Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

1 Introduction

The Monge-Ampère equations come from geometric problems, fluid mechanics and other applied subjects. For example, in [51], Trudinger and Wang pointed out that Monge-Ampère equation can describe reflector shape design, or Weingarten curvature. In recent years, increasing attention has been paid to the study of the Monge-Ampère equations by different methods (see [3, 4, 7, 11, 13, 22, 23, 25, 34, 37, 38, 42, 45, 47, 52, 56, 57, 60]). In particular in [62], Zhang and Wang studied the following Monge-Ampère equation

where Ω is a bounded convex domain in ℝ^N (N ≥ 1) with smooth boundary, and D² u denotes the Hessian of u, det D² u is Monge-Ampère operator. Applying the argument of moving plane, the authors firstly verified that any solution on a ball is radially symmetric. Then the authors showed there exists a critical radius such that when the radius of a ball is smaller than this critical value there exists a solution, and the converse is also true.

Recently, using standard approximation arguments, Philippis and Figalli [44] showed that convex Alexandrov solutions of

with 0 < λ ≤ f ≤ Λ, are Wloc2,1 .

In [59], Zhang and Du proved sharp conditions of K(x) and f(u) on the existence of strictly convex solutions to the Monge-Ampère problem

where M[u] = det (u_xixj) is the Monge-Ampère operator, and Ω is a smooth, bounded, strictly convex domain in ℝ^N (N ≥ 2). The approach is largely based on the construction of suitable sub- and super-solutions.

This result of problem (1.3) has also been improved in Zhang and Feng [58], where it is actually verified that problem (1.3) admits a strictly convex solution if and only if f satisfies Keller-Osserman type condition. The asymptotic behavior of strictly convex solutions to (1.3) is also considered under weaker conditions than previous references. However, the most significant one is that if f does not satisfy Keller-Osserman type condition, the authors obtained the existence results of strictly convex solutions under appropriate conditions on K(x). The proof combines the sub-supersolution method with non-standard arguments and Karamata regular variation theory.

At the same time, we notice that many authors have paid more attention to various of system problems, for example, see [2, 10, 14, 15, 16, 20, 21, 29, 30, 32, 35, 40, 46, 49, 50]. Specially, Lair and Wood [36] analyzed the existence of entire positive solutions of system

where p and q are nonnegative, continuous, c-positive, and satisfy the decay conditions ∫0∞ tp(t)dt < ∞ and ∫0∞ tq(t)dt < ∞ for α and β greater than unity, and ∫0∞ tp(t)dt = ∞ and ∫0∞ tq(t)dt = ∞ if neither α nor β is greater than one. In [11], Cîrstea and Rădulescu generalized the results of Lair and Wood [36] to the following system

for p, q ∈ Cloc0,α (Rⁿ) under the condition

for all c > 0.

Recently, by utilizing the same method as papers [11], Ghergu and Rădulescu [8] improved the results of problem (1.4). Very recently, Mavinga and Pardo [41] provided a-priori L^∞ bounds for classical positive solutions of semilinear elliptic systems

where 1 < p, q < ∞ and α, β > 0. Applying moving planes method, Rellich-Pohozaev type identities for systems, and local and global bifurcation techniques, they proved the existence of positive solutions for system (1.5).

On Monge-Ampère system problems, only a few results can be found, for example see Wang, An [53] and Wang [54, 55]. Specially, Zhang and Qi [61] considered the following elliptic system coupled by Monge-Ampère equations

where Ω is a smooth, bounded and strictly convex domain in R^N, N ≥ 2, α > 0, β > 0. When Ω is the unit ball in R^N, applying the index theory of fixed points for completely continuous operators, the authors obtained the existence, uniqueness results and nonexistence of radial convex solutions under some corresponding assumptions on α, β. When α > 0, β > 0 and αβ = N² they also considered a corresponding eigenvalue problem in more general domains.

Moreover, we notice that many authors have paid more attention to existence and uniqueness problems, for example, see [1, 5, 6, 9, 17, 18, 19, 27, 28, 31, 39, 48]. Specially, we would like to mention some results of Hai and Shivaji [33] and Mohammed, Rădulescu and Vitolo [43]. In [33], Hai and Shivaji considered the following p-Laplacian equation

where Δ_p u = div(|∇ u|^p−2∇ u), p > 1, λ is a positive parameter, Ω is the open unit ball in R^N. By the sub-supersolution method together with sharp estimates near the boundary, the authors proved the existence and uniqueness of positive radial solutions to problem (1.6).

In [43], Mohammed, Rădulescu and Vitolo considered the infinite boundary value problem

where Ω ⊆ R^N is a bounded open set with C² boundary ∂ Ω, H[u] := H(x, u, Du, D²u) is a fully nonlinear uniformly elliptic operator. The authors obtained existence and uniqueness results of solution to problem (1.7) by using the maximum principle. They also studied the asymptotic boundary estimates of solutions to problem (1.7).

