Existence of positive global radial solutions to nonlinear elliptic systems

Abstract

In this article we obtain global positive and radially symmetric solutions to the system of nonlinear elliptic equations $$ \operatorname{div}\big(\phi_j(|\nabla u|) \nabla u\big) +a_j(x)\phi_j(|\nabla u|) |\nabla u| =p_j(x)f_j(u_1(x),\dots,u_k(x))\,, $$ and in particular to Laplace's equation $$ \Delta u_j(x) =p_j(x)f_j(u_1(x),\dots,u_k(x))\,, $$ where \(j=1,\dots,k\), \( x\in\mathbb{R}^N\), \(N\geq 3\), \(\Delta \) is the Laplacian operator, and \(\nabla\) is the gradient. Also we state conditions for solutions to be bounded, and to be unbounded. With an example we illustrate our results. See also https://ejde.math.txstate.edu/special/02/p1/abstr.html