\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 50, 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 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/50\hfil Existence of positive radial solutions] {Existence of positive radial solutions for quasilinear elliptic equations and systems} \author[Z. Zhang \hfil EJDE-2016/50\hfilneg] {Zhijun Zhang} \address{Zhijun Zhang \newline School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, China} \email{chinazjzhang2002@163.com, zhangzj@ytu.edu.cn} \thanks{Submitted November 23, 2015. Published February 17, 2016.} \subjclass[2010]{35J55, 35J60, 35J65} \keywords{Quasilinear elliptic equation; radial solutions; existence} \begin{abstract} Under simple conditions on $f$ and $g$, we show that existence of positive radial solutions for the quasilinear elliptic equation \[ \operatorname{div}(\phi_1(|\nabla u|) \nabla u)=a(|x|)f(u) \quad x\in \mathbb{R}^N, \] and for the system \begin{gather*} \operatorname{div}(\phi_1(|\nabla u|) \nabla u)=a(|x|)f(v) \quad x\in \mathbb{R}^N, \\ \operatorname{div}(\phi_2(|\nabla v|) \nabla v) =b(|x|)g(u)\quad x\in \mathbb{R}^N\,. \end{gather*} \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The purpose of this article is to study the existence of positive radial solutions to the quasilinear elliptic equation \begin{equation}\label{e1.1} \Delta_{\phi_1}u:=\operatorname{div} (\phi_1(|\nabla u|) \nabla u)=a(|x|)f(u),\quad x\in \mathbb{R}^N, \end{equation} and for the system \begin{equation}\label{e1.2} \begin{gathered} \operatorname{div}(\phi_1(|\nabla u|) \nabla u)=a(|x|)f(v),\quad x \in \mathbb{R}^N, \\ \operatorname{div}(\phi_2(|\nabla v|) \nabla v)=b(|x|)g(u), \quad x \in \mathbb{R}^N. \end{gathered} \end{equation} In this article by a solution we mean a solution on the entire domain, as opposed to a local solution. To emphasize this property some authors use entire solution, while others use global solution. We assume the following assumptions: \begin{itemize} \item[(A1)] $a, b: \mathbb{R}^N \to [0,\infty)$ are continuous; \item[(A2)] $f, g: [0,\infty)\to [0,\infty)$ are continuous and increasing, \item[(A3)] $\phi_i\in C^1((0, \infty), (0, \infty))$ ($i=1, 2$) satisfy $(t \phi_i(t))'>0$, for all $t>0$; \item[(A4)] there exist $p_i, q_i > 1$ such that $$ p_i \leq \frac {t \Psi_i'(t)}{\Psi_i(t)} \leq q_i, \quad \forall t > 0, $$ where $\Psi_i(t) = \int_0^t s \phi_i(s)ds$, $t>0$; \item[(A5)] there exist $k_i, l_i > 0$ such that $$ k_i \leq \frac {t \Psi_i ''(t)}{\Psi_i'(t)} \leq l_i, \quad \forall t > 0. $$ \end{itemize} The function $\phi_1$ appears in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics, see e.g., Benci, Fortunato and Pisani \cite{BFP}, Cencelj, Repov\v{s} and Virk \cite{CE}, Fuchs and Li \cite{FL}, Fuchs and Osmolovski \cite{FO}, Fukagai and Narukawa \cite{FN1}, R\u{a}dulescu \cite{RA1} and \cite{RA2}, R\u{a}dulescu and Repov\u{s} \cite{RA3}, Repov\u{s} \cite{RE}, Zhang and Yuan \cite{ZY} and