\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 71, 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 2012 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2012/71\hfil Bounded and large radially symmetric solutions] {Bounded and large radially symmetric solutions for some $(p,q)$-Laplacian stationary systems} \author[A. B. Dkhil, N. Zeddini \hfil EJDE-2012/71\hfilneg] {Adel Ben Dkhil, Noureddine Zeddini} % in alphabetical order \address{Universit\'e Tunis El Manar, Facult\'e des Sciences de Tunis, D\'epartement de Math\'ema\-tiques, Campus Universitaire, 2092 Tunis, Tunisia} \email[Adel Ben Dkhil]{Adel.Bendekhil@ipein.rnu.tn} \address{King Abdulaziz University, Branch Rabigh, College of Sciences and Arts, Department of Mathematics P.O. Box 344, Rabigh 21911, Kingdom of Saudi Arabia.\newline Universit\'e Tunis El Manar, Facult\'e des Sciences de Tunis, D\'epartement de Math\'emati\-ques, Campus Universitaire, 2092 Tunis, Tunisia} \email[Noureddine Zeddini]{noureddine.zeddini@ipein.rnu.tn} \thanks{Submitted January 27, 2012. Published May 7, 2012.} \subjclass[2000]{34C11, 35B07, 35B09, 35J47, 35J92} \keywords{Radial positive solutions; bounded solutions; large solutions; \hfill\break\indent quasilinear elliptic systems} \begin{abstract} This article concerns radially symmetric positive solutions of sec\-ond-order quasilinear elliptic systems. In terms of the growth of the variable potential functions, we establish conditions such that the solutions are either bounded or blow up at infinity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Existence and nonexistence of solutions of second-order quasilinear elliptic systems of the form \begin{equation} \label{S1} \begin{gathered} \operatorname{div}(|\nabla u|^{p-2}\nabla u)=\varphi(|x|) g_1(v)g_2(u), \quad\text{in } \mathbb{R}^n, \\ \operatorname{div}(|\nabla v|^{q-2}\nabla v)=\psi(|x|) f_1(u)f_2(v), \quad\text{in } \mathbb{R}^n, \end{gathered} \end{equation} have been intensively studied in the previous few years. See, for example, \cite{AS, CA, HSh, GM, OuTa, RHM, TTer, YW, ZYa} and the reference therein. Problem \eqref{S1} arises in the theory of quasiregular and quasiconformal mappings as well as in the study of non- Newtonian fluids. In the latter case, the pair $(p, q)$ is a characteristic of the medium. Media with $(p, q) > (2, 2)$ are called dilatant fluids and those with $(p, q) < (2, 2)$ are called pseudoplastics. If $(p, q) = (2, 2)$, they are Newtonian fluids. When $p = q = 2$ system \eqref{S1} becomes \begin{equation} \label{S2} \begin{gathered} \Delta u=\varphi(|x|) g_1(v)g_2(u), \quad\text{in } \mathbb{R}^n, \\ \Delta v=\psi(|x|) f_1(u)f_2(v), \quad\text{in } \mathbb{R}^n, \end{gathered} \end{equation} for which the existence and non-existence of positive radial entire large or bounded solutions has been extensively studied. When $f_2=g_2=1$, $g_1(v)=v^{\alpha}$, $f_1(u)=u^{\beta}$, $0<\alpha\leq \beta$, Lair and Wood \cite{LW} considered the existence and nonexistence of entire positive radial solutions to \eqref{S2} under the conditions of integrability or nonintegrability of the functions $r\to r\varphi(r)$ and $r \to r\psi(r)$ on $(0,\infty)$. Their results were extended by C\^{i}rstea and R\u{a}dulescu \cite{CR1}, Wang and Wood \cite{WW}, Ghergu and R\^{a}dulescu \cite{GR}, Peng and Song \cite{PS}, Ghanmi, M\^{a}agli, R\^{a}dulescu and Zeddini \cite{GMRZ}, Li, Zhang, Zhang \cite{LZZ} and Zhang \cite{ZZ}. Many generalizations of these results have been extended to system \eqref{S1}. See, for example, \cite{TTer, ZYa}. Our purpose is to generalize the results of \cite{GMRZ, ZZ} to systems \eqref{S1} under the hypotheses that the radial potentials $\varphi$, $\psi$ are nonnegative continuous functions on $(0,\infty)$ and the nonlinearities $f_i,g_i$ ($i=1,2)$ are nonnegative, continuous and nondecreasing on $[0,\infty)$. In all the results, we establish in this paper we study only positive radial solutions in the sense of distributions, especially because of the physical meaning of the corresponding unknowns. To discuss the existence of positive radial solutions to this class of nonlinear systems, we are first concerned with the following two systems of differential equations \begin{equation} \label{S3} \begin{gathered} \frac{1}{A}(A\phi_p(y'))'=\varphi(t) g_1(z) g_2(y), \quad\text{in } (0,\infty), \\ \frac{1}{B}(B\phi_q(z'))'=\psi(t) f_1(y) f_2(z), \quad \text{in } (0,\infty), \\ y(0)=a> 0,\quad z(0)=b>0,\\ \lim_{t\to 0}A(t)\phi_p(y'(t))=\lim_{t\to 0}B(t)\phi_q(z'(t))=0, \end{gathered} \end{equation} and \begin{equation} \label{S4} \begin{gathered} \frac{1}{A}(A\phi_p(y'))'=\varphi(t) g_1(z) g_2(y)\,,\quad\text{in } (0,\infty), \\ \frac{1}{B}(B\phi_q(z'))'=\psi(t) f_1(y) f_2(z) ,, \quad\text{in } (0,\infty), \\ y(\infty)=\lim_{t\to\infty}y(t)=c>0,\quad z(\infty)=\lim_{t\to\infty}z(t)=d>0,\\ \lim_{t\to 0}A(t)\phi_p(y'(t))=\lim_{t\to 0}B(t)\phi_q(z'(t))=0\,, \end{gathered} \end{equation} where $p,q>1$, $\phi_k(x)=|x|^{k-2}x$ for $k=p,q$ and $A,B$ are continuous functions in $[0,\infty)$, differentiable and positive in $(0,\infty)$ and satisfy the following growth hypotheses: $$ \int_0^1\Big[\frac{1}{A(t)}\int_0^tA(s) \,ds\Big]^{1/(p-1)}\,dt<\infty, \quad \int_0^1\Big[\frac{1}{B(t)}\int_0^tB(s)\,ds\Big]^{1/(q-1)}\,dt<\infty \,. $$ In particular, these assumptions are fulfilled if $A$ and $B$ are nondecreasing. In the sequel, we denote by $p'=\frac{p}{p-1}$, $q'=\frac{q}{q-1}$ and we remark that $\phi_k$ is a multiplicative function for $k=p,q$. Namely $\phi_k(xy)=\phi_{k}(x)\phi_{k}(y)$ for $x>0$ and $y>0$. Moreover $\phi_{p'}$ and $\phi_{q'}$ are respectively the inverse functions of $\phi_{p}$ and $\phi_{q}$. For any nonnegative measurable functions $\varphi$ in $(0,\infty)$, we define \begin{gather*} K_p\varphi(t)=\int_0^t\phi_{p'}\Big(\frac{1}{A(r)} \int_0^rA(s)\varphi(s) ds\Big) dr\,,\\ S_q\varphi(t)=\int_0^t\phi_{q'}\left(\frac{1}{B(r)} \int_0^rB(s)\varphi(s) ds\right) dr\,, \\ G_p\varphi(t)=\int_t^{\infty}\phi_{p'} \Big(\frac{1}{A(r)}\int_0^rA(s)\varphi(s) ds\Big) dr\,,\\ H_q\varphi(t)=\int_t^{\infty}\phi_{q'} \Big(\frac{1}{B(r)}\int_0^rB(s)\varphi(s) ds\Big) dr\,. \end{gather*} Finally, we define for $\beta>0$ the function $F_{\beta}$ on $[\beta,\infty)$ by \[ F_{\beta}(t)=\int_{\beta}^{t}\frac{ds}{\phi_{p'}(g_1(s)g_2(s)) +\phi_{q'}(f_1(s)f_2(s))} \] and we note that $F_{\beta}$ has an inverse function $F_{\beta}^{-1}$ on $[\beta, \infty)$. \section{Main results} We are first concerned with the existence of a positive solution of the system \eqref{S3}. For this purpose, we assume that $\varphi,\psi,f_i ,g_i$ ($i=1,2$) satisfy the following hypotheses. \begin{itemize} \item[(H1)] $\varphi , \psi: (0,\infty)\to [0,\infty)$ are continuous functions satisfying \begin{gather*} \int_0^1\Big[\frac{1}{A(t)}\int_0^tA(s)\varphi(s) \,ds\Big]^{1/(p-1)}\,dt<\infty,\\ \int_0^1\Big[\frac{1}{B(t)}\int_0^tB(s)\psi(s) \,ds\Big]^{1/(q-1)}\,dt<\infty . \end{gather*} \item[(H2)] The functions $f_i$, $g_i$: $[0,\infty)\to [0,\infty)$ are nondecreasing continuous, positive on $(0,\infty)$. \end{itemize} Our existence result for \eqref{S3} is the following \begin{theorem}\label{thm1} Under the hypotheses {\rm (H1)--(H2)} and \begin{itemize} \item[(H3)] $K_p\varphi(t) +S_q\psi(t)0$, \end{itemize} System \eqref{S3} has a positive solution $(y,z) \in \left(C([0,\infty))\cap C^1((0,\infty))\right)^2$ satisfying for each $t\in [0,\infty)$ \begin{gather*} a+\phi_{p'}(g_1(b)g_2(a))K_p\varphi(t)\leq y(t) \leq F_{a+b}^{-1}[K_p\varphi(t)+S_q\psi(t)], \\ b+\phi_{q'}(f_1(a)f_2(b))S_q\psi(t)\leq z(t) \leq F_{a+b}^{-1}[K_p\varphi(t)+S_q\psi(t)]. \end{gather*} \end{theorem} As a consequence of this result we obtain the following \begin{corollary} \label{coro1} Under the hypotheses {\rm (H1)--(H3)} and \begin{itemize} \item[(H4)] $K_p\varphi(\infty)<\infty$ and $S_q\psi(\infty)<\infty$, \end{itemize} System \eqref{S3} has a positive bounded solution $(y,z) \in \left(C([0,\infty))\cap C^1((0,\infty))\right)^2$. \end{corollary} \begin{corollary} \label{coro2} Under the hypotheses {\rm (H1)--(H3)} and \begin{itemize} \item[(H5)] $K_p\varphi(\infty)=S_q\psi(\infty)=\infty$, \end{itemize} System \eqref{S3} has a positive solution $(y,z) \in \left(C([0,\infty))\cap C^1((0,\infty))\right)^2$ satisfying $\lim_{t\to \infty}y(t)= \lim_{t \to \infty}z(t)=\infty$. \end{corollary} Next, we investigate the existence of positive solution to \eqref{S4}. \begin{theorem} \label{thm2} Under hypotheses {\rm (H1), (H2), (H4)} and \begin{itemize} \item[(H6)] There exist $c>0$ and $d>0$ such that $$ c-\phi_{p'}(g_1(d) g_2(c)) K_p\varphi(\infty)>0,\quad d-\phi_{q'}(f_1(c) f_2(d)) S_q\psi(\infty)>0\,, $$ \end{itemize} Problem \eqref{S4} has a positive bounded solution $$ (y,z) \in \left(C([0,\infty))\cap C^1((0,\infty))\right) \times\left(C([0,\infty))\cap C^1((0,\infty))\right) $$ satisfying, for each $t\in [0,\infty)$, \begin{gather*} c-\phi_{p'}(g_1(d) g_2(c))G_p\varphi(t)\leq y(t)\leq c,\\ d-\phi_{q'}(f_1(c) f_2(d))H_q\psi(t)\leq z(t)\leq d. \end{gather*} \end{theorem} \begin{remark} \rm Let $g_1(t)=t^{{\alpha}_1}$, $g_2(t)=t^{{\alpha}_2}$, $f_1(t)=t^{\beta_1}$ and $f_2(t)=t^{{\beta}_2}$ with $\alpha_i, \beta_i\geq 0$. Then, the condition (H6) is satisfied for infinitely many positive real numbers $c,d$ if $\alpha_1\beta_1\neq (p-1-\alpha_2)(q-1-\beta_2)$. \end{remark} Now, we give our existence results for \eqref{S1}. \begin{theorem}\label{thm3} Assume that {\rm (H2)} is satisfied and that {\rm (H1)} and {\rm (H3)} are satisfied with $A(t)=B(t)=t^{n-1}$. Then \eqref{S1} has infinitely many positive continuous radial solutions $(u,v)$. Moreover, \begin{itemize} \item If $$ \int_0^\infty\phi_{p'}\Big(\frac{1}{r^{n-1}}\int_0^rs^{n-1}\varphi(s) ds\Big)dr =\int_0^\infty\phi_{q'}\Big(\frac{1}{r^{n-1}}\int_0^rs^{n-1}\psi(s) ds\Big)dr =\infty, $$ then these solutions are large; i.e., $\lim_{x\to \infty}u(x)=\lim_{x\to \infty}v(x)=\infty$. \item If $$ \int_0^\infty\phi_{p'}\Big(\frac{1}{r^{n-1}}\int_0^rs^{n-1}\varphi(s) ds\Big)dr <\infty $$ and $$ \int_0^\infty\phi_{q'}\left(\frac{1}{r^{n-1}}\int_0^rs^{n-1}\psi(s) ds\right)dr <\infty, $$ then $u$ and $v$ are bounded. \end{itemize} \end{theorem} Next, we replace hypothesis (H3) by hypothesis (H6) to obtain the existence of positive continuous bounded radial solutions to \eqref{S1}. \begin{theorem}\label{thm4} Let $f_i$, $g_i$, satisfying {\rm (H2)} and assume that {\rm (H1), (H4), (H6)} are satisfied with $A(t)=B(t)=t^{n-1}$. Then \eqref{S1} has a positive radial bounded solution $(u,v)$ with $$ \lim_{|x|\to\infty}u(x)={\rm const}>0,\qquad\lim_{|x|\to\infty}v(x)={\rm const}>0. $$ \end{theorem} \section{Proof of main results} \subsection*{Proof of Theorem \ref{thm1}} Let $(y_k)_{k\geq 0}$ and $(z_k)_{k\geq 0}$ be sequences of positive continuous functions defined on $[0,\infty)$ by \begin{gather*} y_0(t)=a,\quad z_0(t)=b, \\ y_{k+1}(t)=a+\int_0^t\phi_{p'}\Big(\frac{1}{A(r)}\int_0^rA(s) \varphi(s)g_1(z_k(s))g_2(y_k(s)) ds\Big) dr \\ z_{k+1}(t)=b+\int_0^t\phi_{q'}\Big(\frac{1}{B(r)}\int_0^rB(s) \psi(s)f_1(y_k(s))f_2(z_k(s)) ds\Big) dr. \end{gather*} Clearly $y_k, z_k\in C([0,\infty))\cap C^1((0,\infty))$ and positive, so we deduce from the monotonicity of $f_i$, $g_i$, $\phi_{p'}$ and $\phi_{q'}$ that $(y_k)_{k\geq 0}$ and $(z_k)_{k\geq 0}$ are nondecreasing sequences and for each $k\in \mathbb{N}$, the functions $t\to y_k(t)$ and $t\to z_k(t)$ are nondecreasing. Hence, for each $t\in (0,\infty)$, \begin{align*} &y_{k+1}'(t)\\ &= \phi_{p'}\Big(\frac{1}{A(t)}\int_0^tA(s)\varphi(s)g_1(z_k(s))g_2(y_k(s))ds\Big) \\ &\leq \phi_{p'}(g_1(z_k(t))g_2(y_k(t)))\phi_{p'} \Big(\frac{1}{A(t)}\int_0^tA(s)\varphi(s)ds\Big)\\ &\leq \phi_{p'}(g_1(z_{k+1}(t)+y_{k+1}(t))g_2(y_{k+1}(t)+z_{k+1}(t)))\phi_{p'} \Big(\frac{1}{A(t)} \int_0^tA(s)\varphi(s)ds\Big)\\ &\leq [\phi_{p'}((g_1(z_{k+1}(t)+y_{k+1}(t))g_2(y_{k+1}(t)+z_{k+1}(t)))\\ &\quad + \phi_{q'}((f_1(z_{k+1}(t)+y_{k+1}(t))f_2(y_{k+1}(t)+z_{k+1}(t)))] \phi_{p'}\Big(\frac{1}{A(t)} \int_0^tA(s)\varphi(s)ds\Big) \end{align*} Which implies, by putting $w_k=y_k+z_k$, that \begin{align*} &\frac{y_{k+1}'(t)}{\phi_{p'}((g_1(w_{k+1}(t))g_2(w_{k+1}(t))) +\phi_{q'}((f_1(w_{k+1}(t))f_2(w_{k+1}(s)))}\\ &\leq\phi_{p'}\Big(\frac{1}{A(t)}\int_0^tA(s)\varphi(s)ds\Big) , \end{align*} Similarly, we have \begin{align*} &\frac{z_{k+1}'(t)}{\phi_{p'}((g_1(w_{k+1}(t))g_2(w_{k+1}(t))) +\phi_{q'}((f_1(w_{k+1}(t))f_2(w_{k+1}(t)))}\\ &\leq\phi_{q'}\Big(\frac{1}{B(t)}\int_0^tB(s)\psi(s)ds\Big) \end{align*} Consequently, \[ \int_0^t\frac{w_{k}'(s) ds}{\phi_{p'}(g_1(w_{k}(t))g_2(w_{k}(s)))+ \phi_{q'}((f_1(w_{k}(s))f_2(w_{k}(s)))}\leq K_p\varphi(t)+S_q\psi(t), \] which gives \[ \int_{a+b}^{w_k(t)}\frac{ds}{\phi_{q'}(f_1(s)f_2(s)) +\phi_{p'}(g_1(s)g_2(s))}\leq K_p\varphi(t)+S_q\psi(t). \] Namely $$ F_{a+b}(y_k(t)+z_k(t))\leq K_p\varphi(t)+S_q\psi(t). $$ Which by hypothesis (H3) implies $$ y_k(t)+z_k(t)\leq F_{a+b}^{-1}(K_p\varphi(t)+S_q\psi(t)). $$ Therefore, the sequences $(y_k)_{k\geq 0}$ and $(z_k)_{k\geq 0}$ converge locally uniformly to two functions $y$ and $z$ that satisfy for each $t\in [0,\infty)$, \begin{gather*} y(t)=a+\int_0^t\phi_{p'} \Big(\frac{1}{A(r)}\int_0^rA(s)\varphi(s)g_1(z(s))g_2(y(s)) ds\Big) dr,\\ z(t)=b+\int_0^t\phi_{q'}\Big(\frac{1}{B(r)}\int_0^rB(s)\psi(s)f_1(y(s))f_2(z(s)) ds \Big) dr \end{gather*} Hence, $y,z \in C([0,\infty))\cap C^1((0;\infty))$ and $(y,z)$ is a solution of \eqref{S3} satisfying \begin{gather*} a+\phi_{p'}(g_1(b)g_2(a))K_p\varphi(t)\leq y(t)\leq F_{a+b}^{-1}(K_p\varphi(t)+S_q\psi(t)), \\ b+\phi_{q'}(f_1(a)f_2(b))S_q\psi(t)\leq z(t)\leq F_{a+b}^{-1}(K_p\varphi(t)+S_q\psi(t)). \end{gather*} To state another corollary of Theorem \ref{thm1}, we consider two continuous functions $h,k:[0,\infty)\to [0,\infty)$ and study the existence of positive solutions for the system \begin{equation} \label{S6} \begin{gathered} \frac{1}{A}(A\phi_p(y'))'+h(y)|y'|^p=\varphi(t) g_1(z) g_2(y), \quad\text{in } (0,\infty), \\ \frac{1}{B}(B\phi_q(z'))'+k(z)|z'|^q=\psi(t) f_1(y) f_2(z), \quad \text{in } (0,\infty), \\ y(0)=a> 0,\quad z(0)=b>0,\\ \lim_{t\to 0}A(t)\phi_p(y'(t))=\lim_{t\to 0}B(t)\phi_q(z'(t))=0. \end{gathered} \end{equation} To this aim, we define \[ \rho_1(t)=\int_0^t\exp \Big(\frac{1}{p-1}\int_0^\zeta h(s) ds\Big)d\zeta, \quad \rho_2(t)=\int_0^t\exp \Big(\frac{1}{q-1}\int_0^\zeta k(s) ds\Big)d\zeta. \] Clearly $\rho_1$, $\rho_2$ are bijections from $[0,\infty)$ to itself. Let $M_1$, $M_2$, $N_1$ and $N_2$ be the functions defined on $[0,\infty)$ by $M_1 \circ \rho_2=g_1$, $M_2\circ \rho_1={{({\rho_1}')^{p-1}}}g_2$, $N_1 \circ \rho_1=f_1$ and $N_2\circ \rho_2={{({\rho_2}')^{q-1}}}f_2$. \begin{corollary} \label{coro3} Under the hypotheses {\rm (H1), (H2)} and \begin{itemize} \item[(H3')] for all $t>0$, \[ K_p\varphi(t)+S_q\psi(t) <\int_{_{\rho_1(a)+\rho_2(b)}}^{\infty} \frac{dt}{\phi_{p'}(M_1(t)M_2(t))+\phi_{q'}(N_1(t)N_2(t))}, \] \end{itemize} System \eqref{S6} has a positive solution $(y,z) \in \big(C([0,\infty))\cap C^1((0,\infty))\big) \times\big(C([0,\infty))\cap C^1((0,\infty))\big)$. Moreover, when $K_p\varphi(\infty)<\infty$ and $S_q\psi(\infty)<\infty$, $y$ and $z$ are bounded; when $K_p\varphi(\infty)=S_q\psi(\infty)=\infty$, $\lim_{t\to \infty}y(t)= \lim_{t \to \infty}z(t)=\infty$. \end{corollary} \begin{proof} Put $Y=\rho_1(y)$ and $Z=\rho_2(z)$. Then $(y,z)$ is a solution of \eqref{S6} if and only if $(Y,Z)$ is a solution of \begin{gather*} \frac{1}{A}(A\phi_p(Y'))'=\varphi\,M_1(Z) M_2(Y), \quad\text{in } (0,\infty), \\ \frac{1}{B}(B\phi_q(Z'))'=\psi\,N_1(Y) N_2(Z), \quad\text{in } (0,\infty), \\ Y(0)=\rho_1(a)> 0,\quad Z(0)=\rho_2(b)>0,\\ \lim_{t\to 0}A(t)\phi_p(Y'(t))=\lim_{t\to 0}B(t)\phi_q(Z'(t))=0, \end{gather*} So the result follows from Theorem \ref{thm1}. \end{proof} Next, we aim to prove Theorem \ref{thm2}. We note that the proof established in \cite{GMRZ} for the case $p=q=2$ and $g_2=f_2=1$ can not be adapted. So we will use a fixed point argument. \begin{proof}[Proof of Theorem \ref{thm2}] Let $C_0([0,\infty))=\{ \omega \in C([0,\infty), {\mathbb{R}}): \lim_{t \to \infty}|\omega(t)|=0\}$. Clearly $C_0([0,\infty))$ is a Banach space endowed with the uniform norm ${\|\omega\|}_{\infty}=\sup_{t \in [0,\infty)}|\omega (t)|$. To apply the Schauder fixed point theorem, we put $c_1=\phi_{p'}(g_1(d)g_2(c))K_p\varphi(\infty)$, $d_1=\phi_{q'}(f_1(c)f_2(d))S_q\psi(\infty)$ and we consider the nonempty closed convex set \[ \Lambda=\{(\omega,\,\theta) \in (C_0([0,\,\infty)))^2: -c_1\leq \omega \leq 0 \mbox{ and }-d_1\leq \theta \leq 0 \}. \] Consider the operator $T$ defined on $\Lambda$ by $T(\omega,\theta)= (\widetilde{\omega},\widetilde{\theta})$, where \begin{align*} \widetilde{\omega}(t) &=-G_p(\varphi\,g_1(\theta+d) g_2(\omega+c))(t)\\ &=-\int_t^{\infty}\phi_{p'}\Big(\frac{1}{A(r)}\int_0^rA(s)\varphi(s) g_1(\theta(s)+d)g_2(\omega(s)+c) ds\Big)dr \\ \widetilde{\theta}(t) &=-H_q(\psi\,f_1(\omega+c) f_2(\theta+d))(t)\\ &= -\int_t^{\infty}\phi_{q'}\left(\frac{1}{B(r)}\int_0^rB(s)\psi(s) f_1(\omega(s)+c)f_2(\theta(s)+d) ds\right)dr. \end{align*} First, we show that $T\Lambda \subset \Lambda$. Let $(\omega,\theta) \in \Lambda$, then using hypotheses (H1), (H2) and (H4) we deduce that $(\widetilde{\omega},\widetilde{\theta}) \in C([0,\,\infty))$. Moreover, since $\lim_{t\to \infty }G_p\varphi(t)=\lim_{t\to \infty }G_q\psi(t)=0$, it follows that $\lim_{t \to \infty}|\widetilde{\omega}(t)|=\lim_{t \to \infty} |\widetilde{\theta}(t)|=0$. Which implies that $\widetilde{\omega},\widetilde{\theta} \in C_0([0,\,\infty))$. Using again the monotonicity of $f_i,g_i$ we deduce that $(\widetilde{\omega},\widetilde{\theta}) \in \Lambda$ and consequently $T\Lambda \subset \Lambda$. Secondly, we will prove that $T\Lambda$ is relatively compact in $(C_0([0,\infty)))^2$. Clearly $T\Lambda$ is uniformly bounded in $(C_0([0,\infty)))^2$. Let us prove that $T\Lambda$ is equicontinuous on $[0,\infty)$ and satisfy the property $\lim_{t\to \infty}\,\sup_{(\omega,\,\theta)\in \Lambda } \,|\widetilde{\omega}(t)|+|\widetilde{\theta}(t)|=0$ known as equidecay property to $0$ at infinity. Let $t_1,t_2\in [0,\infty]$ with $t_10, \\ \frac{1}{t^{n-1}}(t^{n-1}\phi_p(z'))'=\psi(t) f_1(y)f_2(z), \quad t>0, \\ y'(0)=0,\quad z'(0)=0. \end{gathered} \end{equation} Hence the result follows from Theorem \ref{thm1} with $A(t)=B(t)=t^{n-1}$. Since infinitely many positive real numbers $a,b$ can be chosen in \eqref{S3}, then we can construct an infinitude of positive radial solutions to \eqref{S1}. This completes the proof. \end{proof} Next, we consider some continuous functions $\lambda,\mu: [0,\infty)\to[0,\infty)$ and $\varphi,\psi:(0,\infty) \to [0,\infty)$ satisfying: \begin{itemize} \item[(H7)] \begin{gather*} \int_0^1\phi_{p'}\Big(r^{1-n} \exp\Big(-\int_0^r\lambda(\zeta) \,d\zeta\Big) \int_0^rs^{n-1}\exp\Big(\int_0^s\lambda(\zeta) d\zeta\Big)\varphi(s) ds\Big) dr<\infty, \\ \int_0^1\phi_{q'}\Big(r^{1-n} \exp\Big(-\int_0^r\mu(\zeta) d\zeta\Big) \int_0^rs^{n-1}\exp\Big(\int_0^s\mu(\zeta) d\zeta\Big)\psi(s) ds\Big) dr<\infty. \end{gather*} \end{itemize} and we define \begin{gather*} K_p^{\lambda} \varphi(t)=\int_0^t\phi_{p'} \Big(\frac{1}{\exp\big(\int_0^r\lambda(s) ds\big)r^{n-1}} \int_0^r\exp\Big(\int_0^s\lambda(\varsigma) d\varsigma\Big)s^{n-1}\varphi(s) ds \Big)dr, \\ S_q^{\mu} \psi(t)=\int_0^t\phi_{q'} \Big(\frac{1}{\exp\Big(\int_0^r\mu(s) ds\Big)r^{n-1}} \int_0^r\exp\Big(\int_0^s\mu(\varsigma) d\varsigma\Big)s^{n-1}\psi(s) ds\Big)dr. \end{gather*} \begin{corollary}\label{coro4} Let $f_i,g_i$ satisfying {\rm (H2)} and let $\lambda,\mu: [0,\infty)\to[0,\infty)$ and $\varphi,\psi:(0,\infty) \to [0,\infty)$ be continuous functions satisfying {\rm (H7)}. Assume further that \begin{itemize} \item[(H8)] there exist $a,b>0$ such that $K_p^{\lambda} \varphi(t)+ S_q^{\mu} \psi(t)0$, \end{itemize} then the problem \begin{equation} \label{S8} \begin{gathered} \operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda(|x|)|\nabla u|^{p-1}=\varphi(|x|) g_1(v)g_2(u), \quad\text{in } \mathbb{R}^n, \\ \operatorname{div}(|\nabla v|^{q-2}\nabla v)+\mu(|x|)|\nabla v|^{q-1}=\psi(|x|) f_1(u)f_2(v), \quad\text{in } \mathbb{R}^n, \end{gathered} \end{equation} has infinitely many positive radial solutions $(u,v)$. Moreover, \begin{itemize} \item[(i)] If $K_p^{\lambda} \varphi(t)<\infty=S_q^{\mu} \psi(t)=\infty$, then these solutions are large. \item[(ii)] If $K_p^{\lambda} \varphi(t)<\infty$ and $S_q^{\mu} \psi(t)0, \\ \frac{1}{t^{n-1}}(t^{n-1}\phi_q(z'))'+\mu(t)\phi_q(z')=\psi(t) f_1(y)f_2(z), \quad t>0,\\ y'(0)=0,\quad z'(0)=0, \end{gathered} \end{equation} has infinitely many positive solutions $(y,z) \in (C([0,\infty))\times C^1((0,\infty)))^2$. Put $u(x)=y(t)$, $v(x)=z(t)$, with $t=|x|$. Then $(u,v)$ are positive solutions of \eqref{S8}. \end{proof} \begin{thebibliography}{00} \bibitem{AS} J. Ali, R. Shivaji; \emph{Positive solutions for a class of p-Laplacian systems with multiple parameters}, J. Math. Anal. Appl. 335 (2007), 1013-1019. \bibitem{CA} C. Azizieh, P. Cl\'{e}ment, E. Mitidieri; \emph{Existence and a priori estimates for positive solutions of p-Laplace systems}, Journal of Differential Equations 184 (2002), 422-442. \bibitem{BOM1} I. Bachar, S. Ben Othman, H. M\'{a}agli; \emph{ Radial solutions for