\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 274, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/274\hfil Multiple solutions] {Multiple solutions for an indefinite Kirchhoff-type equation with sign-changing potential} \author[H. Liu, H. Chen \hfil EJDE-2015/274\hfilneg] {Hongliang Liu, Haibo Chen} \address{Hongliang Liu \newline Department of Mathematics, Central South University, Changsha, 410083 Hunan, China} \email{math\_lhliang@163.com} \address{Haibo Chen (corresponding author)\newline Department of Mathematics, Central South University, Changsha, 410083 Hunan, China} \email{math\_chb@csu.edu.cn} \thanks{Submitted December 20 2014. Published October 21, 2015.} \subjclass[2010]{35J15, 35J20, 38E05} \keywords{Multiple solutions; Kirchhoff-type equation; sign-changing potential; \hfill\break\indent Morse theory; variational methods} \begin{abstract} In this article, we study a Kirchhoff-type equation with sign-changing potential on an infinite domain. Using Morse theory and variational methods, we show the existence of two and of infinitely many nontrivial solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and statement of main results} In this article, we study the existence of multiple solutions for the nonlinear Kirchhoff-type equation \begin{equation} -\Big(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\Big) \Delta u+V(x)u=f(x,u),\quad\text{in }\mathbb{R}^{N}, \label{e1.1} \end{equation} where $N\geq 2$ and parameters $a>0$, $b\geq 0$ and the potential $V$ satisfies the condition \begin{itemize} \item[(A1)] $V\in C(\mathbb{R}^{N})\cap L^\infty(\mathbb{R}^{N})$, $V(x)\leq \bar{V}\in (0,\infty)$ for all $x\in \mathbb{R}^{N}$ and there exists a constant $l_0>0$ such that \begin{equation} \int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)|u|^{2}]dx \geq l_0\int_{\mathbb{R}^{N}}[\bar{V}-V(x)]|u|^{2}dx, \quad\forall u\in H^1(\mathbb{R}^{N}).\label{e1.2} \end{equation} \end{itemize} From this condition, we see that $V(x)$ is allowed to be sign-changing and we consider the increasing sequence $\lambda_1\leq \lambda_2\leq\lambda_3\leq\dots$ of minimax values defined by \begin{equation} \lambda_n:=\inf_{V\in \mathcal{V}_{n}}\sup_{u\in V,u\neq 0} \frac{\int_{\mathbb{R}^{N}}\left(|\nabla u|^2+V u^2\right)dx}{\int_{\mathbb{R}^{N}} u^2dx}, \end{equation} where $\mathcal{V}_{n}$ denotes the family of n-dimensional subspaces of $C_0^\infty(\mathbb{R}^{N})$. Denote $ \lambda_\infty=\lim_{n\to\infty}\lambda_n$. Then $\lambda_\infty$ is the bottom of the essential spectrum of $-\Delta+V$ if it is finite and for every $n\in \mathbb{N}$ the inequality $\lambda_n<\lambda_\infty$ implies that $\lambda_n$ is an eigenvalue of $-\Delta+V$ of finite multiplicity \cite{M.Reed}. Throughout this paper, we assume there exists $k\geq 1$ such that \begin{equation} \lambda_k<0<\lambda_{k+1}. \label{e1.3} \end{equation} Problem \eqref{e1.1} has been widely studied in recent years. For instance, by using