A complete description of the asymptotic behavior at infinity of positive radial solutions to Δ² u = u^α in \mathbf Rⁿ

2018/03/30 by Quốc Anh Ngô, Ngô, Quôc Anh, Van Hoang Nguyen +3
Mathematics · Engineering · Computer Science · #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1803.11520

Abstract

We consider the biharmonic equation Δ² u = u^α in \mathbf Rⁿ with n \geqslant 1. It was proved that this equation has a positive classical solution if, and only if, either α\leqslant 1 with n \geqslant 1 or α\geqslant (n+4)/(n-4) with n \geqslant 5. The asymptotic behavior at infinity of all positive radial solutions was known in the case α\geqslant (n+4)/(n-4) and n \geqslant 5. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case α\leqslant 1 with n \geqslant 1; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is higher order, we propose a new approach which relies on a representation formula and asymptotic analysis arguments. We believe that the approach introduced here can be conveniently applied to study other problems with higher order operators.

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