\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 24, pp. 1--13.\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-2014/24\hfil Signed radial solutions] {Signed radial solutions for a weighted $p$-superlinear problem} \author[S. Herr\'on, E. Lopera \hfil EJDE-2014/24\hfilneg] {Sigifredo Herr\'on, Emer Lopera} % in alphabetical order \address{Sigifredo Herr\'on \newline Escuela de Matem\'aticas, Universidad Nacional de Colombia Sede Medell\'in, Apartado A\'ereo 3840, Medell\'{\i}n, Colombia} \email{sherron@unal.edu.co} \address{Emer Lopera \newline Universidad Nacional de Colombia Sede Medell\'in, Medell\'{\i}n, Colombia} \email{edlopera@unal.edu.co} \thanks{Submitted October 1, 2013. Published January 14, 2014.} \subjclass[2000]{35J92, 35J60} \keywords{Quasilinear weighted elliptic equations; radial solutions; \hfill\break\indent one-signed solutions; shooting method} \begin{abstract} We study the existence of one-signed radial solutions for weighted semipositone problems where $\Delta_p$ operator is involved and the nonlinearity is $p$-superlinear at infinity and has only two zeros. We establish the existence of at least two one-signed solutions when the weight is small enough. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let us consider the existence of one-signed radial solutions for the boundary-value problem \begin{equation} \begin{gathered} -\Delta_pu=K(\| x\|) f(u),\quad x\in B_1(0) \\ u=0,\quad \| x\| =1, \end{gathered} \label{1} \end{equation} where $B_1(0) \subset\mathbb{R}^N$ is the unit ball and $2\leq p0$, where $v_0$ and $u_0$ are the only zeros of $f$. \item[(F2)] $f(0^{-}) :=\lim_{t\to0^{-}}f(t)$ is a positive number and $f(0^{+}):=\lim_{t\to0^{+}}f(t) $ is a negative number. \item[(F3)] $\lim_{|\alpha|\to\infty}\frac{f(\alpha) }{\varphi_p(\alpha) } =+\infty$. \item[(F4)] There exist constants $k\in(0,1) $ and $\theta>N-p$ such that for all $\delta\geq \theta$ \[ \lim_{|\alpha|\to\infty}\Big( \frac{\varphi_p(\alpha) }{f(\alpha) }\Big) ^{\frac{N}{p}}\Big(\delta F(k\alpha) -\frac{N-p}{p}\alpha f(\alpha)\Big) =+\infty, \] where $F(t) =\int_0^{t}f(s) ds$. \end{itemize} For the weight $K$, hereafter we will assume that \begin{itemize} \item[(K1)] $K\in C([0,1],\mathbb{R}^{+}) $ and $K$ is differentiable in $(0,1)$. \item[(K2)] $r\mapsto N+r\frac{K'(r) }{K(r) }>\theta$, is non-increasing in $(0,1) $ and $\lim_{r\to0^{+}}\frac{K'(r) }{K(r)}$ exists in $\mathbb{R}$. \end{itemize} A function satisfying the statements (K1) and (K2) will be called an admissible weight. The aim of this article is to prove that under hypotheses (F1)--(F4) for the nonlinearity and (K1)--(K2) for the weight, problem \eqref{2} has at least two solutions, provided that the weight is small enough. Moreover, one of them is positive and the other one is negative. In order to prove the existence of positive solution, we modify our nonlinearity in the following way \[ f^{+}(t) :=\begin{cases}f(t) &\text{if } t>0\\ f(0^{+}) &\text{if } t=0\\ 0 &\text{if } t<0. \end{cases} \] Similar modifications vanishing the positive part of $f$ give us a