Positive Solutions of Two-Point Boundary Value Problems for Monge-Ampère Equations

This paper considers the following boundary value problem: ((−u󸀠(t))n)󸀠 = ntn−1f(u(t)), 0 < t < 1, u󸀠(0) = 0, u(1) = 0, where n > 1 is odd.We establish the method of lower and upper solutions for some boundary value problems which generalizes the above equations and using this method we present a necessary and sufficient condition for the existence of positive solutions to the above boundary value problem and some sufficient conditions for the existence of positive solutions.

After Kutev's works, the Monge-Ampère equation has attracted a growing attention in recent years because of its important role in several areas of applied mathematics.For instance, in [2], Hu and Wang obtained some sufficient conditions for the existence and multiplicity of the positive solutions for problem (1) and Dai [3] and Wang [4] discussed the unilateral global bifurcation results for the problem with () =   + ().And there are some results on the existence and multiplicity and nonexistence of nontrivial radial convex solutions of systems of Monge-Ampère equations (see [5][6][7][8]) also.We notice that the nonlinearity () is continuous at  = 0 in the above works.For the case that () is singular at  = 0, there are fewer results for BVP (1).But some interesting results are presented for BVP (2) in [9][10][11][12] where () is singular at  = 0. We also refer to [7,8,13,14] and references therein for further discussions regarding solutions of the Monge-Ampère equations.
One goal in this paper is to consider the existence of positive solutions under the conditions that  > 1 and () is singular at  = 0. Compared to the results in [10], we do not need the monotonicity of .
Our paper is organized as follows.Section 2 lists some lemmas.In Section 3, we obtain a theorem on the upper and lower solutions which is an extension of this method to the class of problem we consider.Section 4 is devoted to our main results on some sufficient conditions for positive solutions of the BVP (1).
Define a set  ⊆  by 2

Journal of Function Spaces
It can be easily verified that  is indeed a normal cone in  (see [2]).
Lemma 1 (see [15,16]).Let  be a Banach space, and let  ⊆  be a bounded, closed, and convex set.Assume that  :  →  is a continuous compact operator.Then,  has at least one fixed point in .
Remark 2. Obviously, assume that  is a Banach space,  ⊆  is a bounded, closed, and convex set and let  :  →  be a continuous compact operator.Then,  has at least one fixed point in  also.
Proof.Since V  () is decreasing in [0, 1], we have which implies Then, for  ∈ [0, 1], we have That is, The proof is complete.
Remark 4. The idea of the proof of Lemma 3 comes from Lemma 2.2 in [2].Now, we list some conditions for convenience: Throughout this paper, we always assume that condition ( 2 ) holds; that is to say,  is odd.

Upper and Lower Solutions
Consider the following two-boundary value problem: where  :  →  is continuous and  ⊆ .Now, we give following definitions.
then () is called a lower solution of the BVP (11).
Similarly, we can define an upper solution of BVP (11).
then () is called an upper solution of BVP (11).
Then, the necessary and sufficient condition of the existence of the positive solution in  1 [0, 1] ∩  2 (0, 1) for BVP (1) is Proof.
Remark 11.Idea of the proof comes from [19].