\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 87, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/87\hfil Existence of solutions] {Existence of solutions to p-Laplace equations with logarithmic nonlinearity} \author[J. Mo, Z. Yang\hfil EJDE-2009/87\hfilneg] {Jing Mo, Zuodong Yang} % in alphabetical order \address{Jing Mo \newline Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Jiangsu Nanjing 210097, China} \email{jingshuihailang@163.com} \address{Zuodong Yang \newline Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Jiangsu Nanjing 210097, China. \newline College of Zhongbei, Nanjing Normal University, Jiangsu Nanjing 210046, China} \email{zdyang\_jin@263.net} \thanks{Submitted February 23, 2009. Published July 10, 2009.} \thanks{Supported by grants 10871060 from the National Natural Science Foundation of China \hfill\break\indent and 08KJB110005 from the Natural Science Foundation of Educational Department, \hfill\break\indent Jiangsu Province, China} \subjclass[2000]{35B20, 35B65, 35J65} \keywords{Existence; logarithmic nonlinearity; supersolution; subsolution} \begin{abstract} This article concerns the the nonlinear elliptic equation $$ -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u) =\log u^{p-1}+\lambda f(x,u) $$ in a bounded domain $\Omega \subset \mathbb{R}^{N}$ with $N\geq 1$ and $u=0$ on $\partial\Omega$. By means of a double perturbation argument, we obtain a nonnegative solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this paper we consider the existence of solutions to the problem \begin{equation} \begin{gathered} -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u) =\log{u^{p-1}}+\lambda f(x,u),\quad\text{in } \Omega,\\ u>0,\quad\text{in } \Omega,\\ u=0,\quad\text{on } \partial\Omega, \end{gathered} \label{eq1.1} \end{equation} where $\Omega$ is a bounded $C^2$ domain in $\mathbb{R}^N$, $N\geq 2$, $12$ are called dilatant fluids and those with $p<2$ are called pseudo-plastics. If $p=2$, they are Newtonian fluids. When $p=2$, the existence of bounded positive solutions were proved by Deng [3]. When $p\not =2$, the problem becomes more complicated since certain nice properties inherent to the case $p=2$ seem to be lost or at least difficult to be verified. The main differences between $p=2$ and $p\neq 2$ can be found in [8,9]. In recent years, the existence and uniqueness of the positive solutions for the quasilinear eigenvalue problem \begin{equation} \begin{gathered} \mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+\lambda f(x,u)=0 \quad\text{in }\Omega ,\\ u(x)=0 \quad\text{on }\partial\Omega, \end{gathered} \label{eq1.2} \end{equation} with $\lambda>0$, $p>1$ on a bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 2$ have been studied by many authors see [9,10,19] and the references