Existence of entire solutions to a fractional Liouville equation in ℝⁿ

2015/02/09 by Hyder, Ali
#35 #53 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1502.02685

Abstract

We study the existence of solutions to the problem (-Δ)^(n)/(2)u = Qe^nu\quadin ℝⁿ, V := ∫_ℝⁿe^nudx lt; ∞, where Q=(n-1)! or Q=-(n-1)!. Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension n≥ 3 we show that to a certain extent the asymptotic behavior of u and the constant V can be prescribed simultaneously. Furthermore if Q=-(n-1)! then V can be chosen to be any positive number. This is in contrast to the case n=3, Q=2, where Jin-Maalaoui-Martinazzi-Xiong showed that necessarily V≤ |S³|, and to the case n=4, Q=6, where C-S. Lin showed that V≤ |S⁴|.

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