a variant version of fountain theorem, Liu and He \cite{W.Liu} studied the existence of infinitely many high energy solutions of \eqref{e1.1}. Wu \cite{X.Wu} investigated the existence of nontrivial solutions and infinitely many high energy solutions of \eqref{e1.1} via a symmetric mountain pass theorem. Sun and Wu \cite{J.Sun} applied variational methods to study the existence and the non-existence of nontrivial solutions of \eqref{e1.1} and explored the concentration of solutions. Li and Ye \cite{G.Li} considered \eqref{e1.1} with pure power nonlinearities $f(x,u)=|u|^{p-1}u$ in $\mathbb{R}^3$. By using a monotonicity trick and a new version of global compactness lemma, they verified that the problem has a positive ground state solution which can be viewed as a partial extension of \cite{X.He2} where the authors studied the existence and concentration behavior of positive solutions of \eqref{e1.1}. For other interesting results on the related Kirchhoff equations, we refer to \cite{B.Cheng,G.Figueiredo2015AA,X.He1,Y.Li,H.Liu2015BVPs,A.Mao,G.Bisci2014PEMS, G.Bisci2015RLMA,G.Bisci2014AAFM,K.Perera,L.Xu2014BVPs,J.Zhang2014JMAA,J.Zhang2014AMC} and the references therein. It is well known that the Morse theory \cite{K.Chang} and variational methods \cite{J.Mawhin} are two useful tools in studying the existence and multiplicity of solutions for the variational problem (see, e.g.\cite{M.Jiang2012,H.Liu2015AML,H.Liu2014CMA,Su2006,M.Sun2014,J.Zhang}). However, to the best of our knowledge, there is only one paper \cite{D.Liu}, in which the authors considered the problem in a domain $\Omega\subset\mathbb{R}^N$ with smooth boundary $\partial\Omega$, dealing with the Kirchhoff-type problem by using Morse theory up to now. Inspired by the above facts, the aim of this paper is to study the multiple solutions of \eqref{e1.1} with sign-changing potential by using Morse theory and variational methods. Before stating our main results we need to make some assumptions on the nonlinearity $f$. \begin{itemize} \item[(A2)] $f\in C^1(\mathbb{R}^{N}\times\mathbb{R})$ and there exist $p\in(2,2^*)$ and $c_1>0$ such that \begin{equation} |f(x,t)|\leq c_1(1+|t|^{p-1}),\quad\forall (x,t)\in \mathbb{R}^{N}\times\mathbb{R}. \label{e1.4} \end{equation} \item[(A3)] There exists $00$ such that \begin{equation} C_1\|u\|_{H^1(\mathbb{R}^{N})}^2 \leq \int_{\mathbb{R}^{N}}[|\nabla u|^{2}+V(x)u^{2}]dx \leq C_2\|u\|_{H^1(\mathbb{R}^{N})}^2,\quad\forall u \in E. \label{e2.2} \end{equation} \end{lemma} Recall that $H^1(\mathbb{R}^{N})$ is a Hilbert space with the norm \eqref{e2.1} and is continuous embedded into $L^p(\mathbb{R}^{N})$ for any $p\in [2,2^*]$. By Lemma \ref{lem2.1}, for any $p\in [2,2^*]$, there exists an imbedding constant $\gamma_s\in(0,\infty)$ such that \begin{equation} \|u\|_s\leq\gamma_s\|u\|,\quad\forall u\in E. \label{e2.3} \end{equation} From (A3), we can choose $l_0>0$ and $\bar{V}\in (h,\lambda_\infty)$ such that $\bar{V}\notin \{\lambda_i|1\leq i<+\infty\}$ and \eqref{e1.2} holds. Let $E^-$ be the space spanned by the eigenfunctions with corresponding eigenvalues less than $\bar{V}$. Then, $E^-$ is finite dimensional subspace of $E$. Let $E^+$ be the orthogonal complement space of $E^-$ in $E$. Since $E$ is a Hilbert space, we have $E=E^+\bigoplus E^-$. So, for every $u\in E$, we have a unique decomposition $u=u^++u^-$ with $u^+\in E^+$ and $u^-\in E^-$. By $\bar{V}\notin \{\lambda_i|1\leq i<+\infty\}$ and Lemma \ref{lem2.1}, there exists an equivalent norm of $E$, still denoted by $\|\cdot\|$, such that \begin{equation} \int_{\mathbb{R}^{N}}|\nabla u|^2+\int_{\mathbb{R}^{N}} V(x)u^2 -\bar{V}\int_{\mathbb{R}^{N}}u^2 =\|u^+\|^2-\|u^-\|^2. \label{e2.4} \end{equation} Let $E$ be a real Banach space and $J\in C^{1}(E,\mathbb{R})$. \begin{definition}[\cite{K.Chang}]\label{def2.1}\rm Let $u$ be an isolated critical point of $J$ with $J(u)=c$, for $c\in \mathbb{R}$, and let $U$ be a neighborhood of $u$, containing the unique critical point. We call \[ C_q(J,u):=H_q(J^c\cap U,J^c\cap U\setminus\{u\}),\quad q=0,1,2,\dots, \] the $q$th critical group of $J$ at $u$, where $J^c:=\{u\in E:J(u)\leq c\}$, $H_q(\cdot,\cdot)$ stands for the $q$th singular relative homology group with integer coefficients. \end{definition} We say that $u$ is a homological nontrivial critical point of $J$ if at least one of its critical groups is nontrivial. \begin{proposition}[\cite{T.Bartsch}] \label{prop2.1} Let $0$ be a critical point of $J$ with $J(0)=0$. Assume that $J$ has a local linking at $0$ with respect to $E=E_1\oplus E_2$, $m=\dim E_1<\infty$, that is, there exists $\rho>0$ small such that \begin{equation} J(u)\leq 0,\quad u\in E_1, \|u\|\leq \rho, \quad J(u)>0,\quad u\in E_2,\; 0<\|u\|\leq\rho. \end{equation} Then $C_m(J,0)\not\cong 0$; that is, $0$ is a homological nontrivial critical point of $J$. \end{proposition} \begin{definition} \label{def2.2} \rm We say that $J\in C^{1}(E,\mathbb{R})$ satisfies $(PS)$-condition if any sequence $\{u_n\}$ in $E$ such that \[ J(u_n)\to c,\quad J'(u_n)\to 0,\quad\text{as } n\to\infty, \] has a convergent subsequence. \end{definition} \begin{proposition}[\cite{J.Zhang}] \label{pro2.2} Assume that $J$ satisfies the $(PS)$-condition and is bounded from below. If $J$ has a critical point that is homological nontrivial and is not the minimizer of $J$. Then $J$ has at least three critical points. \end{proposition} \begin{proposition}[\cite{Z.Liu}] \label{pro2.3} Let $X$ be a Banach space, $J\in C^{1}(X,\mathbb{R})$. Assume that $J$ satisfies $(PS)$-condition, is even and bounded from below, and $J(0)=0$. If for any $m\in \mathbb{N}$, there exists a $k$-dimensional subspace $X^m$ of $X$ and $\rho_m>0$ such that $\sup_{X^m\cap S_{\rho_m}}J<0$, where $S_{\rho_m}=\{u\in X|\|u\|=\rho_m\}$, then at least one of the following conclusions holds. \begin{itemize} \item[(i)] There exists a sequence of critical points $\{u_m\}$ satisfying $J(u_m)<0$ for all $m$ and $\|u_m\|\to 0$ as $m\to\infty$. \item[(ii)] There exists $r>0$ such that for any $00$ and $\|u_n\|\to\infty$ such that $J(u_n)\leq C$ as $n\to \infty$. For all $(x,u)\in \mathbb{R}^{N}\times\mathbb{R}$, we deduce from \eqref{e1.5} that \begin{equation} \frac{1}{2}h u^2\geq F(x,u)>0. \label{e3.3} \end{equation} Now, we choose $h<\bar{V}<\lambda_\infty$ and $l_0>0$ such that $\bar{V}\notin\{\lambda_i|1\leq i<+\infty\}$ and \eqref{e1.2} holds. Then, applying \eqref{e2.4}, \eqref{e3.1} and \eqref{e3.3} yields \begin{equation} \begin{aligned} J(u_n) &= \frac{1}{2}\int_{\mathbb{R}^{N}}[a|\nabla u_n|^{2} +V(x)u_n^{2}-\bar{V}u_n^2] +\frac{b}{4}\Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^{2}\Big)^{2}\\ &\quad+\int_{\mathbb{R}^{N}}\big[\frac{1}{2}\bar{V}u_n^{2}-F(x,u_n)\big]\\ &\geq \frac{1}{2}\min\{a,1\}\left(\|u_n^+\|^2-\|u_n^-\|^2\right)+\frac{b}{4} \Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^{2}\Big)^{2}\\ &\geq \frac{1}{2}\min\{a,1\}\big(\|u_n^+\|^2-\|u_n^-\|^2\big). \end{aligned}\label{e3.4} \end{equation} Let $v_n:=u_n/\|u_n\|$. By $\|u_n\|\to\infty$, $J(u_n)\leq C$ and \eqref{e3.4}, we have \begin{equation} \|v_n^+\|^2\leq\|v_n^-\|^2+o(1). \label{e3.5} \end{equation} Going if necessary to a subsequence, we may assume that $v_n\rightharpoonup v$ in $E$ and $v_n(x)\to v(x)$ a.e. in $\mathbb{R}^{N}$. If $v=0$, then by the finite dimension of $E^-$, we deduce that $v_n^-\to 0$ in $E$. This and \eqref{e3.5} yield $v_n\to 0$ in $E$. It is a contradiction, because for every $n$, we have $\|v_n\|=1$. Therefore, $v^-\neq 0$ and then $v\neq 0$. Then it deduces from Fatou's lemma that \begin{equation} \begin{aligned} \liminf_{n\to \infty}\frac{b}{4\|u_n\|^{4}} \Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^2 \Big)^{2} &=\liminf_{n\to \infty}\frac{b}{4} \Big(\int_{\mathbb{R}^{N}}|\nabla v_n|^2 \Big)^{2}\\ &\geq\frac{b}{4}\Big(\int_{\mathbb{R}^{N}}|\nabla v|^2 \Big)^{2}>0. \end{aligned}\label{e3.6} \end{equation} Since $\|u_n\|\to\infty$ and $J(u_n)\leq C$, we have \begin{equation} \|u_n\|^{-4}J(u_n)\to 0,\quad\text{as } n\to\infty. \label{e3.7} \end{equation} Hence, multiplying both sides of the following inequality by $\|u_n\|^{-4}$ and letting $n\to \infty$, \begin{equation} J(u_n) \geq \frac{1}{2}\min\{a,1\}\left(\|u_n^+\|^2-\|u_n^-\|^2\right) +\frac{b}{4} \Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^{2}\Big)^{2}, \label{e3.8} \end{equation} From \eqref{e3.6} and \eqref{e3.7} we obtain \[ 0\geq\frac{b}{4}\Big(\int_{\mathbb{R}^{N}}|\nabla v|^2 \Big)^{2} >0. \] It is a contradiction. Therefore, we prove that $J$ is coercive in $E$. Consequently, $J$ is bounded form below in $E$. The proof is complete. \end{proof} \begin{lemma} \label{lem3.2} Assume that {\rm (A1)-(A3)} hold. Then $J$ satisfies the $(PS)$-condition. \end{lemma} \begin{proof} Let $\{u_n\}$ be a $(PS)_c$ sequence, i.e., $J(u_n)\to c$ and $J'(u_n)\to 0$ in $E^*$, as $n\to\infty$. Lemma \ref{lem3.1} shows that $J$ is coercive. Then $J(u_n)\to c$ implies that $\{u_n\}$ is bounded. By \eqref{e3.2} and $J'(u_n)\to 0$, we have \begin{equation} \begin{aligned} &o(\|u_n\|)\\ &=\langle J'(u_n),u_n\rangle\\ &=a\int_{\mathbb{R}^{N}}|\nabla u_n|^2 +\int_{\mathbb{R}^{N}}V(x)u_n^2 +b\Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^2 \Big)^2 -\int_{\mathbb{R}^{N}}f(x,u_n)u_n \\ &\geq \min\{a,1\}\big(\|u_n^+\|^2-\|u_n^-\|^2\big) +b \Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^{2}\Big)^{2}\\ &\quad+\int_{\mathbb{R}^{N}}[hu_n^2-f(x,u_n)u_n]. \end{aligned}\label{e3.9} \end{equation} Then we deduce from \eqref{e3.9} that \begin{equation} \begin{aligned} &o(\|u_n\|)+\min\{a,1\}\|u_n^-\|^2\\ &\geq \min\{a,1\}\|u_n^+\|^2 +b\Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^{2}\Big)^{2} +\int_{\mathbb{R}^{N}}[hu_n^2-f(x,u_n)u_n]. \end{aligned}\label{e3.10} \end{equation} Up to a subsequence, we may assume $u_n\rightharpoonup u$ in $E$. Then we have that $u$ is a critical point of $J$. It follows that \begin{equation} \begin{aligned} 0&=\langle J'(u),u\rangle\\ &\geq \min\{a,1\}\left(\|u^+\|^2-\|u^-\|^2\right) +b\Big(\int_{\mathbb{R}^{N}}|\nabla u|^{2}\Big)^{2} +\int_{\mathbb{R}^{N}}[hu^2-f(x,u)u], \end{aligned} \end{equation} which implies \begin{equation} \begin{aligned} &\min\{a,1\}\|u^-\|^2\\ &\geq \min\{a,1\}\|u^+\|^2+b \Big(\int_{\mathbb{R}^{N}}|\nabla u|^{2}\Big)^{2} +\int_{\mathbb{R}^{N}}[hu^2-f(x,u)u]. \end{aligned} \label{e3.11} \end{equation} Since $E^-$ is a finite dimensional subspace of $E$, we get $u_n^-\to u^-$, and then $\|u_n^-\|^2\to\|u^-\|^2$. This together with \eqref{e3.10} and \eqref{e3.11} imply \begin{equation} \begin{aligned} &\lim_{n\to\infty}\Big[ \min\{a,1\}\|u_n^+\|^2+b \Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^{2}\Big)^{2} +\int_{\mathbb{R}^{N}}[hu_n^2-f(x,u_n)u_n]\Big]\\ & =\min\{a,1\}\|u^+\|^2+b \Big(\int_{\mathbb{R}^{N}}|\nabla u|^{2}\Big)^{2} +\int_{\mathbb{R}^{N}}[hu^2-f(x,u)u]. \end{aligned}\label{e3.12} \end{equation} An easy calculation, using (A3) and Fatou's lemma, shows that \begin{equation} \begin{aligned} & \liminf_{n\to\infty}\Big[ b \Big(\int_{\mathbb{R}^{N}}|\nabla u_n|^{2}\Big)^{2} +\int_{\mathbb{R}^{N}}[hu_n^2-f(x,u_n)u_n]\Big]\\ & \geq b\Big(\int_{\mathbb{R}^{N}}|\nabla u|^{2}\Big)^{2} +\int_{\mathbb{R}^{N}}[hu^2-f(x,u)u]. \end{aligned}\label{e3.13} \end{equation} Combining \eqref{e3.12} with \eqref{e3.13} gives that $\lim_{n\to\infty}\|u_n^+\|^2=\|u^+\|^2$. It follows that $u_n\to u$ in $E$. Thus, we completed the proof. \end{proof} Now, we are in a position to calculate the critical groups of $J$ at $0$. \begin{lemma} \label{lem3.3} Assume that {\rm (A1)--(A3)} hold. Then there exists $m\in \mathbb{N}$ with $m\geq k$ such that $C_m(J,0)\not\cong 0$. \end{lemma} \begin{proof} Let $E_1=E^-$ and $E_2=E^+$. Then $m=dim(E^-)\geq k$. On one hand, from \eqref{e2.3}, \eqref{e3.1}, \eqref{e3.3} and Lemma \ref{lem2.1}, for any $u\in E_1$, we