Fukagai and Narukawa \cite{FN2}. Positive solutions to \eqref{e1.1} were first considered by Santos, Zhou and Santos \cite{SAN}. Some classical examples of $\phi_1$-Laplacian functions are: \begin{itemize} \item[(1)] when $\phi_1(t)\equiv 2$, $\Psi_1(t)=t^2$, $t>0$, $\Delta_{\phi_1} u =\Delta u$ is the Laplacian operator. In this case, $p_1=q_1=2$ in (A4), and $k_1=l_1=1$ in (A5); \item[(2)] when $\phi_1(t)=pt^{p-2}$, $\Psi_1(t)=t^p$, $t>0$, $p>1$, $\Delta_{\phi_1} u =\Delta_p u$ is the $p$-Laplacian operator. In this case, $p_1=q_1=p$ in (A4), and $k_1=l_1=p-1$ in (A5); \item[(3)] when $\phi_1(t)=pt^{p-2}+qt^{q-2}$, $\Psi_1(t)=t^p+t^q$, $t>0$, $10$, $p>1/2$, $p_1=\min\{2, 2p \}$, $q_1=\max\{2, 2p \}$ in (A4), and $k_1=\min\{1, 2p-1\}$, $l_1=\max\{1, 2p-1 \}$ in (A5); \item[(5)] when $\phi_1(t)= \frac {p(\sqrt{1+t^2}-1)^{p-1}}{\sqrt{1+t^2}}$, $\Psi_1(t)=(\sqrt{1+t^2}-1)^p$, $t>0$, $p>1$, $p_1=p$, $q_1=2p$ in (A4), and $k_1=p-1$, $l_1=2p-1$ in (A5); \item[(6)] when $\phi_1(t)=p t^{p-2} (\ln (1+t))^q +\frac {q t^{p-1} (\ln (1+t))^{q-1}}{1+t}$, $\Psi_1(t)=t^{p} (\ln (1+t))^q$, $t>0$, $p>1$, $q>0$, $p_1=p$, $q_1=p+q$ in (A4), and $k_1=p-1$, $l_1=p+q-1$ in (A5). \end{itemize} We say that $u\in C^1(\mathbb{R}^N)$ is a solution of \eqref{e1.1} if $$ \int_{\mathbb{R}^N} \phi_1(|\nabla u|) \nabla u \nabla \psi dx =-\int_{\mathbb{R}^N}a(x)f(u)\psi dx,\quad \forall \psi\in C_0^\infty(\mathbb{R}^N). $$ When $\lim_{|x|\to \infty}u(x)=+\infty$, we say that $u$ is a large solution to equation \eqref{e1.1}. For convenience, we denote by \begin{gather}\label{e1.3} h_i^{-1} \text{ the inverses of } h_i(t)=t \phi_i(t),\quad t>0; \\ \label{e1.4} I_{i\rho}(\infty):=\lim_{r\to \infty}I_{i\rho}(r),\quad I_{i\rho}(r):=\int_0^r h_i^{-1}(\Lambda_\rho(t))dt,\quad r\geq 0, \end{gather} where $\rho\in C([0, \infty), [0, \infty))$ and \begin{gather}\label{e1.5} \Lambda_\rho(t):=t^{1-N} \int_{0}^{t} s^{N-1}\rho(s) ds, \quad t>0; \\ \label{e1.6} \theta_i(t):= \min\{t^{p_i}, t^{q_i}\}, \quad \Theta_i(t) := \max\{t^{p_i}, t^{q_i}\}, \quad t \geq 0; \\ \label{e1.7} \theta_i^{-1}(t):= \min\{t^{1/{p_i}}, t^{1/{q_i}}\}, \quad \Theta_i^{-1}(t):= \max\{t^{1/{p_i}}, t^{1/{q_i}}\}, \quad t \geq 0; \end{gather} and, for an arbitrary $\alpha>0$ and $t\geq \alpha$, \begin{gather}\label{e1.8} H_{1\alpha}(\infty):=\lim_{t\to \infty}H_{1\alpha}(t),\quad H_{1\alpha}(t):=\int_{\alpha}^t\frac {d\tau}{\Theta_1^{-1}(f(\tau))}; \\ \label{e1.9} H_{2\alpha}(\infty):=\lim_{t\to \infty}H_{2\alpha}(t),\quad H_{2\alpha}(t):=\int_{\alpha}^t\frac {d\tau}{\Theta_1^{-1}(f(\tau)) +\Theta_2^{-1}(g(\tau))}. \end{gather} We see that for $t>\alpha$, \begin{gather*} H_{1\alpha}'(t)=\frac {1}{\Theta_1^{-1}(f(t))}>0,\\ H_{2\alpha}'(t)=\frac {1}{\Theta_1^{-1}(f(t))+\Theta_2^{-1}(g(t))}>0, \end{gather*} and that $H_{1\alpha}, H_{2\alpha}$ have the inverse functions $H_{1\alpha}^{-1}$ and $H_{2\alpha}^{-1}$ on $[0,H_{1\alpha}(\infty))$ and $[0, H_{2\alpha}(\infty))$, respectively. First, let