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

2010/10/11 by Marie-Françoise Bidaut-Véron, Bidaut-Véron, Marie-Françoise, Marta García‐Huidobro +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.1010.2127

openalex publication_date 2010/10/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this work we study the nonnegative solutions of the elliptic system Δu=|x|^av^δ, Δv=|x|^bu^μ in the superlinear case μδ>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 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.

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