JournalsrlmVol. 27, No. 1pp. 1–14

Existence of entire solutions to a fractional Liouville equation in Rn\mathbb R^n

  • Ali Hyder

    Universität Basel, Switzerland
Existence of entire solutions to a fractional Liouville equation in $\mathbb R^n$ cover

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Abstract

We study the existence of solutions to the problem

(Δ)n2u=QenuinRn,V:=Rnenudx<,(-\Delta)^{\frac{n}{2}}u=Qe^{nu} \quad \mathrm{in }\mathbb R^n, \quad V: = \int_{\mathbb R^n}e^{nu} dx < \infty,

where Q=(n1)!Q=(n-1)! or Q=(n1)!Q=-(n-1)!. Extending the works of Wei–Ye and Hyder–Martinazzi to arbitrary odd dimension n3n \geq 3 we show that to a certain extent the asymptotic behavior of uu and the constant VV can be prescribed simultaneously. Furthermore if Q=(n1)!Q=-(n-1)! then VV can be chosen to be any positive number. This is in contrast to the case n=3n = 3, Q=2Q = 2, where Jin–Maalaoui–Martinazzi–Xiong showed that necessarily VS3V \le |S^3|, and to the case n=4n = 4, Q=6Q = 6, where C.-S. Lin showed that VS4V \le |S^4|.

Cite this article

Ali Hyder, Existence of entire solutions to a fractional Liouville equation in Rn\mathbb R^n. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 27 (2016), no. 1, pp. 1–14

DOI 10.4171/RLM/718