\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 102, 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 (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/102\hfil Existence of positive solutions] {Existence of positive solutions for a singular $p$-Laplacian Dirichlet problem} \author[W. Zhou \hfil EJDE-2008/102\hfilneg] {Wenshu Zhou} \address{Wenshu Zhou \newline Department of Mathematics, Dalian Nationalities University, 116600, China} \email{pdezhou@126.com, wolfzws@163.com} \thanks{Submitted December 17, 2007. Published July 30, 2008.} \thanks{Supported by grants 20076209 from Dalian Nationalities University, and 10626056 \hfill\break\indent from Tianyuan Fund} \subjclass[2000]{34B18} \keywords{$p$-Laplacian; singularity; positive solution; regularization technique} \begin{abstract} By a argument based on regularization technique, upper and lower solutions method and Arzel\'a-Ascoli theorem, this paper shows sufficient conditions of the existence of positive solutions of a Dirichlet problem for singular $p$-Laplacian. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} This paper shows the existence of positive solutions for the singular $p$-Laplacian equation \big(\phi_p(u')\big)'-\lambda\frac{|u'|^p}{u^m}+f(t, u')=0,\quad 01$,$\lambda$and$m$are positive constants, and$f$is a continuous function. We call$u \in C^1[0,1]$is a solution if$u>0$in$(0, 1)$,$|u'|^{p-2}u' \in C^1(0,1)$, and it satisfies \eqref{e1}--\eqref{e2}. Such equation arises in the studies of some degenerate parabolic equations and in Non-Newtonian fluids; see \cite{a2,b1,b2,b3,y2}. The interesting feature of \eqref{e1} is the lower term both is singular at$u=0$and depends on the first derivative. Recently, the one-dimensional singular p-Laplacian differential equations without dependence on the first derivative have been studied extensively, see \cite{a1,j1,y1} and references therein. When it depends on the first derivative, however, it has not received much attention, see \cite{j2,l1,o1,w1}. Recently, the authors \cite{y3}, considered the equation \begin{equation*} \big(\phi_p(u')\big)'-\lambda\frac{|u'|^p}{u}+g(t)=0,\quad 00$ and $g \in C[0, 1]$ with $g >0$ on $[0, 1]$. In the present paper we extend the result and obtain the sufficient conditions of existence. Our argument is based on regularization technique, upper and lower solutions method and Arzel\'a-Ascoli theorem. In addition, an example is also given to illustrate our main result. \section{Main result} The following hypotheses will be adopted in this section: \begin{itemize} \item[(H1)] $1 \leq m 0$, $\beta \in [0, 1)$ such that $f(t, r) \leq \alpha +\beta|r|^{p-1}$, for all $(t, r)\in [0, 1]\times \mathbb{R}$. \item[(H3)] $\lambda>\inf_{r\geq 1}{H}(r)$, where $H(r):\mathbb{R}^+\to \mathbb{R}^+$ is defined by \begin{equation*} H(r)=\alpha r^{m-p}+\beta r^{m-1}. \end{equation*} \end{itemize} \begin{remark} \label{rmk1} \rm Let $m \in (1, p)$ and define \begin{equation*} X_0=\Big(\frac{\alpha(p-m)}{\beta(m-1)}\Big)^{1/(p-1)};\quad X_*=\begin{cases} X_0,& X_0 \geq1\\ 1.