us review the model \begin{equation}\label{e1.10} \Delta u =a(|x|)f(u), \quad x \in \mathbb{R}^N. \end{equation} For $a(x)\equiv 1$ on $\mathbb{R}^N$: when $f$ satisfies (A2), Keller \cite{KE} and Osserman \cite{OS} supplied a necessary and sufficient condition \begin{equation}\label{e1.11} \int_1^\infty \frac{dt}{\sqrt{2F(t)}}=\infty, \quad F(t)=\int_0^t f(s) ds, \end{equation} for the existence of positive radial large solutions to \eqref{e1.10}. For $N\geq3$, $f(u)=u^\gamma$, $\gamma\in (0, 1]$, and $a$ satisfies (A1) with $a(x)=a(|x|)$, Lair and Wood \cite{LW1} first showed that equation \eqref{e1.10} has infinitely many positive radial large solutions if and only if \begin{equation}\label{e1.12} \int_0^\infty r a(r)dr=\infty. \end{equation} The above results have been extended by many authors and in many contexts, see, for instance, \cite{BZ,BR1,BR2,DGG,EMH,LW2,LAIR2,TZ,YANG,YZ} and the references therein. Next we review the system \begin{equation}\label{e1.13} \begin{gathered} \Delta u=a(x)f(v),\quad x \in \mathbb{R}^N, \\ \Delta v=b(x)g(u),\quad x \in \mathbb{R}^N. \end{gathered} \end{equation} When $N\geq3$, $f(v)=v^{\gamma_1}$, $g(u)=u^{\gamma_2}$, $0<\gamma_1\leq \gamma_2$, and $a(x)=a(|x|)$, $b(x)=b(|x|)$, Lair and Wood \cite{LW3} have considered the existence and nonexistence of positive radial solutions to system \eqref{e1.13}. For further results, see for instance, \cite{AM,BS,BEN1,BEN2,CR,GMRZ,LAIR3,LAIR4,LZZ,WW,Z1,Z2} and the references therein. Now we return to equation \eqref{e1.1}. Recently, Santos, Zhou and Santos \cite{SAN} considered the existence of positive radial and nonradial large solutions to equation $$ \operatorname{div}(\phi_1(|\nabla u|) \nabla u) =a(x)f(u), \quad x\in \mathbb{R}^N. $$ A basic result read as follows. \begin{lemma}[{\cite[Corollary 1.2]{SAN}}] \label{Lem1.1} Let {\rm (A3)--(A5)} hold, $f$ satisfy {\rm (A2)}, and $a$ satisfy {\rm (A1)} with $a(x)=a(|x|)$ for $x\in \mathbb{R}^N$. If $$ I_{1a}(\infty)=\infty, $$ then \eqref{e1.1} admits a sequence of symmetric radial large solutions $u_m(|x|) \in C^1(\mathbb{R}^N)$ with $u_m(0)\to \infty$ as $m \to \infty$ if and only if $f$ satisfies $$ \int_1^\infty\frac {dt}{\Psi_1^{-1}(F(t))}=\infty, $$ where $\Psi_1^{-1}$ is the inverse of $\Psi_1$ which is given in {\rm (A4)}. \end{lemma} Inspired by the above works, by using a monotone iterative method and Arzela-Ascoli theorem, we show existence of positive radial solutions to equation \eqref{e1.1} and system \eqref{e1.2} under simple conditions on $f$ and $g$. Our main results for equation \eqref{e1.1} read as follows. \begin{theorem}\label{Thm1.1} Let {\rm (A1)--(A5)} hold. If \begin{itemize} \item[(A6)] $H_{1\alpha}(\infty)=\infty$, \end{itemize} then \eqref{e1.1} has a positive radial solution $u \in C^1(\mathbb{R}^N)$. Moreover, if $I_{1a}(\infty)<\infty$, then $u$ is bounded, and $\lim _{r\to \infty} u(r)=\infty$ provided $I_{1a}(\infty)=\infty$. \end{theorem} \begin{theorem}\label{Thm1.2} Under