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In this work we study the nonnegative solutions of the elliptic system \[ \Delta u=|x|^{a}v^{\delta},\qquad\Delta v=|x|^{b}u^{\mu}% \] in the superlinear case $\mu\delta>1,$ which blow up near the boundary of a domain of $\mathbb{R}^{N},$ or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension $N$. We also show the existence of infinitely many solutions blowing up at $0.$ Furthermore, we show that there exists a global positive solution in $\mathbb{R}^{N}\backslash\left\{ 0\right\} ,$ large at $0,$ and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation \[ \Delta^{2}u=\left\vert x\right\vert ^{b}\left\vert u\right\vert ^{\mu}. \] Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, introduced in [4], combined with the nonradial upper estimates of [5].

Large solutions of elliptic systems of second order and applications to the biharmonic equation