LARGE SOLUTIONS OF ELLIPTIC SYSTEMS OF SECOND ORDER AND APPLICATIONS TO THE BIHARMONIC EQUATION

Abstract

In this work we study the nonnegative solutions of the elliptic system

Delta U - vertical bar x vertical bar(a)v(delta), Delta v - vertical bar x vertical bar(b)u(mu)

in the superlinear case mu delta > 1, which blow up near the boundary of a domain of 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 R-N \ {0}, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation

Delta(2)u = vertical bar x vertical bar(b) vertical bar u vertical bar(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].