assumptions {\rm(A1)--(A5)} and \begin{itemize} \item[(A7)] $I_{1a}(\infty)0$ sufficiently small such that (A7) holds provided $I_{1a}(\infty)<\infty$ and $H_{1\alpha}(\infty)<\infty$. \end{remark} \begin{remark}\label{Rmk1.2} \rm For $f(s)=s^{\gamma_1}$ with $s\geq 0$, $\gamma_1>0$, since $\Theta_1^{-1}(t) =\frac{1}{p_1}$, $t\geq 1$, one can see that when $\gamma_1>p_1$, $H_{1\alpha}(\infty)<\infty$, and $H_{1\alpha}(\infty)=\infty$ provided $\gamma_1 \leq p_1$, where $p_1$ is given as in (A4). \end{remark} \begin{remark}\label{Rmk1.3} \rm For $f(s)=(1+s)^{\gamma_1} (\ln (1+s))^{\mu_1}$ with $s\geq 0$, $\mu_1, \gamma_1>0$, one can see that when $\gamma_1>p_1$ or $\gamma_1=p_1$ and $\mu_1>p_1$, $H_{1\alpha}(\infty)<\infty$, and $H_{1\alpha}(\infty)=\infty$ provided $\gamma_1 < p_1$ or $\gamma_1 = p_1$ and $\mu_1 \leq p_1$. \end{remark} \begin{remark}\label{Rmk1.4} \rm For $f(s)=\exp (c_1 s)$, $s\geq 0$, $c_1>0$, one can see that $H_{1\alpha}(\infty)<\infty$. \end{remark} Our main results for system \eqref{e1.2} are as follows. \begin{theorem}\label{Thm1.3} Let {\rm (A1)--(A5)} hold. If \begin{itemize} \item[(A8)] $H_{2\alpha}(\infty)=\infty$, \end{itemize} then \eqref{e1.2} has a positive radial solution $(u,v)$ in $ C^1(\mathbb{R}^N)\times C^1(\mathbb{R}^N)$. Moreover, when $I_{1a}(\infty)+I_{2b}(\infty)<\infty$, $u$ and $v$ are bounded; when $I_a(\infty)=I_b(\infty)=\infty$, $\lim _{r\to \infty} u(r)=\lim _{r\to \infty}v(r)=\infty$. \end{theorem} \begin{theorem}\label{Thm1.4} Under hypotheses {\rm (A1)--(A5)} and \begin{itemize} \item[(A9)] $$ I_{1a}(\infty)+I_{2b}(\infty)0$, when $\gamma_1>p_1$ or $\gamma_2> p_2$, $H_{2\alpha}(\infty)<\infty$, and $H_{2\alpha}(\infty)=\infty$ provided $\gamma_1\leq p_1$ and $\gamma_2\leq p_2$, where $p_1$ and $p_2$ are given as in (A4). \end{remark} \begin{remark}\label{Rmk1.7} \rm For $f(s)=(1+s)^{\gamma_1} (\ln (1+s))^{\mu_1}$, $g(s)=(1+s)^{\gamma_2} (\ln (1+s))^{\mu_2}$, $s\geq 0$, $\gamma_i, \mu_i>0$ ($i=1, 2$), when $\gamma_1>p_1$ or $\gamma_2> p_2$; or $\gamma_1=p_1$ and $\eta_1> p_1$; or $\gamma_2=p_2$ and $\eta_2> p_2$, $H_{2\alpha}(\infty)<\infty$, and $H_{2\alpha}(\infty)=\infty$ provided $\gamma_10$, one can see that $H_{2\alpha}(\infty)<\infty$. \end{remark} \section{Proof of Theorems \ref{Thm1.1} and \ref{Thm1.2}} \begin{lemma}[{\cite[Lemma 2.2]{SAN}}] \label{Lem2.1} Let {\rm (A3)--(A5)} hold, $\theta_i, \Theta_i$ and $\theta_i^{-1}, \Theta_i^{-1}$ ($i=1, 2$) be given as in \eqref{e1.6} and \eqref{e1.7}. We have \begin{itemize} \item[(i)] $\theta_i$, $\Theta_i$, $\theta_i^{-1}$ and $\Theta_i^{-1}$ are strictly increasing on $(0, \infty)$; \item[(ii)] $\theta_i^{-1}(\beta)h_i^{-1} (t)\leq h_i^{-1}(\beta t)\leq \Theta_i^{-1}(\beta)h_i^{-1}(t)$, for all $\beta, t>0$. \end{itemize} \end{lemma} Let us consider the initial value problem \begin{equation}\label{e2.1} \begin{gathered} \big(r^{N-1}\phi_1(u'(r))u'(r)\big)'=r^{N-1}a(r)f(u),\quad