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

TL;DR

This paper investigates large solutions of elliptic systems with power nonlinearities, characterizing their behavior near boundaries and points, and applies findings to sign-changing solutions of the biharmonic equation.

Contribution

It provides a detailed analysis of large solutions for elliptic systems, including existence, behavior near singularities, and applications to biharmonic equations, using a novel dynamical approach.

Findings

Precise boundary behavior of large solutions in any dimension

Existence of infinitely many solutions blowing up at zero

Application to sign-changing solutions of biharmonic equations

Abstract

In this work we study the nonnegative solutions of the elliptic system \Delta u=|x|^{a}v^{\delta}, \Delta v=|x|^{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=|x|^{b}|u|^{\mu}. 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.