Qualitative properties for solutions to conformally invariant fourth order critical systems

Critical curves in conformally invariant statistical systems

We consider critical curves — conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit ...

Fourth Order Equations of Critical Sobolev Growth. Energy Function and Solutions of Bounded Energy in the Conformally Flat Case

In 1983, Paneitz [23] introduced a conformally fourth order operator defined on 4-dimensional Riemannian manifolds. Branson [1] generalized the definition to n-dimensional Riemannian manifolds, n ≥ 5. Such operators have a geometrical meaning. While the conformal Laplacian is associated to the scalar curvature, the Paneitz-Branson operator is associated to a notion of Q-curvature. Possible refe...

On Positive Solutions to Semi-linear Conformally Invariant Equations on Locally Conformally Flat Manifolds

In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn, [g]). We prove that, on a closed and locally conformally flat manifold with...

Qualitative behavior of global solutions to some nonlinear fourth order differential equations

We study global solutions to a fourth order semilinear ordinary differential equation. We determine sufficient conditions on the nonlinearity that ensure global continuation of the solutions. Furthermore, we discuss their qualitative behaviors such as oscillations and boundedness. Mathematics Subject Classification: 34C11, 34A12.

Multiple Nodal Solutions for Some Fourth-Order Boundary Value Problems via Admissible Invariant Sets