\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 272, pp. 1--8.\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-2016/272\hfil Hessian systems with convection terms] {Existence of positive entire radial solutions to a $(k_1,k_2)$-Hessian systems with convection terms} \author[D.-P. Covei \hfil EJDE-2016/272\hfilneg] {Dragos-Patru Covei} \address{Dragos-Patru Covei \newline Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st district, postal code 010374, postal office 22, Romania} \email{coveidragos@yahoo.com} \thanks{Submitted August 23, 2016. Published October 10, 2016.} \subjclass[2010]{35J25, 35J47, 35J96} \keywords{Entire solution; large solution; elliptic system} \begin{abstract} In this article, we prove two new results on the existence of positive entire large and bounded radial solutions for nonlinear system with gradient terms \begin{gather*} S_{k_1}(\lambda (D^{2}u_1) )+b_1(| x| ) | \nabla u_1|^{k_1} =p_1(| x| ) f_1(u_1,u_2) \quad\text{for }x\in \mathbb{R}^{N}, \\ S_{k_2}(\lambda (D^{2}u_2) ) +b_2(| x| ) | \nabla u_2|^{k_2} =p_2(| x| ) f_2(u_1,u_2) \quad\text{for }x\in \mathbb{R}^{N}, \end{gather*} where $S_{k_i}(\lambda (D^{2}u_i) ) $ is the $k_i$-Hessian operator, $b_1$, $p_1$, $f_1$, $b_2$, $p_2$ and $f_2$ are continuous functions satisfying certain properties. Our results expand those by Zhang and Zhou \cite{ZZ}. The main difficulty in dealing with our system is the presence of the convection term. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The purpose of this article is to present new results concerning the nonlinear Hessian system with convection terms \begin{equation} \begin{gathered} S_{k_1}(\lambda (D^{2}u_1) ) +b_1(| x| ) | \nabla u_1|^{k_1} =p_1(| x| ) f_1(u_1,u_2) , x\in \mathbb{R}^{N}, \\ S_{k_2}(\lambda (D^{2}u_2) ) +b_2(| x| ) | \nabla u_2|^{k_2} =p_2(| x| ) f_2(u_1,u_2) , x\in \mathbb{R}^{N}, \end{gathered} \label{11} \end{equation} where $N\geq 3$, $b_1$, $p_1$, $f_1$, $b_2$, $p_2$, $f_2$ are continuous functions satisfying certain properties, $k_1,k_2\in \{1,2,\dots ,N\} $ and \begin{equation*} S_{k_i}(\lambda ) =\sum_{1\leq i_1<\dots 0,\; a_1\geq 0,\; a_1\geq 0, \\ P_1(\infty ) =\lim_{r\to \infty }P_1(r) ,\quad P_2(\infty ) =\lim_{r\to \infty }P_2(r) , \quad F_{1,2}(\infty ) =\lim_{s\to \infty }F_{1,2}(s) . \end{gather*} We will always assume that the variable weights functions $b_1,b_2,p_1,p_2$ and the nonlinearities $f_1$, $f_2$ satisfy: \begin{itemize} \item[(A1)] $b_1,b_2:[ 0,\infty ) \to [ 0,\infty) $ and $p_1,p_2:[ 0,\infty ) \to [0,\infty ) $ are spherically symmetric continuous functions (i.e., $p_i(x) =p_i(| x| ) $ and $b_i(x) =b_i(| x|) $ for $i=1,2$); \item[(A2)] $f_1$, $f_2:[ 