therein when $f$ is strictly increasing on $\mathbb{R}^{+}$, $f(0)=0,\lim_{s\to 0^{+}}{f(s)/{s^{p-1}}}=0$ and $f(s)\leq\alpha_{1}+\alpha_{2}{s^{\mu}}$, $0<\mu0$. It was shown in [10] that there exists at least two positive solutions for equation (1.2) when $\lambda>0$ is sufficiently large. If $\liminf_{s \to 0^{+}}(f(s)/(s^{p-1}))> 0,f(0)=0$ and the monotonicity hypothesis $(f(s)/(s^{p-1}))^{'}< 0$ holds for all $s>0$, it was proved in [9] that the problem (1.2) has a unique positive solution when $\lambda$ is sufficiently large. For $p=2$, some results to a semilinear elliptic equation with logarithmic nonlinearity \begin{equation} \begin{gathered} -{\Delta u}=\log u +h(x)u^{q},\quad\text{in } B_{R},\\ u>0,\quad\text{in } B_{R},\\ u=0,\quad\text{on } \partial B_{R}; \end{gathered} \label{eq1.3} \end{equation} and \begin{equation} \begin{gathered} -{\Delta u}=\chi_{\{u>0\}}(\log u+\lambda\ f(x,u)),\quad\text{in } \Omega,\\ u\geq 0, \quad\text{in};\; \Omega,\\ u=0, \quad\text{on } \partial\Omega, \end{gathered}\label{eq1.4} \end{equation} have been extensively studied. (See, for example, [15,19] and their references.) In [19], the authors obtained a positive radial solution $u\in C^{2}(\overline{B}_{R}\backslash \{0\})\bigcap C(\overline{B}_{R})$ of (1.3) by means of a double perturbation argument. In [15], the authors study the problem (1.4), which obtain a maximal solution $u_\lambda\geq 0$ for every $\lambda>0$ and prove its global regularity $C^{1,\gamma}(\overline{\Omega})$. Motivated by the results of the above cited papers, we shall attempt to treat such equation \eqref{eq1.1}, the results of the semilinear equations are extended to the quasilinear ones. We can find the related results for $p=2$ in [15]. In this paper, the authors obtained the maximal solution $u_{\lambda}\geq 0$ for every $\lambda>0$ and proved its global regularity $C^{1,\gamma}(\overline{\Omega})$. Our strategy in the study of (1.1) is to use the sub-super solution method and the mountain pass lemma. The paper is organized as follows. In section 2, we obtain a subsolution of (1.1) by adopting a double perturbation argument. Section 3 is dedicated to prove the existence of a supersolution of (1.1) by the mountain pass lemma. In section 4, we shall use the results of Section 2 and 3 to obtain a solution for the problem (1.1) by using the sub-super solution method which proves our main result. Some regularity properties of the solution of (1.1) are studied in section 5. In this problem, the function $f$ satisfies the following hypothesis: \begin{itemize} \item[(H1)] $f:\Omega\times[0,+\infty)$ is measurable in $x\in\Omega$ with $f$ is continuous; \item[(H2)] $f$ is nondecreasing, $f\neq0$; \item[(H3)] $\lim_{s\to \infty}{f(x,s)/{s^\beta}}=0$, $f(x,s)/{s^\beta}$ is decreasing