\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 175, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/175\hfil Quasilinear inequalities] {Positivity and nonexistence of solutions for quasilinear inequalities} \author[X. Li \hfil EJDE-2016/175\hfilneg] {Xiaohong Li} \address{Xiaohong Li \newline College of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, Shandong 264005, China} \email{xh0535@sina.com} \thanks{Submitted October 30, 2015. Published July 6, 2016.} \subjclass[2010]{35J60, 35J70} \keywords{Quasilinear inequality; positivity property; nonexistence theorem; \hfill\break\indent comparison principle} \begin{abstract} We prove a positivity property and a nonexistence theorem for weak solutions of quasilinear differential equalities in $\mathbb{R}^N$. To obtain our results, we use a comparison principle. Also we establish a criterium for the existence of positive radial solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this article, we consider the positivity nonexistence for weak solutions of the anisotropic divergence structure quasilinear differential inequality \begin{equation} \label{1.1} L (u)= -\operatorname{div}_L (a(x)A(| \nabla_L u | )\nabla_L u)\geq h(x) f(u)\quad \text{in } \mathbb{R}^N, \end{equation} where $A$ and $f$ satisfy \begin{itemize} \item[(A1)] $A\in C(0,\infty)$ and $a,h \in C(\mathbb{R}, \mathbb{R})$; \item[(A2)] $f: \mathbb{R}\to\mathbb{R}$ is continuous function and nondecreasing on $(-\infty, 0]$ satisfying $f(t)>0$ for $t<0$. \end{itemize} and $L$ belongs to a wide class of anisotropic quasilinear operator including \[ \operatorname{div}_L(| \nabla_L u |^{p-2}\nabla_L u), \quad \operatorname{div}_L\Big(\frac{\nabla_L u}{\sqrt{1+| \nabla_L u |^{2}}}\Big), \quad \operatorname{div}_L\Big(\frac{| \nabla_L u |^{p-2}\nabla_L u} {(1+| \nabla_L u |^{s})^k}\Big), \] with $p>1$, $s>0$, $k\geq0$. Recently, this kind of problems have received a great attention. Tools based different forms of the maximum principle like the moving planes method or moving spheres method, nonlinear capacitary estimates and Pohozaev type identities, energy methods and Harnack inequality type argument, have been proved to be very successful for solving interesting problems related to be applications and to the general theory of partial differential equations. We refer to \cite{VP2001}-\cite{SZ2002} and the references therein for some recent contributions. In the special case of \eqref{1.1}, given by $a(x)= h(x)\equiv 1$, the positivity results and nonexistence theorems are of great interest. For the Euclidean gradient case of \eqref{1.1}, that is $\nabla_L u =\nabla u $, D'Ambrosio and Mitidieri \cite{DM20102} proved the positivity results and nonexistence theorems of weak solution $W_{{\rm loc}}^{1,p}$ in $\mathbb{R}^N$. Both in \cite{DM2008} and \cite{DM20101}, the authors obtained the positivity results of solution $C^1$ solution of \eqref{1.1} in the Heisenberg setting and Carnot Groups. An interesting discussion on the