\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 179, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/179\hfil Kirchhoff-type equations] {Spectrum, global bifurcation and nodal solutions to Kirchhoff-type equations} \author[X. Cao, G. Dai \hfil EJDE-2018/179\hfilneg] {Xiaofei Cao, Guowei Dai} \address{Xiaofei Cao (corresponding author)\newline Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian 223003, China} \email{caoxiaofei258@126.com} \address{Guowei Dai \newline School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China} \email{daiguowei@dlut.edu.cn} \thanks{Submitted July 4, 2017. Published November 5, 2018.} \subjclass[2010]{34C23, 47J10, 34C10} \keywords{Bifurcation; spectrum; nonlocal problem; nodal solution} \begin{abstract} In this article, we consider a Dancer-type unilateral global bifurcation for the Kirchhoff-type problem \begin{gather*} -\Big(a+b\int_0^1 | u'|^2\,dx\Big)u'' =\lambda u+h(x,u,\lambda)\quad\text{in } (0,1),\\ u(0)=u(1)=0. \end{gather*} Under natural hypotheses on $h$, we show that $(a\lambda_k,0)$ is a bifurcation point of the above problem. As applications we determine the interval of $\lambda$, in which there exist nodal solutions for the Kirchhoff-type problem \begin{gather*} -\Big(a+b\int_0^1 | u'|^2\,dx\Big) u'' =\lambda f(x,u)\quad\text{in } (0,1),\\ u(0)=u(1)=0, \end{gather*} where $f$ is asymptotically linear at zero and is asymptotically 3-linear at infinity. To do this, we also establish a complete characterization of the spectrum of a nonlocal eigenvalue problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} We consider the unilateral global bifurcation phenomenon for the problem \begin{equation}\label{bp} \begin{gathered} -\Big(a+b\int_0^1 | u'|^2\,dx\Big)u''=\lambda u+h(x,u,\lambda)\quad\text{in } (0,1),\\ u(0)=u(1)=0, \end{gathered} \end{equation} where $a>0$, $b>0$ are real constants, $\lambda$ is a parameter and $h:(0,1)\times \mathbb{R}^2\to\mathbb{R}$ is a continuous function satisfying \begin{equation}\label{a2} \lim_{ s\to0}\frac{h(x,s,\lambda)}{s}=0 \end{equation} uniformly for all $x\in(0,1)$ and $\lambda$ on bounded sets. Problem \eqref{bp} is related to the stationary problem of a model introduced by Kirchhoff to describe the transversal oscillations of a stretched string \cite{K}. Problem \eqref{bp} received much attention only after Lions \cite{L} proposed an abstract framework to it. Some important and interesting results can be found, for example, in \cite{AP,CCS,DS,DS1,HZ}. Recently, many mathematicians have studied problem \eqref{bp} by variational method, see \cite{CKW,CW,GMS,LLS,MR,MZ,MRS,MR2,PZ,ST,ZP,ZZR} and the references therein. The study of Kirchhoff-type equations has already been extended to the case involving the $p$-Laplacian and $p(x)$-Laplacian. We refer the readers to \cite{APS,CF,DH,DJ,DL,DM,D2,D1,F,PXZ} on this subject. A distinguishing feature of problem \eqref{bp} is that the first equation contains a nonlocal coefficient $a+b\int_0^1 | u'|^{2}\,dx$, and hence the equation is no longer a pointwise equation. Moreover, the first equation of problem \eqref{bp} with $h\equiv0$ is not homogeneous. So problem \eqref{bp} is a fully nonlinear problem which raises some essential difficulties to the study of this kind of problems. It is well known that the problem \begin{gather*} -u''=\lambda u\quad\text{in } (0,1),\\ u(0)=u(1)=0 \end{gather*} possesses infinitely many eigenvalues $0<\lambda_1<\lambda_2<\dots<\lambda_k\to+\infty$, all of which are simple. The eigenfunction $\varphi_k$ corresponding to $\lambda_k$ has exactly $k-1$ simple zeros in $(0,1)$. Let $S_k^+$ denote the set of functions in $E:=C_0^1[0,1]$ which have exactly $k-1$ interior