On a power-type coupled system with mean curvature operator in Minkowski space

We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space {M(u)+vα=0in B,M(v)+uβ=0in B,u|∂B=v|∂B=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{in } B, \\ u|_{\partial B}=v|_{\partial B}=0, \end{cases} $$\end{document} where M(w)=div(∇w1−|∇w|2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1-|\nabla w|^{2}}} )$\end{document} and B is a unit ball in RN(N≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N} (N\geq 2)$\end{document}. We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.


Introduction and main results
Consider the Dirichlet problem of a quasilinear differential system of the type these are related to maximal and constant mean curvature spacelike hypersurfaces having the property that the trace of the extrinsic curvature is zero, respectively, constant (see [12]). It is known (see [1]) that the study of spacelike submanifolds of codimension one in the flat Minkowski space L N+1 (L N+1 := {(x, t) : x ∈ R N , t ∈ R} endowed with the Lorentzian metric N j=1 (dx j ) 2 -(dt) 2 , where (x, t) are the canonical coordinates in R N+1 ) with prescribed mean extrinsic curvature, can lead to the type where is a bounded domain in R N and the nonlinearity H : × R → R is continuous. The existence and multiplicity of positive solutions of problem (1.2) have been discussed in the last two decades by several authors (see [1-7, 15, 16, 20, 22, 23]) in connection with various configurations of H. If is a bounded domain and H is a bounded function defined on × R, Bartnik and Simon [1] proved that the problem (1.2) has a strictly spacelike solution. In particular, if = B R := {x ∈ R N : |x| < R} with R > 0, Bereanu, Jebelean and Torres [2,3] established some existence/nonexistence and multiplicity results for positive radial solutions of problem (1.2) via a Leray-Schauder degree argument and critical point theory. In [6,7,15,20], by using the bifurcation method, the authors studied the existence, multiplicity, and the global behavior of positive solutions of problem (1.2) with H = λf (x, v) on the unit ball. However, to the authors' best knowledge, the study of the Dirichlet problem of a quasilinear differential system with mean curvature operator M seems to be in its early stages, we refer the reader to [10-13, 16, 17, 21] and the references therein. For instance, Gurban et al. [11] investigated the following two-parameter problem By using the fixed-point index, they obtained the following results: nondecreasing with respect to both s, t and satisfy for every r ∈ (0, 1], g i (r, s, t) > 0, ∀s, t > 0, g 1 (r, ξ , 0) = g 2 (r, 0, ξ ) = 0, ∀ξ > 0 where b ∈ (0, 1), 0 < α < 1b are constants. Then, there exist λ * 1 > 0 < λ * 2 such that for all λ 1 > λ * 1 and λ 2 > λ * 2 , problem (1.3) has at least one positive radial solution. Note that (1.1) is a special case of (1.3) and Theorem A does not cover the case where λ 1 = λ 2 = 1.
In 2015, Zhang and Qi [24] studied the following system coupled by Monge-Ampère equations: where is a ball in R N , N ≥ 2, α > 0, β > 0, det D 2 u stands for the determinant of the Hessian matrix ( ∂ 2 u ∂x i ∂x j ) of u. By reducing it to a system coupled by ODEs and using the fixed-point index, they obtained the existence, uniqueness results and nonexistence of radial convex solutions under some corresponding assumptions on α, β.
Motivated by these studies, the main objective of this paper is to investigate the existence/nonexistence and uniqueness of positive radial solutions for system (1.1) on the unit ball B mainly by the fixed-point index in a cone in the same way as in [24]. Our results are completely new and complementary to the results of [11].
We obtain: This paper is organized as follows: In Sect. 2, some preliminaries are given; in Sect. 3, we obtain the main results.

Preliminaries
In order to present the existence results of positive radial solutions for system (1.1), setting r = |x| and u(|x|) = u(r), v(|x|) = v(r), the system (1.1) reduces to the homogeneous mixed boundary-value problem: By a solution of (2.1) we mean a couple of nonnegative functions which satisfies problem (2.1). Here and below, · stands for the usual sup-norm on The following lemma is a direct consequence of [ In particular, for 0 < s 1 ≤ 1 we have Define P to be a cone in C by For each u ∈ P, we define two solution operators T i : P → P (i = 1, 2) as follows: From [19], we know that each operator T i , i = 1, 2 is a nonnegative concave function, this combines with Lemma 2.1, we have T i : P → P is a completely continuous operator. Define a composite operator T = T 1 • T 2 , which is also completely continuous from P to itself. This implies from (2.2) and (2. then v 1 must be a nonzero fixed point of T in P. Hence, our task is to search for nonzero fixed points of T. (i) For any x > θ , there exist θ 1 , θ 2 > 0 such that

Lemma 2.3 ([8]) Let E be a Banach space and K a cone in E. For r
(ii) For any αu 0 ≤ x ≤ βu 0 and t ∈ (0, 1), there exists some η > 0 such that Then, A is called u 0 -sublinear.

Proof of main results
Proof of Theorem 1.1 By Lemma 2.2 and the definition of T 2 , for each u ∈ P, we have By using the same method, we can obtain Hence, we have which implies that 1 . On the other hand, for each u ∈ P, we have In the same way, we can obtain Moreover, Now, let us consider the case of αβ < 1. Since lim s→0 small enough such that for every u ∈ P satisfying u = R 1 , we have 2 u αβ-1
Moreover, by (3.2), we know that there exists R 2 > R 1 , and for each u ∈ P satisfying u = R 2 it holds that Tu < u .
Therefore, i(T, P R 2 \P R 1 , P) = 1, which implies that T has a fixed-point u 1 ∈ P R 2 \P R 1 .
Remark 3.1 Bereanu et al. [3] studied the existence and multiplicity of positive radial solutions for the problem where q > 1. By using the Leray-Schauder degree argument and critical point theory, they obtained a sharper result: there exists > 0 such that problem (3.3) has zero, at least one or at least two positive radial solutions according to λ ∈ (0, ), λ = or λ > . In [11], Gurban et al. investigated the existence of positive radial solutions for the Dirichlet problem of a quasilinear differential system of type where p 1 , q 2 are nonnegative, while q 1 , p 2 are positive exponents. By using the fixed-point index, they proved that there exist λ * i > 0, such that for all λ > λ * i , system (3.4) has a positive radial solution (u, v). We note that the relationship between and 1 (λ * i and 1) is still uncertain. For the same reason, in this paper, if αβ > 1, it is difficult to obtain any results in the case of λ i = 1, i = 1, 2.
Next, we can further prove that the positive radial solution obtained in Theorem 1.
Secondly, we prove that for any θ 1 u 0 ≤ x ≤ θ 2 u 0 and ξ ∈ (0, 1), there exists some η > 0 such that From the definition of T 1 and T 2 , it is easy to obtain Moreover, for 0 < αβ < 1, there exists η > 0 such that which implies that T satisfies Definition 2.4(ii). Then, T is u 0 -sublinear and T has at most one fixed-point in K by Lemma 2.5. Therefore, the system (1.1) has a unique positive radial solution.
Finally, we prove the nonexistence results.
From the proof of Theorem 1.1, we know that for any u ∈ P, T 1 (u) ≤ u α and T 2 (u) ≤ u β . Let u = v 0 , then combining this with the concavity property of v 0 , we obtain that T 1 (v 0 ) < v 0 α and T 2 (v 0 ) < v 0 β . Moreover, which is a contradiction. Therefore, the system (1.1) has no positive radial solution.