r>0, \\ u(0)= \alpha,\quad u'(0)=0, \end{gathered} \end{equation} by a simple calculation, \begin{equation}\label{e2.2} u'(r)=h_1^{-1}\big( r^{1-N} \int_{0}^{r} s^{N-1}a(s)f(u(s)) ds\big),\quad r> 0, \quad u(0)=\alpha, \end{equation} and thus \begin{equation}\label{e2.3} u(r)=\alpha+\int_{0}^{r}h_1^{-1}\big(t^{1-N} \int_{0}^{t} s^{N-1}a(s)f(u(s)) ds\big)dt,\quad r\geq 0. \end{equation} Note that solutions in $C[0, \infty)$ to problem \eqref{e2.3} are solutions in $C^1[0, \infty)$ to problem \eqref{e2.1}. Let $\{u_{m}\}_{m\geq 1}$ be the sequence of positive continuous functions defined on $[0,\infty)$ by \begin{equation}\label{e2.4} \begin{gathered} u_{0}(r)=\alpha, \\ u_m(r)=\alpha+\int_{0}^{r}h_1^{-1}\big(t^{1-N} \int_{0}^{t} s^{N-1}a(s)f(u_{m-1}(s)) ds\big)dt,\quad r\geq 0. \end{gathered} \end{equation} Obviously, \begin{equation}\label{e2.5} u_m'(r)=h_1^{-1}\big(r^{1-N} \int_{0}^{r} s^{N-1}a(s)f(u_{m-1}(s)) ds\big),\quad r>0, \end{equation} and, for all $ r\geq 0$ and $m\in {\mathbb{N}}$, $u_{m}(r)\geq \alpha$, and $u_0\leq u_1$. Then (A1)--(A3) and Lemma \ref{Lem2.1} yield $u_1(r)\leq u_2(r)$ for all $r\geq 0$. Continuing this line of reasoning, we obtain that the sequence $\{u_m\}$ is non-decreasing on $[0, \infty)$. Moreover, we obtain by (A1)--(A3) and Lemma \ref{Lem2.1} that for each $r>0$, \begin{align*} u_m'(r)&=h_1^{-1}\big(r^{1-N} \int_{0}^{r} s^{N-1}a(s)f(u_{m-1}(s)) ds\big)\\ &\leq h_1^{-1}\big(f(u_{m}(r)) r^{1-N} \int_{0}^{r} s^{N-1}a(s) ds\big)\\ &\leq \Theta_1^{-1}(f(u_{m}(r))) h_1^{-1}\big(r^{1-N} \int_{0}^{r} s^{N-1}a(s) ds\big), \end{align*} and \[ \int_{a}^{u_m (r)}\frac {d\tau}{\Theta_1^{-1}(f(\tau))}\leq I_{1a}(r). \] Consequently, for an arbitrary $R>0$, \begin{equation}\label{e2.6} H_{1\alpha}(u_m(r))\leq I_{1a}(r)\leq I_{1a}(R), \quad \forall r\in [0,R]. \end{equation} (i) When (A6) holds, we see that \begin{equation}\label{e2.7} H_{1\alpha}^{-1}(\infty)=\infty,\quad u_m(r)\leq H_{1\alpha}^{-1}(I_{1a}(r))\leq H_{1\alpha}^{-1} (I_{1a}(R)),\quad \forall r\in [0, R], \end{equation} i.e., the sequence $\{u_m\}$ is bounded on $[0, R]$ for an arbitrary $R>0$. It follows from \eqref{e2.5} that $\{u_m'\}$ is bounded on $[0,R]$. By the Arzela-Ascoli theorem, $\{u_m\}$ has a subsequence converging uniformly to $u$ on $[0, R]$. Since $\{u_m\}$ is non-decreasing on $[0, \infty), $ we see that $\{u_m\}$ itself converges uniformly to $u$ on $[0, R]$. By the arbitrariness of $R$, we see that $u$ is a positive radial solution to equation \eqref{e1.1}. Moreover, when $I_{1a}(\infty)<\infty$, we see by \eqref{e2.7} that $$ u(r)\leq H_{1\alpha}^{-1}(I_{1a}(\infty)),\quad \forall r\geq 0; $$ when $I_{1a}(\infty)=\infty,$ we see by (A2) and Lemma \ref{Lem2.1} that $$ u(r)\geq \alpha +\theta_1^{-1}(f(\alpha))I_{1a}(r), \quad \forall r\geq0. $$ Thus $\lim _{r\to \infty} u(r)=\infty$. \smallskip \noindent (ii) When (A7) holds, we see by \eqref{e2.6} that \begin{equation}\label{e2.8} H_{1\alpha}(u_m(r))\leq