negative solution. The case $p=2$ was studied by Castro and Shivaji in \cite{CS}. Recently, Hakimi and Zertiti, \cite{HZ}, following the ideas in \cite{CS} obtained existence of positive solutions for a more general nonlinearity than Castro and Shivaji. Both works considered a constant weight in the semilinear case. We emphasize that in this work we deal with the case $2\leq pp-1$, in exterior domains. They employed a global continuation theorem and fixed point index theory based on a weighted space as the underlying space. By using this approach they obtained multiplicity of positive solutions depending on a certain real parameter $\mu$. A one dimensional weighted $p$-Laplacian problem is presented in \cite{NT}. Sharp conditions for the existence of solutions with prescribed numbers of zeros in terms of the ratio $f (s)/s^{ p-1}$ at zero and at infinity were established there. Their technique was based on the shooting method together with the qualitative theory for half-linear differential equations. Other results can be found in \cite{AP, CH, CHV, GMY, Z} and some references therein. Our main tool for solving problem \eqref{2} is the shooting method. Hence, we start considering the auxiliary initial value problem \begin{equation} \begin{gathered} {}[\varphi_p(u'(r)) ]'+\frac{N-1}{r}\varphi_p(u'(r)) +K(r) f(u(r)) =0,\quad 00$. \subsection{Pohozaev Identity} In this subsection, we present a Pohozaev identity as well as some consequences. The proof of this identity is quite standard but it is presented in the appendix for the sake of completeness. Let us define the Energy associated to problem \eqref{3} by \[ E(t,\alpha,K) :=\frac{|u'(t,\alpha,K) |^{p}}{p'K(t) }+F^+(u(t,\alpha,K)). \] Also, we define \[ H(t,\alpha,K) :=tK(t) E(t,\alpha,K)+\frac{N-p}{p}\varphi_p(u'(t,\alpha,K)) u(t,\alpha,K) . \] Suppose that $u(\cdot,\alpha,K) $ is a solution of \eqref{3}. Then, a Pohozaev type identity takes place \begin{equation}\label{pohozaev} \begin{aligned} & t^{N-1}H(t,\alpha,K) -s^{N-1}H(s,\alpha,K) \\ & =\int_{s}^{t}r^{N-1}K(r) \Big[\Big( N+r\frac{K'(r) }{K(r) }\Big) F^+(u) -\frac{N-p}{p}f^+(u) u\Big]dr, \end{aligned} \end{equation} for $0\leq s\leq t\leq1$. We shall use this version of Pohozaev identity as follows. For $s=0$ and $t=t_0$, we get \[ t_0^{N-1}H(t_0,\alpha,K) =\int_0^{t_0} r^{N-1}K(r) \Big[\Big(N+r\frac{K'(r) }{K(r) }\Big) F^+(u) -\frac{N-p}{p}f^+(u) u\Big]dr. \] Since $f^+$ and $F^+$ are nonnegative and increasing functions on the interval $[u_1,\infty) $, then for all $r\in[0,t_0]$, \begin{align*} \Big(N+r\frac{K'(r) }{K(r) }\Big) F^+(u) -\frac{N-p}{p}uf^+(u) & \geq\delta_0F^+(u) -\frac{N-p}{p}uf^+(u)\\ &\geq \delta_0F^+(k\alpha) -\frac{N-p}{p}\alpha f^+(\alpha) . \end{align*} In consequence, \[ t_0^{N-1}H(t_0,\alpha,K) \geq\frac{\eta}{N} \Big(\delta_0F^+(k\alpha) -\frac{N-p}{p}\alpha f^+(\alpha)\Big) t_0^N. \] From this and \eqref{5} we find that \begin{equation} t_0^{N-1}H(t_0,\alpha,K) \geq\frac{\eta C_1^N} {N\bar{\lambda}^{N/p}}\Big(\delta_0F^+(k\alpha) -\frac {N-p}{p}\alpha f^+(\alpha)\Big) \Big(\frac{\varphi _p(\alpha) }{f^+(\alpha) }\Big) ^{N/p}. \label{7} \end{equation} We claim that for each number $\delta\geq\theta$, there is a positive constant $B_{\delta}$, such that \begin{equation} \delta F^+(s) -\frac{N-p}{p}sf^+(s) \geq-B_{\delta },\quad \text{for all }s\in\mathbb{R}. \label{7a} \end{equation} In fact, (F4) guarantees existence of $C_{\delta}>0$ satisfying \[ \delta F^+(s) -\frac{N-p}{p}sf^+(s) \geq\delta F^+(ks) -\frac{N-p}{p}sf^+(s) \geq0, \] for all $s>C_{\delta}$ and all $s<0$. On the other hand, if we set $M_{\delta}:=\sup_{s\in[0,C_{\delta}]}|f^+(s) |$, then \begin{align*} \delta F^+(s) -\frac{N-p}{p}sf^+(s) &=\int _0^{s}\Big(\delta f^+(t) -\frac{N-p}{p}f^+(s)\Big) dt\\ & \geq-M_{\delta}\int_0^{s}\Big(\delta+\frac{N-p}{p}\Big)dt\\ &=-M_{\delta}s(\delta+\frac{N-p}{p}) \\ & \geq-M_{\delta}C_{\delta}\Big(\delta+\frac{N-p}{p}\Big) =:-B_{\delta }. \end{align*} Now, replacing $t_0$ by $s$ in Pohozaev identity \eqref{pohozaev}, and using the estimate \eqref{7a} with $\delta_0$ and $\delta_1$ we obtain \begin{align*} &t^{N-1}H(t,\alpha,K) \\ &=t_0^{N-1}H(t_0,\alpha,K) +\int_{A}r^{N-1}K(r) \Big[(N+r\frac{K'(r) }{K(r) }) F^+(u) -\frac{N-p}{p}f^+(u)u\Big]dr\\ &\quad +\int_{B}r^{N-1}K(r) \Big[(N+r\frac{K'(r) }{K(r) }) F^+(u)-\frac{N-p}{p}f^+(u) u\Big]dr\\ &\geq t_0^{N-1}H(t_0,\alpha,K) +\int_{A} r^{N-1}K(r) \Big[\delta_0F^+(u) -\frac{N-p} {p}f^+(u) u\Big]dr\\ & \quad +\int_{B}r^{N-1}K(r) \Big[\delta_1F^+( u) -\frac{N-p}{p}f^+(u) u\Big]dr \\ & \geq t_0^{N-1}H(t_0,\alpha,K) -B_{\delta_0} \int_{A}r^{N-1}K(r) dr-B_{\delta_1}\int_{B} r^{N-1}K(r) dr\\ & \geq t_0^{N-1}H(t_0,\alpha,K) -2\frac{\bar{\lambda}B_{\delta_1}}{N}. \end{align*} Here $A:=\{ r\in[t_0,t]:F^+(u(r,\alpha,K) \geq0) \} $ and $B:=\{ r\in[ t_0,t]:F^+(u(r,\alpha,K) <0) \} $. Then we reach the estimate \begin{equation} t^{N-1}H(t,\alpha,K) \geq t_0^{N-1}H(t_0 ,\alpha,K) -2\bar{\lambda}M_NC_N\Big(1+\frac{N-p}{Np}\Big), \label{8} \end{equation} where we have used the fact that $\delta_1=N$ (it is remarkable that $\delta_1$ does not depend on $K$). \section{Main Result}\label{prooftheorem} In this section we shall prove our main theorem. Before, we will establish three preliminary Lemmas. \begin{lemma}\label{lemma1} There exists a positive real number $\lambda_2$ with the following property. For every admissible weight $K$, with $\|K\|_{\infty}\equiv\bar{\lambda}<\lambda_2$, there is a real number $\underline{\alpha}>u_1/k$, such that for all $\alpha\geq\underline{\alpha}$ and all $t\in[0,1]$, $|u(t,\alpha,K) |^{p}+|u'(t,\alpha,K) |^{p}>0$. \end{lemma} \begin{proof} Given a weight $K$ with the properties (K1) and (K2), there is $\underline{\alpha}>u_1/k$ such that for all $\alpha\geq \underline{\alpha}$ \[ \Big(\delta_0F^+(k\alpha) -\frac{N-p}{p}\alpha f^+( \alpha)\Big) \Big(\frac{\varphi_p(\alpha) }{f^+(\alpha) }\Big) ^{N/p}\geq\frac{1}{\eta}. \] From \eqref{7} and \eqref{8} we have, for $t\geq t_0$ \begin{equation}\label{lema1-numerada1} \begin{aligned} t^{N-1}H(t,\alpha,K) & \geq t_0^{N-1}H(t_0 ,\alpha,K) -2\bar{\lambda}M_{\delta_1}C_{\delta_1} \Big(1+\frac{N-p}{Np}\Big) \\ & \geq\frac{\eta