nonexistence of solutions of \eqref{1.1} for anisotropic case in $\mathbb{R}^N$ is given by L. D'Ambrosio in \cite{D2010}, who used the test function method developed by Mitidieri and Pohozaev (see \cite{MP1998}--\cite{MP2001}). In particular, both in \cite{CDM2008} and \cite{DM2006}, the authors proved positivity results by integral representation formulae, when $L$ is the Laplacian operator or the polyharmonic operator in the Euclidean setting or, more generally $L$ is a sub elliptic Laplacian on a Carnot group and $f$ is nonnegative. Motivated by the above works, in the present paper, we obtain the positivity property and the nonexistence Theorem for $W_{L,{\rm loc}}^{1,p}$ solutions of anisotropic quasilinear differential equality in $\mathbb{R}^N$. General results on the $W_{L,{\rm loc}}^{1,p}$ solutions for \eqref{1.1} are considered, which is based on a comparison principle and the analysis of an ordinary differential equation. For the main results of the paper we shall present some preliminaries (see \cite{DM2005}-\cite{DM20102}). In this paper $ \nabla $ and $| \cdot|$ stands respectively for usual gradient in $\mathbb{R}^N$ and the Euclidean norm. Let $\mu \in C(\mathbb{R}^N, \mathbb{R}^m)$ be a matrix $\mu:=(\mu_{ij})$, $i=1,\ldots, m$, $j=1,\ldots, N$. For $i=1,\ldots, m$, let $X_i$ and its formal adjoint $X_i^*$ be defined as \begin{equation} \label{2.1} X_i:=\sum^N_{j=1}\mu_{ij}(\xi)\frac{\partial}{\partial\xi_j},\quad X_i^*:=-\sum^N_{j=1}\frac{\partial}{\partial\xi_j}(\mu_{ij}(\xi)\cdot), \end{equation} where $\nabla_L$ and $\nabla_L^*$ are the vector field defined by % \begin{equation} \label{2.2} \nabla_L:=(X_1,\ldots,X_m)^T=\mu\nabla, \quad \nabla_L^*:=(X_1^*,\ldots,X_m^*)^T. \end{equation} For any vector field $h=(h_1,\ldots,h_m)^T \in C^1(\Omega,\mathbb{R}^m)$, we shall use the notation $\operatorname{div}_L(h):=\operatorname{div}(\mu^Th)$; that is \begin{equation} \label{2.3} \operatorname{div}_L(h)=-\sum^l_{j=1}X_j^*h=-\nabla_L^*\cdot h. \end{equation} Let $\delta:=(\delta_1, \ldots, \delta_N)$ be an $N$-uple of positive real numbers. Let $R>0$, we shall denote by $\delta_R $ the anisotropic dilation $\delta_R: \mathbb{R}^N \to \mathbb{R}^N$ defined by \begin{equation} \label{2.4} \delta_R(x)=\delta_R(x_1,\ldots, x_N):=(R^{\delta_1}x_1,\ldots,R^{\delta_N}x_N). \end{equation} The Jacobian of such a transformation is $J(\delta_R)=R^Q$ where $Q=\delta_1+\delta_2+\ldots+\delta_N$. A nonnegative continuous function $H: \mathbb{R}^N \to \mathbb{R}^+$ is called a homogeneous norm if (i) $H(\xi)=0$ if and only if $\xi=0$, and (ii) it is homogeneous of degree 1 with respect to $\delta_R $, i.e., $H(\delta_R(\xi))=RH(\xi)$. Notice that if $H$ is a homogeneous norm differentiable a.e, then $| \nabla_L H|$ is homogeneous of degree 0 with respect to $\delta_R$; hence $| \nabla_L H|$ is bounded. For the rest of this article, we shall fix a homogeneous norm $H$ differentiable away from 0. We set \begin{equation} \label{2.4b} \psi:=| \nabla_L