I_{1a}(\infty)0,\\ \big(r^{N-1}\phi_2(v'(r))v'(r)\big)'=r^{N-1}b(r)g(u), \quad r>0, \\ u(0)=v(0)=\alpha/2,\quad u'(0)=v'(0)=0, \end{gather*} which is equivalent to \begin{gather*} u(r)=\alpha/2+\int_{0}^{r}h_1^{-1}\big(t^{1-N} \int_{0}^{t} s^{N-1}a(s)f(v(s)) ds\big)dt,\quad r\geq 0, \\ v(r)=\alpha/2+\int_{0}^{r}h_2^{-1}\big(t^{1-N} \int_{0}^{t} s^{N-1}b(s)g(u(s)) ds\big)dt,\quad r\geq 0. \end{gather*} Let $\{u_{m}\}_{m\geq 1}$ and $\{v_{m}\}_{m\geq 0}$ be the sequences of positive continuous functions defined on $[0,\infty)$ by \begin{gather*} v_{0}(r)=\alpha/2, \\ u_{m}(r)=\alpha/2+\int_{0}^{r}h_1^{-1}\big(t^{1-N} \int_{0}^{t} s^{N-1}a(s)f(v_{m-1}(s)) ds\big)dt,\quad r\geq 0,\\ v_{m}(r)=\alpha/2+\int_{0}^{r}h_2^{-1}\big(t^{1-N} \int_{0}^{t} s^{N-1}b(s)g(u_{m}(s)) ds\big)dt,\quad r\geq 0. \end{gather*} Obviously, for all $ r\geq 0$ and $m\in {\mathbb{N}}$, $u_{m}(r)\geq \alpha/2$, $v_{m}(r)\geq \alpha/2$ and $v_0\leq v_1$. Assumptions (A1)--(A3) and Lemma \ref{Lem2.1} yield $u_1(r)\leq u_2(r)$, for all $r\geq 0$, then $v_1(r)\leq v_2(r)$, for all $r\geq 0$. Continuing this line of reasoning, we obtain that the sequences $\{u_m\}$ and $\{v_m\}$ are increasing on $[0, \infty)$. Moreover, by (A1)-(A3) and Lemma \ref{Lem2.1} that for each $r>0$, we obtain \begin{align*} u_m'(r)&=h_1^{-1}\big(r^{1-N} \int_{0}^{r} s^{N-1}a(s)f(v_{m-1}(s)) ds\big)\\ &\leq h_1^{-1}\big(f(v_{m}(r))r^{1-N}\int_{0}^{r} s^{N-1}a(s) ds\big)\\ &\leq \Theta_1^{-1}(f(v_m(r)))h_1^{-1}\big(r^{1-N} \int_{0}^{r} s^{N-1}a(s) ds\big)\\ &\leq \Theta_1^{-1}(f(u_m(r)+v_m(r))) (h_1^{-1}(\Lambda_a(r))+h_2^{-1}(\Lambda_b(r))); \end{align*} and \begin{align*} v_m'(r)&=h_2^{-1}\big(r^{1-N} \int_{0}^{r} s^{N-1}b(s)g(u_{m}(s)) ds\big)\\ &\leq \Theta_2^{-1}(g(u_m(r)))h_2^{-1}\big(r^{1-N} \int_{0}^{r} s^{N-1}b(s) ds\big)\\ &\leq \Theta_2^{-1}(g(u_m(r)+v_m(r))) \big(h_1^{-1}(\Lambda_a(r))+h_2^{-1}(\Lambda_b(r))\big). \end{align*} Consequently, \begin{align*} u_m'(r)+v_m'(r) &\leq \big(\Theta_1^{-1}(f(v_m(r)+u_m(r)))\\ &\quad +\Theta_2^{-1}(g(v_m(r)+u_m(r)))\big) \big(h_1^{-1}(\Lambda_a(r))+h_2^{-1}(\Lambda_b(r))\big), \quad r>0, \end{align*} and \begin{equation}\label{e3.1} \begin{gathered} \int_{a}^{u_m(r)+v_m(r)} \frac {d\tau}{\Theta_1^{-1}(f(\tau))+\Theta_2^{-1}(g(\tau))}\leq I_{1a}(r)+I_{2b}(r),\quad r>0, \\ H_{2\alpha}(u_m(r)+v_m(r))\leq I_{1a}(r)+I_{2b}(r), \quad \forall r\geq 0. \end{gathered} \end{equation} The remaining proofs are similar to that for Theorems \ref{Thm1.1} and \ref{Thm1.2}. Here we omit their proof. \subsection*{Acknowledgment} This work is supported in part by NSF of China under grant 11571295. \begin{thebibliography}{99} \bibitem{AM} R. Alsaedi, H. M\^{a}agli, V. R\u{a}dulescu, N. Zeddini; \emph{Positive bounded solutions for semilinear elliptic systems with indefinite weights in the half-space}, Electronic J. Diff. Equations 2015 (2015), No. 177, 1-8. \bibitem{BZ} I. Bachar, N. Zeddini; \emph{On the existence of positive solutions