0,\infty ) \times [ 0,\infty ) \to [ 0,\infty ) $ are continuous and increasing. \end{itemize} Here is a first result. \begin{theorem} \label{thm1} We assume that $F_{1,2}(\infty ) =\infty $ and \textrm{(A1)}, hold. Furthermore, if $f_1$ and $f_2$ satisfy \textrm{(A2)}, then system \eqref{11} has at least one positive radial solution $(u_1,u_2) \in C^{2}([ 0,\infty) ) \times C^{2}([ 0,\infty ) ) $ with central value in $(a_1,a_2) $. Moreover, the following hold: \begin{itemize} \item[(1)] If $P_1(\infty ) +P_2(\infty ) <\infty $, then $\lim_{r\to \infty}u_1(r) <\infty $ and $\lim_{r\to \infty}u_2(r) <\infty $. \item[(2)] If $P_1(\infty ) =\infty $ and $P_2(\infty) =\infty $, then \[ \lim_{r\to \infty}u_1(r) =\infty \quad\text{and}\quad \lim_{r\to \infty}u_2(r) =\infty . \] \end{itemize} \end{theorem} In the same spirit we also have, our next result. \begin{theorem}\label{thm2} Assume that the hypotheses \textrm{(A1)} and \textrm{(A2)} are satisfied. If $P_1(\infty ) +P_2(\infty ) 0, \\ &\frac{C_{N-1}^{k_2-1}}{r^{N-1}}[\frac{r^{N-k_2}}{k_2}e^{\int_{0}^{r} \frac{1}{C_{00}}t^{k_2-1}b_2(t) dt}(u_2')^{k_2}]'\\ &=e^{\int_{0}^{r}\frac{1}{C_{00}}t^{k_2-1}b_2(t) dt}p_2(r) f_2(u_1,u_2) , r>0, \end{aligned}\\ u_1'(r) \geq 0\quad \text{and}\quad u_2'(r) \geq 0\quad\text{for }r\in [ 0,\infty ) , \\ u_1(0) =a_1\quad \text{and}\quad u_2(0) =a_2, \end{gathered} \label{ss} \end{equation} has a solution. Therefore, at least one solution of \eqref{ss} can be obtained using successive approximation by defining the sequences $\{u_1^{m}\} _{m\geq 0}$ and $\{ u_2^{m}\} _{m\geq 0}$ on $[ 0,\infty ) $ in the following way \begin{equation} \begin{gathered} u_1^{0}=a_1, \quad u_2^{0}=a_2\quad\text{for }r\geq 0 \\ u_1^{m}(s) =a_1+\int_{0}^{r}\Big[ B_1^{-}(t) \int_{0}^{t}B_1^{+}(s) f_1(u_1^{m-1}(s) ,u_2^{m-1}(s) ) ds\Big] ^{1/k_1}dt, \\ u_2^{m}(s) =a_2+\int_{0}^{r}\Big[ B_2^{-}(t) \int_{0}^{t}B_2^{+}(s) f_2(u_1^{m-1}(s) ,u_2^{m-1}(s) ) ds\Big] ^{1/k_2}dt. \end{gathered} \label{recs} \end{equation} It is easy to see that $\{ u_1^{m}\} _{m\geq 0}$ and $\{ u_2^{m}\}_{m\geq 0}$ are non-decreasing on $[ 0,\infty ) $. Indeed, we consider \begin{align*} u_1^{1}(r) &= a_1+\int_{0}^{r}\Big[B_1^{-}(t) \int_{0}^{t}B_1^{+}(s) f_1(u_1^{0}(s),u_2^{0}(s) ) ds\Big]^{1/k_1}dt \\ &= a_1+\int_{0}^{r}\Big[B_1^{-}(t) \int_{0}^{t}B_1^{+}(s) f_1(a_1,a_2) ds \Big]^{1/k_1}dt \\ &\leq a_1+\int_{0}^{r}\Big[B_1^{-}(t) \int_{0}^{t}B_1^{+}(s) f_1(u_1^{1}(s),u_2^{1}(s) ) ds\Big]^{1/k_1}dt=u_1^{2}(r) . \end{align*} This implies that $u_1^{1}(r) \leq u_1^{2}(r) $ which further produces $u_1^{2}(r) \leq u_1^{3}(r) $. Continuing, an induction argument applied to \eqref{recs} show that for any $r\geq 0$ we have \begin{equation*} u_1^{m}(r) \leq u_1^{m+1}(r) \quad\text{and}\quad u_2^{m}(r) \leq u_2^{m+1}(r) \quad \text{for any } m\in \mathbb{N} \end{equation*} i.e., $\{ u_1^{m}\} _{m\geq 0}$ and $\{ u_2^{m}\}_{m\geq 0}$ are non-decreasing on $[ 0,\infty ) $. By their monotonicity, we have the inequalities \begin{gather} C_{N-1}^{k_1-1}\{\frac{r^{N-k_1}}{k_1}e^{\int_{0}^{r}\frac{1}{C_{0}} t^{k_1-1}b_1(t) dt}[ (u_1^{m}) '] ^{k_1}\}' \leq B_1^{+}(r) f_1(u_1^{m},u_2^{m}) , \label{gen1} \\ C_{N-1}^{k_2-1}\{\frac{r^{N-k_2}}{k_2}e^{\int_{0}^{r}\frac{1}{C_{00}} t^{k_2-1}b_2(t) dt}[(u_2^{^{m}}) ']^{k_2}\}' \leq B_2^{+}(r) f_2(u_1^{m},u_2^{m}) . \label{gen2} \end{gather} After integration from $0$ to $r$, an easy calculation yields \begin{equation} \begin{aligned} &(u_1^{m}(r) ) '\\ &\leq \Big(B_1^{-}( r) \int_{0}^{r}B_1^{+}(t) f_1(u_1^{m}( t) ,u_2^{m}(t) ) dt\Big)^{1/k_1} \\ &\leq \Big(B_1^{-}(r) \int_{0}^{r}B_1^{+}(t) f_1(u_1^{m}(t) +u_2^{m}(t) ,u_1^{m}(t) +u_2^{m}(t) ) dt\Big)^{1/k_1} \\ &\leq (f_1^{1/k_1}+f_2^{1/k_2})(u_1^{m}(r) +u_2^{m}(r) ,u_1^{m}(r) +u_2^{m}(r) ) (B_1^{-}(r) \int_{0}^{r}B_1^{+}(t) dt)^{1/k_1}. \end{aligned} \label{exin} \end{equation} As before, exactly the same type of conclusion holds for $(u_2^{m}(r) ) '$: \begin{equation} \begin{aligned} (u_2^{m}(r) ) ' &\leq \Big(B_2^{-}(r) \int_{0}^{r}B_2^{+}(z) f_2(u_1^{m}(z) ,u_2^{m}(z) ) dz\Big)^{1/k_2} \\ &\leq (f_1^{1/k_1}+f_2^{1/k_2})( u_1^{m}+u_2^{m},u_1^{m}+u_2^{m}) \Big(B_2^{-}(r) \int_{0}^{r}B_2^{+}(t) dt\Big)^{1/k_2}. \end{aligned} \label{exin2} \end{equation} Summing the inequalities \eqref{exin} and \eqref{exin2}, we obtain \begin{equation} \frac{(u_1^{m}(r) +u_2^{m}(r) )'}{(f_1^{1/k_1}+f_2^{1/k_2})(u_1^{m}(r) +u_2^{m}(r) ,u_1^{m}(r) +u_2^{m}(r) ) } \leq P_1'(r) +P_2'(r) . \label{mat} \end{equation} Integrating from $0$ to $r$, we obtain \begin{equation*} \int_{a_1+a_2}^{u_1^{m}(r) +u_2^{m}(r) } \frac{1}{f_1^{1/k_1}(t,t) +f_2^{1/k_2}(t,t) } dt\leq P_1(r) +P_2(r) . \end{equation*} We now have \begin{equation} F_{1,2}(u_1^{m}(r) +u_2^{m}(r) ) \leq P_1(r) +P_2(r) , \label{ints} \end{equation} which will play a basic role in the proof of our main results. The inequalities \eqref{ints} can be reformulated as \begin{equation} u_1^{m}(r) +u_2^{m}(r) \leq F_{1,2}^{-1}( P_1(r) +P_2(r) ) . \label{int} \end{equation} This can be easily seen from the fact that $F_{1,2}$ is a bijection with the inverse function $F_{1,2}^{-1}$ strictly increasing on $[ 0,\infty) $. So, we have found upper bounds for $\{ u_1^{m}\} _{m\geq 0}$ and $\{ u_2^{m}\}_{m\geq 0}$ which are dependent of $r$. We are now ready to give a complete proof of the Theorems \ref{thm1} and \ref{thm2}. \begin{proof}[Proof