H| \end{equation} and for $R>0$, we define $B_R$ as the ball of radius $R>0$ generated by the norm $H$; that is, $B_R:=\{x:H(x)1$. Throughout this paper we shall denote by \begin{gather*} W^{1,p}_{L}(\Omega )=\{ u\in L^p(\Omega): \ | \nabla_L u | \in L^p(\Omega) \}, \\ W^{1,p}_{L,{\rm loc}}(\Omega )=\{ u\in L^p_{{\rm loc}}(\Omega): \ | \nabla_L u | \in L^p_{{\rm loc}}(\Omega) \}. \end{gather*} Notice that when $\mu=I_n$, where $I_n$ is the identity matrix, then $ W^{1,p}_{L}(\Omega )=W^{1,p}(\Omega )$ and $ W^{1,p}_{L,{\rm loc}}(\Omega )=W^{1,p}_{{\rm loc}}(\Omega )$. \begin{definition} \label{def1.1} \rm We shall say that $u \in W^{1,p}_{L,{\rm loc}}(\Omega )$ satisfies \eqref{1.1} in the weak sense and for any nonnegative test function $\varphi \in C^1_0(\Omega)$ such that \begin{equation} \label{2.7} \int_{\Omega }a(x)A( |\nabla_L u| ) \nabla_L u \cdot \nabla_L \varphi\,dx \geq \int_{\Omega } h(x) f(u) \varphi \,dx \end{equation} holds. \end{definition} \begin{definition} \label{def1.2} \rm Let $d:\Omega\to \mathbb{R}$ be a nonnegative non constant measurable function. For $\alpha\neq0$, $d^{\alpha}$ is called an $L_p$--harmonic function if \[ L_pd^{\alpha}=\operatorname{div}_L(|\nabla_L d^\alpha|^{p-2}\nabla_L d^{\alpha})=0 \] in the weak sense; that is, for every nonnegative $\varphi\in C^1_0(\Omega)$, we have \begin{equation} \label{2.8} \int_{\Omega}|\nabla_L d^\alpha|^{p-2}\nabla_L d^{\alpha}\cdot \nabla_L \varphi =\alpha | \alpha | ^{p-2} \int_{\Omega}d^{(\alpha-1)(p-2)}|\nabla_L d|^{p-2} \nabla_L d\cdot \nabla_L \varphi=0, \end{equation} where $d^{(\alpha-1)(p-1)}|\nabla_L d|^{p-2} \in L^1_{\rm loc}(\Omega)$. \end{definition} The main result in this paper is the following theorems. \begin{theorem} \label{thm1.1} Let $d^{\alpha}$ be defined as Definition \ref{def1.2} and $h( x )\geq M\bigl| \nabla_L d \bigr| ^p$, where $p>1$ and $M$ is a positive constant. Assume {\rm (A1), (A2)}, $a(x)>M$ and \begin{equation} \label{2.9} \int_{-\infty}^{+\infty}\Big( \int_t^{+\infty} f(s)ds\Big)^{-1/p} dt =+\infty \end{equation} hold. Let $A$ satisfy (i) $A(t)\geq t^{p-2}$ or (ii) $A(t)\leq t^{p-2}$, and let $ tA(t)$ be strictly increasing for $t>0$. If $u$ is a $W^{1,p}_{L,{\rm loc}}$ solution of \eqref{1.1}, then $u\geq0$. \end{theorem} \begin{remark} \label{rmk1.1} \rm The condition that $t\longmapsto tA(t)$ is strictly increasing is a minimal requirement for ellipticity of \eqref{1.1}. Furthermore, it allows singular and degenerate behavior of the operator $A$ at $t=0$. \end{remark} \begin{remark} \label{rmk1.2} \rm Similar results have been proved in \cite{DM2008}--\cite{DM20102}, when $a(x)=h(x)\equiv1$, under different conditions. In particular, if we set $d(x)=| x |$, then the condition $h( x )\geq M\bigl| \nabla_L d \bigr| ^p$ reduces to $h( x )\geq M$ in the Euclidean case. \end{remark} \begin{theorem} \label{thm1.2} Assume that $d^{\alpha}$ is defined as Definition \ref{def1.2} and $h( x )\geq M\bigl| \nabla_L d \bigr| ^p$, where $p>1$ and $M$ is a positive constant. Let $f:\mathbb{R}\to\mathbb{R}$ be a positive, non-increasing and continuous function satisfying \eqref{2.9}. Then \eqref{1.1} has no solutions. \end{theorem} \section{Proofs of Theorems \ref{thm1.1} and \ref{thm1.2}} To prove theorem \ref{thm1.1}, we establish some preliminary results in this section. Now, we shall prove a comparison lemma that it is useful when considering solutions of inequalities of the form \begin{gather} \label{3.1} \operatorname{div}_L \left(a( x )A_1(| \nabla_L u | )\nabla_L u\right)\geq g_1( x, u)\quad \text{in } \Omega, \\ \label{3.2} \operatorname{div}_L (MA_2(| \nabla_L v | )\nabla_L v)\leq g_2( x, v)\quad \text{in } \Omega, \end{gather} where $M$ is a positive constant. Here, for $i=1,2$, $A_i$ is a continuous function such that $A_i>0$ for $t>0$ and $g_i:\Omega\times \mathbb{R} \to \mathbb{R}$ is continuous. In a similar manner to Definition \ref{def1.1}, we can define the solution of equalities \eqref{3.1} and \eqref{3.2}. \begin{definition}\label{def3.1} \rm We shall say that $u \in W^{1,p}_{L,{\rm loc}}(\Omega )$ satisfies \eqref{3.1} (resp. \eqref{3.2}) in the weak sense, and for any nonnegative test function $\varphi \in C^1_0(\Omega)$ such that \begin{equation} \label{3.1'} -\int_{\Omega }a(x)A_1( |\nabla_L u| ) \nabla_L u \cdot \nabla_L \varphi\,dx \geq \int_{\Omega } g_1( x, u) \varphi \,\,dx \end{equation} (resp. \begin{equation} \label{3.2'} -\int_{\Omega }a(x)A_2( |\nabla_L u| ) \nabla_L u \cdot \nabla_L \varphi\,dx \leq \int_{\Omega } g_2( x, u) \varphi \,dx\big) \end{equation} holds. \end{definition} \begin{lemma}[Comparison principle] \label{lem3.1} Let $\Omega$ be a bounded open set. Let $u$ and $v$ be respectively solutions of \eqref{3.1} and \eqref{3.2} of class $W^{1,p}_{L,{\rm loc}}(\Omega )$. Assume that $a( x ) \geq M$ and \begin{itemize} \item[(i)] for any $x \in \Omega$, $t\geq s$ there holds $g_1( x, t)\geq g_2( x, s)$, $g_1( x, \cdot)$ is not decreasing. \item[(ii)] (1) $A_1(t)\geq A_2(t)$ for $t>0$ and the function $tA_2(t)$ is strictly increasing for $t>0$; or (2) $A_1(t)\leq A_2(t)$ for $t>0$ and the function $tA_1(t)$ is strictly increasing for $t>0$; \item[(iii)] $u\leq v$ on $\partial \Omega$. \end{itemize} Then $u\leq v$ in $ \Omega$. \end{lemma} \begin{remark} \label{rmk3.1} \rm If $a(x)\equiv1$, $ M=1$ and $A_1=A_2$ in \eqref{3.1} and \eqref{3.2}, similar results to Lemma \ref{lem3.1} have been proved under different conditions. \cite{DM2005,DM2008,DM20101,NU19971,NU19972,PS2007}. In Lemma \ref{lem3.1}, we extend some results of \cite{DM2008,DM20101}, which hold for $C^1$ solutions, to the large class of $W^{1,p}_{L,{\rm loc}}$ solutions. \end{remark} \begin{proof}[Proof of Lemma \ref{lem3.1}] Let $\epsilon>0$ be fixed and set $v_\epsilon=v+\epsilon$. It is a simple to check that the function $v_\epsilon$ satisfies the inequality $$ \operatorname{div}_L (MA_2(| \nabla_L v | )\nabla_L v)\leq g_2( x, v_\epsilon)\quad \text{in } \Omega, $$ Therefore, for any