for a class of semilinear elliptic equations}, Nonlinear Anal. 52 (2003), 1239-1247. \bibitem{BS} S. Barile, A. Salvatore; \emph{Existence and multiplicity results for some Lane-Emden elliptic systems: subquadratic case}, Adv. Nonlinear Anal. 4 (2015), 25-35. \bibitem{BR1} N. Belhaj Rhouma, A. Drissi, W. Sayeb; \emph{Nonradial large solutions for a class of nonlinear problems}, Complex Var. Elliptic Equations 59 (5) (2014), 706-722. \bibitem{BR2} N. Belhaj Rhouma, A. Drissi; \emph{Large and entire large solutions for a class of nonlinear problems}, Appl. Math. Comput. 232 (2014), 272-284. \bibitem{BEN1} A. Ben Dkhil, N. Zeddini; \emph{Bounded and large radially symmetric solutions for some $(p, q)$-Laplacian stationary systems}, Electronic J. Diff. Equations 2012 (2012), No. 71, 1-9. \bibitem{BEN2} A. Ben Dkhil; \emph{Positive solutions for nonlinear elliptic systems}, Electronic J. Diff. Equations 2012 (2012), No. 239, 1-10. \bibitem{BFP} V. Benci, D. Fortunato, L. Pisani; \emph{Solitons like solutions of a Lorentz invariant equation in dimension 3}, Rev. Math. Phys. 10 (1998), 315-344. \bibitem{CE} M. Cencelj, D. Repov\v{s}, Z. Virk; \emph{Multiple perturbations of a singular eigenvalue problem}, Nonlinear Anal. 119 (2015), 37-45. \bibitem{CR} F. C\^{\i}rstea, V. R\u adulescu; \emph{Entire solutions blowing up at infinity for semilinear elliptic systems}, J. Math. Pures Appl. 81 (2002), 827-846. \bibitem{DGG} L. Dupaigne, M. Ghergu, O. Goubet, G. Warnault; \emph{Entire large solutions for semilinear elliptic equations}, J. Diff. Equations 253 (2012), 2224-2251. \bibitem{EMH} K. El Mabrouk, W. Hansen; \emph{Nonradial large solutions of sublinear elliptic problems}, J. Math. Anal. Appl. 330 (2007), 1025-1041. \bibitem{FL} M. Fuchs, G. Li; \emph{Variational inequalities for energy functionals with nonstandard growth conditions}, Abstr. Appl. Anal. 3 (1998), 41-64. \bibitem{FO} M. Fuchs, V. Osmolovski; \emph{Variational integrals on Orlicz-Sobolev spaces}, Z. Anal. Anwendungen 17 (1998), 393-415. \bibitem{FN1} N. Fukagai, K. Narukawa; \emph{Nonlinear eigenvalue problem for a model equation of an elastic surface}, Hiroshima Math. J. 25 (1995), 19-41. \bibitem{FN2} N. Fukagai, K. Narukawa; \emph{On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems}, Ann. Mat. Pura Appl. 4 (2007), 539-564. \bibitem{GMRZ} A. Ghanmi, H. M\^{a}agli, V. R\u{a}dulescu, N. Zeddini; \emph{Large and bounded solutions for a class of nonlinear Schr\"{o}dinger stationary systems}, Analysis and Applications 7(4) (2009), 1-14. \bibitem {KE} J. B. Keller; \emph{On solutions of} $\Delta u=f(u),$ Commun. Pure Appl. Math. 10 (1957), 503-510. \bibitem{LW1} A. V. Lair, A. W. Wood; \emph{Large solutions of semilinear elliptic problems}, Nonlinear Anal. 37 (1999), 805-812. \bibitem{LW2} A. V. Lair, A. W. Wood; \emph{Large solutions of sublinear elliptic equations}, Nonlinear Anal. 39 (2000), 745-753. \bibitem{LW3} A. V. Lair, A. W. Wood; \emph{Existence