have \begin{equation} \begin{aligned} J(u) &\leq -\frac{1}{2}\max\{a,1\}\|u^-\|^2+\frac{b}{4} \Big(\int_{\mathbb{R}^{N}}|\nabla u|^{2}\Big)^{2}+\frac{1}{2} \bar{V}\int_{\mathbb{R}^{N}}u^{2}\\ &\quad -\int_{\mathbb{R}^{N}}F(x,u)\\ &\leq -\frac{1}{2}\max\{a,1\}\|u^-\|^2+\frac{b}{4} \Big(\int_{\mathbb{R}^{N}}|\nabla u|^{2}\Big)^{2}+\frac{1}{2} \bar{V}\int_{\mathbb{R}^{N}}u^{2}\\ &\leq -\frac{1}{2}\max\{a,1\}\|u^-\|^2+\frac{b}{4} \Big(\int_{\mathbb{R}^{N}}|\nabla u|^{2}+u^2 \Big)^{2}+\frac{1}{2} \bar{V}\|u\|_2^2\\ &\leq -\frac{1}{2}\max\{a,1\}\|u^-\|^2+\frac{b}{4} \|u\|_{H^{1}(\mathbb{R}^{N})}^4+\frac{1}{2}\bar{V}\|u\|_2^2\\ &\leq -\frac{1}{2}\max\{a,1\}\|u^-\|^2+C_1\|u\|^4+C_2\|u\|_2^2. \end{aligned} \label{e3.14} \end{equation} Since $E_1$ is a finite dimensional subspace and all norms on a finite dimensional space are equivalent, we deduce from \eqref{e3.14} that \[ J(u)\leq -\frac{1}{2}\max\{a,1\}\|u^-\|^2+C_1\|u^-\|^4-C_2\|u^-\|^2, \] which implies that $J(u)\leq 0$, if $\|u\|$ small. On the other hand, for any $u\in E_2$, \eqref{e3.4} shows that \[ J(u)\geq \frac{1}{2}\min\{a,1\}\|u^+\|^2, \] which implies that $J(u)> 0$, if $\|u\|$ is small. The above arguments shows that $J$ has a local linking at $0$ with respect to $E=E^-\oplus E^+$. Clearly, it follows from \eqref{e3.1} that $J(0)=0$. Therefore, by Proposition \ref{prop2.1}, we get that there exists $m\in \mathbb{N}$ such that $C_m(J,0)\not\cong 0$. That is, $0$ is a homological nontrivial critical point of $J$. The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] From Lemmas \ref{lem3.1} and \ref{lem3.2}, we know that $J$ is bounded from below and satisfies $(PS)$-condition. Lemma \ref{lem3.3} shows that $0\in E$ is a homologically nontrivial critical point of $J$ but not a minimizer. Then by virtue of Proposition \ref{pro2.2}, we get that problem \eqref{e1.1} has two nontrivial solutions. The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] By (A4) and \eqref{e3.1}, one can easily check that functional $J$ is even and satisfies $J(0)=0$. Lemma \ref{lem3.1} and Lemma \ref{lem3.2} show that $J$ is bounded from below in $E$ and satisfies the $(PS)$-condition. For any $m\in \mathbb{N}$ and $m\geq k$, $\rho_m>0$, let $S_{\rho_m}=\{u\in X:\|u\|=\rho_m\}$. Then for any $ u\in S_{\rho_m}$, it deduces from (A3) that \begin{equation} J(u)\leq \frac{1}{2}\max\{a,1\}\left(\|u^-\|^2-\|u^-\|^2\right)+\frac{b}{4} \|u\|_{H^1(\mathbb{R}^{N})}^4+\frac{1}{2} \bar{V}\|u\|_2^2. \label{e3.15} \end{equation} Note that $E^-:=X^m$ is a $m-$dimensional subspace of $E$. Since all norms are equivalent on a finite dimensional space, for $ u\in X^m\cap S_{\rho_m}$, it follows from \eqref{e3.15} that \begin{equation} \sup_{X^m\cap S_{\rho_m}}J(u) \leq -\frac{1}{2}\max\{a,1\}\|u^-\|^2+C_3 \|u^-\|^4-C_4\|u^-\|^2, \end{equation} which implies that \[ \sup_{X^m\cap S_{\rho_m}}J(u)< 0, \] if $\rho_m>0$ is sufficiently small. Moreover, if there exists $r>0$ such that for any $0