where $0<\beta0$, such that $\sup_{0<\varepsilon<1}{\|u^{\varepsilon}\|_{L^{\infty}}}\leq C_{1}$. \end{lemma} \begin{proof} We denote $$ h_{\varepsilon}(s)=\log{\frac{s^{p}+\varepsilon s+\varepsilon}{s+\varepsilon}}. $$ Assume by contradiction that there exists a sequence $\varepsilon_{j}\to 0$ as $j\to \infty$, and $\|u^{\varepsilon j}\|_{L^{\infty}}\to \infty$ as $j\to \infty$, where $u^{\varepsilon j}$ solves (2.1), for each $j\in \mathbb{N}$, we set $$ \alpha_{j}=\|u^{\varepsilon j}\|_{L^{\infty}},\quad \beta_{j}=\inf_{s\geq 0}h_{\varepsilon j}(s),\quad \Omega_{j}=|\beta_{j}|\Omega, \quad \widetilde{x}=x/|\beta_{j}| $$ and define $$ U^{\varepsilon j}(x)=u^{\varepsilon j}(\widetilde{x})/\alpha_{j}, \quad x\in\Omega_{j}, $$ clearly, $\|U^{\varepsilon j}\|_{L^{\infty}(\Omega_{j})}=1$ for all $j\in \mathbb{N}$. On the other hand $$ -\mathop{\rm div}(|\nabla U^{\varepsilon j}(\widetilde{x})|^{p-2}\nabla U^{\varepsilon j}(\widetilde{x}))=\frac{h_{\varepsilon j}(u^{\varepsilon j}(\widetilde{x}))+\lambda f(\widetilde{x},u^{\varepsilon j}(\widetilde{x}))}{(\alpha_{j})^{p-1}|\beta_{j}|^{p-1}} $$ As a result, $\|U^{\varepsilon j}\|_{C(\overline{\Omega}_{j})}\to 0$ as $j\to \infty$, which contradicts $\|U^{\varepsilon j}\|_{L^{\infty}(\Omega_{j})}=1$. \end{proof} We shall prove that (2.1) has a solution. First we find a supersolution which is independent on $\varepsilon$. Clearly $\underline{u}=0$ is a subsolution of (2.1). Then our solution $u^{\varepsilon}\geq 0$. \begin{lemma} \label{lem2.3} Suppose $f$ satisfies {\rm (H1)--(H3)}, then for each $\lambda>0$, there is a supersolution $\bar{u}$ of (2.1) for $0<\varepsilon<1$. \end{lemma} \begin{proof} First consider the solution $Y$ of the problem \begin{equation} \begin{gathered} -\mathop{\rm div}(|\nabla Y|^{p-2}\nabla Y)=1,\quad\text{in } \Omega,\\ Y=0,\quad\text{on } \partial\Omega, \end{gathered} \label{eq2.2} \end{equation} Since $Y$ is bounded in $\Omega$, we choose $\theta>0$ such that $\theta\|Y\|_{L^{\infty}}\leq1$, next we fix $M>0$ and $c_{1}>0$ in a such way that (see $(H_{3}))$, $f(x,u)\leq\theta u^{\beta} $ for all $u\geq M$ and $f(x,u)\leq c_{1}$ for all $u\leq M$. In fact we may choose $\theta<\frac{\beta+1}{2pC(N,p)^{\beta+1}}$. We fix $k>0$ such that $$ k^{p-1}-\log(k^{p-1}\|Y\|_{L^{\infty}}^{p-1}+1) \geq \lambda\theta k^{p-1}\|Y\|_{L^{\infty}}^{p-1} $$ and $$ k^{p-1}-\log(M^{p-1}+1)\geq c_{1}, $$ setting $\bar{u^{\varepsilon}}=\bar{u}=kY$, we obtain a supersolution of (2.1) for all $0<\varepsilon<1$. Indeed, recall the definition of $h_{\varepsilon}$, if $u\geq M$, we have \begin{align*} -\Delta_{p}\bar{u}-h_{\varepsilon}(\bar{u}) &=k^{p-1}-h_{\varepsilon}(\bar{u})\\ &\geq k^{p-1}-\log(\bar{u}^{p-1}+1) \\ &=k^{p-1}-\log(k^{p-1}Y^{p-1}+1)\\ &\geq k^{p-1}-\log(k^{p-1}\|Y\|_{L^{\infty}}^{p-1}+1)\\ & \geq \lambda\theta k^{p-1}\|Y\|_{L^{\infty}}^{p-1}\\ &\geq \lambda\theta\bar{u}^{p-1} \geq\lambda\theta\bar{u}^{\beta}\\ &\geq \lambda f(x,\bar{u}). \end{align*} Whenever $\bar u\leq M$, we obtain \begin{align*} -\Delta_{p}\bar{u}-h_{\varepsilon}(\bar{u}) &= k^{p-1}-h_{\varepsilon}(\bar{u}) \geq k^{p-1}-\log(\bar{u}^{p-1}+1)\\ &\geq k^{p-1}-\log(M^{p-1}+1) \geq c_{1}\\ &\geq \lambda f(x,\bar{u}). \end{align*} Consequently, $\bar{u}^{\varepsilon}=kY$ is a supersolution of \eqref{eq2.1} for all $\varepsilon>0$. \end{proof} \begin{lemma} \label{lem2.4} Let $0<\varepsilon<\varepsilon_{0}$ and $\lambda>0$ be fixed. Then the problem (2.1) has a solution $u^{\varepsilon}>0$. \end{lemma} \begin{proof} Let $\varepsilon>0$ be fixed and $$ F_{\varepsilon}(x,u)=\log{\frac{u^{p}+\varepsilon u+\varepsilon}{u+\varepsilon}}+\lambda f(x,u)+a_{\varepsilon} u $$ where the constant $a_{\varepsilon}$ is fixed in such a way that $u\to F_{\varepsilon}(x,u)$ is increasing on $[\underline{u}^{\varepsilon},\bar{u}^{\varepsilon}]$ uniformly in $x\in \Omega$. Starting with $u_{0}=\bar{u}^{\varepsilon}$, we define the sequence $\{u_{n}\}$ of (unique) solution of the problem \begin{equation} \begin{gathered} -\mathop{\rm div}(|\nabla u_{n}|^{p-2}\nabla u_{n})+a_{\varepsilon}u_{n}=F_{\varepsilon}(x,u_{n-1}), \quad\text{in } \Omega,\\ u_{n}=0,\quad\text{on } \partial\Omega, \end{gathered}\label{eq2.3} \end{equation} Then we have $\underline{u}^{\varepsilon}\leq \ldots,\leq u_{n+1}\leq u_{n}\ldots \leq u_{0}=\bar{u}^{\varepsilon}$. In fact, it follows by the comparison principle in lemma 2.1 applied to the problem \begin{equation} \begin{gathered} -\mathop{\rm div}(|\nabla u_{0}|^{p-2}\nabla u_{0})+a_{\varepsilon}u_{0}\geq -\mathop{\rm div}(|\nabla u_{1}|^{p-2}\nabla u_{1})+a_{\varepsilon}u_{1},\quad\text{in } \Omega,\\ u_{0}\geq u_{1},\quad\text{on } \partial\Omega, \end{gathered} \label{eq2.4} \end{equation} that $u_{0}\geq u_{1}\geq 0$. Similarly, $u^{\varepsilon}\leq u_{1}$ in $\Omega$. There is a function $u^{\varepsilon}$ defined by pointwise limit $$ u^{\varepsilon}(x)=\lim_{n\to \infty}u_{n}(x),x\in \Omega. $$ By a standard bootstrap argument, we may take the $\lim n \to \infty$, so we conclude that $u$ satisfies (2.1). \end{proof} \begin{lemma} \label{lem2.5} The pointwise $ u(x)=\lim_{\varepsilon\to 0}u^{\varepsilon}(x)(x\in \Omega)$ is the subsolution of (1.1), in other words \begin{equation} \int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla\varphi dx +\int_{\Omega}(-\log u^{p-1})\varphi dx \leq \int_\Omega \lambda f(x,u)\varphi dx\label{eq2.5} \end{equation} for all $\varphi\in C_{0}^{\infty}(\Omega)$ with $\varphi\geq 0$ in $\Omega$. \end{lemma} \begin{proof} Let $\varphi\in C_{0}^{\infty}(\Omega)$ with $\varphi\geq 0$ in $\Omega$, $\lambda>0$ and recall