of entire large positive solutions of semilnear elliptic systems}, J. Diff. Equations 164 (2000), 380-394. \bibitem{LAIR1} A. V. Lair; \emph{Nonradial large solutions of sublinear elliptic equations}, Appl. Anal. 82 (2003), 431-437. \bibitem{LAIR2} A. V. Lair; \emph{Large solutions of semilinear elliptic equations under the Keller-Osserman condition}, J. Math. Anal. Appl. 328 (2007), 1247-1254. \bibitem{LAIR3} A. V. Lair; \emph{A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems}, J. Math. Anal. Appl. 365 (2010), 103-108. \bibitem{LAIR4} A. V. Lair; \emph{Entire large solutions to semilinear elliptic systems}, J. Math. Anal. Appl. 382 (2011), 324-333. \bibitem{LZZ} H. Li, P. Zhang, Z. Zhang; \emph{A remark on the existence of entire positive solutions for a class of semilinear elliptic systems}, J. Math. Anal. Appl. 365 (2010), 338-341. \bibitem {OS} R. Osserman; \emph{On the inequality} $\Delta u\geq f(u)$, Pacific J. Math. 7 (1957), 1641-1647. \bibitem {RA1} V. R\u{a}dulescu; \emph{Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications}, vol. 6, Hindawi Publ. Corp., 2008. \bibitem {RA2} V. R\u{a}dulescu; \emph{Nonlinear elliptic equations with variable exponent: old and new}, Nonlinear Anal. 121 (2015), 336-369. \bibitem {RA3} V. R\u{a}dulescu, D. Repov\v{s}; \emph{Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis}, CRC Press, Taylor and Francis Group, Boca Raton FL, 2015. \bibitem {RE} D. Repov\v{s}; \emph{Stationary waves of Schr\"{o}dinger-type equations with variable exponent}, Anal. Appl. 13 (2015), 645-661. \bibitem {SAN} C. A. Santos, J. Zhou, J. A. Santos; \emph{Necessary and sufficient conditions for existence of blow-up solutions for elliptic problems in Orlicz-Sobolev spaces}, arXiv preprint arXiv:1601.01267, 2016. \bibitem {TZ} S. Tao, Z. Zhang; \emph{On the existence of explosive solutions for a class of semilinear elliptic problems}, Nonlinear Anal. 48 (2002), 1043-1050. \bibitem {WW} X. Wang, A. W. Wood; \emph{Existence and nonexistence of entire positive solutions of semilinear elliptic systems}, J. Math. Anal. Appl. 267 (2002), 361-362. \bibitem{YANG} H. Yang; \emph{On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in} $\mathbb{R}^N$, Comm. Pure Appl. Anal. 4 (2005), 197-208. \bibitem{YZ} D. Ye, F. Zhou; \emph{Invariant criteria for existence of bounded positive solutions}, Discrete Contin. Dyn. Syst. 12 (3) (2005), 413-424. \bibitem{Z1} Z. Zhang; \emph{Existence of entire positive solutions for a class of semilinear elliptic systems}, Electronic J. Diff. Equations 2010 (2010), No. 16, 1-5. \bibitem{Z2} Z. Zhang, Y. Shi, Y. Xue; \emph{Existence of entire solutions for semilinear elliptic systems under the Keller-Osserman condition}, Electronic J. Diff. Equations 2011 (2011), No. 39, 1-9. \bibitem{ZY} Zhitao Zhang, R. Yuan; \emph{Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign}, Adv. Nonlinear Anal. 4 (2015), 59-72. \end{thebibliography} \end{document}