_2, 0,s_1\}$.
\end{itemize}
We look for radial nodal solution of \eqref{e1.3}, namely for $u=u(r)$ verifying
\begin{equation}\label{e1.4}
\begin{gathered}
\big(r^{N-1}\varphi_p(u')\big)'+\lambda r^{N-1} a(r)f(u)=0,\quad
\text{a.e. } r\in I,\\
u'(0)=u(1)=0, 
\end{gathered}
\end{equation}
where $r=|x|$ with $x\in B$.

The rest of this article is arranged as follows. 
In Section 2, we establish the unilateral global bifurcation results 
from infinity of \eqref{e1.1}. 
In Section 3, we study the global behavior of the components of 
nodal solutions of problem \eqref{e1.3}.

\section{Unilateral global bifurcation from infinity}

Let $E:=\{u\in C^1(\bar{I})|u'(0)=u(1)=0\}$ with the norm 
$\|u\|= \max_{r\in\bar{I}} |u(r)| +\max_{r\in\bar{I}}|u'(r)|$.
Let $S_k^+$ denote the set of functions in $E$ which have
exactly $k-1$ interior nodal zeros in $I$ and are positive near
$r=0$, and set $S_k^-=-S_k^+$ and $S^k=S_k^+\cup S_k^-$. It is clear
that $S_k^+$ and $S_k^-$ are disjoint and open in $E$. We also let
$\phi_k^\nu=\mathbb{R}\times S_k^\nu$ and $\phi_k=\mathbb{R} \times
S_k$ 
under the product topology, where $\nu\in\{+, -\}$. We
use $\mathscr{S}$ to denote the closure of the set of nontrivial
solutions of \eqref{e1.6} in $\mathbb{R}\times E$. We add the points
$\{(\lambda,\infty)|\lambda \in \mathbb{R}\}$ to space $\mathbb{R}
\times E$.

\begin{lemma}[{\cite[Theorem 1.5.3]{Peral}}] \label{lem2.1} 
Assume {\rm (H1)} holds. Then the problem
\begin{equation}\label{e1.5}
\begin{gathered}
\big(r^{N-1}\varphi_p(u')\big)'+\lambda r^{N-1}a(r)\varphi_p(u)=0,\quad
\text{a.e. } r\in I,\\
u'(0)=u(1)=0 
\end{gathered}
\end{equation}
has a sequence of simple eigenvalues $\lambda_k$ with $\lambda_k\to\infty$ 
as $k\to\infty$, and the corresponding  eigenfunctions $\varphi_k$ have 
exactly $k-1$ simple zeros, and each $\lambda_k(p)$ depends continuously 
on $p$.
\end{lemma}

Let $\lambda_k$ denote the $k$-th eigenvalue of problem \eqref{e1.5}. 
The main result of this section is the following theorem.

\begin{theorem}\label{thm2.2}
Let assumption \eqref{e1.2} hold. Then there exists a connected component
$\mathcal{D}_k^\nu$  of $\mathscr{S}\cup(\lambda_k\times\{\infty\})$, 
containing $\lambda_k\times\{\infty\}$. 
Moreover if $\Lambda\subset\mathbb{R}$ is an interval such that 
$\Lambda\cap(\cup_{k=1}^\infty\lambda_k)=\lambda_k$ and $ \mathcal{U}$ 
is a neighborhood of $\lambda_k\times\{\infty\}$ whose projection on 
$\mathbb{R}$ lies in $\Lambda$ and whose projection on $E$ is bounded away from
$0$, then either
\begin{itemize}
\item[(1)]  $\mathcal{D}_k^\nu - \mathcal{U}$ is bounded in
$\mathbb{R}\times E$ in which case $\mathcal{D}_k^\nu - \mathcal{U}$
meets $\mathscr{R}=\{(\lambda, 0)|\lambda\in \mathbb{R}\}$, or

\item[(2)] $\mathcal{D}_k^\nu - \mathcal{U}$ is unbounded.
\end{itemize}
If (2) occurs and $\mathcal{D}_k^\nu - \mathcal{U}$ has a bounded projection 
on $\mathbb{R}$, then $\mathcal{D}_k^\nu - \mathcal{U}$ meets
 $\lambda_j\times\{\infty\}$ for some $j\neq k$.
\end{theorem}