nodal (i.e. non-degenerate) zeros in $(0,1)$ and are positive near $x=0$, 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 $E$. Finally, let $\Phi_k^{\pm}=\mathbb{R}\times S_k^{\pm}$ and $\Phi_k=\mathbb{R}\times S_k$ under the product topology. \begin{theorem} \label{thm1.1} The pair $(a\lambda_k,0)$ is a bifurcation point of problem \eqref{bp}. Moreover, there are two distinct unbounded continua in $\mathbb{R}\times H_0^1(0,1)$, $\mathscr{C}_k^+$ and $\mathscr{C}_k^-$, consisting of the bifurcation branch $\mathscr{C}_k$ emanating from $(a\lambda_k, 0)$, such that $\mathscr{C}_{k}^\nu\subseteq (\{(a\lambda_k,0)\}\cup\Phi_k^\nu)$, $\nu\in\{+,-\}$. \end{theorem} We shall prove Theorem \ref{thm1.1} in Section 2. If $b=0$, the conclusions of Theorem \ref{thm1.1} are well known. However, the case of $b>0$ is nontrivial because problem \eqref{bp} is nonlinear. So the standard bifurcation theory cannot be used directly here. In order to apply the Dancer unilateral global bifurcation theorem, we find a skillful transformation which can convert problem \eqref{bp} to a desired form. This technique can also be used to deal with other similar problems. To find more detailed information of $\mathscr{C}_{k}^\nu$, in Section 3 we study the eigenvalue problem \begin{equation}\label{kee} \begin{gathered} -\Big(\int_0^1| u'|^2\,dx\Big) u''=\mu u^3\quad \text{in }(0,1),\\ u(0)=u(1)=0. \end{gathered} \end{equation} We shall establish a complete characterization of the spectrum of problem \eqref{kee}. \begin{theorem} \label{thm1.2} The set of all eigenvalues of problem \eqref{kee} satisfy \[ 0<\mu_1<\mu_2<\cdots<\mu_k<\cdots\to +\infty. \] Every $\mu_k$ is simple and the corresponding one-dimensional space of solutions of problem \eqref{kee} with $\mu=\mu_k$ is spanned by a function having precisely $k$ bumps in $(0,1)$. Each $k$-bump solution is constructed by the reflection and compression of the eigenfunction $\varphi_1$ associated with $\mu_1$. \end{theorem} Note that problem \eqref{kee} is nonlinear and nonlocal. So the Pr\"{u}fer-type transformation method cannot be used to get the desired results. We use variational method combined with reflection-compression technique to prove Theorem \ref{thm1.2}. We believe that the first trying of this way has significance to nonlinear eigenvalue problems. In Section 4, we describe $\mathscr{C}_{k}^\nu$ more detailed for problem \eqref{bp} with $h(x,s,\lambda)=\lambda f(x,s)-\lambda s$, i.e., the problem \begin{equation}\label{fp} \begin{gathered} -\Big(a+b\int_0^1 | u'|^2\,dx\Big) u''=\lambda f(x,u)\quad\text{in } (0,1),\\ u(0)=u(1)=0. \end{gathered} \end{equation} We assume that $f$ satisfies the following assumptions: \begin{itemize} \item[(A1)] $f:(0,1)\times \mathbb{R}\to\mathbb{R}$ is a continuous function such that $f(x,s)s>0$ for all $x\in (0,1)$ and any $s\neq0$. \item[(A2)] there exist $f_0$, $f_\infty\in (0,+\infty)$ such that \[ \lim_{s\to 0}\frac{f(x,s)}{a s}=f_0,\quad \lim_{ | s|\to +\infty}\frac{f(x,s)}{b s^3}=f_\infty \] uniformly with respect to $x\in (0,1)$. \end{itemize} Our last main theorem reads as follows. \begin{theorem} \label{thm1.3} Assume that {\rm (A1)--(A2)} are satisfied. Then for \[ \lambda\in\Big(\frac{\lambda_k}{f_0},\frac{\mu_k}{f_\infty}\Big) \cup \Big(\frac{\mu_k}{f_\infty},\frac{\lambda_k}{f_0}\Big), \] problem \eqref{fp} possesses at least two solutions $u_k^+$ and $u_k^-$ such that $u_k^+$ has exactly $k-1$ simple zeros in $(0,1)$ and is positive near 0, and $u_k^-$ has exactly $k-1$ simple zeros in $(0,1)$ and is negative near $0$. \end{theorem} We end this section by introducing some notation convention which will be used later in this paper. Let $X$ be the usual Sobolev space $H_0^1(0,1)$ with the norm $\| u\|=(\int_0^1 | u'|^2\,dx)^{1/2}$. For a measurable set $A$ of $\mathbb{R}^N$, we denote its measure by $| A|$. Also, denote by $c$ and $c_{i}$, $i\in \mathbb{N}$, some generic positive constants (the exact value may be different from line to line). \section{Global bifurcation} Firstly, we consider the auxiliary problem \begin{equation}\label{ap} \begin{gathered} -u''=f(x)\quad \text{in } (0,1),\\ u(0)=u(1)=0. \end{gathered} \end{equation} It is well known that problem \eqref{ap} possesses a unique weak solution for each $f\in L^1(0,1)$. Let us denote by $G(f)$ the unique solution of problem \eqref{ap}. Then $G:L^1(0,1)\to X$ is a linear continuous operator. Since the embedding of $X\hookrightarrow C[0,1]$ is compact, the restriction of $G$ to $X$ is a completely continuous operator. \begin{theorem} \label{thm2.1} The pair $(a\lambda_k,0)$ is a bifurcation point of problem \eqref{bp}. Moreover, there are two distinct continua in $\mathbb{R}\times X$, $\mathscr{C}_k^+$ and $\mathscr{C}_k^-$, consisting of the bifurcation branch $\mathscr{C}_k$ emanating from $(a\lambda_k,0)$, which contain $\{(a\lambda_k,0)\}$ and satisfy either $\mathscr{C}_k^+$ and $\mathscr{C}_k^-$ are both unbounded or $\mathscr{C}_k^+\cap\mathscr{C}_k^-\neq \{(a\lambda_k,0)\}$. \end{theorem} \begin{proof} Clearly, the pair $(\lambda,u)$ is a solution of problem \eqref{bp} if and only if $(\lambda,u)$ satisfies \begin{equation}\label{oe} u=G(\frac{1}{a+b\| u\|^2}(\lambda u+h(x,u,\lambda))). \end{equation} Let \[ Lu=\frac{1}{a}G(u),\quad \widetilde{H}(\lambda,u)=\frac{1}{a+b\| u\|^2} G(h(x,u,\lambda)) -\frac{\lambda b\| u\|^2}{a(a+b\| u\|^2)}G(u). \] Clearly, $L{:}X\to X$ is linear completely continuous, $\widetilde{H}{:}\mathbb{R}\times X\to X$ is compact. Moreover, it is easy to see that $a\lambda_k$ is simple characteristic value of $L$. Then equation \eqref{oe} is equivalent to \[ u=\lambda Lu+\widetilde{H}(\lambda,u). \] Let $\widetilde{h}(x,u,\lambda)=\max_{0\leq | s|\leq u}| h(x,s,\lambda)|$ for all $x\in(0,1)$ and $\lambda$ on bounded sets. Then $\widetilde{h}$ is nondecreasing with respect to $u$ and \begin{equation}\label{eg0+1} \lim_{ u\to 0^+}\frac{\widetilde{h}(x,u,\lambda)}{u}=0. \end{equation} Further it follows from \eqref{eg0+1} that \begin{equation}\label{egn0} \frac{h(x,u,\lambda)}{\| u\|} \leq\frac{ \widetilde{h}(x,| u|,\lambda)}{c\| u\|_\infty} \leq \frac{\widetilde{h}(x,\| u\|_\infty,\lambda)}{c\| u\|_\infty}\to0\quad \text{as } \| u\|\to 0, \end{equation} uniformly for $x\in(0,1)$ and $\lambda$ belonging to a bounded set, where $c>0$ is the best embedding constant of $X\hookrightarrow C[0,1]$, $\| u\|_\infty=\max_{x\in[0,1]} | u(x)|$. It follows that $\widetilde{H}=o(\| u\|)$ for $u$ near $0$ uniformly on bounded $\lambda$ intervals. \cite[Theorem 2]{Dancer0} shows the desired conclusions. \end{proof} For the regularity of weak solution, we have the following result. \begin{proposition} \label{prop2.1} Any weak solution $u\in X$ of problem \eqref{bp} is also a classical solution, i.e., $u\in C^{2}(0,1)\cap C^{1,\alpha}[0,1]$ satisfying \eqref{bp} and $u(0)=u(1)=0$. \end{proposition} \begin{proof} Let \[ f(x)=\frac{1}{a+b\| u\|^2}(\lambda u+h(x,u,\lambda)). \] Then it is easy to see that $f\in L^2(0,1)$. By \cite[Theorem 8.12]{GT}, we know that $u\in W^{2,2}(0,1)$. Furthermore, by the general Sobolev embedding theorem \cite[p. 270]{E}, we get $u\in C^{1,\alpha}[0,1]$ for some $\alpha \in(0,1)$. According to the definition of weak solution, we have \[ -\Big(a+b\int_0^1 | u'|^2\,dx\Big)u''=\lambda u+h(x,u,\lambda) \] in the sense of distribution, i.e., \begin{equation}\label{fe00} -\Big(a+b\int_0^1 | u'|^2\,dx\Big)u'' =\lambda