A note on a class of higher order conformally covariant equations
Sun-Yung Alice Chang Wenxiong Chen
Discrete & Continuous Dynamical Systems - A 2001, 7(2): 275-281 doi: 10.3934/dcds.2001.7.275
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: non-uniqueness variational method. Higher order semilinear elliptic equations conformally covariant equations
A Liouville problem for the Sigma-2 equation
Sun-Yung Alice Chang Yu Yuan
Discrete & Continuous Dynamical Systems - A 2010, 28(2): 659-664 doi: 10.3934/dcds.2010.28.659
We show that any global convex solution to the Sigma-2 equation must be quadratic.
keywords: Sigma-2 equation Legendre-Lewy transformation Evans-Krylov-Safonov theory.
Principal curvature estimates for the convex level sets of semilinear elliptic equations
Sun-Yung Alice Chang Xi-Nan Ma Paul Yang
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1151-1164 doi: 10.3934/dcds.2010.28.1151
We give a positive lower bound for the principal curvature of the strict convex level sets of harmonic functions in terms of the principal curvature of the domain boundary and the norm of the boundary gradient. We also extend this result to a class of semi-linear elliptic partial differential equations under certain structure condition.
keywords: semilinear elliptic equation. Curvature estimate level sets

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