Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

A note on a class of higher order conformally covariant equations

Pages: 275 - 281, Volume 7, Issue 2, April 2001      doi:10.3934/dcds.2001.7.275

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Sun-Yung Alice Chang - Department of Mathematics, Princeton University and UCLA, United States (email)
Wenxiong Chen - Department of Mathematics, Southwest Missouri State University, United States (email)

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.

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

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.

Revised: September 2000;      Available Online: January 2001.