\begin{proof} 
If $(\lambda,u)\in \mathscr{S}$ with $\|u\|\neq0$, dividing \eqref{e1.6} 
by $\|u\|^2$ and setting $w = u/\|u\|^2$ yield
\begin{equation}\label{e2.1}
\begin{gathered}
-\big(r^{N-1}\varphi_p(w')\big)'
=\lambda \big(r^{N-1}a(r)\varphi_p(w)\big)+r^{N-1}
\frac{g(r,u;\lambda)}{\|u\|^{2(p-1)}},\quad\text{a.e. } r\in I,\\
w'(0)=w(1)=0. 
\end{gathered}
\end{equation}
Define
$$
f(r,w; \lambda)=\begin{cases}
\|w\|^{2(p-1)}r^{N-1}g(r, w/\|w\|^2;\lambda), &\text{if } w\neq 0,\\
0, &\text{if } w= 0, 
\end{cases}
$$
Clearly, \eqref{e2.1} is equivalent to
\begin{equation}\label{e2.2}
\begin{gathered}
-\big(r^{N-1}\varphi_p(w')\big)'
 =\lambda \big(r^{N-1}a(r)\varphi_p(w)\big)+f(r,w;\lambda),\quad
\text{a.e. } r\in I,\\
w'(0)=w(1)=0. 
\end{gathered}
\end{equation}
It is obvious that $(\lambda,0)$ is always the solution of \eqref{e2.2}. 
By simple computation, we can show that assumption \eqref{e1.2} implies
$$
f(r,w;\lambda)= o(|w|^{p-1})
$$
near $w=0$, uniformly for all $ r\in I$ and on bounded $\lambda$ intervals.

Now applying \cite[Theorem 3.2]{Dai} to problem \eqref{e2.2}, we
have the connected component $\mathcal{C}_k^\nu$ of
$\mathscr{S}\cup(\lambda_k\times\{0\})$, containing
$\lambda_k\times\{0\}$ is unbounded and lies in
$\phi_k^\nu\cup(\lambda_k\times\{0\})$. Under the inversion $w \to
w/\|w\|^2=u, \mathcal{C}_k^\nu ŠÍ\to \mathcal{D}_k^\nu$ satisfying
problem \eqref{e1.6}. Clearly, $\mathcal{D}_k^\nu$ satisfies the
conclusions of this theorem.
\end{proof}

By \cite[Lemma 6.4.1]{Lopez} and using the similar argument, we can prove  
\cite[Corollary 1.8]{Rabinowitz} with obvious changes. Also we have the following theorem.


\begin{theorem} \label{thm2.3} 
There exists a neighborhood $\mathcal{N} \subset \mathcal{U}$ of 
$\lambda_k\times\{\infty\}$ such that 
$(\lambda, u)\in(\mathcal{D}_k^\nu \cap\mathcal{N})
\setminus\{(\lambda_k\times\{\infty\})\}$ implies 
$(\lambda, u)=(\lambda_k + o(1), \alpha\varphi_k + w)$, where 
$\varphi_k$ is the eigenfunction corresponding to $\lambda_k$ with 
$\|\varphi_k\|= 1, \alpha >0 (\alpha<0)$ and $\|w\|= o(|\alpha|)$ at 
$|\alpha|=\infty$.
\end{theorem}

\begin{remark}\label{rmk2.4} \rm
Note that Theorem \ref{thm2.3} implies that 
$(\mathcal{D}_k^\nu\cap\mathcal{N})\subset
\left(\phi_k^\nu\cup(\lambda_k\times\{\infty\})\right)$. 
However, it need not be the case that 
$\mathcal{D}_k^\nu\subset\left(\phi_k^\nu\cup(\lambda_k\times\{\infty\})\right)$ 
even in the case of $p = 2$ (see the example in \cite{Rabinowitz}).
\end{remark}

\section{Global behavior of the components of nodal solutions}

Let $\xi,\eta\in C(\mathbb{R},\mathbb{R})$ be such that
$$
f(u)=f_0\varphi_p(u)+\xi(u), \quad
f(u) = f_\infty\varphi_p(u)+\eta(u)
$$
with
$$
\lim_{|u|\to 0}\frac{\xi(u)}{\varphi_p(u)}=0,\quad
\lim_{|u|\to\infty}\frac{\eta(u)}{\varphi_p(u)}=0.
$$
Let us consider
\begin{equation}\label{e3.1}
\begin{gathered}
-\big(r^{N-1}\varphi_p(u')\big)'=\lambda r^{N-1}a(r)f_0\varphi_p(u)
+\lambda r^{N-1}a(r)\xi(u),\quad\text{a.e. } r\in I,\\
u'(0)=u(1)=0 
\end{gathered}
\end{equation}
as a bifurcation problem from the trivial solution $u\equiv0$, and
\begin{equation}\label{e3.2}
\begin{gathered}
-\big(r^{N-1}\varphi_p(u')\big)'=\lambda r^{N-1}a(r)f_\infty\varphi_p(u)
+\lambda r^{N-1}a(r)\eta(u),\quad\text{a.e. } r\in I,\\
u'(0)=u(1)=0 
\end{gathered}
\end{equation}
as a bifurcation problem from infinity.

