POSITIVE RADIAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS IN R^n

By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guarantee the existence of bounded and unbounded radial solutions and consider the nonexistence of positive solution in Rn.

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR,UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de- rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this： if interior gra- dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.

ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPèRE EQUATION

In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampere measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampere equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.

POSITIVE SOLUTIONS AND BIFURCATION OF FULLY NONLINEAR ELLIPTIC EQUATIONS INVOLVING SUPER－CRITICAL SOBOLEV EXPONENTS

This paper discuss the existence of bifurcation point of positive solutions for the fully nonlinear elliptic equations involving super-critical Soboley exponent which include semilinear, MongeAmpere and Hessian equations as its examples, by setting abstract bifurcation theorem via the topological degree theory.

A Liouville Theorem for Möbius Invariant Equations

In this paper, we classify Mobius invariant differential operators of second orderin two-dimensional Euclidean space, and establish a Liouville type theorem forgeneral Mobius invariant elliptic equations. The equationsare naturally associ-ated with a continuous family of convex cones Γ_(p) in R^(2), with parameter p∈[1,2],joining the half plane Γ_(1) := {(λ_(1),λ_(2)) : λ_(1)+λ_(2)> 0} and the first quadrant Γ_(2) := {(λ_(1),λ_(2)) : λ_(1),λ_(2)> 0}. Chen and C. M. Li established in 1991 a Liouvilletype theorem corresponding to Γ_(1) under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville typetheorem we establish in this paper for Γ_(p),1 < p ≤ 2, does not require any additionalassumption on the solution as for Γ_(1). This is reminiscent of the I iouville type theo-rems in dimensions n≥3 established by Caffarelli, Gidas and Spruck in 1989 andby A.B. Li and Y. Y. Li in 2003-2005, where no additional assumption was neededeither. On the other hand, there is a striking new phenomena in dimension n=2 that Γ_(p) ,for p=1 is a sharp dividing line for such uniqueness result to hold without anyfurther assumption on the solution. In dimensions n≥3, there is no such dividing line.

SYMMETRY OF TRANSLATING SOLUTIONS TO MEAN CURVATURE FLOWS

First, we review the authors＇ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.