\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 54, 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 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/54\hfil Existence of radial positive solutions] {Existence of radial positive solutions vanishing at infinity for asymptotically homogeneous systems} \author[A. Djellit, M. Moussaoui, S. Tas\hfil EJDE-2010/54\hfilneg] {Ali Djellit, Mohand Moussaoui, Saadia Tas} % in alphabetical order \address{Ali Djellit \newline University of Annaba, Faculty of Sciences, Department of Mathematics, BP 12, 23000 Annaba, Algeria} \email{a\_djellit@hotmail.com} \address{Mohand Moussaoui \newline Ecole Centrale de Lyon, Department of Mathematics, 36 Rue de Collongue, 69130 Ecully, France} \email{mohand.moussaoui@ec-lyon.fr} \address{Saadia Tas \newline University of Bejaia, Faculty of Sciences and Engineering Sciences, Department of Mathematics, 06000 Bejaia, Algeria} \email{tas\_saadia@yahoo.fr} \thanks{Submitted November 10, 2009. Published April 19, 2010.} \subjclass[2000]{35P65, 35P30} \keywords{$p$-Laplacian operator; nonvariational system; blow up method; \hfill\break\indent Leray-Schauder topological degree} \begin{abstract} In this article we study elliptic systems called asymptotically homogeneous because their nonlinearities may not have polynomial growth. Using the Gidas-Spruck Blow-up method, we obtain a priori estimates, and then using Leray-Schauder topological degree theory, we obtain radial positive solutions vanishing at infinity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} We study asymptotically homogeneous systems involving nonlinearities which may not have polynomial growth. More precisely, we establish the existence of radial positive solutions vanishing at infinity, the so-called fundamental states, for systems of the form \begin{equation} \label{SNV} \begin{gathered} -\Delta _{p}u=a_{11}(|x|) f_{11}(u) +a_{12}(|x|) f_{12}(v) \quad\text{in }\mathbb{R}^{N} \\ -\Delta _{q}v=a_{21}(|x|) f_{21}(u) +a_{22}(|x|) f_{22}(v) \quad\text{in }\mathbb{R}^{N} \end{gathered} \end{equation} Here $10$ if for all $\sigma >0$, we have $\lim_{s\to +\infty} \frac{ \varphi (\sigma s) }{\varphi (s) }=\sigma ^{\rho }$ (respect. $\lim_{s\to 0} \frac{\varphi (\sigma s) }{\varphi (s) }=\sigma ^{\rho }$). As an example, we have the function $\varphi (s) =|s| ^{\alpha -2}s(\ln (1+|s|)) ^{\beta }$ with $\alpha >1$ and $\beta >1-\alpha $. It is asymptotically homogeneous at infinity of order $\alpha -1$ and at the origin of order $\alpha +\beta -1$. \begin{proposition}[\cite{GGM}] \label{prop1} Let $\varphi :\mathbb{R}\to \mathbb{R}$ be a continuous, odd, asymptotically homogeneous at infinity (respect. at the origin) of order $\rho $ such that $t\varphi (t) >0$ for all $t\neq 0$ and $\varphi (t)\to infty$ as $t\to \infty$, then \begin{itemize} \item[(i)] For all $\varepsilon $ $\in ] 0,\rho [$, there exists $t_{0}>0$ such that $\forall t\geq t_{0}$ (respect. $0\leq t\leq t_{0}$), $c_1t^{\rho -\varepsilon }\leq \varphi (t) \leq c_{2}t^{\rho +\varepsilon }$; $c_1,c_{2}$ are positive constants. Moreover $\forall s\in [ t_{0},t] : (\rho +1-\varepsilon) \varphi (s) \leq (\rho +1+\varepsilon) \varphi (t) $. \item[(ii)] If $(w_n) ,(t_n) $ are real sequences such that $w_n\to w$ and $t_n\to +\infty $ (respect. $t_n\to 0$) then $\lim_{n\to +\infty } \frac{\varphi (t_nw_n) }{\varphi (t_n) } =w^{\rho }$. \end{itemize} \end{proposition} We assume that both the coefficients $a_{ij}$ and the functions $f_{ij}$ verify smooth conditions; explicitly: \begin{itemize} \item[(H1)] For all $i,j=1,2$, the coefficient $a_{ij}:[ 0,+\infty [ \to ] 0,+\infty [ $ is continuous and satisfies $\exists \theta _{11},\theta _{12}>p$; $\exists \theta _{21},\theta _{22}>q$; there exists $R>0$ such that $a_{ij}(\xi) =O(\xi ^{-\theta _{ij}})$ for all $\xi >R$ and $\tilde{a}_{i}=\min_{r\in [ 0,R] } a_{ij}(r) >0$; $i,j=1,2;i\neq j$. \item[(H2)] For all $i,j=1,2$, the function $f_{ij}: \mathbb{R}\to \mathbb{R}$ is continuous, odd such that $sf_{ij}(s) >0$ for all $s\neq 0$ and $\lim_{s\to +\infty} f_{ij}(s) =+\infty $. \item[(H3)] For all $i,j=1,2$, $f_{ij}$ is asymptotically homogeneous at the infinity of order $\delta _{ij}$ satisfying $\frac{\delta _{12}\delta _{21}}{(p-1) (q-1) }>1$, $\alpha _1\delta _{11}-\alpha _1(p-1)-p<0$, $\alpha_2\delta _{22}-\alpha_2(q-1) -q<0$ and $\max (\beta _1,\beta_2) \geq 0$ where $\alpha _1=\frac{p(q-1) +\delta _{12}q}{\delta _{12}\delta _{21}-( p-1) (q-1) },\alpha_2=\frac{q(p-1) +\delta _{21}p}{\delta _{12}\delta _{21}-(p-1) (q-1)}$, $\beta _1=\,\alpha _1-\frac{N-p}{p-1}$, $\beta_2=\alpha_{2}-\frac{N-q}{q-1}$. \item[(H4)] For all $i,j=1,2$, $f_{ij}$ is asymptotically homogeneous at the origin of order $\bar{\delta }_{ij} $ with $\bar{\delta }_{11}$, $\bar{\delta }_{12}>p-1$, $\bar{\delta }_{21}$, $\bar{\delta }_{22}>q-1$. \end{itemize} To show the existence result, it is necessary to state some lemmas. \begin{lemma} \label{lem1} Let $u\in C^{1}([ 0,+\infty [) \cap C^{2}(] 0,+\infty [) $ be a positive solution of the problem $$ -(r^{N-1}|u'(r) |^{p-2}u'(r)) '\geq 0 $$ in $[ 0,+\infty [ $ such that $u(0) >0$ and $u'(0) \leq 0$, then \begin{itemize} \item[(i)] $u(r) >0$ and $u'(r) \leq 0$ for all $r\geq 0$. Moreover, if $u'(s) =0$ for all $s>0$ then $u'(r) =0$ for all $r\in [0,s]$. \item[(ii)] The function $M_{p}$ defined by $M_{p}(r) =ru'(r)+\frac{N-p}{p-1}u(r)$, $r\geq 0$, is nonnegative and nonincreasing. In particular, the function $r\mapsto r^{\frac{N-p}{p-1}}u(r)$ is nondecreasing in $[0,+\infty [$. \end{itemize} \end{lemma} \begin{proof} To show (i), let us consider a nontrivial positive solution $u$ of problem $$ -(r^{N-1}|u'(r) |^{p-2}u'(r)) '\geq 0\quad\text{in }[ 0,+\infty [ . $$ Integrating from $s$ to $r$, we obtain $r^{N-1}|u'(r) | ^{p-2}u'(r) \leq s^{N-1}|u'(s) |^{p-2}u'(s) $, for $0r_{0}$ and $u'(r) =0$, $u(r) =u(0)$ for $0\leq r\leq r_{0}$. So $u$ is non increasing and $u(0)>0$. Let us show (ii). Since $u$ is a positive solution of the problem $$ -(r^{N-1}|u'(r) | ^{p-2}u'(r)) '\geq 0\quad\text{in }[ 0,+\infty [ , $$ we have $-r^{N-1}(p-1) |u'(r)| ^{p-2}u''(r) -(N-1) r^{N-2}|u'(r) | ^{p-2}u'(r) \geq 0$. In other words $ru''(r) +\frac{N-1}{p-1}u'(r)\leq 0$, or $(ru'(r)) '+\frac{N-p}{p-1} u'(r)\leq 0$. This yields that $M_{p}$ is non increasing. To show that $M_{p}(r) \geq 0$ for