Applying \cite[Theorem 3.2]{Dai} to \eqref{e3.1}, we have that for 
each integer $k\geq1$, there exists a continuum $\mathcal{C}_{k,0}^\nu$, 
of solutions of \eqref{e1.4} joining $(\lambda_k/f_0, 0)$ to infinity, 
and $(\mathcal{C}_{k,0}^\nu\backslash\{(\lambda_k/f_0, 0)\})\subseteq\phi_k^\nu$. 
Applying Theorem \ref{thm2.2} to \eqref{e3.2}, we can show that for
each integer $k\geq1$, there exists a continuum $\mathcal{D}_{k,\infty}^\nu$ 
of solutions of \eqref{e1.4} meeting $(\lambda_k/f_\infty,\infty)$. 
Moreover, Theorem \ref{thm2.3} imply that 
$$
(\mathcal{D}_{k,\infty}^\nu\backslash\{(\lambda_k/f_\infty,\infty)\})
\subseteq\phi_k^\nu.
$$
Next, we shall show that these two components are disjoint under the 
assumption (H3). Hence the essential role is played
by the fact of whether $f$ possesses zeros in $\mathbb{R}\backslash\{0\}$.


\begin{theorem}\label{thm3.1}
 Let {\rm (H1)-(H3)} hold. Then
\begin{itemize}
\item[(i)] for $(\lambda, u)\in(\mathcal{C}_{k,0}^+\cup\mathcal{C}_{k,0}^-)$, 
we have that $s_2< u(r)<s_1$ for all $r\in\bar{I}$;

\item[(ii)] for $(\lambda, u)\in (\mathcal{D}_{k,\infty}^+
 \cup \mathcal{D}_{k,\infty}^-)$, we have that either 
$\max_{r\in\bar{I}}u(r)>s_1$ or $\min_{r\in\bar{I}}u(r)<s_2$.
\end{itemize}
\end{theorem}

\begin{proof} 
Suppose on the contrary that there exists 
$(\lambda, u)\in(\mathcal{C}_{k,0}^+\cup\mathcal{C}_{k,0}^-
\cup\mathcal{D}_{k,\infty}^+\cup \mathcal{D}_{k,\infty}^-)$ 
such that either $\max\{u(r)|r\in\bar{I}\}=s_1$ or
$\min\{u(r)|r\in\bar{I}\}= s_2$. Let $0<\tau_1<\cdots<\tau_k=1$ 
denote the zeros of $u$. We only treat the case of 
$\max\{u(r)|r\in\bar{I}\}=s_1$ because the proof for the case of
 $\min\{u(r)|r\in\bar{I}\}=s_2$ can be given similarly. In this case, 
there exists $j\in\{1,\cdots, k,\}$ such that $\max\{u(r)|r\in\bar{I}\}=s_1$ 
and $0\leq u(r) \leq s_1$ for all $r\in[\tau_j, \tau_{j+1}]$.

We claim that there exists $0 < m<\infty$ such that 
$f(s) \leq m\varphi_p(s_1-s)$ for any $s \in[0,s_1]$. 

Clearly, the claim is true for the case of $s = 0$ or $s = s_1$ by  (H3). 
Suppose on the contrary that there exists $s_0\in(0,s_1)$ such that
$$
f(s_0)>m\varphi_p(s_1-s_0)
$$
for any $m > 0$. It follows that $m < f (s_0)/\varphi_p(s_1-s_0)$. 
This contradicts the arbitrariness of $m$.

Now, let us consider the problem
\begin{gather*}
-(r^{N-1}\varphi_p((s_1-u)'))'+\lambda r^{N-1}m a(r)\varphi_p(s_1-u) \\
=\lambda r^{N-1}m a(r)\varphi_p(s_1-u)-\lambda  r^{N-1}a(r)f(u),
\quad r\in (\tau_j,\tau_{j+1}),\\
s_1-u(\tau_j)>0,\quad s_1-u(\tau_{j+1})>0.
\end{gather*}
It is obvious that $f(s) \leq m\varphi_p(s_1-s)$ for any $s\in [0, s_1]$ implies
\begin{gather*}
-(r^{N-1}\varphi_p((s_1-u)'))'+\lambda r^{N-1} m a(r)\varphi_p(s_1-u)\geq 0,\quad
r\in (\tau_j,\tau_{j+1}),\\
s_1-u(\tau_j)>0,\quad s_1-u(\tau_{j+1})>0.
\end{gather*}
The strong maximum principle of \cite{Montenego} implies that
 $s_1 > u(r)$ in $[\tau_j, \tau_{j+1}]$. This is a contradiction.
\end{proof}

\begin{remark}\label{rmk3.2} \rm
If $N=1$, then Theorems \ref{thm2.2}, \ref{thm2.3} and \ref{thm3.1} 
correspond to the main results  in \cite{DaiML}. 

In \cite{AJI}, they needed  $f\in C^1(\mathbb{R}^+, \mathbb{R})$, 
while in this article, we  need just $f\in C(\mathbb{R}, \mathbb{R})$. 
Furthermore, they studied the existence of branches of positive 
solutions, while we have the existence of branches of sign-changing solutions. 
So we have extended the results in \cite{AJI,DaiML}.
\end{remark}

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\end{document}