the definition of $h_{\varepsilon}$. For each $0<\varepsilon<\varepsilon_{0}$, we have \begin{equation} \int_{\Omega}|\nabla u^{\varepsilon}|^{p-2}\nabla u^{\varepsilon}\nabla\varphi dx=\int_{\Omega}h(u^{\varepsilon})\varphi+\int_{\Omega}\lambda f(x,u^{\varepsilon})\varphi dx \label{eq2.6} \end{equation} The dominated convergence theorem implies \begin{equation} \lim_{\varepsilon \to 0}\int_{\Omega}\lambda f(x,u^{\varepsilon})\psi dx=\int_\Omega \lambda f(x,u)\varphi dx \label{eq2.7} \end{equation} Analogously, \begin{equation} \lim_{\varepsilon \to 0}\int_{\Omega}|\nabla u^{\varepsilon}|^{p-2}\nabla u^{\varepsilon}\nabla\varphi dx=\int_{\Omega}|\nabla u|^{p-2}\nabla u \nabla \varphi dx \label{eq2.8} \end{equation} Since $$ \liminf_{\varepsilon\to 0}{-h_{\varepsilon}(u^{\varepsilon})}\geq -\log (u^{\varepsilon})^{p-1}, $$ from the Fatou's Lemma, it follows that \begin{equation} \liminf_{\varepsilon\to 0}\int_{\Omega}{-h_{\varepsilon} (u^{\varepsilon})\varphi} dx \geq \int_{\Omega}{-\log (u^{\varepsilon})^{p-1}\varphi} dx \label{eq2.9} \end{equation} Letting $\varepsilon \to 0$ in (2.6) and using (2.7)-(2.8), we obtain (2.5). \end{proof} \section{Supersolutions for (1.1)} In this section we use that that \begin{itemize} \item[(F1)] $\log u^{p-1}\leq u^{q-1}\quad\text{for all }u>0,q>p$. \end{itemize} As in lemma 2.3, we only consider the case $u\geq M$ and $f(x,u)\leq \theta u^{\beta},0<\beta0$ such that $I(u)\geq a$ if $\|u\|=r$; \item[(3)] There exists an element $v\in H$ with $\|v\|>r,I(v)\leq 0$. \end{itemize} Define $$ \Gamma:=\{g\in C[0,1]; H:g(0)=0,g(1)=1\}. $$ Then $c=\inf_{g\in\Gamma}\max_{0\leq t\leq 1}I[g(t)]$ is a critical value of $I$. \end{lemma} In the following, we define the space $D^{1,p}(\Omega)$ as the closure of the set $C_{c}^{\infty}(\Omega)$ with the norm $$ \|u\|_{D^{1,p}(\Omega)}=\Big(\int_{\Omega}|\nabla u|^{p} dx\Big)^{1/p}. $$ \begin{lemma} \label{lem3.2} There exists a solution $u$ of the problem (3.2). \end{lemma} To prove the existence of a solution of (3.2), we will apply the mountain pass lemma to the energy functional \begin{equation} J(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p} dx -\frac{1}{q}\int_{\Omega}u^{q}dx-\frac{\lambda \theta }{\beta+1} \int_{\Omega}u^{\beta+1}dx \label{eq3.2} \end{equation} The facts that $D^{1,p}(\Omega)$ is a Banach space (reflexive) and that $J \in C^{1}(D^{1,p}(\Omega),\mathbb{R})$ satisfies the Palais-Smale condition are basic results (see [2]). It remains to see the two following points to prove that the functional $J$ has a mountain pass geometry: \begin{itemize} \item[(C1)] There exists $R>0$ and $a>0$ such that $\|u\|_{D^{1,p}(\Omega)}=R$ implies $J(u)\geq a$; \item[(C2)] There exists $u_{0}\in D^{1,p}(\Omega)$ such that $\|u_{0}\|_{D^{1,p}(\Omega)}>R$ and $J(u_{0})p>\beta+1).