On the existence of solutions for an elliptic system of equations with arbitrary order growth

Let B _R be the ball centered at the origin with radius R in ? ^N (N≥2). In this paper we study the existence of solution for the following elliptic system $$\left\{ \begin{gathered} - \Delta u + \lambda u = \frac{p} {{p + q}}\kappa \left( {\left| x \right|} \right)u^{p - 1} v^q , x \in B_R , \hfill \\ - \Delta v + \mu v = \frac{q} {{p + q}}\kappa \left( {\left| x \right|} \right)u^p v^{q - 1} , x \in B_R , \hfill \\ u > 0, v > 0, x \in B_R , \hfill \\ \frac{{\partial u}} {{\partial v}} = 0, \frac{{\partial v}} {{\partial v}} = 0, x \in \partial B_R \hfill \\ \end{gathered} \right.$$ where λ > 0, µ > 0 p ≥ 2, q ≥ 2, ν is the unit outward normal at the boundary ?B _R . Under certain assumptions on κ(|x|), using variational methods, we prove the existence of a positive and radially increasing solution for this problem without growth conditions on the nonlinearity.