CONVEX SOLUTIONS OF SYSTEMS ARISING FROM MONGE-AMPÈRE EQUATIONS

We establish two criteria for the existence of convex solutions to a boundary value problem for weakly coupled systems arising from the Monge-Ampere equations. We shall use fixed point theorems in a cone.


Introduction
In this paper we consider the existence of convex solutions to the Dirichlet problem for the weakly coupled system where N ≥ 1.A nontrivial convex solution of (1.1) is negative on [0,1).Such a problem arises in the study of the existence of convex radial solutions to the Dirichlet problem for the system of the Monge-Ampère equations where B = {x ∈ R N : |x| < 1} and detD 2 u i is the determinant of the Hessian matrix ( ∂ 2 u i ∂xm∂xn ) of u i .For how to reduce (1.2) to (1.1), one may see Hu and the author [5].

H. Wang
The Dirichlet problem for a single unknown variable Monge-Ampère equations in general domains in R n may be found in Caffarelli, Nirenberg and Spruck [1].Kutev [7] investigated the existence of strictly convex radial solutions of (1.3) when f (−u) = (−u) p .Delano [3] treated the existence of convex radial solutions of (1.3) for a class of more general functions, namely λ exp f (|x|, u, |∇u|).
The author [10] and Hu and the author [5] showed that the existence, multiplicity and nonexistence of convex radial solutions of (1.3) can be determined by the asymptotic behaviors of the quotient f (u) u N at zero and infinity.In this paper we shall establish the existence of convex radial solutions of the weakly coupled system (1.1) in superlinear and sublinear cases.First, introduce the notation x N , and x N .We shall show that if (1.1) is superlinear, or f 0 = g 0 = 0 and f ∞ = g ∞ = ∞, (1.1) is sublinear, or f 0 = g 0 = ∞ and f ∞ = g ∞ = 0, then (1.1) has a convex solution.
Our main results are: 1) has a convex solution.

Preliminaries
With a simple transformation v i = −u i , i = 1, 2 (1.1) can be brought to the following equation (2.4) Now we treat positive concave classical solutions of (2.4).
The following well-known result of the fixed point index is crucial in our arguments.2,4,6]).Let E be a Banach space and K a cone in E. For r > 0, In order to apply Lemma 2.1 to (2.4), let X be the Banach space Also, define, for r a positive number, Ω r by Let T : K → X be a map with components (T 1 , T 2 ), which are defined by It is straightforward to verify that (2.4) is equivalent to the fixed point equation The following lemma is a standard result due to the concavity of u.We prove it here only for completeness.
EJQTDE Spec.Ed.I, 2009 No. 26 Proof Note, from the definition of Similarly, We define new functions f (t), ĝ(t Note that f0 = lim t→0 t N and ĝ0 , ĝ∞ can be defined similarly.