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

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In this work we study the nonnegative solutions of the elliptic system Δu=|x|ᵃvδ, Δv=|x|ᵇuμ in the superlinear case μδ>1, which blow up near the boundary of a domain of Rⁿ, 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 Rⁿ{0, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation Δ² u=|x|ᵇ|u|μ. Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, combined with nonradial upper estimates.