\label{eq3.3} \end{equation} So \begin{align*} J(u) &\geq \frac{\varepsilon}{p}-\frac{2\lambda \theta }{\beta+1}C(N,p)^{\beta+1}\varepsilon ^{\beta+1}\\ &\geq \frac{\varepsilon}{p}-\frac{2\lambda \theta }{\beta+1}C(N,p)^{\beta+1}\varepsilon\\ &=\varepsilon(\frac{1}{p}-\frac{2\lambda \theta }{\beta+1}C(N,p)^{\beta+1}). \end{align*} Finally, when $$ \theta<(\beta+1)(2pC(N,p)^{\beta+1})^{-1} $$ if we take two constant $R=\varepsilon>0$ and $a=\varepsilon(\frac{1}{p}-\frac{2\lambda \theta }{\beta+1}C(N,p)^{\beta+1})>0$, the functional $J$ satisfies the condition (C1). Let $u\in C_{0}^{\infty}(\Omega)$ fixed such that $u>0$ in $\Omega$, $u\geq 0$ on $\partial \Omega$. \begin{equation} J(ku)=\frac{k^{p}}{p}\int_{\Omega}|\nabla u|^{p} dx-\frac{k^{q}}{q}\int_{\Omega}u^{q}dx-\frac{\lambda \theta k^{\beta+1} }{\beta+1}\int_{\Omega}u^{\beta+1}dx \label{eq3.4} \end{equation} for all $k>0$. As $q>p>1$, we obtain $J(ku)\to -\infty$ when $k\to \infty$. So putting $u^{0}=ku$, there exists some $k$ great enough that $\|u^{0}\|_{D^{1,p}(\Omega)}>R$ and $J(u^{0})0,v>0$ on $ \Omega$ be such that $$ -\Delta_{p}u\leq \psi_{1}(x,u) \mbox{ and }-\Delta_{p}v\geq \psi_{2}(x,v) \quad\text{on } \Omega . $$ If $u\leq v$ on $\partial\Omega$ and $\psi_{1}(x,u)$ (or $\psi_{2}(x,u)$) belongs to $L^{1}(\Omega)$, then $u\leq v$ on $\Omega$. \end{lemma} \begin{lemma} \label{lem4.2} $\underline{u}<\overline{u}$ in $\Omega$. \end{lemma} \begin{proof} From section 2 and section 3, we know that $$ -\Delta_{p}\underline{u }\leq \log \underline{u}^{p-1}+\lambda f(x,\underline{u}) $$ and $$ -\Delta_{p}\overline{u}\geq \log \overline{u}^{p-1}+\lambda f(x,\overline{u}) $$ in weak sense. From (F1) and (F2), we know that $$ \log u^{p-1}+\lambda f(x,u)0 $$ where we choose $b$ in such a way that the function $u\mapsto G(x,u)$ is increasing in $u$ on $[\underline{u },\overline{u}]$ for all $x\in \Omega$. \end{proof} \begin{theorem} \label{thm4.3} There exists a solution for (1.1). \end{theorem} \begin{proof} As noted above we start with $u_{0}=\overline{u}$. We define the sequence $\{u_{n}\}$ of (unique) solution of the problems \begin{equation} \begin{gathered} -\mathop{\rm div}(|\nabla u_{n}|^{p-2}\nabla u_{n})+b u_{n}=G(x,u_{n-1}),\quad\text{in } \Omega\\ u_{n}=0,\quad\text{on } \partial\Omega \end{gathered} \label{eq4.1} \end{equation} we apply the comparison principle in lemma 2.1 to the problem \begin{equation} \begin{gathered} -\mathop{\rm div}(|\nabla u_{0}|^{p-2}\nabla u_{0})+b u_{0}\geq -\mathop{\rm div}(|\nabla u_{1}|^{p-2}\nabla u_{1})+b u_{1},\quad\text{in } \Omega\\ u_{0}\geq u_{1},\quad\text{on } \partial\Omega\\ \end{gathered} \label{eq4.2} \end{equation} it follows that $u_{0}\geq u_{1}\geq 0$, similarly, $\underline{u}\leq u_{1}$ in $\Omega$. So $\underline{u }\leq \ldots,\leq