Radial symmetry of positive entire solutions of a fourth order elliptic equation with a singular nonlinearity

2018/04/23 by Guo, Zongming, Wei, Long, Zhou, Feng
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1804.08215

Abstract

The necessary and sufficient conditions for a regular positive entire solution u of the biharmonic equation: -Δ² u=u^-p in \R^N (N ≥ 3), pgt;1 to be a radially symmetric solution are obtained via the moving plane method (MPM) of a system of equations. It is well-known that for any a>0, \eqref0.1 admits a unique minimal positive entire radial solution \underline u_a (r) and a family of non-minimal positive entire radial solutions u_a (r) such that u_a (0)=\underline u_a (0)=a and u_a (r) ≥ \underline u_a (r) for r ∈ (0, ∞). Moreover, the asymptotic behaviors of \underline u_a (r) and u_a (r) at r=∞ are also known. We will see in this paper that the asymptotic behaviors similar to those of \underline u_a (r) and u_a (r) at r=∞ can determine the radial symmetry of a general regular positive entire solution u of \eqref0.1. The precisely asymptotic behaviors of u (x) and -Δu (x) at |x|=∞ need to be established such that the moving-plane procedure can be started. We provide the necessary and sufficient conditions not only for a regular positive entire solution u of \eqref0.1 to be the minimal entire radial solution, but also for u to be a non-minimal entire radial solution.

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