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

Ali Hyder

doi:10.4171/rlm/718

JournalsrlmVol. 27, No. 1pp. 1–14

Existence of entire solutions to a fractional Liouville equation in $R^{n}$

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)^{\frac{n}{2}} u = Q e^{n u} in R^{n}, V := \int_{R^{n}} e^{n u} d x < \infty,

where $Q = (n - 1)!$ or $Q = - (n - 1)!$ . Extending the works of Wei–Ye and Hyder–Martinazzi to arbitrary odd dimension $n \geq 3$ 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 $V \leq ∣ S^{3} ∣$ , and to the case $n = 4$ , $Q = 6$ , where C.-S. Lin showed that $V \leq ∣ S^{4} ∣$ .

Cite this article

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

DOI 10.4171/RLM/718