Inspired by the above works, we are interested in the existence and uniqueness of nontrivial radial convex solutions to the following Monge-Ampère equation

Here λ > 0 is a parameter, D² u denotes the Hessian of u, det(D² u) is Monge-Ampère operator, Ω = {x ∈ R^N : |x| < 1}.

We also give new existence results for system

λ_i (i = 1, 2, …, n) are positive parameters. Our main tool is the eigenvalue theory in cones. However, based on the idea of decoupling method we will investigate composite operators. Besides, the exactly determined intervals of positive parameter λ₁ × λ₂ × ⋯ × λ_n are established. Here we extend the study in Zhang and Qi [61] from a power-type coupled system to a more general system with n parameters, and also generalize that in Hai and Shivaji [33] from a quasilinear problem to a fully nonlinear system problem. Meanwhile, we obtain some new existence results by defining composite operators and using the eigenvalue theory in cones. Moreover, we also analyze the asymptotic behavior of nontrivial radial convex solutions to system (S_λ1,⋯,n).

The rest of the present article is organized as follows. In Section 2, we give some preliminary and known results to be used for the proof of our main results, which mainly includes the eigenvalue theory. Section 3 is devoted to establish the uniqueness result of nontrivial radial convex solution to problem (P_λ). In Section 4, we analyze the exactly determined intervals of positive parameter λ_i (i ∈ {1, 2, …, n}) in which we get the existence and nonexistence results of nontrivial radial convex solutions for system (S_λ1,⋯,n). In Section 5, as an application, some new sufficient conditions for the existence of nontrivial radial convex solutions of the power-type system of Monge-Ampère equations are given. Finally in Section 6, we will discuss the asymptotic behavior of nontrivial radial convex solutions to system (S_λ1,⋯,n) on the parameters λ₁, λ₂, ⋯, λ_n.

2 Preliminaries

Let us search radial convex solutions of (P_λ) and (S_λ1,⋯,n). It is similar to that of Appendix A.2 of [24], one can convert (P_λ) and (S_λ1,⋯,n) to the following equation

and system

Solutions of (2.1) and (2.2) equal to fixed points of certain operators, and we can handle more general equations and systems. Equivalently, we can search positive concave solutions for convenience by letting v = − u and v_i = − u_i(i = 1, 2, …, n), and then we can transform (2.1) and (2.2) to

and

For i = 1, 2, …, n, we assume that f_i satisfies

1. f_i ∈ C(R₊, R₊), R₊ = [0, +∞);

   and f satisfies

2. f ∈ C(R₊, R₊) is strictly increasing and C¹ on (0, ∞);

3. limx→+∞f(x)xN=0,limx→0+f(x)xN>0;
4. limx→+∞xf′(x)f(x)<N,limx→0+xf′(x)<∞.

Let E = C[0, 1]. Then E is a real Banach space with the norm ∥ ⋅ ∥ defined by

Let P ⊂ E be

where

It is easy to see that P is a normal cone of E.

For v ∈ P, define T_i : P → E (i = 1, 2, …, n) to be

It follows from Lemma 2.2 in [44] that T_i (i = 1, 2, …, n) maps P into itself. Moreover, for i = 1, 2, …, n, T_i are completely continuous by standard arguments.

Define a composite operator T1~ = T₁T₂ ⋯ T_n, which is also completely continuous from P to itself. For the case λ₁ = λ₂ = … = λ_n = 1, similar to what Zhang and Qi [61] pointed out that

solves system (S_λ1,⋯,n) if and only if (v₁, v₂, …, v_n) belongs to

satisfying

This shows that if v₁ ∈ P ∖ {0} is a fixed point of T1~ , define v₂ = T₂ v₃, …, v_n = T_n v₁, then v₁ ∈ P ∖ {0} such that

solves system (S_λ1,⋯,n); on the contrary, if

solves system (S_λ1,⋯,n), then v₁ is certainly a nonzero fixed point of T1~ in P. Therefore the task of this paper is to search nonzero fixed points of operator T1~ .

We can also define other composite operators

which have the same meaning as T1~ .

Next, we consider the existence results of system (S_λ1,⋯,n) by using a completely different method from that of [2, 8, 10, 11, 14, 15, 16, 20, 21, 29, 30, 32, 35, 36, 40, 41, 46, 49, 50, 53, 54, 55, 61], namely the eigenvalue theory:

3 Uniqueness of positive concave solution for (P_λ)

In this section, we apply sharp estimates near the boundary to establish the uniqueness results of positive concave solution to system (P_λ).

The next result provides sharp upper and lower estimates for positive concave solutions of (P_λ).

4 New existence and nonexistence results

In this section, we apply Theorems 2.1 to establish the existence and nonexistence of positive radial concave solutions for system (S_λ1,⋯,n).

For i = 1, 2, …, n, let