A note on a class of higher order conformally covariant equations

Sun-Yung Alice Chang; Wenxiong Chen

doi:10.3934/dcds.2001.7.275

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2001, Volume 7, Issue 2: 275-281. Doi: 10.3934/dcds.2001.7.275

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A note on a class of higher order conformally covariant equations

Sun-Yung Alice Chang¹ and
Wenxiong Chen^2,

1.
Department of Mathematics, Princeton University and UCLA
2.
Department of Mathematics, Southwest Missouri State University

Revised: September 2000

Published: April 2000

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Abstract

In this paper, we study the higher order conformally covariant equation

$(- \Delta )^{\frac{n}{2}} w = (n -1)! e^{n w} x \in R^n$

for all even dimensions n.
Let

$\alpha = \frac{1}{|S^n|} \int_{R^n} e^{n w} dx .$

We prove, for every $0 < \alpha < 1$, the existence of at least one solution. In particular, for $ n = 4$, we obtain the existence of radial solutions.

Keywords:

Higher order semilinear elliptic equations,
conformally covariant equations,
non-uniqueness,
variational method.

Mathematics Subject Classification: 35J60.

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