u_{n+1}\leq u_{n}\ldots \leq u_{0} =\bar{u}$. There is a function $u$ defined by pointwise limits $$ u(x)=\lim_{n\to \infty}u_{n}(x),x\in \Omega. $$ We see that $\underline{u}\leq u\leq \overline{u},x\in \Omega$. By a standard bootstrap argument, we may take the $\lim n \to \infty$. The function $u(x)$ is in fact a solution of (1.1). \end{proof} \section{Regularity Properties of the Solution} In this section, we study some regularity properties of the solution to (1.1). Firstly, we state the following lemma, due to DiBenedetto [5], which is the local regularity for the elliptic equation. \begin{lemma} \label{lem5.1} Let $u\in W^{1,p}_{\rm loc}(\Omega)\bigcap L^{\infty}_{\rm loc}(\Omega)$ be a local weak solution of $-\Delta_{p}u=b(x,r)$ in $\Omega$, an open domain in $\mathbb{R}^{N}$, where $b(x,r)$ is measurable in $x\in \Omega$ and continuous in $r\in \mathbb{R}$ such that $|b(x,r)|\leq \gamma$ on $\Omega\times\mathbb{R}$. Given a sub-domain compact $\Omega'\subset\subset\Omega$, there exists positive constants $C_{0},C_{1}$ and $\alpha\in(0,1)$, depending only upon $N,p,\gamma,M=\mbox{ess}\sup_{\Omega'}|u|$ and $\mathop{\rm dist}(\Omega',\Omega)$ such that $\|\nabla u(x)\|_{\infty,\Omega'}\leq C_{0}$ and $x\mapsto \nabla u(x)$ is locally H\"older continuous in $\Omega'$; i.e., \begin{equation} |u_{x_{i}}(x)-u_{x_{i}}(y)|\leq C_{1}|x-y|^{\alpha},\quad x,y\in\Omega',i=1,2,\dots ,N\label{eq5.1} \end{equation} \end{lemma} \begin{theorem} \label{thm5.2} Assume $f$ satisfies {\rm (H1--H2)}. For the solution $u$ of (1.1) there holds: \begin{itemize} \item[(1)] $u\in C^{1,\alpha}(\Omega)$ where $0<\alpha<1$; \item[(2)] There exists $\underline{\lambda}>0$ such that, for each $\lambda\geq \underline{\lambda}$, the solution to (1.1) is positive in $\Omega$; \item[(3)] Let $\lambda_{1} $ be the first eigenvalue of $-\Delta_{p}$ in $W^{1,p}_{0}(\Omega)$. There exists $\theta>0$ such that, if $\lambda_{1}(\Omega)<\theta$, then $u>0$ for all $\lambda>0$. \end{itemize} \end{theorem} \begin{proof} (1) Since we have got the weak solution of (1.1), $u\in W^{1,p}_{0}(\Omega)$. From the interior $C^{1,\alpha}$ estimate in lemma 5.1, we conclude that $|\nabla u|\in C^{\alpha}(\Omega)$ for some $\alpha\in(0,1)$ and we find that $u\in C^{1,\alpha}(\Omega)$ for $\alpha\in(0,1)$. (2) We just need to find a strictly positive subsolution. Let $Y$ be the solution of (2.2) and $\phi$ be the solution of the following problem \begin{equation} \begin{gathered} -\mathop{\rm div}(|\nabla \phi|^{p-2}\nabla \phi) =\lambda f(x,\delta^{\nu}(x)),\quad\text{in } \Omega,\\ \phi=0,\quad\text{on } \partial\Omega, \end{gathered} \label{eq5.2} \end{equation} where $\delta(x)=\mathop{\rm dist}(x,\partial\Omega)$ is the distance function independently of $\lambda$, and $\nu>1$ will be fixed