all $r\geq 0$, we use a contradiction argument. Indeed, assume that there exists $r_1>0$ such that $M_{p}(r_1) <0$. Since $M_{p}$ is non increasing, for all $r>r_1$, $M_{p}(r) \leq M_{p}(r_1)$ or $u'(r)+\frac{N-p}{p-1}\frac{u(r)}{r}\leq \frac{ M_{p}(r_1) }{r}$. On the other hand $u(r) >0$, $\frac{N-p}{p-1}>0$, hence $u'(r)\leq \frac{M_{p}(r_1) }{r}$. Consequently $u(r) -u(r_1) \leq M_{p}(r_1) \ln \frac{r}{r_1}$, $r>r_1$. It follows immediately that $\lim_{r\to +\infty }u(r)=-\infty $. This contradicts $u$ begin positive. In particular $$ \frac{M_{p}(r) }{ru(r) }\geq 0\quad \forall r>0. $$ Finally, we obtain $\frac{u'(r)}{u(r)}+\frac{N-p}{p-1}\frac{1}{r}\geq 0$. In other words, $$ (\ln r^{\frac{N-p}{p-1}}u(r)) '\geq 0. $$ This implies that the function $r\mapsto r^{\frac{N-p}{p-1}}u(r)$ is non decreasing. \end{proof} The study of the function $M_{p}$ is essential and help us to estimate $u(r)$. \begin{lemma} \label{lem2} If {\rm (H1)} is satisfied, then the operators $T_{h}$ and $S_{\lambda }$ are compact. \end{lemma} The proof of the above lemma follows the same argument as in \cite[Lemma 6]{DT}, and is omitted. We remark that the ground states of \eqref{SNV} are precisely the fixed points of the operator $T_{0}$. Now, we show a nonexistence result related to a ``limit'' system. \begin{theorem} \label{thm1} Under hypotheses {\rm (H1)-(H3)}, the system \begin{equation} \label{SL} \begin{gathered} -\Delta _{p}u=a_{12}(|x|) |v| ^{\delta_{12}-1}v\quad \text{in }\mathbb{R}^{N} \\ -\Delta _{q}v=a_{21}(|x| ) |u| ^{\delta _{21}-1}u\quad \text{in }\mathbb{R}^{N} \end{gathered} \end{equation} has no non-trivial radial positive solutions; in particular \eqref{SL} has no ground state. \end{theorem} \begin{proof} Let us argue by contradiction. Let $(u,v) $ be a radial positive solution of System \eqref{SL}. Then $(u,v)$ satisfies the differential system \begin{equation} \label{Sinfty} \begin{gathered} -(r^{N-1}|u'(r) |^{p-2}u'(r)) '=r^{N-1}a_{12}(r) (v(r)) ^{\delta _{12}}\quad\text{in }[ 0,+\infty [ \,,\\ -(r^{N-1}|v'(r) |^{q-2}v'(r)) '=r^{N-1}a_{21}(r) (u(r)) ^{\delta _{21}}\text{ in }[ 0,+\infty [\,, \\ u'(0) =v'(0) =0 \end{gathered} \end{equation} Hence, \begin{gather} -(r^{N-1}|u'(r) | ^{p-2}u'(r)) '\geq r^{N-1}\overset {\mathbb{s}}{a}_1v^{\delta _{12}} \,, \label{e1}\\ -(r^{N-1}|v'(r) | ^{q-2}v'(r)) '\geq r^{N-1}\overset {\mathbb{s}}{a}_{2}u^{\delta _{21}}\,. \label{e2} \end{gather} First, consider the case $\beta _1>0$ or $\beta_2>0$. Integrating both \eqref{e1} and \eqref{e1} from $0$ to $r$ and taking into account that $u'(r) <0$, $v'(r) <0$ for all $r>0$, we obtain \begin{gather*} -u'(r) \geq (\frac{\tilde{a}_1}{N}) ^{\frac{1}{p-1}} r^{\frac{1}{p-1}}v^{\frac{^{\delta _{12}}}{p-1}},\\ -v'(r) \geq (\frac{\tilde{a}_{2}}{N}) ^{\frac{1}{q-1}} r^{\frac{1}{q-1}}u^{\frac{^{\delta _{21}}}{q-1}}. \end{gather*} By Lemma \ref{lem1}, we have $M_{p}\geq 0$, $M_{q}\geq 0$, thus \begin{gather*} 0\geq -ru'(r)-\frac{N-p}{p-1}u(r)\geq (\frac{\tilde{a}_1}{N}) ^{\frac{1}{p-1}}r^{\frac{p}{p-1}}v^{\frac{ ^{\delta _{12}}}{p-1}}-\frac{N-p}{p-1}u(r),\\ 0\geq -rv'(r)-\frac{N-q}{q-1}v(r)\geq (\frac{\tilde{a}_{2}}{N}) ^{\frac{1}{q-1}}r^{\frac{q}{q-1}}u^{\frac{ ^{\delta _{21}}}{q-1}}-\frac{N-q}{q-1}v(r). \end{gather*} This yields \begin{gather} u(r)\geq Cr^{\frac{p}{p-1}}v^{\frac{^{\delta _{12}}}{p-1}}, \label{e3}\\ v(r)\geq Cr^{\frac{q}{q-1}}u^{\frac{^{\delta _{21}}}{q-1}}. \label{e4} \end{gather} Combining these two inequalities, we have \begin{gather} u(r)\leq Cr^{-\alpha _1}, \label{e5} \\ v(r)\leq Cr^{-\alpha_2}. \label{e6} \end{gather} Since $r^{\frac{N-p}{p-1}}u(r)$ and $r^{\frac{N-q}{q-1}}v(r)$ are nondecreasing, for all $r>r_{0}>0$, \begin{gather} u(r)\geq Cr^{-\frac{N-p}{p-1}}, \label{e7}\\ v(r)\geq Cr^{-\frac{N-q}{q-1}}. \label{e8} \end{gather} Inequalities \eqref{e5}-\eqref{e8} imply either $r^{\beta _1}\leq C$ or $r^{\beta_2}\leq C$. This yields a contradiction. Suppose now that $\beta _1=0$ (we may prove in a similar manner for $\beta_2=0)$. Integrating with respect to $r$ the first equation of System \eqref{Sinfty} from $r_{0}>0$ to $r$, we obtain $$ r^{N-1}|u'(r)| ^{p-1}-r_{0}^{N-1}|u'(r_{0})| ^{p-1} \geq \tilde{a}_1\underset{r_{0}}{ \overset{r}{\int }}s^{N-1}v^{\delta _{12}}(s) ds. $$ Then \eqref{e4} yields $$ v^{\delta _{12}}(s)\geq Cs^{\frac{\delta _{12}q}{q-1}}u^{\frac{^{\delta _{12}\delta _{21}}}{q-1}}(s). $$ Consequently, $$ r^{N-1}|u'(r)| ^{p-1}\geq C\int_{r_0}^r s^{N-1 +\frac{\delta _{12}q}{q-1}}u^{\frac{^{\delta _{12}\delta _{21}}}{q-1}}(s) ds. $$ Taking into account inequality \eqref{e7} and the fact that $\beta_1=0$, we have \[ r^{N-1}|u'(r)| ^{p-1}\geq C\underset{r_{0}}{\overset{ r}{\int }}s^{N-1+\frac{\delta _{12}q}{q-1}-\frac{N-p}{p-1}\frac{^{\delta _{12}\delta _{21}}}{q-1}}\,ds=C\int_{r_0}^r s^{-1}\,ds=C\ln \frac{r}{r_{0}}. \] On the other hand, $M_{p}(r) \geq 0$ for $r>0$ implies $(\frac{N-p}{p-1}) ^{p-1}u^{p-1}(r)\geq r^{p-1}| u'(r)| ^{p-1}$. Hence $$ u^{p-1}(r)\geq Cr^{p-1}|u'(r)| ^{p-1}\geq Cr^{p-N}\ln \frac{r}{r_{0}}. $$ Then we write $$ r^{\frac{N-p}{p-1}}u(r)\geq C\big(\ln \frac{r}{r_{0}}\big) ^{\frac{1}{p-1 }}. $$ This together with \eqref{e5} yields a contradiction. \end{proof} We now show that the eventual radial positive solutions of System \eqref{SAH} are bounded. \begin{theorem} \label{thm2} Assume {\rm (H1)-(H4)}. If $(u,v)$ is a ground state of \eqref{SAH}. then there exists a constant $C>0$ (independent of $u$ and $v$) such that $\|(u,v) \|_{X}\leq C$. \end{theorem} \begin{proof} Let $(u,v) $ be a ground state of \eqref{SAH} for $h=0$, then $(u,v) $ satisfies the system \begin{equation} \label{Sr} \begin{gathered} -(r^{N-1}|u'(r) |^{p-2}u'(r)) ' =r^{N-1}a_{11}(r) f_{11}(u(r)) + r^{N-1}a_{12}(r) f_{12}(v(r)) \\\text{in }[ 0,+\infty [, \\ -(r^{N-1}|v'(r) |^{q-2}v'(r)) ' =r^{N-1}a_{21}(r) f_{21}(u(r)) +r^{N-1}a_{22}(r) f_{22}(v(r)) \\text{in }[ 0,+\infty [\,, \\ u'(0) =v'(0) =0,\quad \lim_{r\to +\infty } u(r)=\lim_{r\to +\infty } v(r)=0 \end{gathered} \end{equation} Assume now that there exists a sequence $(u_n,v_n)$ of positive solutions of \eqref{Sr} such that $\|u_n\|_{\infty }\to \infty$ as $n\to +\infty$ or $\|v_n\|_{\infty }\to \infty$ as $n\to +\infty$. Taking $\gamma _n=\|u_n\|_{\infty }^{\frac{1}{\alpha _1}} +\|v_n\|_{\infty }^{\frac{1}{\alpha_2}}$, and using (H3), we have $\alpha _1>0$ and $\alpha_2>0$. So $\gamma _n\to +\infty $ as $n\to +\infty $. Now we introduce the transformations \[ y=\gamma _nr,\quad w_n(y)=\frac{u_n(r) }{\gamma_n^{\alpha _1}},\quad z_n(y)=\frac{v_n(r) }{\gamma_n^{\alpha_2}}. \] Observe that for all $y\in [ 0,+\infty [$, $0\leq w_n(y)\leq 1$, $0\leq z_n(y)\leq 1$. Furthermore it is easy to see that for any $n$ the pair $(w_n,z_n)$ is a solution of the system \begin{equation} \label{Sy} \begin{gathered} \begin{aligned} &-(y^{N-1}|w_n'(y) | ^{p-2}w_n'(y)) '\\ &=y^{N-1}a_{11}(\frac{y}{\gamma _n}) \frac{f_{11}( \gamma _n^{\alpha _1}w_n(y)) }{\gamma _n^{\alpha _1(p-1) +p}} +y^{N-1}a_{12}(\frac{y}{\gamma _n}) \frac{f_{12}(\gamma _n^{\alpha_2}z_n(y)) }{\gamma _n^{\alpha _1(p-1) +p}}\quad \text{in }[ 0,+\infty [\,, \\ &-(y^{N-1}|z_n'(y) |^{q-2}z_n'(y)) '\\ &=y^{N-1}a_{21}(\frac{y}{\gamma _n}) \frac{f_{21}( \gamma _n^{\alpha _1}w_n(y)) }{\gamma _n^{\alpha _2(q-1) +q}} +y^{N-1}a_{22}(\frac{y}{\gamma _n}) \frac{f_{22}(\gamma _n^{\alpha_2}z_n(y)) }{\gamma _n^{\alpha _2(q-1) +q}}\quad \text{in }[ 0,+\infty [\,, \end{aligned}\\ w_n'(0) =z_n'(0) =0, \quad \lim_{r\to +\infty } w_n(r)=\lim_{r\to +\infty } z_n(r)=0\,. \end{gathered} \end{equation} Let $R>0$ be fixed. We claim that $(w_n')$ and $(z_n') $ are bounded in $C([ 0,R]) $. Indeed passing to a subsequence of $(w_n') $ (denoted again $(w_n')$) assume that $\|w_n'\|_{C([ 0,R]) }\to +\infty $ as $n\to +\infty $. Hence there exists a sequence $(y_n) $ in $[0,R] $ such that for all $A>0$, there exists $n_{0}\in \mathbb{N}$ such that for all $n\geq n_{0}$, $|w_n'(y_n)| >A$. Integrating with respect to $y$ the first equation of System \eqref{Sy}, we obtain \begin{align*} &|w_n'(y_n) | ^{p-1}\\ &=\frac{1}{y_n^{N-1}}\int_0^{y_n} \Big(y^{N-1}a_{11}(\frac{y}{\gamma _n}) \frac{f_{11}(\gamma _n^{\alpha _1}w_n(y)) }{\gamma _n^{\alpha _1(p-1) +p}}+y^{N-1}a_{12}(\frac{y}{\gamma _n}) \frac{f_{12}(\gamma _n^{\alpha_2}z_n(y)) }{ \gamma _n^{\alpha _1(p-1) +p}}\Big) dy. \end{align*} From the fact that $f_{1j}$, $j=1,2$, are asymptotically homogeneous at the infinity together with part (i) of Proposition \ref{prop1}, we arrive to the statement: for all $\varepsilon \in [ 0,\delta _{1j}[$, there exists $c_{1j}^{1},c_{1j}^{2}>0$, $s_{0}>0$ such that for all $s\geq s_{0}$ $$ c_{1j}^{1}s^{\delta _{1j}-\varepsilon }\leq f_{1j}(s) \leq c_{1j}^{2}s^{\delta _{1j}+\varepsilon }. $$ Since $(w_n) $ and $(z_n) $ are bounded, we conclude that \begin{gather*} c_{11}^{1}\gamma _n^{\alpha _1(\delta _{11}-\varepsilon) -\alpha _1(p-1) -p}\leq \frac{ f_{11}(\gamma _n^{\alpha _1}w_n(y)) }{\gamma _n^{\alpha _1(p-1) +p}}\leq \,c_{11}^{2}\gamma _n^{\alpha _1(\delta _{11}+\varepsilon) -\alpha _1(p-1) -p} ,\\ c_{12}^{1}\gamma _n^{\alpha_2(\delta _{12}-\varepsilon) -\alpha _1(p-1) -p}\leq \frac{ f_{12}(\gamma _n^{\alpha_2}z_n(y)) }{\gamma _n^{\alpha _1(p-1) +p}}\leq \,c_{12}^{2}\gamma _n^{\alpha _2(\delta _{12}+\varepsilon) -\alpha _1(p-1) -p}. \end{gather*} By choosing $\varepsilon $ sufficiently small, the assumption (H3) yields $$ \frac{f_{11}(\gamma _n^{\alpha _1}w_n(y) ) }{\gamma _n^{\alpha _1(p-1) +p}} \to 0\quad\text{and}\quad \frac{f_{12}(\gamma_n^{\alpha_2}z_n(y)) }{\gamma _n^{\alpha _1(p-1) +p}} \to