of Theorem \ref{thm1} completed] When $F_{1,2}(\infty) =\infty $ it follows that the sequences $\{ u_1^{m}\}_{m\geq 0}$ and $\{ u_2^{m}\} _{m\geq 0}$ are bounded and equicontinuous on $[ 0,c_{0}] $ for arbitrary $c_{0}>0$. By the Arzela-Ascoli theorem, $\{ (u_1^{m},u_2^{m}) \} _{m\geq 0}$ has a subsequence converging uniformly to $(u_1,u_2) $ on $[0,c_{0}] \times [ 0,c_{0}] $. Since $\{u_1^{m}\} _{m\geq 0}$ and $\{ u_2^{m}\} _{m\geq 0}$ are non-decreasing on $[ 0,\infty ) $ we see that $\{ (u_1^{m},u_2^{m}) \} _{m\geq 0}$ itself converges uniformly to $(u_1,u_2) $ on $[ 0,c_{0}] \times [0,c_{0}] $. At the end of this process, we conclude by the arbitrariness of $c_{0}>0$, that $(u_1,u_2) $ is positive entire solution of system \eqref{ss}. The solution constructed in this way will be radially symmetric. Since the radial solutions of the ordinary differential equations system \eqref{ss} are solutions \eqref{11} it follows that the radial solutions of \eqref{11} with $u_1(0) =a_1$, $u_2(0) =a_2$ satisfy \begin{gather} u_1(r) = a_1+\int_{0}^{r}\Big(B_1^{-}(y) \int_{0}^{y}B_1^{+}(t) f_1(u_1(t) ,u_2(t) ) dt\Big)^{1/k_1}dy, \label{eq1} \\ u_2(r) = a_2+\int_{0}^{r}\Big(B_2^{-}(y) \int_{0}^{y}B_2^{+}(t) f_2(u_1(t),u_2(t) ) dt\Big)^{1/k_2}dy, \label{eq2} \end{gather} for all $r\geq 0$. Next, it is easy to verify that the Cases 1. and 2. occur. \smallskip \noindent\textbf{Case 1.} When $P_1(\infty ) +P_2(\infty ) <\infty$, it is not difficult to deduce from \eqref{eq1} and \eqref{eq2} that \begin{equation*} u_1(r) +u_2(r) \leq F_{1,2}^{-1}(P_1(\infty ) +P_2(\infty ) ) <\infty \quad \text{for all }r\geq 0, \end{equation*} and so $(u_1,u_2) $ is bounded. We next consider: \smallskip \noindent\textbf{Case 2.} When $P_1(\infty ) =P_2(\infty ) =\infty$, we observe that \begin{equation} \begin{aligned} u_1(r) &= a_1+\int_{0}^{r}\Big(B_1^{-}(t) \int_{0}^{t}B_1^{+}(s) f_1(u_1(s),u_2(s) ) ds\Big)^{1/k_1}dt \\ &\geq a_1+f_1^{1/k_1}(a_1,a_2) \int_{0}^{r}\Big(B_1^{-}(t) \int_{0}^{t}B_1^{+}(s) ds\Big)^{1/k_1}dt \label{i1} \\ &= a_1+f_1^{1/k_1}(a_1,a_2) P_1(r) . \end{aligned} \end{equation} The same computations as in \eqref{i1} yields \begin{equation*} u_2(r) \geq a_2+f_2^{1/k_2}(a_1,a_2) P_2(r) . \end{equation*} and passing to the limit as $r\to \infty $ in \eqref{i1} and in the above inequality we conclude that \begin{equation*} \lim_{r\to \infty }u_1(r) =\lim_{r\to \infty }u_2(r) =\infty , \end{equation*} which yields the result. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2} completed] In view of the above analysis, the proof can be easily deduced from \begin{equation*} F_{1,2}(u_1^{m}(r) +u_2^{m}(r) ) \leq P_1(\infty ) +P_2(\infty )