C_1^N}{N\bar{\lambda}^{N/p}}\frac{1}{\eta}-2\bar {\lambda}M_{N}C_{N}\Big(1+\frac{N-p}{Np}\Big) \\ & =\bar{\lambda}\Big(\frac{C_1^N}{N\bar{\lambda}^{(N-p)/p}} -2M_{N}C_{N}(1+\frac{N-p}{Np})\Big). \end{aligned} \end{equation} Now, there exists $\lambda_2>0$ such that \begin{equation}\label{lema1-numerada2} \frac{C_1^N}{N\lambda^{(N-p)/p}}-2M_{N}C_{N}(1+\frac{N-p}{Np})>0, \end{equation} for every $\lambda\in (0, \lambda_2)$. We fix a weight $K$ so that $||K||_\infty:=\bar{\lambda}<\lambda_2$. Thus, from \eqref{lema1-numerada1} and \eqref{lema1-numerada2} we have $H(t,\alpha, K) >0$ for $t\in [t_0, 1]$. Then, for all $t\in[t_0,1]$, $|u(t,\alpha,K) |^{p}+|u'(t,\alpha,K) |^{p}>0$. Clearly, $|u(t,\alpha,K) |^{p}+|u'(t,\alpha,K) |^{p}>0$ holds for $t\in[0,t_0]$. \end{proof} \begin{lemma}\label{lemma2} There exists a real number $\lambda_1\in(0, \lambda_2]$ with the following property. For every admissible weight $K$, with $\bar{\lambda}<\lambda_1$, we have $u(t,\underline{\alpha},K) \geq u_0$ for all $t\in[0,1]$. Here, $\underline{\alpha}$ is a number obtained in the preceding Lemma. \end{lemma} \begin{proof} Given a weight $K$, set \[ t_1:=\sup\{ t\in[0,1]:u(r,\underline{\alpha},K) \geq u_0\text{ for all }r\in(0,t) \} . \] We observe that $t_0$ belongs to the previous set. Since $f^+$ is nonnegative on the interval $[u_0,\infty) $, we see from \eqref{4}, that \[ \varphi_p(u'(t,\underline{\alpha},K)) =-t^{1-N}\int_0^{t}r^{N-1}K(r) f^+(u(r)) dr\leq0,\quad \text{for all }t\in[0,t_1]. \] Therefore, $u$ is decreasing on $[0,t_1]$. Besides, for all $t\in[0,t_1]$, \begin{align*} |\varphi_p(u'(t,\underline{\alpha},K)) | &\leq t^{1-N}\int_0^{t}r^{N-1}K(r) f^+(u(r,\underline{\alpha},K)) dr\\ & \leq\bar{\lambda}f^+(u(0,\underline{\alpha},K)) t^{1-N}\int_0^{t}r^{N-1}dr\\ &\leq\frac{\bar{\lambda}f^+(\underline{\alpha}) }{N}t\\ &\leq\frac{\bar{\lambda}f^+(\underline{\alpha}) }{N}. \end{align*} Hence \[ |u'(t,\underline{\alpha},K)| \leq\varphi_{p'}\Big(\frac{\bar{\lambda}f^+(\underline{\alpha}) }{N}\Big) . \] Now, fix $\lambda_1\leq\min\{ \lambda_2,\frac{N}{f^+( \underline{\alpha}) }\varphi_p(\underline{\alpha}-u_0) \} $, then $\varphi_{p'}(\lambda_1) \leq\varphi_{p'}(\frac{N}{f^+(\underline {\alpha}) }) (\underline{\alpha}-u_0) $. It follows that if $\bar{\lambda}<\lambda_1$, we have $|u'(r,\underline{\alpha},K) |\leq\underline{\alpha}-u_0$ for all $t\in[0,t_1]$. An application of the mean value theorem, allows us to choose a real number $\xi\in(0,t_1) $ such that \[ u(t_1,\underline{\alpha},K) -u(0,\underline{\alpha},K) =u'(\xi,\underline{\alpha},K) t_1 \geq-(\underline{\alpha}-u_0) t_1. \] If we assume that $t_1<1$, then $u(t_1,\underline{\alpha},K)>u_0$, contradicting the definition of $t_1$. This completes the proof. \end{proof} \begin{lemma}\label{lemma3} For a given admissible weight $K$ with $\bar{\lambda}<\lambda_1$, there exists $\alpha_1\geq\underline{\alpha}$ such that $u(t,\alpha_1,K) <0$ for some $t\in[0,1]$. \end{lemma} \begin{proof} We argue by contradiction. Suppose that there exists a suitable weight $K$, such that for all $\alpha\geq\underline{\alpha}$ and all $t\in[0,1]$, $u(t,\alpha,K) \geq0$. Without lost of generality we can assume that $u(t,\alpha,K) >0$ for all $t\in[0,1) $. Let $\bar{t}=\bar{t}(\alpha) $ be the supremum of the set \[ V:=\{ t\in[0,1]:u(\cdot,\alpha,K) \text{ is decreasing on }[0,t]\}. \] $V$ is a nonempty set because it contains $t_0$. On the other hand, in view of the inequalities \eqref{7}, \eqref{8} and hypothesis (F4), we can fix a real number $\alpha_1$ such that for all $\alpha\geq$ $\alpha_1$ and all $t\in[0,1]$, $t^{N-1}H(t,\alpha,K) >0$. Now, we claim that $u'(t,\alpha,K)\neq0$ for all $t\in(0,1]$. For if $u'(t_1,\alpha,K) =0$ for some $t_1\in(0,1]$ then the differential equation in \eqref{2} would imply that $f^+(u(t_1,\alpha,K))=0$. Hence $u(t_1,\alpha,K) =u_0$, but \[ 00\quad \text{for }0u_1/k$ such that \begin{equation} \frac{f^+(x) }{\varphi_p(x) }\geq\mu/ {\bar{\lambda}},\quad \text{for all }x\geq\alpha_0. \label{14} \end{equation} From the corresponding integral formulas for the solutions $u$ and $v$ (cf. \eqref{4} and \eqref{eigenvalue_problem}) we have \begin{gather} [r^{N-1}\varphi_p(u') ]'\varphi_p(v) =-r^{N-1}K(r) f^+(u) \varphi_p(v), \label{10} \\ [r^{N-1}\varphi_p(v') ]'\varphi_p(u) =-\mu r^{N-1}\varphi_p(v) \varphi_p(u) . \label{11} \end{gather} Let $t_1$ be the supremum of the set $A:=\{ t\in[0,\rho] :v'u(r) \leq u'v(r) \text{ for all }r\in(t,\rho]\} $, which is a nonempty set due to the fact that $0<-v'u(\rho) +u'v(\rho)$. Certainly \begin{equation} v'u(t_1) =u'v(t_1) . \label{12} \end{equation} We claim that there exists a real number $t$ on the interval $[t_1,\rho]$ that satisfies the inequality $u(t,\alpha,K) <\alpha_0$. The proof of this claim will be carried out arguing by contradiction. Assume that $u(t,\alpha,K) \geq \alpha_0$ for all $t\in$ $[t_1,\rho]$. Integrating by parts \eqref{10} and \eqref{11}, subtracting the resulting equations and taking into account \eqref{12}, we see that \begin{equation} \label{13} \begin{aligned} -\rho^{N-1}\varphi_p(v'u) (\rho) &=(p-1) \int_{t_1}^{\rho}r^{N-1}(| u'v|^{p-2}-|v'u|^{p-2})u'v'dr\\ &\quad +\int_{t_1}^{\rho}r^{N-1}\Big(\mu-K(r) \frac{f^+(u) }{\varphi_p(u) }\Big) \varphi_p(uv) dr. \end{aligned} \end{equation} Since $|u'v|(t) \leq|uv'|(t) $ for all $t\in(t_1,\rho) $ and $p-2\geq0$, then \[ |u'v|^{p-2}(t) -|uv'|^{p-2}(t) \leq0. \] In consequence, the first term of \eqref{13} is nonpositive. On the other hand, from \eqref{14} we see that \[ \mu-\bar{\lambda}\frac{f^+(u) }{\varphi_p(u) }\leq0. \] Hence, the second term of \eqref{13} is also nonpositive. This is impossible since $v'u(\rho) <0$. This proves the claim. Therefore, $u(t_2,\alpha,K) =\alpha_0$ for some $t_2\in(0,\rho)$. Since $\alpha>u_{0\text{ }}$ and $u$ is decreasing, the estimate \begin{equation} u(t,\alpha,K) \leq\alpha_0,\text{ for all }t\in[t_2,1]\label{15} \end{equation} holds. Because $F^+$ is increasing on $[u_0,\infty) $ and $u(t,\alpha,K) \geq\alpha k>u_1$ for all $t\in[0,t_0]$ then \[ E(t,\alpha,K) \geq F^+(u(t,\alpha,K))\geq F^+(k\alpha) . \] On the other hand, for $t\in(t_0,1]$, since $u(t) u'(t) \leq0$, then \begin{align*} t^NK(t) E(t,\alpha,K) & \geq t^{N-1}H(t,\alpha,K) \\ & \geq\frac{\eta C_1^N}{N\bar{\lambda}^{N/p}} \Big(\delta_0F^+(k\alpha) -\frac{N-p}{p}\alpha f^+(\alpha)\Big) \Big(\frac{\varphi_p(\alpha) }{f^+(\alpha) }\Big)^{\frac{N}{p}}\\ &\quad -2\bar{\lambda}M_{N}C_{N}\Big(1+\frac{N-p}{Np}\Big) . \end{align*} Thus, $K(t) E(t,\alpha,K) \to+\infty$ as $\alpha\to+\infty$, uniformly in $t\in[0,1]$. Set $\alpha\geq\alpha_0$ such that \[ K(t) E(t,\alpha,K) \geq\bar{\lambda}F^+( \alpha_0) +(p') ^{p-1}\alpha_0^{p}\quad \text{for all }t\in[0,1]. \] Relation \eqref{15} implies $F^+(\alpha_0) -F^+(u(t,\alpha,K)) \geq0$ for all $t$ in $[t_2,1]$, thus \[ |u'(t,\alpha,K) |\geq p'\alpha_0,\quad \text{for all }t\in[t_2,1]. \] Define $\tau=t_2+\frac{1}{p'} $ and observe that $(t_2,\tau) \subseteq(0,1) $. The mean value theorem applied to $u$ on the interval $[t_2,\tau]$, leads to the equation \[ u(\tau,\alpha,K) -u(t_2,\alpha,K) =u'(\xi,\alpha,K) \frac{1}{p'}, \] for some $\xi\in(t_2,\tau) $. On the other hand, $u(t_2,\alpha,K) =\alpha_0$ and $u'(\xi,\alpha,K)\frac{1}{p'}\leq-\alpha_0$, thus $u(\tau,\alpha,K) \leq0$, which is absurd. \end{proof} \begin{proposition}\label{the_proposition} Under hypotheses {\rm (F1)--(F4)} and {\rm (K1)--(K2)}, there exists a positive number $\lambda_0$ such that if $\|K\| _{\infty}<\lambda_0$ then problem \eqref{1} has at least one positive decreasing radial solution, with radial negative derivative in $\| x\| =1$. \end{proposition} \begin{proof} Set $\lambda_0=\lambda_1$ from Lemma \ref{lemma2} and \underline{$\alpha$} given by Lemma \ref{lemma1}. According to Lemma \ref{lemma3}, there exists $\alpha_1>\underline{\alpha}$ with its corresponding solution being negative in some point on $[0,1]$. Let $\tilde{\alpha}$ be the supremum of the set $\mathcal{A}$ defined as \[ \{ \alpha\in[\underline{\alpha},\alpha_1]:u(t,\alpha,K) \geq0 \text{ for all }t\in[0,1]\}. \] This supremum makes sense because of $\underline{\alpha}$ belongs to $\mathcal{A}$ (which is implied by Lemma \ref{lemma2}). Due to the continuous dependence of $u$ on $\alpha$, $\mathcal{A}$ is closed. Thus, $\tilde{\alpha}$ belongs to $\mathcal{A}$ and therefore $u(t,\tilde{\alpha},K) \geq0$, for all $t\in[0,1]$. Moreover, $\tilde{\alpha}<\alpha_1$. Now, we will see that $u(\cdot,\tilde{\alpha},K) $ is a solution needed. (i) $u(t,\tilde{\alpha},K) >0$ for all $t\in[0,1) $. Arguing by contradiction, if $u(\tau,\tilde{\alpha},K) =0$ for some $\tau\in(0,1) $, then by Lemma \ref{lemma1}, $u'(\tau,\tilde{\alpha},K) \neq0$. Hence, there exists a $\tau_1\in(0,1) $, such that $u(\tau_1,\tilde{\alpha},K) <0$. This is not possible. (ii) $u(1,\tilde{\alpha},K) =0$. If $u(1,\tilde{\alpha},K) >0$, then $u(\cdot,\tilde{\alpha},K)>0$ on the compact set $[0,1]$. For the continuous dependence in the initial data, there is some $\alpha$, $\tilde{\alpha}<\alpha<\alpha_1$, such that $u(\cdot,\alpha,K) >0$ on $[0,1]$. This contradicts the definition of $\tilde{\alpha}$. (iii) $u'(1,\tilde{\alpha},K) <0$. Due to the previous steps, $u'(1,\tilde{\alpha},K) \leq0$. Now, the