nonnegative test function $\varphi \in C^1_0(\Omega)$ we have $$ -\int_{\Omega }MA_2( |\nabla_L v| ) \nabla_L v \cdot \nabla_L \varphi\,dx \leq \int_{\Omega } g_2( x, v_\epsilon) \varphi \,dx $$ By subtraction we obtain \begin{equation} \label{3.3} \begin{aligned} &-\int_\Omega \left( a( x )A_1(| \nabla_L u | )\nabla_L u) -MA_2(| \nabla_L v | )\nabla_L v\right)\cdot \nabla_L\varphi\,dx \\ &\geq \int_\Omega(g_1( x, u)-g_2( x,v_\epsilon))\varphi\,dx \end{aligned} \end{equation} We choose the nonnegative $\varphi=((u-v_\epsilon)^+)^2$ as test function in \eqref{3.3}. Obviously, $\varphi\in W_{L}^{1,p}(\Omega)$ and $\varphi$ has compact support since $u-v_\epsilon<0$ on $\partial\Omega$. Then, we obtain \begin{equation} \label{3.4} \begin{split} &-2\int_\Omega \left( a( x )A_1(| \nabla_L u | )\nabla_L u)-MA_2(| \nabla_L v | )\nabla_L v\right)\cdot (\nabla_Lu-\nabla_Lv)(u-v_\epsilon)^+\,dx\\ &\geq \int_\Omega(g_1( x, u)-g_2( x, v_\epsilon))((u-v_\epsilon)^+)^2\,dx \end{split} \end{equation} Since $a( x ) \geq M>0$, we have \begin{equation} \label{3.5} \begin{aligned} &\left( a( x )A_1\left(| \nabla_L u | \right))\nabla_L u)-MA_2(| \nabla_L v | )\nabla_L v\right)\cdot (\nabla_Lu-\nabla_Lv) \\ &=a( x )A_1(| \nabla_L u | )| \nabla_L u |^2 +MA_2(| \nabla_L v | )| \nabla_L v |^2\\ &\quad -(a( x )A_1(| \nabla_L u | )+MA_2(| \nabla_L v | ))(\nabla_L u\cdot\nabla_L v) \\ &=\left(a( x ) A_1(| \nabla_L u | )| \nabla_L u | -MA_2(| \nabla_L v | )| \nabla_L v |) (| \nabla_L u |-| \nabla_L v |\right) \\ &\quad +(a( x )A_1(| \nabla_L u | ) +MA_2(| \nabla_L v | ))(| \nabla_L u || \nabla_L v | -\nabla_L u\cdot\nabla_L v)\\ & \geq M(\left( A_1(| \nabla_L u | )| \nabla_L u | -A_2(| \nabla_L v | )| \nabla_L v |) (| \nabla_L u |-| \nabla_L v |\right) \\ &\quad +(A_1(| \nabla_L u | ) +A_2(| \nabla_L v | ))(| \nabla_L u || \nabla_L v | -\nabla_L u\cdot\nabla_L v)) \\ &=:M(I_1+I_2) . \end{aligned} \end{equation} Since $A_i(t)>0$ for $t>0$, where $i=1,2$, we have $I_2\geq0$. First we consider the case (ii) (1). From $A_1(t)\geq A_2(t)$ for $t>0$ and the function $tA_2(t)$ is increasing for $t>0$, we obtain % \begin{equation} \label{3.6} \begin{split} I_1&=\left( A_1(| \nabla_L u | )| \nabla_L u | -A_2(| \nabla_L v | )| \nabla_L v |) (| \nabla_L u |-| \nabla_L v |\right)\\ &\geq \left( A_2(| \nabla_L u | )| \nabla_L u | -A_2(| \nabla_L v | )| \nabla_L v |) (| \nabla_L u |-| \nabla_L v |\right) \geq0. \end{split} \end{equation} Therefore, $\int_\Omega M(I_1+I_2) (u-v_\epsilon)^+dx\geq 0$. Since $g_1( x, \cdot)$ is not decreasing, \begin{equation} \label{3.7} \begin{split} &\int_\Omega(g_1( x, u)-g_2( x, v))((u-v_\epsilon)^+)^2\,dx\\ &\geq (g_1( x, u)-g_1( x, v_\epsilon))((u-v_\epsilon)^+)^2\,dx \geq0, \end{split} \end{equation} and then combining \eqref{3.4} with \eqref{3.5}, we obtain \begin{equation} \label{3.12} \begin{split} 0&\geq\int_\Omega M(I_1+I_2)(u-v_\epsilon)^+dx\\ &\geq\int_\Omega ( a( x )A_1(| \nabla_L u | )\nabla_L u)-MA_2(| \nabla_L v | )\nabla_L v)\cdot (\nabla_Lu-\nabla_Lv)(u-v_\epsilon)^+\,dx\\ &\geq \int_\Omega(g_1( x, u)-g_2( x, v_\epsilon))((u-v_\epsilon)^+)^2\,dx \geq 0; \end{split} \end{equation} that is, \begin{equation} \label{3.8} \int_\Omega (I_1+I_2) (u-v_\epsilon)^+dx= 0. \end{equation} The following proof is by contradiction. Assume that $\varphi=u-v-\epsilon>0$ for $x \in \Omega$, then we have $I_1=I_2=0$ by \eqref{3.8}. We claim that $\nabla_L u = \nabla_L v$. Indeed, If $\nabla_L u \not= \nabla_L v$, by $I_2=0$, we obtain \begin{equation} \label{3.9} | \nabla_L u || \nabla_L v |=\nabla_L u\cdot\nabla_L v \end{equation} and \begin{equation} \label{3.10} \begin{split} (| \nabla_L u |-| \nabla_L v |)^2 &=| \nabla_L u |^2-2| \nabla_L u | | \nabla_L v |+ | \nabla_L v |^2\\ & =| \nabla_L u |^2-2 \nabla_L u \cdot \nabla_L v + | \nabla_L v |^2\\ &=( \nabla_L u - \nabla_L v )^2, \end{split} \end{equation} which implies $| \nabla_L u |\not =| \nabla_L v |$. Moreover from $I_1=0$ and the monotonicity of $tA_2$, we obtain \begin{equation} \label{3.11} \begin{split} 0&=(A_1(| \nabla_L u | )| \nabla_L u | -A_2(| \nabla_L v | )| \nabla_L v |)(| \nabla_L u |-| \nabla_L v |)\\ &\geq(A_2(| \nabla_L u | )| \nabla_L u | -A_2(| \nabla_L v | )| \nabla_L v)(| \nabla_L u |-| \nabla_L v |) | >0, \end{split} \end{equation} which is a contradiction. Thus, we have $\nabla_L u = \nabla_L v$, which implies $\nabla_L \varphi=\nabla_L ((u - v_\epsilon)^+)^2=0$; that is, $\varphi=((u - v_\epsilon)^+)^2\equiv constant$ in $\Omega$. Since $\varphi\in W^{1,p}_{L,{\rm loc}}(\Omega )$, we have $\varphi=((u - v_\epsilon)^+)^2\equiv 0$ in $\Omega$, that is $u - v_\epsilon\leq 0$, which is a contradiction of our assumption. Thus, $u \leq v+\epsilon$ in $\Omega$. Letting $\epsilon\to 0$ completes the proof. Now, we consider the case (ii)(2), which proof is the same as that of (ii)(1). By virtue of \eqref{3.6} and \eqref{3.12}, we only to replace \eqref{3.6} and \eqref{3.12} by the following inequalities \begin{equation} \begin{split}\label{3.13} I_1&=\left( A_1(| \nabla_L u | )| \nabla_L u | -A_2(| \nabla_L v | )| \nabla_L v |) (| \nabla_L u |-| \nabla_L v |\right)\\ &\geq \left( A_1(| \nabla_L u | )| \nabla_L u | -A_1(| \nabla_L v | )| \nabla_L v |) (| \nabla_L u |-| \nabla_L v |\right) \geq0 \end{split} \end{equation} and \begin{equation} \label{3.14} \begin{split} & 0=(A_1(| \nabla_L u | )| \nabla_L u | -A_2(| \nabla_L v | )| \nabla_L v |)(| \nabla_L u |-| \nabla_L v |)\\ &\geq(A_1(| \nabla_L u | )| \nabla_L u | -A_1(| \nabla_L v | )| \nabla_L v)(| \nabla_L u |-| \nabla_L v |) | >0 \end{split} \end{equation} respectively in the proof of (1). The proof is complete. \end{proof} \begin{lemma}[{\cite[Lemma2.18]{DM2005}}] \label{lem3.2} Let $p>1$, $\alpha\in\mathbb{R}^N$, $\alpha \not= 0$. $d^{\alpha} \in C^2(\Omega)$ be a positive $L_p$-harmonic function; that is, $L_pd^{\alpha}=0$ in the weak sense of \eqref{2.8}. Let $u(x):=\phi(d(x))$, we have \begin{equation} \label{3.15} \begin{split} L_p u &=(p-1)\bigl| \nabla_L d \bigr| ^p \bigl| \nabla_L \phi'(d) \bigr|^{p-2} \big(\phi'(d)+\frac{1-\alpha}{d}\phi'(d)\big) \\ &=\bigl| \nabla_L d \bigr| ^p d^{(p-1)(\alpha-1)} \Big(d^{(p-1)(1-\alpha)} \bigl| \phi'(d) \bigr|^{p-2}\phi'(d) \Big)' \end{split} \end{equation} \end{lemma} \begin{remark} \label{rmk3.2} \rm Some special cases of Lemma \ref{lem3.2} are the following. In the Euclidean case, we set $d(x)=| x |:=r$, then \eqref{3.15} reduces to $$ L_p u=(p-1){| \phi'(r) |}^{p-2} \Big(\phi''(r)+\frac{N-1}{p-1}\frac{\phi'(r)}{r}\Big), $$ which is discussed in \cite{GR2003,NU19971,NU19972} for $p\not=2$ and \cite{O1960} for $p=2$. In the Heisenberg setting studied in \cite{DM2008}, we set \[ d(x)=| x |_H= \Big( \sum_{i=1}^n(\xi_i^2+\eta_i^2)^2+\tau^2\Big)^{1/4}:=r, \] then \eqref{3.15} reduces to \begin{equation} \label{3.16} L_p u=(p-1)\psi^p{| \phi'(r) |}^{p-2} \Big(\phi''(r)+\frac{Q-1}{r}\phi'(r)\Big), \end{equation} where $\psi=\bigl|\nabla_L| x |_H\bigr|$. In the Carnot group considered in \cite{DM2008,DM20101,DM20102}, setting \begin{equation} \label{3.17} d(x):=N_p= \begin{cases} \Gamma(x)^\frac{p-1}{p-Q}, & p>1, \; p\not=Q, \\ \exp(-\Gamma(x)), & p=Q, \end{cases} \end{equation} where $\Gamma $ is the fundamental solution of the quasilinear operator \[ L_p u=\operatorname{div}_L(|\nabla_L u|^{p-1}\nabla_L u) \] at the origin, we obtain \eqref{3.16} with $r=N_p$ and $\psi=|\nabla_L N_p|$. \end{remark} \begin{lemma} \label{lem3.3} Let $p>1$ and let $g$ be a continuous and non-decreasing function on $[0,+\infty)$ satisfying $g(t)>0$ for $t>0$. If \begin{equation} \label{3.18} \int_1^{+\infty}\Big( \int_1^t g(s)ds\Big)^{-1/p} dt=\infty, \end{equation} then for any $c>0$ and $\sigma>0$, there exists $R>0$ and a function $\phi$ satisfying \begin{equation} \label{3.19} \left( r^{\sigma}| \phi'(r)|^{p-2} \phi'(r)\right)' =r^{\sigma}g(\phi(r)), \quad \phi(0)=c, \quad \phi'(0)=0, \end{equation} where $\phi$ is increasing on $[0,R]$ and $\phi\to\infty$ as $r\to R$. \end{lemma} \begin{remark} \label{rmk3.3} \rm For the proof of Lemma \ref{lem3.3}, we can see \cite{GR2003,NU19971,O1960}. Osserman \cite{O1960} proved the result in the case $p=2$. On the other hand, Naito and Usami \cite{NU19971} obtained the result when $\sigma=N-1$, and then Ghergu and R\v{a}dulescu \cite{GR2003} given a generalization proof of Lemma \ref{lem3.3}. \end{remark} \begin{proof}[Proof of Theorem \ref{thm1.1}] Let $\sigma=(p-1)(1-\alpha)>0$, and let $\phi$ be a solution of \eqref{3.19} such that $\phi(r)\to+\infty$ as $r\to R$. We set $v(x):=\phi(d(x))$, where $d(x)$ satisfy the condition of Lemma \ref{lem3.2}. By Lemma \ref{lem3.3} then $v$ satisfies \begin{equation} \label{3.20} \begin{split} &\operatorname{div}_L(M|\nabla_L v|^{p-1}\nabla_L v) \\ &=M\bigl| \nabla_L d \bigr| ^p d^{(p-1)(\alpha-1)} \Big(d^{(p-1)(1-\alpha)} \bigl| \phi'(d) \bigr|^{p-2}\phi'(d) \Big)'\\ &=M\bigl| \nabla_L d \bigr| ^p g(v):=g_2(x,v), \end{split} \end{equation} $\phi(d(x))\to+\infty$ as $d(x)\to R$, and $v(0)=c$ in $\Omega_R=\{x: d(x)