u+h(x,u,\lambda) \quad \text{in } (0,1)\setminus I_0 \end{equation} for some $I_0\subset (0,1)$ which satisfies $| I_0|=0$. Let $I:=(0,1)$. Then the equation \eqref{fe00} follows that $u\in C^2(I\setminus I_0)$ and \begin{equation}\label{rlmt0} u''(x)=-f(x), \quad x\in I\setminus I_0. \end{equation} For any $x_0\in I_0$, it is easy to see that equation \eqref{rlmt0} implies the existence of $\underset{x\to x_0}\lim u''(x)$. \cite[Proposition 1]{DMW} follows that $u''(x_0)=\underset{x\to x_0}\lim u''(x)$. By the arbitrary property of $x_0$, we get that $u\in C^2(I)$ and satisfies \eqref{bp}. Clearly, one has $u(0)=u(1)=0$. \end{proof} \begin{lemma} \label{lem2.1} If $(\lambda, u)$ is a solution of \eqref{bp} and $u$ has a double zero, then $u \equiv 0$. \end{lemma} \begin{proof} Let $u$ be a solution of problem \eqref{bp} and $x^*\in[0, 1]$ be a double zero. We note that \[ u(x)=\frac{-1}{a+b\| u\|^2}\int_{x^*}^{x}\int_{x^*}^s (\lambda u+h(\tau,u,\lambda))\,d\tau\,ds. \] Firstly, we consider $x\in[0, x^*]$. Then \begin{align*} | u(x)| &\leq \frac{1}{a}\int_{x}^{x^*}|\lambda u+h(\tau,u,\lambda)|\,d\tau, \\ &\leq \frac{1}{a}\int_x^{x^*} \Big(| \lambda|+|\frac{h(\tau,u(\tau),\lambda)}{u(\tau)}|\Big) | u(\tau)|\,d\tau. \end{align*} In view of \eqref{a2}, for any $\varepsilon>0$, there exists a constant $\delta>0$ such that \[ | h(x,s,\lambda)|\leq \varepsilon| s| \] uniformly with respect to all $x\in(0,1)$ and fixed $\lambda$ when $| s|\in[0,\delta]$. Hence, \[ | u(x)|\leq \int_x^{x^*} \frac{1}{a}\Big(| \lambda|+\varepsilon+\max_{s\in[\delta, \| u\|_\infty]}\big|\frac{h(\tau,s,\lambda)}{s}\big|\Big) | u(\tau)|\,d\tau. \] By the Gronwall-Bellman inequality \cite{Bre}, we get $u \equiv 0$ on $[0, x^*]$. Similarly, we can get $u \equiv 0$ on $[x^*, 1]$ and the proof is complete. \end{proof} \begin{lemma} \label{lem2.2} We have $\mathscr{C}_k^\nu \cap (\mathbb{R}\times \{0\}) =\{(a\lambda_k,0)\}$ if $\mathscr{C}_k^\nu \subseteq(\Phi_k^\nu\cup\{(a\lambda_k,0)\})$. \end{lemma} \begin{proof} By Proposition \ref{prop2.1} follows that $\mathscr{C}_k^\nu\in \mathbb{R}\times E$. Suppose, on the contrary, if there exists $(\mu_m,u_m)\to(a\lambda_j,0)$ when $m\to+\infty$ with $(\mu_m,u_m)\in \mathscr{C}_k^\nu$, $u_m\in S_k^\nu$, $u_m \not\equiv 0$ and $j\neq k$. Let $v_m :=u_m/\| u_m\|$, then $v_m$ should be a solution of the problem \begin{equation}\label{ewgm} v=G\Big(\frac{1}{a+b\| u_m\|^2} \Big(\mu_m v+\frac{h(x,u_m,\mu_m)}{\| u_m(x)\|}\Big)\Big). \end{equation} By \eqref{egn0}, \eqref{ewgm} and the compactness of $G$ we obtain that for some convenient subsequence $v_m\to v_0$ as $m\to+\infty$. Now $v_0$ verifies the equation \[ -v''= \lambda_j v \] and $\| v_0\| = 1$. Hence $v_0\in S_j$ which is an open set in $X$, and as a consequence for some $m$ large enough, $v_m \in S_j$, and this is a contradiction. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] By \cite[Lemma 1.24]{R2} there exists a bounded open neighborhood $\mathscr{O}_k$ of $(a\lambda_k,0)$ such that $(\mathscr{C}_{k}^\nu\cap\mathscr{O}_k) \subseteq(\Phi_k^\nu\cup\{(a\lambda_k, 0)\})$ or $(\mathscr{C}_{k}^\nu\cap\mathscr{O}_k) \subseteq(\Phi_k^{-\nu}\cup\{(a\lambda_k, 0)\})$. Without loss of generality, we assume that $(\mathscr{C}_{k}^\nu\cap\mathscr{O}_k) \subseteq(\Phi_k^\nu\cup\{(a\lambda_k, 0)\})$. Next, we show that $\mathscr{C}_k^\nu\subseteq(\Phi_k^\nu\cup\{(a\lambda_k,0)\})$. Suppose that $\mathscr{C}_k^\nu\not\subseteq(\Phi_k^\nu\cup\{(a\lambda_k,0)\})$. Then there exists $(\mu, u)\in \mathscr{C}_k^\nu\cap(\mathbb{R}\times \partial S_k^\nu)$ such that $(\mu, u) \neq(a\lambda_k, 0)$ and $(\mu_n,u_n)\to(\mu, u)$ with $(\mu_n,u_n)\in \mathscr{C}_k^\nu \cap(\mathbb{R}\times