&X_0< 1 \end{cases} \end{equation*} Then $\inf_{s\geq 1}H(s)=H(X_*)$. Indeed, since $\lim_{s\to 0^+}H(s)=\lim_{s \to +\infty}H(s)=+\infty$, $H(s)$ must reach a minimum at some $s \in (0,\infty)$ satisfying $H'(s)=0$. Solving it gives $s=X_0$ and hence, $\inf_{s>0}H(s)=H(X_0)$. Since $H'(s) \geq 0$ for all $s \geq X_0$, we see that $\inf_{s \geq 1}H(s)=H(X_0)$ if $X_0 \geq 1$, and $\inf_{s\geq 1}H(s)=H(1)$ if $X_0 < 1$. \end{remark} The main result of this paper is stated as follows. \begin{theorem} \label{thm1} Under Assumptions {\rm (H1)--(H3)}, problem \eqref{e1}--\eqref{e2} has at least one solution. \end{theorem} \begin{remark} \label{rmk2} \rm If $m=1$ and $f\equiv 1$ (taking $\alpha=1,\beta=0$), then $\inf_{s\geq 1}H(s)=0$. Clearly, Theorem \ref{thm1} is an extension of the corresponding result of \cite{y3}. \end{remark} \subsection*{Proof of Theorem \ref{thm1}} Let $\epsilon \in (0, 1)$, and define $H_\epsilon(t, v,\xi): (0, 1)\times\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ by $$H_\epsilon(t, v,\xi)= \lambda \frac{|\xi|^p}{[I_\epsilon(v)]^m}-f(t,\xi),$$ where $I_\epsilon(v) =v+\epsilon$ if $v \geq 0$, $I_\epsilon(v) =\epsilon$ if $v < 0$. By (H2) and using the inequality: $a^{p-1} \leq a^p+1$, for all $a \geq 0$, we have \begin{equation*} %\label{0} |H_\epsilon(t, v,\xi)| \leq \frac{\lambda}{\epsilon^{m}}|\xi|^p + \alpha+\beta|\xi|^{p-1} \leq \big( \frac{\lambda}{\epsilon^{m}} +\alpha +\beta \big)\mathcal{H}(|\xi|) \end{equation*} for all $(t, v, \xi) \in (0, 1)\times\mathbb{R}\times\mathbb{R}$, where $\mathcal{H}(s)=1+s^p$ for $s \geq 0$. Denote $\mathfrak{M}=\{u \in C^1(0, 1); |u'|^{p-2}u' \in C^1(0, 1)\}$, and define $\mathscr{L}_\epsilon: \mathfrak{M}\to C(0, 1)$ by \begin{equation*} ( \mathscr{L}_\epsilon u)(t)=-\big(\phi_p(u')\big)'+H_\epsilon(t,u,u'),\quad 0\delta$that for$\delta_0=\frac{\lambda-\delta}{2}>0$, there exists$S^* \geq 1$such that$H(S^*)<\delta+\delta_0<\lambda$. \begin{lemma} \label{lem1} There exists a constant$\epsilon_0\in(0, 1)$, such that for any$\epsilon \in (0, \epsilon_0)$,$U_\epsilon=S^*(t+\epsilon)$is an upper solution of \eqref{01}. \end{lemma} \begin{proof} Noticing$U_\epsilon \geq \epsilon$in$(0,1)$and$m \geq 1, we have \begin{align*} \mathscr{L}_\epsilon U_\epsilon &=-\big(|U_\epsilon'|^{p-2}U_\epsilon'\big)' +\lambda\frac{|U_\epsilon'|^p}{(U_\epsilon+\epsilon)^m}- f(t, U_\epsilon') \\ &=\frac{\lambda {S^*}^{p-m}}{(t+\epsilon+\epsilon/S^*)^m}- f(t, S^*) \\ & \geq\frac{\lambda {S^*}^{p-m}}{(1+\epsilon+\epsilon/S^*)^m}- \alpha-\beta {S^*}^{p-1}\\ &= {S^*}^{p-m}[\lambda-H(S^*)] +r_\epsilon,\quad 0 H(S^*), there exists a constant $\epsilon_0 \in (0, 1)$, such that for any $\epsilon \in (0, \epsilon_0)$ there holds ${S^*}^{p-m} [\lambda-H(S^*) ] +r_\epsilon \geq 0$. So that we obtain ${\mathscr{L}_\epsilon}U_\epsilon \geq 0$ in $(0, 1)$ for all $\epsilon \in (0, \epsilon_0)$. The lemma follows. \end{proof} \begin{lemma} \label{lem2} Let $W= C \Phi^{\alpha}$, where $\alpha=\frac{p}{p-m}$, $\Phi(t)$ is defined by $$\Phi(t)= \frac{p-1}{p}\big[\big(\frac{1}{2}\big)^{p/(p-1)}-\big| \frac{1}{2}-t\big|^{p/(p-1)}\big], \quad 0 \leq t\leq 1,$$ and $C \in (0,1)$ such that $C\alpha <1$ and $(C\alpha)^{p-1} +\lambda C^{p-m}\alpha^p \leq \min_{[0, 1] \times [-1,1]}f(s, r)$. Then $W$ is a lower solution of problem \eqref{01}. \end{lemma} \begin{proof} It is easy to check that $\Phi$ has the following properties: \begin{itemize} \item[(a)] $\Phi>0$ in $(0, 1)$, $\Phi \in C^1[0,1]$. \item[(b)] $(|\Phi'|^{p-2} \Phi')'=-1$ in $(0, 1)$, $\Phi(1)=\Phi(0)=0$. \item[(c)] $\Phi(t)\leq t$, $|\Phi'(t)|\leq 1$, for all $t\in [0,1]$. \end{itemize} Using these properties of $\Phi$, by some calculations, we have \begin{align*} \mathscr{L}_\epsilon W &=-\big(|W'|^{p-2}W\big)' +\lambda \frac{|W'|^p}{(W+\epsilon)^m}- f(t, W')\\ &\leq-\big(|W'|^{p-2}W\big)' +\lambda \frac{|W'|^p}{W^m}- f(t, W')\\ &=-(C\alpha)^{p-1}\Phi^{(\alpha-1)(p-1)}\big(|\Phi'|^{p-2}\Phi'\big)'\\ &\quad -(C\alpha)^{p-1}(\alpha-1)(p-1)\Phi^{(\alpha-1)(p-1)-1}|\Phi'|^p\\ &\quad +\lambda C^{p-m}\alpha^p|\Phi'|^p- f(t, C\alpha \Phi^{\alpha-1}\Phi')\\ &\leq (C\alpha)^{p-1} \Phi^{(\alpha-1)(p-1)} +\lambda C^{p-m}\alpha^p|\Phi'|^p- \min_{[0, 1]\times [-1, 1]}f(s, r)\\ &\leq(C\alpha)^{p-1} + \lambda C^{p-m}\alpha^p - \min_{[0, 1]\times [-1, 1]}f(s, r) \leq 0,\quad 00,\quad t\in(0,1). Hence $u_\epsilon$ satisfies $$\label{07} -\big(|u_\epsilon'|^{p-2}u_\epsilon'\big)' +\lambda\frac{|u_\epsilon'|^p}{(u_\epsilon+\epsilon)^m} - f(t, u_\epsilon')= 0,\quad t \in (0, 1).$$ \begin{lemma} \label{lem3} For all $\epsilon \in (0, \epsilon_0)$, we have $$\label{31} |u_{\epsilon}'(t) | \leq [\alpha (1-\beta)^{-1}]^{1/(p-1)},\quad \forall t \in [0,1].$$ \end{lemma} \begin{proof} Noticing that $u_{\epsilon}(1)=u_{\epsilon}(0)=0$ and $u_{\epsilon} \geq0$ on $[0, 1]$, we have $$\label{12} u_{\epsilon}'(0)\geq0 \geq u_{\epsilon}'(1).$$ From \eqref{07}, we obtain $$\label{08} \big(|u_\epsilon'|^{p-2}u_\epsilon'\big)'+ \alpha +\beta|u_\epsilon'|^{p-1} \geq 0,\quad t \in (0, 1).