the p-Laplacian equation}, Nonlinear Analysis 70 (2009), 2198-2205. \bibitem{BP} J. F. Bonder, J. P. Pinasco; \emph{Estimates for eigenvalues of quasilinear elliptic systems. Part II}, J. Differential Equations 245 (2008), 875-891. \bibitem{CR1} F. Cirstea, V. Radulescu; \emph{Entire solutions blowing up at infinity for semilinear elliptic systems}, J. Math. Pures Appl. 81 (2002), 827-846. \bibitem{GMRZ} A. Ghanmi, H. M\^{a}agli, V. Radulescu, N. Zeddini; \emph{Large and bounded solutions for a class of nonlinear Schr\"{o}dinger stationary systems}, Analysis and Applications, Vol. 7, No. 4 (2009), 391-404. \bibitem{GR} M. Ghergu, V. R\u adulescu; \emph{Explosive solutions of semilinear elliptic systems with gradient term}, RACSAM Revista Real Academia de Ciencias (Serie A, Matem\'aticas) 97 (2003), 467-475. \bibitem{GGS} M.H. Giga, Y. Giga, J. Saal; \emph{Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions}, Progress in Nonlinear Differential Equations and Their Applications, Birkh\"{a}user, Vol 79 (2010). \bibitem{HSh} D.D. Hai, R. Shivaji; \emph{An existence result on positive solutions for a class of p-Laplacian systems}, Nonlinear Analysis 56 (2004), 1007-1010. \bibitem{L1} A. V. Lair; \emph{A necessary and sufficient condition for exitence of large solutions to sublinear elliptic systems}, J. Math. Anal. Appl 365 (2010), 103-108. \bibitem{LW} A. V. Lair, A. W. Wood; \emph{Existence of entire large positive solutions of semilinear elliptic systems}, J. Differential Equations 164 (2000), 380-394. \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{GM} J. Garcia Meli\'{a}n; \emph{Large solutions for an elliptic system of quasilinear equations} J. Diff. Eqns. 245 (2008), 3735-3752. \bibitem{OuTa} Zeng-Qi Ou, Chun-Lei Tang; \emph{Resonance problems for the p-Laplacian systems}, J. Math. Anal. Appl. 345 (2008), 511-521. \bibitem{PS} Y. Peng, Y. Song; \emph{Existence of entire large positive solutions of a semilinear elliptic system}, Appl. Math. Comput. 155 (2004), 687-698. \bibitem{RHM} S.H. Rasouli, Z. Halimi, Z. Mashhadban; \emph{A remark on the existence of positive weak solution for a class of (p,q)-Laplacian nonlinear system with sign-changing weight}, Nonlinear Analysis, 73 (2010), 385-389. \bibitem{TTer} T. Teramoto; \emph{On positive radial entire solutions of second order quasilinear elliptic systems}, J. Math. Anal. Appl, 282 (2003), 531-552. \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), 361-368. \bibitem{YW} Guoying Yang, Mingxin Wang; \emph{Existence of multiple positive solutions for a p-Laplacian system with sign-changing weight functions}, Computers and Mathematics with Applications 55 (2008), 636-653. \bibitem{ZYa} Z. Yang; \emph{Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems}, J. Math. Anal. Appl 288 (2003), 768-783. \bibitem{ZZ} Z. Zhang; \emph{Existence of entire positive solutions for a class of semilinear elliptic systems}, Electron. Journal of Diffrential Equation, Vol. 2010 (2010), No. 16. pp. 1-5. \end{thebibliography} \end{document}