c_1\quad \text{as } n\to +\infty $$ where $c_1$ is positive constant. So there exists $n_1\in \mathbb{N}$ such that for any $n\geq n_1$, we have $$ |w_n'(y_n) | ^{p-1}\leq \frac{a_{12}(0) }{y_n^{N-1}}c_1 \int_0^{y_n} y^{N-1}dy=\frac{c_1}{N}a_{12}(0) y_n \leq \frac{Rc_1}{N}a_{12}(0) \equiv c. $$ Setting $n\geq \max (n_{0},n_1)$, we have $A<|w_n'(y_n) | \leq c$. This contradicts the fact that $A$ may be infinitely large. Similarly we prove that $(z_n') $ is bounded in $C([ 0,R])$. Consequently $(w_n) $ and $(z_n) $ are equicontinuous in $C([ 0,R])$. By Arz\'{e}la-Ascoli theorem, there exists a subsequence of $(w_n) $ denoted again $(w_n) $ (respect. $(z_n) $) such that $w_n\to w$ (respect. $z_n\to z$) in $C([ 0,R])$. On the other hand, $$ \|w_n\|_{\infty }^{\frac{1}{\alpha _1}} +\|z_n\|_{\infty }^{\frac{1}{\alpha_2}}=1, $$ this implies that the real-valued sequences $(\|w_n\|_{\infty }) $\ and $(\|z_n\|_{\infty }) $ are bounded. Hence there exist subsequences denoted again $(\|w_n\|_{\infty })$ and $(\|z_n\|_{\infty }) $ such that $\|w_n\|_{\infty }\to w_{0}$, $\|z_n\|_{\infty }\to z_{0}$ and $w_{0}^{\frac{1}{\alpha _1}}+z_{0}^{\frac{1}{\alpha_2}}=1$. In view of the uniqueness of the limit in $C([ 0,R])$, we get $\|w\|_{\infty }^{\frac{1}{\alpha _1}}+\| z\|_{\infty }^{\frac{1}{\alpha_2}}=1$. This implies that $(w,z) $ is not identically null. Integrating from $0$ to $y\in [0,R] $, the first and the second equation of System \eqref{Sy}, we obtain \begin{gather} w_n(0) -w_n(y) =\int_0^y (g_n(s)) ^{\frac{1}{p-1}}ds, \label{e9}\\ z_n(0) -z_n(y) =\int_0^y (h_n(s)) ^{\frac{1}{q-1}}ds\,. \label{e10} \end{gather} Clearly $g_n(y) $ and $h_n(y) $ are defined by \begin{gather*} g_n(y) =\frac{1}{y^{N-1}}\int_0^y \Big(s^{N-1}a_{11}(\frac{s}{\gamma _n}) \frac{f_{11}( \gamma _n^{\alpha _1}w_n(s)) }{\gamma _n^{\alpha _1(p-1) +p}} +s^{N-1}a_{12}(\frac{s}{\gamma _n}) \frac{f_{12}(\gamma _n^{\alpha_2}z_n(s)) }{ \gamma _n^{\alpha _1(p-1) +p}}\Big) ds \\ h_n(y) =\frac{1}{y^{N-1}}\int_0^y \Big(s^{N-1}a_{21}(\frac{s}{\gamma _n}) \frac{f_{21}( \gamma _n^{\alpha _1}w_n(s)) }{\gamma _n^{\alpha _2(q-1) +q}} +s^{N-1}a_{22}(\frac{s}{\gamma _n}) \frac{f_{22}(\gamma _n^{\alpha_2}z_n(s)) }{ \gamma _n^{\alpha_2(q-1) +q}}\Big) ds. \end{gather*} Compiling Proposition \ref{prop1} and (H3), we obtain \begin{gather*} \frac{f_{11}(\gamma _n^{\alpha _1}w_n(s)) }{ \gamma _n^{\alpha _1(p-1) +p}} \to 0, \quad \frac{f_{22}(\gamma _n^{\alpha_2}z_n(s)) }{\gamma _n^{\alpha_2(q-1) +q}} \to 0,\\ \frac{f_{12}(\gamma _n^{\alpha_2}z_n(s)) }{ \gamma _n^{\alpha _1(p-1) +p}}=\frac{f_{12}(\gamma _n^{\alpha_2}) }{\gamma _n^{\alpha _1(p-1) +p}} \frac{f_{12}(\gamma _n^{\alpha_2}z_n(s)) }{ f_{12}(\gamma _n^{\alpha_2}) } \to cz^{\delta _{12}}(s), \\ \frac{f_{21}(\gamma _n^{\alpha _1}w_n(s)) }{ \gamma _n^{\alpha_2(q-1) +q}}=\frac{f_{21}(\gamma _n^{\alpha _1}) }{\gamma _n^{\alpha_2(q-1) +q}} \frac{f_{21}(\gamma _n^{\alpha _1}w_n(s)) }{ f_{21}(\gamma _n^{\alpha _1}) } \to cw^{\delta _{21}}(s), \end{gather*} as $n\to\infty$. By the Lebesgue theorem on dominated convergence, it follows that \begin{gather*} g_n(y) \to \frac{c}{y^{N-1}}\int_0^y s^{N-1}a_{12}( 0) z^{\delta _{12}}(s) ds, \\ h_n(y) \to \frac{c}{y^{N-1}}\int_0^y s^{N-1}a_{21}( 0) w^{\delta _{21}}(s) ds, \end{gather*} as $n\to\infty$. Passing to the limit in \eqref{e9} and \eqref{e10}, we arrive to \begin{gather*} w(0) -w(y) =c\int_0^y \frac{1}{\tau ^{N-1}} \Big(\int_0^\tau s^{N-1}a_{12}(0) z^{\delta _{12}}(s) ds\Big) ^{ \frac{1}{p-1}}d\tau, \\ z(0) -z(y) =c\int_0^y \frac{1}{\tau ^{N-1}} \Big(\int_0^\tau s^{N-1}a_{21}(0) w^{\delta _{21}}(s) ds\Big) ^{ \frac{1}{q-1}}d\tau . \end{gather*} In this way $w\geq 0$, $z\geq 0$, $w, z\in C^{1}([ 0,R]) \cap C^{2}(] 0,R])$ and satisfy the system \begin{equation} \label{SR} \begin{gathered} -(y^{N-1}|w'(y) |^{p-2}w'(y)) '=ca_{12}( 0) y^{N-1}(z(y)) ^{\delta _{12}}\quad \text{in }[ 0,R] \\ -(y^{N-1}|z'(y) |^{q-2}z'(y)) '=ca_{21}( 0) y^{N-1}(w(y)) ^{\delta _{21}}\quad \text{in }[ 0,R] \\ w'(0) =z'(0) =0 \end{gathered} \end{equation} If we use the same arguments on $[0,R^{\ast }]$ where $R^{\ast }>R$, we obtain a solution $(w^{\ast },z^{\ast }) $ of System \eqref{SR} with $R^{\ast }$ in stead of $R$, which coincide with $(w,z) $ in $[0,R]$. To this end, we indefinitely extend $(w,z) $ to $[ 0,+\infty [ $. By Lemma \ref{lem1} we have $w(y)>0$, $z(y) >0$, for all $y\geq 0$. The pair $(w,z)$ also satisfies System \eqref{SR}. In other words $(w,z) $ is a radial positive solution of \eqref{Sinfty}. This contradicts Theorem \ref{thm1}. \end{proof} \begin{lemma} \label{lem3} Under assumptions {\rm (H1)-(H4)}, there exists $h_{0}>0$ such that the problem $(u,v) =T_{h}(u,v) $ has no solution for $h\geq h_{0}$. \end{lemma} \begin{proof} Suppose by contradiction that there is a solution $(u,v) \in X$ of the above problem. Then $(u,v) $ satisfies system \begin{equation} \label{SAH1} \begin{gathered} -(r^{N-1}|u'(r) | ^{p-2}u'(r)) '=r^{N-1}a_{11}(r) f_{11}(|u(r) |) + r^{N-1}a_{12}(r) [ f_{12}(|v(r) |) +h ] \\ \text{in }[ 0,+\infty [ \,, \\ -(r^{N-1}|v'(r) | ^{q-2}v'(r)) '=r^{N-1}a_{21}(r) f_{21}(|u(r) |) + r^{N-1}a_{22}(r) f_{22}(|v(r) | )\\ \text{in }[ 0,+\infty [\,, \\ u'(0) =v'(0) =0,\quad \lim_{r\to +\infty } u(r) =\lim_{r\to +\infty } v(r)=0 \end{gathered} \end{equation} Assume that there exists a sequence $(h_n)$ $h_n\to +\infty$ as $n\to +\infty$, such that \eqref{SAH1} admits a pair of solutions $(u_n,v_n) $. In accordance with Lemma \ref{lem1}, we have $u_n(r)>0$, $v_n(r)>0,\,u_n'(r)\leq 0$ and $v_n'(r)\leq 0$, for all $n\in \mathbb{N}$. Integrating the first equation of System \eqref{SAH1}, from $R$ to $2R$, $R>0$, we obtain $$ u_n(R)\geq \int_R^{2R} \Big(\eta ^{1-N} \overset{\eta }{\underset{0}{\int }}\xi ^{N-1}a_{12}(\xi) h_nd\xi\Big) ^{\frac{1}{p-1}}d\eta \geq cRh_n^{\frac{1}{p-1}} $$ Here $$ c=\Big(\frac{1}{(2R) ^{N-1}}\overset{R}{ \underset{0}{\int }}\xi ^{N-1}a_{12}(\xi) d\xi\Big) ^{\frac{1 }{p-1}}. $$ Consequently $u_n(R)\geq c Rh_n^{\frac{1}{p-1}}$. Passing to the limit we get $u_n(R)\to +\infty $. On the other hand, integrating the second equation of \eqref{SAH1}, from $R$ to $2R$, we obtain $$ v_n(R)\geq \int_R^{2R} (\eta ^{1-N} \overset{\eta }{\underset{0}{\int }}\xi ^{N-1}a_{21}(\xi) f_{21}(u_n(\xi )) d\xi) ^{\frac{1}{q-1}}d\eta \geq cR(f_{21}(u_n(R))) ^{\frac{1}{q-1}} $$ By hypothesis (H3) and