fact that $u'(1,\tilde{\alpha},K) $ is nonzero follows from Lemma \ref{lemma1}. (iv) The proof that $u(\cdot,\alpha,K) $ is decreasing is contained in the proof of Lemma \ref{lemma3}. Therefore the proposition is proved. \end{proof} \begin{proof}[Proof of theorem \ref{maintheorem}] The existence of the positive solution is a consequence of previous proposition. Let \[ f^{-}(t) :=\begin{cases} -f(-t) &\text{if } t>0\\ -f(0^{-}) & \text{if } t=0\\ 0 &\text{if } t<0. \end{cases} \] A straightforward application of the Proposition \ref{the_proposition} with $f^{-}$ gives us a negative solution with the desired properties. \end{proof} Now, we exhibit some examples of functions $f$ and $K$ satisfying conditions (F1)--(F4) and (K1)--(K2). Let $f(t)=t^q-t^{q-1}-1$, for $t>0$ and $f(t)=1+(-t)^{q-1}-(-t)^q$ for $t<0$, where $1\frac{N-p}{p}, \end{equation} from which follows (F4). If we consider the same nonlinearity $f$ with $10$ is fixed and we choose $\frac{(N-p)(q+1)}{p}<\theta \theta$, we obtain (K2). Other example is given by $f(t)=t^q\ln t-1$ for for $t>0$ and $f(t)=1-(-t)^q\ln (-t)$ for $t<0$, where $20$ is fixed, is an admissible weight. \section{Appendix} This section we establish the Pohozaev identity as well as the existence of local solution to problem \eqref{3}. \begin{proposition}[Pohozaev Identity] Assume that $u(t,\alpha,K) $ is a solution of the initial value problem \eqref{3}, then for all $0\leq s\leq t\leq1$, \begin{equation} \label{4.3} \begin{aligned} & t^{N-1}H(t,\alpha,K) -s^{N-1}H(s,\alpha,K)\\ & =\int_{s}^{t}r^{N-1}K(r) \Big[\Big( N+r\frac{K'(r) }{K(r) }\Big) F( u) -\frac{N-p}{p}f(u) u\Big]dr\,. \end{aligned} \end{equation} \end{proposition} \begin{proof} It is easy to see that our ordinary differential equation can be written as \begin{equation}\label{appendix18} [r^{N-1}\varphi_p(u') ]'=-r^{N-1}K(r) f(u). \end{equation} Multiplying \eqref{appendix18} by $u$ and integrating on $[s, t]$, by parts, we obtain \[ t^{N-1}\varphi_p(u') u-s^{N-1}\varphi_p( u') u=\int_{s}^{t}r^{N-1}\varphi_p(u') u'dr -\int_{s}^{t}r^{N-1}K(r) f(u) u\,dr. \] Then \begin{equation} \int_{s}^{t}r^{N-1}|u'|^{p}dr =b(s,t) +\int_{s}^{t}r^{N-1}K(r) f(u)u\,dr,\label{4.2} \end{equation} where $b(s,t) =t^{N-1}\varphi_p(u'(t)) u(t)-s^{N-1}\varphi_p(u'(s))u(s) $. Now, multiplying \eqref{appendix18} by $ru'$ and integrating by parts, we have \begin{equation} t^N\varphi_p(u') u'-s^N\varphi _p(u') u'=\int_{s}^{t}r^{N-1} \varphi_p(u') (ru^{\prime\prime}+u') dr-\int_{s}^{t}r^NK(r) f(u) u'dr.\label{4.1} \end{equation} From \eqref{appendix18}, we realize that \[ \varphi_p(u') (ru^{\prime\prime}+u') =\frac{p-N}{p-1}|u'|^{p}-\frac{K(r)}{p-1}rf(u) u'. \] Then, from \eqref{4.1}, it follows that \[ a(s,t) -\int_{s}^{t}r^{N-1}\frac{p-N}{p-1}| u'|^{p}dr-\int_{s}^{t}r^N\frac{K(r)}{p-1}f(u) u'dr =-\int_{s}^{t}r^NK(r) f(u) u'dr, \] where $a(s,t) :=t^N\varphi_p(u'( t)) u'(t) -s^N\varphi_p( u'(s)) u'(s) $. Therefore, \[ a(s,t) +\frac{N-p}{p-1}\int_{s}^{t}r^{N-1}| u'|^{p}dr=-p'\int_{s}^{t}r^NK(r) f(u) u'dr. \] This equation and \eqref{4.2} imply that \[ a(s,t) +\frac{N-p}{p-1}\Big(b(s,t) +\int_{s}^{t}r^{N-1}K(r) f(u) udr\Big) =-p'\int_{s}^{t}r^NK(r) f(u)u'dr. \] Now, integrating by parts the right hand side of the last equation we obtain \begin{align*} & a(s,t) +\frac{N-p}{p-1}b(s,t) \\ & =\frac{p-N} {p-1}\int_{s}^{t}r^{N-1}K(r) f(u) udr-p'(t^NK(t) F(u)-s^NK(s) F(u)) \\ &\quad +p'\int_{s}^{t}r^{N-1}(NK(r)+rK'(r)) F(u) dr. \end{align*} Taking into account the definitions of $a(s,t) $ and $b(s,t) $, we can see that \begin{align*} & t^N|u'|^{p}-s^N|u'|^{p}+\frac{N-p}{p-1}(t^{N-1}\varphi_p(u^{\prime }) u-s^{N-1}\varphi_p(u') u) \\ & +p'(t^NK(t) F(u) -s^NK(s) F(u)) \\ & =\int_{s}^{t}r^{N-1}\Big[\frac{p-N}{p-1}K(r) f(u) u+\frac{p}{p-1}(NK(r) +rK^{\prime }(r)) F(u) \Big]dr. \end{align*} Consequently, \begin{align*} & \frac{p-1}{p}t^{N-1}\Big[\Big(t|u'| ^{p}+\frac{N-p}{p-1}\varphi_p(u') u+p^{\prime }tK(t) F(u)\Big) \\ & -s^{N-1}\Big(s|u'|^{p}+\frac {N-p}{p-1}\varphi_p(u') u+p'sK( s) F(u)\Big) \Big]\\ & =\int_{s}^{t}r^{N-1}K(r) \Big[\Big( N+r\frac{K'(r) }{K(r) }\Big) F(u) -\frac{N-p}{p}f(u) u\Big]dr. \end{align*} However, \begin{align*} & t^{N-1}\Big(\frac{t}{p'}\{ |u'|^{p}+p'K(t) F(u) \} +\frac{N-p}{p} \varphi_p(u') u\Big) \\ & -s^{N-1}(\frac{s}{p'}\{ |u' |^{p}+p'K(s) F(u) \} +\frac{N-p}{p} \varphi_p(u') u) \\ & =t^{N-1}\Big[tK(t) E(t) +\frac{N-p} {p}\varphi_p(u') u\Big]-s^{N-1}\Big[sK( s) E(s) +\frac{N-p}{p}\varphi_p(u') u\Big]. \end{align*} Hence \eqref{4.3} holds. \end{proof} \begin{proposition} Let $p\geq2$, $\alpha>4u_1/3$ and $K$ be a suitable weight. There exists a positive real number $\varepsilon$ such that \eqref{3} has a unique solution $u(\cdot,\alpha,K) $ on the interval $[0,\varepsilon]$. \end{proposition} \begin{proof} For $\varepsilon>0$ and $R=\alpha/4$, set $Y:=\{ u\in C([0,\varepsilon]:\mathbb{R}):\| u-\alpha\| _{\infty}\leq R\} $. Define $T:Y\to Y$ by \[ (Tu) (s) :=\alpha-\int_0^{s} \varphi_{p'}\Big(r^{1-N}\int_0^{r}t^{N-1}K( t) f(u(t)) dt\Big) dr,\quad 0\leq s\leq\varepsilon. \] This was suggested by \eqref{3.a}. $T$ is well defined if $\varepsilon$ i s small enough, since $f$ is Lipschitz continuous on $[3\alpha/4,5\alpha/4]$. Now, we will see that $T$ is a contraction. Let $u,v$ be elements of $Y$ and $s\in\lbrack0,\varepsilon]$. Applying the mean value theorem we obtain the estimate \begin{align*} & |(Tu) (s) -(Tv) (s) |\\ & \leq\int_0^{s}\varphi_{p'}(r^{1-N}) \Big|\varphi_{p'}\Big(\int_0^{r}t^{N-1}K( t) f(u) dt\Big) -\varphi_{p'}\Big( \int_0^{r}t^{N-1}K(t) f(v) dt\Big) \Big|dr\\ & \leq(p'-1) \int_0^{s}\varphi_{p' }(r^{1-N}) |\xi_{r}|^{p'-2}\Big( \int_0^{r}t^{N-1}K(t) |f(u)-f(v) |dt\Big) dr, \end{align*} where $\xi_{r}$ is a value between the two positive numbers $\int _0^{r}t^{N-1}K(t) f(u) dt$ and $\int_0^{r}t^{N-1}K(t) f(v) dt$. Assume, without loss of generality that $\int_0^{r}t^{N-1}K(t) f(u)dt\leq\xi_{r}$. Then, we have $\frac{\eta}{N}f(\frac{3}{4}\alpha) r^N\leq\xi_{r}$. Because of $p'-2\leq0$, \[ |\xi_{r}|^{p'-2}\leq[\frac{\eta}{N} r^Nf\big(\frac{3}{4}\alpha\big) ]^{p'-2}. \] Thus, \[ |(Tu) (s) -(Tv) ( s) |\leq C_{\alpha}\| u-v\| _{\infty} \int_0^{s}r^{p'-1}dr