# American Institute of Mathematical Sciences

February  2012, 32(2): 411-432. doi: 10.3934/dcds.2012.32.411

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

 1 Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200 Tours, France 2 Departamento de Matemáticas, Pontiﬁcia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile 3 Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

Received  December 2010 Revised  May 2011 Published  September 2011

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].
Citation: Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411
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