S_k^\nu)$. Since $u\in \partial S_k^\nu$, by Lemma \ref{lem2.1}, $u\equiv 0$. Let $w_n :=u_n/\| u_n\|$, then $w_n$ should be a solution of the following problem \begin{equation}\label{ewgm1} w=G\Big(\frac{1}{a+b\| u_n\|^2}\Big(\mu_n w+\frac{h(x,u_n,\mu_n)}{\| u_n(x)\|} \Big)\Big). \end{equation} By \eqref{egn0}, \eqref{ewgm1} and the compactness of $G$ we obtain that for some convenient subsequence $w_n\to w_0\neq 0$ as $n\to+\infty$. Now $w_0$ verifies the equation \[ -w'' = \frac{\mu}{a} w \] and $\| w_0\| = 1$. Hence $\mu = a\lambda_j$, for some $j \neq k$. Therefore, $(\mu_n,u_n)\to (a\lambda_j, 0)$ with $(\mu_n,u_n)\in \mathscr{C}_k^\nu\cap (\mathbb{R}\times S_k^\nu)$. This contradicts Lemma \ref{lem2.2}. Thus, we have that \[ \mathscr{C}_{k}^\nu\subseteq(\Phi_k^\nu\cup\{(a\lambda_k, 0)\}). \] We claim that both $\mathscr{C}_{k}^+$ and $\mathscr{C}_{k}^-$ are unbounded. Without loss of generality, we may suppose that $\mathscr{C}_{k}^-$ is bounded. Therefore, there exists $(\lambda_*,u_*)\in \mathscr{C}_{k}^+\cap\mathscr{C}_{k}^-$ such that $(\lambda_*,u_*)\neq(a\lambda_k,0)$ and $u_*\in S_k^+\cap S_k^-$. This contradicts the definitions of $S_k^+$ and $S_k^-$. \end{proof} \section{Spectrum} By an argument similar to that of Proposition \ref{prop2.1}, we can get the following regularity result. \begin{proposition} \label{prop3.1} Any weak solution $u\in X$ of problem \eqref{kee} is also a classical solution, i.e., $u\in C^{2}(0,1)\cap C^{1,\alpha}[0,1]$ satisfying \eqref{kee} and $u(0)=u(1)=0$. \end{proposition} \begin{lemma} \label{lem3.1} If $(\mu, u)$ is a solution of \eqref{kee} and $u$ has a double zero, then $u \equiv 0$. \end{lemma} \begin{proof} The homogeneity of problem \eqref{kee} implies that it suffices to consider a solution such that $\| u\| = 1$, which therefore is solution of the ordinary differential equation \begin{gather*} -u''=\mu u^3\quad \text{in } (0,1),\\ u(0)=u(1)=0. \end{gather*} Such a nontrivial solution necessarily cannot have a double zero by the uniqueness property of Cauchy problem for \eqref{kee}. \end{proof} \begin{lemma} \label{lem3.2} Each nontrivial solution of \eqref{kee} has a finite number of zeros. \end{lemma} \begin{proof} If a nontrivial solution has an infinite number of zeros, its accumulation point is a double zero, a contradiction. \end{proof} \begin{lemma} \label{lem3.3} $\mu_1(I)$ with $I=(0,1)$ satisfies the strict monotonicity property with respect to the domain $I$, i.e.\ if $J$ is a strict sub interval of $I$, then $\mu_1(I)<\mu_1(J)$. \end{lemma} \begin{proof} Let $\varphi_1$ with $\| \varphi_1\|=1$ be the eigenfunction of \eqref{kee} on $J$ corresponding to $\mu_1(J)$, and denote by $\widetilde{\varphi}_1$ the extension by zero on $I$. Then we have that \[ \frac{1}{\mu_1(J)}=\int_J | \varphi_1|^4\,dx =\int_I | \widetilde{\varphi}_1|^4\,dx <\sup_{u\in X,\| u\|=1}\int_0^1| u|^4\,dx =\frac{1}{\mu_1(I)}. \] The last strict inequality holds from the fact that $\widetilde{\varphi}_1$ vanishes in $I\setminus J$ so cannot be an eigenfunction corresponding to the principal eigenvalue $\mu_1(I)$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] Let $\varphi_1$ be a positive eigenfunction corresponding to $\mu_1$. It follows from the symmetry of \eqref{kee} and \cite[Theorem 1.2]{Dai2016} that $\varphi_1(x)=\varphi_1(1-x)$ for $x\in[0,1]$, i.e.