latter. Since $f(x,\delta^{\nu}(x))$ is not identically zero in $\Omega$, there exists a constant $C>0$ such that $\phi\geq 2C\|Y\|_{L^{\infty}}$. We set $v:=\phi-C\|Y\|_{L^{\infty}}$ and $\underline{u}:=kv^{\nu}$, where $k>0$ to be fixed accordingly. We choose $\Omega'\subset\Omega$ and $\eta_{1},\eta_{2}>0$ such that $$ |\nabla v|^{p}\geq \eta_{1}>0, \quad\text{in}\;\;\;\Omega\backslash \Omega',\quad v\geq\eta_{2}>0 \quad\text{in }\Omega'. $$ Since \begin{align*} &\log(kv^{\nu})^{p-1}-(k\nu)^{p-1}(\nu-1)(p-1)v^{(\nu-1)(p-1)-1}|\nabla v|^{p} \\ &\leq \log(kv^{\nu})^{p-1}-(k\nu)^{p-1}(\nu-1)(p-1)v^{(\nu-1)(p-1)-1} \eta_{1} \\ &\leq 0 \quad\text{in }\Omega\backslash \Omega'. \end{align*} we obtain $\underline{u}=kv^{\nu}$ is strictly positive subsolution for $\underline{\lambda}\leq (k\nu)^{p-1}\|v\|^{(\nu-1)(p-1)}|_{L^{\infty}}$, which proves (2). (3) Similarly as in the above proof, we need to find a positive subsolution for (1.1) with $\lambda=0$. Thus, let $Y$ be the solution of (2.2) and $\varphi_{1}$ be the first eigenfunction associated with $\lambda_{1}$. There exists a constant $C>0$ such that $\varphi_{1}\geq 2C\|Y\|_{L^{\infty}}$. We set $v:=\phi-C\|Y\|_{L^{\infty}}$ and $\underline{u}:=kv^{\nu}$, where $k>0$ to be fixed accordingly. Then if $\nu>1$, we have \begin{align*} -\Delta_{p}\underline{u} &= -(k\nu)^{p-1}(\nu-1)(p-1)v^{(\nu-1)(p-1)-1}|\nabla v|^{p}\\ &\quad +(k\nu)^{p-1}v^{(\nu-1)(p-1)}\lambda_{1}|v+C\|Y\|_{L^{\infty}}|^{p-2}(v +C\|Y\|_{L^{\infty}})\\ &\leq -(k\nu)^{p-1}(\nu-1)(p-1)v^{(\nu-1)(p-1)-1}\eta_{1} \\ &\quad +(k\nu)^{p-1}v^{(\nu-1)(p-1)}\lambda_{1}(v+C\|Y\|_{L^{\infty}})^{p-1}\\ &\leq (k\nu)^{p-1}v^{(\nu-1)(p-1)}\big[\lambda_{1}(\|v\|_{L^{\infty}} +C\|Y\|_{L^{\infty}})^{p-1}-\frac{(\nu-1)(p-1)\eta_{1}}{\|v\|_{L^{\infty}} +C\|Y\|_{L^{\infty}}}\big]. \end{align*} Suppose that $$ \lambda_{1}<\frac{(\nu-1)(p-1)\eta_{1}}{\|v\|_{(L^{\infty}} +C\|Y\|_{L^{\infty}})^{p}}, $$ then $$ (k\nu)^{p-1}v^{(\nu-1)(p-1)}[\lambda_{1}(\|v\|_{L^{\infty}} +C\|Y\|_{L^{\infty}})^{p-1}-\frac{(\nu-1)(p-1)\eta_{1}}{\|v\|_{L^{\infty}} +C\|Y\|_{L^{\infty}}}]\to -\infty $$ So \begin{align*} &-\mathop{\rm div}(|\nabla\underline{u}|^{p-2}\nabla \underline{u})\\ &\leq (k\nu)^{p-1}v^{(\nu-1)(p-1)}[\lambda_{1}(\|v\|_{L^{\infty}} +C\|Y\|_{L^{\infty}})^{p-1}-\frac{(\nu-1)(p-1) \eta_{1}}{\|v\|_{L^{\infty}}+C\|Y\|_{L^{\infty}}}]\\ &\leq \log(kv^{\nu}) \end{align*} for some $k>0$. The proof is complete. \end{proof} \subsection*{Acknowledgments} The authors want to thank the anonymous the referees for their comments and suggestions. \begin{thebibliography}{00} \bibitem{a1} G. Astrita, G. Marrucci; Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974. \bibitem{c1} K. Chaib; Necessary and sufficient conditions of existence for a system invdving the $p$-Laplacian($1