Proposition \ref{prop1}, we have $v_n(R)\geq c(u_n(R)) ^{\frac{\delta _{21}-\varepsilon}{q-1}}$ Operating similarly, we obtain $u_n(R)\geq c(v_n(R)) ^{\frac{\delta _{12}-\varepsilon }{p-1}}$. It follows from the last two inequalities, that \[ (u_n(R)) ^{\frac{(\delta _{12}-\varepsilon) (\delta _{21}-\varepsilon) -(p-1) (q-1) }{(p-1) (q-1) }}\leq \frac{1}{c}. \] This is the desired contradiction since $u_n(R)$ increases to infinitely. \end{proof} \begin{lemma} \label{lem4} There exists $\bar{\rho }>0$ such that for all $\rho \in ] 0,\bar{\rho }[$ and all $(u,v) \in X$ satisfying $\|(u,v) \|=\rho $, the equation $(u,v)=S_{\lambda }(u,v)$ has no solution. \end{lemma} \begin{proof} Assume that there exist $(\rho _n) \in \mathbb{R} _{+}$, $\rho _n\to 0$; $(\lambda _n) \subset [0,1] $ and $(u_n,v_n) \in X$ such that $(u_n,v_n) =S_{\lambda _n}(u_n,v_n) $ with $\|(u_n,v_n) \|=\rho_n$. Taking (H4) into account, \begin{gather*} \|u_n\|_{\infty }\leq c\lambda _n^{\frac{1}{p-1}}\Big( \|u_n\|_{\infty }^{\frac{\bar{\delta } _{11}-\varepsilon }{p-1}}+\|v_n\|_{\infty }^{\frac{\bar {\delta }_{12}-\varepsilon }{p-1}}\Big) \\ \|v_n\|_{\infty }\leq c\lambda _n^{\frac{1}{q-1}}\Big( \|u_n\|_{\infty }^{\frac{\bar{\delta } _{21}-\varepsilon }{q-1}}+\|v_n\|_{\infty }^{\frac{\bar {\delta }_{22}-\varepsilon }{q-1}}\Big) \end{gather*} Adding term by term, we obtain \begin{align*} \|(u_n,v_n) \| &\leq C\Big(\|(u_n,v_n) \|^{\frac{\bar{\delta } _{11}-\varepsilon }{p-1}} +\|(u_n,v_n) \|^{\frac{\bar{\delta }_{12}-\varepsilon }{p-1}}\,, \\ &\quad +\|(u_n,v_n) \|^{\frac{\bar{\delta }_{21} -\varepsilon }{q-1}} +\|(u_n,v_n) \|^{\frac{\bar{\delta }_{22}-\varepsilon }{q-1}}\Big)\,. \end{align*} This implies \begin{align*} 1&\leq C\Big(\|(u_n,v_n) \|^{\frac{\bar{\delta }_{11} -\varepsilon }{p-1}-1} +\|(u_n,v_n) \|^{\frac{\bar{\delta }_{12}-\varepsilon }{p-1}-1}\\ &\quad +\|(u_n,v_n) \|^{\frac{\bar{\delta }_{21}-\varepsilon }{q-1}-1} +\|(u_n,v_n)\|^{\frac{\bar{\delta }_{22}-\varepsilon }{q-1}-1}\Big). \end{align*} The above inequality contradicts the fact that $\|(u_n,v_n) \|=\rho _n\to$ as $n\to +\infty $. \end{proof} \begin{theorem} \label{thm3} Under hypotheses (H1)-(H4), System \eqref{SNV} has positive radial solution. \end{theorem} \begin{proof} To show the existence of ground states for \eqref{SNV} (or \eqref{SAH} with $h=0$), it is sufficient to prove that the compact operator $T_{0}$ admits a fixed point. In view of Theorem \ref{thm2}, the eventual fixed point $(u,v) $ of $T_{0}$ are bounded; explicitly there exists $C>0$ such that $\|(u,v) \|_{X}\leq C$. Let us chose $R_1>C$ and let us designate by $B_{R_1}$ the ball of $X$, centered at the origin with radius $R_1$. To this end, the Leray-Schauder degree $\deg _{LS}(I-T_{h},B_{R_1},0)$ is well defined. It being understood that $I$ denote the identical operator in $X$. Moreover, by Lemma \ref{lem3}, we have $\deg_{LS}(I-T_{h},B_{R_1},0) =0$ for all $h\geq h_{0}$. It follows from the homotopy invariance of the Leray-Schauder degree that \begin{equation*} \deg _{LS}(I-T_{0},B_{R_1},0) =\deg _{LS}( I-T_{h},B_{R_1},0) =0. \end{equation*} On the other hand, by Lemma \ref{lem4}, there exists $0<\rho <\bar{\rho }