\ $\varphi_1$ is even with respect to $1/2$. For any $k\geq 2$, set \[ \varphi_k(x)=\begin{cases} \varphi_1(kx),& x\in[0,1/k],\\ -\varphi_1(kx-1), & x\in[1/k,2/k],\\ \dots\\ (-1)^k\varphi_1(kx-k+1), & x\in[\frac{k-1}{k},1]. \end{cases} \] Then $\varphi_k$ is an eigenfunction of \eqref{kee} associated with the eigenvalue $\mu_k=k^4\mu_1$. On the other hand, let $u=u(x)$ be an eigenfunction of \eqref{kee} associated with some eigenvalue $\mu_*>\mu_1$. According to \cite[Theorem 1.2]{Dai2016}, $u$ changes sign in $(0,1)$. Lemmas \ref{lem3.1} and \ref{lem3.2} imply that $u\in S_k$ for some $k\geq 2$. Without loss of generality, we may assume that $u'(0)>0$. Let \[ 0<\tau_1<\tau_2<\dots<\tau_{k-1}<1 \] denote the zeros of $u$ in $(0,1)$. Without loss of generality, we may assume that $\tau_1\leq 1/k$. Applying Lemma \ref{lem3.3} on $[0,1/k]$, we have that $\mu_*\geq \mu_k$. By \cite[Lemma 2]{Be}, there exist integers $p$ and $q$, $1\leq p\leq k-1$, $1\leq q\leq k-1$, such that \[ \tau_p\leq \frac{1}{q+1}<\frac{1}{q}\leq \tau_{p+1}. \] Applying Lemma \ref{lem3.3} on $[\tau_p,\tau_{p+1}]$, we have that $\mu_*\leq\mu_k$. So we have that $\mu_*=\mu_k$. Furthermore, if $\tau_1<1/k$, we have $\mu_*>\mu_k$; if $\tau_1>1/k$, we have $\mu_*<\mu_k$. Thus we have $\tau_1=1/k$ and $u=c_1\varphi_k(x)$ for $x\in [0,1/k]$. Similarly, we can obtain that $\tau_i=i/k$ and $u=c_i\varphi_k(x)$ for $x\in [(i-1)/k,i/k]$, $2\leq i\leq k-1$. Let us normalize $u$ as $u'(0)=\varphi_k'(0)$. It follows that $c_1=1$. Hence $\varphi_k'(\frac{1}{k})=c_2\varphi_k'(\frac{1}{k})$. So we have $c_2=1$. Similarly, one has $c_i=1$ for all $3\leq i\leq k-1$. Therefore, we have that $u(x)=\varphi_k(x)$, $x\in [0,1]$. \end{proof} \section{Nodal solutions} In this section, we apply Theorems \ref{thm1.1} and \ref{thm1.2} to study the existence of nodal solutions for \eqref{fp}. \begin{proof}[Proof of Theorem \ref{thm1.3}] Let $g:(0,1)\times\mathbb{R}\to \mathbb{R}$ be a continuous function such that \[ f(x,s)=af_0s+g(x,s) \] with \begin{equation}\label{ehac} \lim_{s\to0}\frac{g(x,s)}{as}=0\quad\text{and}\quad \lim_{ | s|\to+\infty}\frac{g(x,s)}{s^3}=bf_\infty \end{equation} uniformly with respect to $x\in(0,1)$. From \eqref{ehac}, we can see that $\lambda g$ satisfies the assumptions of \eqref{a2}. Now, by Theorem \ref{thm1.1}, there are two distinct unbounded continua, $\mathscr{C}_k^+$ and $\mathscr{C}_k^-$ emanating from $(\lambda_k/f_0, 0)$, such that \[ \mathscr{C}_{k}^\nu\subset (\{(\lambda_k/f_0,0)\}\cup\Phi_k^\nu). \] If $\mathscr{C}_{k}^\nu$ is unbounded in the parameter direction, the conclusion is done. Next we assume that $\mathscr{C}_{k}^\nu$ is bounded in the parameter direction. Then it is sufficient to show that $\mathscr{C}_{k}^\nu$ joins $(\lambda_k/f_0,0)$ to $(\lambda_k/f_\infty,\infty)$. Let $(\xi_n, u_n) \in \mathscr{C}_{k}^\nu$ where $u_n\not\equiv 0$ satisfies $\xi_n+\| u_n\|\to+\infty$. Since (0,0) is the only solution of \eqref{fp} for $\lambda = 0$, we have $\mathscr{C}_{k}^\nu\cap(\{0\}\times X)=\emptyset$. It follows that $\xi_n >0$ for all $n \in \mathbb{N}$. Clearly, we have \[ \| u_n\|\to+\infty\quad\text{as } n\to +\infty. \] Let $h{:}(0,1)\times\mathbb{R}\to \mathbb{R}$ be a continuous function such that \[ f(x,s)=bf_\infty s^3+h(x,s) \] with \[ \lim_{| s|\to+\infty}\frac{h(x,s)}{s^3}=0\quad\text{and}\quad \lim_{ s\to0}\frac{h(x,s)}{s}=af_0 \] uniformly with respect to $x\in(0,1)$. Then $(\xi_n,u_n)$ satisfies \begin{equation}\label{evm2} u_n=G\Big(\frac{\xi_n}{a+b\| u_n\|^2}( bf_\infty u_n^3+h(x,u_n))\Big). \end{equation} Dividing the above equation by $\| u_n\|$ and letting $\overline{u}_n=u_n/\| u_n\|$, we obtain \[ \overline{u}_n =G\Big(\frac{\xi_n \| u_n\|^2}{a+b\| u_n\|^2}( bf_\infty \overline{u}_n^3+\frac{h(x,u_n)}{\| u_n\|^3})\Big). \] Let \[ \widetilde{h}(x,u)=\max_{0\leq | s|\leq u}| h(x,s)|\quad \text{for } x\in(0,1). \] Then $\widetilde{h}$ is nondecreasing with respect to $u$. Define \[ \overline{h}(x,u)=\max_{u/2\leq | s|\leq u}| h(x,s)|\quad \text{for } x\in(0,1). \] Then we can see that \[ \lim_{ u\to +\infty}\frac{\overline{h}(x,u)}{u^3}=0\quad\text{and}\quad \widetilde{h}(x,u)\leq \widetilde{h}(x,\frac{u}{2})+\overline{h}(x,u). \] It follows that \[ \limsup_{ u\to +\infty}\frac{\widetilde{h}(x,u)}{u^3} \leq \limsup_{ u\to +\infty}\frac{\widetilde{h}(x,\frac{u}{2})}{u^3} =\limsup_{ u/2\to +\infty}\frac{\widetilde{h}(x,\frac{u}{2})}{8(\frac{u}{2})^3}. \] So we have \begin{equation}\label{eg0+} \lim_{ u\to +\infty}\frac{\widetilde{h}(x,u)}{u^3}=0. \end{equation} Further it follows from \eqref{eg0+} that \[ \frac{h(x,u_n)}{\| u_n\|^3} \leq\frac{\widetilde{h}(x,| u_n|)}{\| u_n\|^3} \leq \frac{\widetilde{h}(x,\| u_n\|_\infty)}{\| u_n\|^3} \leq c^3\frac{\widetilde{h}(x,c\| u_n\|)}{c^3\| u_n\|^3}\to0 \] as $ n\to +\infty$ uniformly for $x\in(0,1)$, where $c>0$ is the best embedding constant of $X\hookrightarrow C[0,1]$. From the compactness of $G$ we obtain \[ -\| \overline{u}\|^2 \overline{u}'' =\overline{\mu}f_\infty \overline{u}^3, \] where $\overline{u}=\underset{n\to+\infty}\lim \overline{u}_n$ and $\overline{\mu}=\underset{n\to+\infty}\lim\xi_n$, again choosing a subsequence and relabeling it if necessary. It is clear that $\| \overline{u}\|=1$ and $\overline{u}\in \mathscr{C}_k^\nu$. Theorem \ref{thm1.2} shows that $\overline{\mu}=\mu_k/f_\infty$. Therefore, $\mathscr{C}_k^\nu$ joins $(\lambda_k/f_0,0)$ to $(\mu_k/f_\infty,\infty)$. \end{proof} \subsection*{Acknowledgments} This research was supported by the NNSF of China (No. 11871129), by the Fundamental Research Funds for the Central Universities (No. DUT17LK05), by the Xinghai Youqing funds from Dalian University of Technology, and by the Natural Science Foundation of Jiangsu Education Committee (No. 18KJB110002). \begin{thebibliography}{99} \bibitem{AP} A. Arosio, S. Pannizi; \emph{On the well-posedness of the Kirchhoff string}, Trans. Amer. Math. Soc., 348 (1996), 305--330. \bibitem{APS} G. Autuori, P. Pucci, M. C. Salvatori; \emph{Asymptotic stability for anisotropic Kirchhoff systems}, J. Math. Anal. Appl., 352 (2009), 149--165. \bibitem{Be} H. Berestycki; \emph{On some nonlinear Sturm-Liouville problems}, J. Differential Equations, 26 (1977), 375--390. \bibitem{Bre} H. Brezis; \emph{Operateurs Maximaux Monotone et Semigroup de Contractions dans les Espase de Hilbert}, Math. Studies, 5, North-Holland, Amsterdam, 1973. \bibitem{CCS} M. M. Cavalcante, V. N. Cavalcante, J. A. Soriano; \emph{Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation}, Adv. Differential Equations, 6 (2001), 701--730. \bibitem{CKW} C. Chen, Y. Kuo, T. Wu; \emph{The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions}, J. Differential Equations, 250 (4) (2011), 1876--1908. \bibitem{CW} B. Cheng, X. Wu; \emph{Existence results of positive solutions of Kirchhoff type problems}, Nonlinear Anal., 71 (10) (2009), 4883--4892. \bibitem{CF} F. J. S. A. Corr\^{e}a, G. M. Figueiredo; \emph{On a elliptic equation of $p$-kirchhoff type via variational methods}, Bull. Austral. Math. Soc., 74 (2006), 263--277. \bibitem{Dai2016} G. Dai; \emph{Eigenvalue, global bifurcation and positive solutions for a class of nonlocal elliptic equations}, Topol. Methods Nonlinear Anal., 48 (2016), 213--233. \bibitem{DH} G. Dai, R. Hao; \emph{Existence of solutions for a $p(x)$-Kirchhoff-type equation}, J. Math. Anal. Appl., 359 (2009), 275--284. \bibitem{DMW} G. Dai, R. Ma, H. Wang; \emph{Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian}, Commun. Pure Appl. Anal., 12 (2013), 2839--2872. \bibitem{DJ} G. Dai, J. Wei; \emph{Infinitely many non-negative solutions for a $p(x)$-Kirchhoff-type problem with Dirichlet boundary condition}, Nonlinear Anal., 73 (2010), 3420--3430. \bibitem{DL} G. Dai, D. Liu; \emph{Infinitely