Elliptic systems in unbounded domains

We study a nonlinear elliptic system of Lane--Emden type \[\left\{ \begin{array}{ll} -\Delta u\ =\ {\rm sgn}(v) |v|^{p-1} & \text{in $\R^{N}$,} \\ -\Delta v\ =\ - {\rm sgn}(u) |u|^{\frac1{p-1}} + f(u) & \text{in $\R^{N}$,} \\ u,\ v\ \rightarrow 0 \quad\text{as}\quad \left\vert x\right\vert \rightarrow +\infty,& \end{array}\right. \] which is equivalent to a fourth order elliptic equation. By using variational methods the existence of radial solutions of the given problem is proved. To this aim, new compact imbeddings are stated