$$ Let $\chi=\phi_p(u_\epsilon')$. Then we obtain from \eqref{08}, $\chi'+ \alpha+\beta|\chi| \geq 0$, $t \in (0, 1)$; i.e., $\big(\int_0^{\chi(t)} \frac{1}{\alpha+\beta|s| }ds+t\big)' \geq 0$, $t \in (0, 1)$. This and \eqref{12} give $1\geq \int_0^{\chi(t)} \frac{1}{\alpha+\beta|s| }ds+t \geq 0$, $t \in [0, 1]$, hence $\big|\int_0^{\chi(t)} \frac{1}{\alpha+\beta|s| }ds\big|\leq 1$, $t \in [0, 1]$. Using the inequality: $|\int_0^{y} \frac{1}{\alpha+\beta|s| }ds| \geq \frac{|y|}{\alpha+\beta|y|}$ ($y \in \mathbb{R}$), we deduce that $|\chi| \leq \alpha+\beta|\chi|$, $t \in [0, 1]$; that is, $|\chi| \leq \alpha(1- \beta)^{-1}$ on $[0, 1]$. The lemma is proved. \end{proof} \begin{lemma} \label{lem4} For each $\delta \in (0,1/2)$, there exists a positive constant $C_\delta$ independent of $\epsilon$, such that for all $\epsilon \in (0, \epsilon_0)$ $$\label{22} |u_\epsilon'(t_2)-u_\epsilon'(t_1)| \leq C_\delta|t_2-t_1|^{\gamma},\quad \forall t_2,t_1 \in [\delta, 1-\delta],$$ where $\gamma=1/(p-1)$ if $p \geq 2$; $\gamma=1$ if $10$ independent of $\epsilon$, such that for all $\epsilon \in (0, \epsilon_0)$ $$\label{30} \big|(|u_\epsilon'|^{p-2} u_\epsilon')'\big| \leq C_\delta,\quad \delta \leq t \leq 1-\delta.$$ Recalling the inequality (see \cite{d1}) \begin{equation*} (|\eta|^{p-2}\eta-|\eta'|^{p-2}\eta') \cdot (\eta-\eta') \geq \begin{cases} C_1 |\eta - \eta '|^p,& p \geq 2\\ C_2 (|\eta|+|\eta'|)^{p-2}|\eta - \eta '|^2,&10$in$(0, 1)$. We now show that$u$satisfies \eqref{e1}. Integrating \eqref{07} over$(t_0, t)$gives \begin{equation*} |u_\epsilon'(t)|^{p-2}u_\epsilon'(t) =\int_{t_0}^t\Big( \lambda\frac{|u_\epsilon'|^p}{(u_\epsilon+\epsilon)^m} - f(s, u_\epsilon')\Big)ds+|u_\epsilon'(t_0)|^{p-2}u_\epsilon'(t_0), \end{equation*} and letting$\epsilon \to 0$and using Lebesgue's dominated convergence theorem yield $$\label{23} |u'(t)|^{p-2}u'(t) =\int_{t_0}^t\Big( \lambda\frac{|u'|^p}{u^m}- f(s, u')\Big)ds+|u'(t_0)|^{p-2}u'(t_0).$$ This shows that$|u'(t)|^{p-2}u'(t) \in C^1(0, 1)$and \eqref{e1} is satisfied. It remains to show that$u \in C^1[0, 1]$. Integrating \eqref{07} over$(0, 1)$and using \eqref{31} and \eqref{12}, we derive that \begin{equation*} \int_0^1 \frac{|u_\epsilon'|^p}{(u_\epsilon+\epsilon)^m}dt \leq \frac{1}{\lambda}\min_{[0, 1]\times[-Y, Y]}f(t, r),\quad Y:=\big(\frac{\alpha}{1-\beta}\big)^{1/(p-1)}, \end{equation*} and letting$\epsilon\to 0$and using Fatou's lemma and \eqref{21}, we obtain \begin{equation*} \int_0^1 \frac{|u'|^p}{u^m}dt \leq \frac{1}{\lambda} \min_{[0, 1]\times[-Y, Y]}f(t, r). \end{equation*} So,$\frac{|u'|^p}{u^m} \in L^1[0,1]$. By \eqref{23}, the function$\omega(t)=|u'(t)|^{p-2}u'(t) = \phi_{p}(u'(t))$is absolutely continuous on$[0,1]$. Since$u'(t)=\phi_q(\omega(t))$($\frac1p + \frac1q =1$),$u' \in C[0,1]$. The proof of Theorem \ref{thm1} is complete. \subsection*{Example} Let$\lambda>4/27$. 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