many positive solutions for a $p(x)$-Kirchhoff-type equation}, J. Math. Anal. Appl., 359 (2009), 704--710. \bibitem{DM} G. Dai, R. Ma; \emph{Solutions for a $p(x)$-Kirchhoff type equation with Neumann boundary data}, Nonlinear Anal. Real World Appl., 12 (2011), 2666--2680. \bibitem{Dancer0} E. N. Dancer; \emph{On the structure of solutions of non-linear eigenvalue problems}, Indiana Univ. Math. J., 23 (1974), 1069--1076. \bibitem{DS} P. D'Ancona, S. Spagnolo; \emph{Global solvability for the degenerate Kirchhoff equation with real analytic data}, Invent. Math., 108 (1992), 247--262. \bibitem{DS1} P. D'Ancona, Y. Shibata; \emph{On global solvability of nonlinear viscoelastic equations in the analytic category}, Math. Methods Appl. Sci., 17 (6) (1994), 477--486. \bibitem{D2} M. Dreher; \emph{The Kirchhoff equation for the $p$-Laplacian}, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217--238. \bibitem{D1} M. Dreher; \emph{The wave equation for the $p$-Laplacian}, Hokkaido Math. J., 36 (2007), 21--52. \bibitem{E} L. C. Evans; \emph{Partial differential equations}, AMS, Rhode Island, 1998. \bibitem{F} X. L. Fan; \emph{On nonlocal $p(x)$-Laplacian Dirichlet problems}, Nonlinear Anal., 72 (2010), 3314--3323. \bibitem{GMS} J. Giacomoni, P. K. Mishra, K. Sreenadh; \emph{Fractional elliptic equations with critical exponential nonlinearity}, Adv. Nonlinear Anal., 5 (2016), 57--74. \bibitem{GT} D. Gilbarg, N. S. Trudinger; \emph{Elliptic partial differential equations of second order}, Springer, Berlin, 2001. \bibitem{HZ} X. He, W. Zou; \emph{Infinitely many positive solutions for Kirchhoff-type problems}, Nonlinear Anal., 70 (2009), 1407--1414. \bibitem{K} G. Kirchhoff; \emph{Mechanik}, Teubner, Leipzig, 1883. \bibitem{LLS} Z. Liang, F. Li, J. Shi; \emph{Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior}, Ann. I. H. Poincar\'{e}-AN, 31 (2014), 155--167. \bibitem{L} J. L. Lions; \emph{On some equations in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations} (Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284--346. \bibitem{MR} T. F. Ma, J. E. Mu\~{n}oz Rivera; \emph{Positive solutions for a nonlinear nonlocal elliptic transmission problem}, Appl. Math. Lett., 16 (2) (2003), 243--248. \bibitem{MZ} A. Mao, Z. Zhang; \emph{Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition}, Nonlinear Anal., 70 (3) (2009), 1275--1287. \bibitem{MRS} G. Molica Bisci, V. Radulescu, R. Servadei; \emph{Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications}, 162. Cambridge University Press, Cambridge, 2016. \bibitem{MR2} G. Molica Bisci, D. Repov\v{s}; \emph{On doubly nonlocal fractional elliptic equations}, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 246--255. \bibitem{PZ} K. Perera, Z. Zhang; \emph{Nontrivial solutions of Kirchhoff-type problems via the Yang index}, J. Differential Equations, 221 (1) (2006), 246--255. \bibitem{PXZ} P. Pucci, M. Xiang, B. Zhang; \emph{Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations}, Adv. Nonlinear Anal., 5 (2016), 27--55. \bibitem{R2} P. H. Rabinowitz; \emph{Some global results for nonlinear eigenvalue problems}, J. Funct. Anal., 7 (1971), 487--513. \bibitem{ST} J. Sun, C. Tang; \emph{Existence and multiplicity of solutions for Kirchhoff type equations}, Nonlinear Anal., 74 (4) (2011), 1212--1222. \bibitem{ZP} Z. Zhang, K. Perera; \emph{Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow}, J. Math. Anal. Appl., 317 (2) (2006), 456--463. \bibitem{ZZR} X. Zhang, B. Zhang, D. Repov\v{s}; \emph{Existence and symmetry of solutions for critical fractional Schr\"{o}dinger equations with bounded potentials}, Nonlinear Anal., 142 (2016), 48--68. \end{thebibliography} \end{document}