Existence and nonexistence of positive radial entire solutions of second order quasilinear elliptic systems (Q5931515)

This paper deals with second-order quasilinear elliptic system of the form NEWLINE\[NEWLINE\begin{cases} \Delta_pu\equiv \text{div} \bigl(|Du|^{p-2}Du \bigr) =H\bigl(|x|\bigr)v^\alpha\\ \Delta_q v\equiv\text{div}\bigl( |Dv|^{q-2} Dv\bigr)= K\bigl(|x|\bigr) u^\beta \end{cases} \quad \text{in }\mathbb{R}^N, \tag{1}NEWLINE\]NEWLINE where \(N\geq 1\), \(p>1\), \(q>1\), \(\alpha\) and \(\beta\) are positive constants satisfying \(\alpha\beta >(p-1)(q-1)\), and \(H,K:[0, \infty)\to [0,\infty)\) are continuous. The author presents sufficient conditions guaranteeing both positive radial entire relations and nonexistence of nonnegative nontrivial radial entire solutions of (1) under some natural restrictions on \(p,q,\alpha\) and \(\beta\). Moreover, the author characterizes the existence property of positive radial entire solutions, when \(H\) and \(K\) behave like positive constant multiples of \(|x|^\gamma\), \(\gamma\in\mathbb{R}\).