Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space

We deal with a multiparameter Dirichlet system having the form \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -\mathcal M(u) = \lambda_1f_1(u,v), & \hbox{in $\Omega$},\\ -\mathcal M(v) = \lambda_2f_2(u,v), & \hbox{in $\Omega$},\\ u|_{\partial\Omega} = 0 = v|_{\partial\Omega}, \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ \mathcal M $\end{document} stands for the mean curvature operator in Minkowski space \begin{document}$ \mathcal M(u) = \mbox{div} \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right), $\end{document} \begin{document}$ \Omega $\end{document} is a general bounded regular domain in \begin{document}$ \mathbb{R}^N $\end{document} and the continuous functions \begin{document}$ f_1,f_2 $\end{document} satisfy some sign and quasi-monotonicity conditions. Among others, these type of nonlinearities, include the Lane-Emden ones. For such a system we show the existence of a hyperbola like curve which separates the first quadrant in two disjoint sets, an open one \begin{document}$ \mathcal{O}_0 $\end{document} and a closed one \begin{document}$ \mathcal{F} $\end{document} , such that the system has zero or at least one strictly positive solution, according to \begin{document}$ (\lambda_1, \lambda_2)\in \mathcal{O}_0 $\end{document} or \begin{document}$ (\lambda_1, \lambda_2)\in \mathcal{F} $\end{document} . Moreover, we show that inside of \begin{document}$ \mathcal{F} $\end{document} there exists an infinite rectangle in which the parameters being, the system has at least two strictly positive solutions. Our approach relies on a lower and upper solutions method - which we develop here, together with topological degree type arguments. In a sense, our results extend to non-radial systems some recent existence/non-existence and multiplicity results obtained in the radial case.

In this paper we deal with non-potential systems of type where M stands for the mean curvature operator in Minkowski space: Problems involving operator M are originated in differential geometry and in theory of relativity, being related to maximal and constant mean curvature spacelike hypersurfaces (spacelike submanifolds of codimension one in the flat Minkowski space 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 ) having the property that the trace of the extrinsic curvature is zero, respectively constant. The importance of spacelike hypersurfaces in the study of different aspects (singularities, gravitational radiation, positivity of mass) from general relativity was emphasized in [10,11,21,22,23]. Also, their geometric interest is motivated by the fact that the spacelike hypersurfaces in the Minkowski space display nice Bernstein-type properties [1,2,8,9].
In recent years, many papers were devoted to the study of Dirichlet problems for a single equation with operator M in a general bounded domain (see e.g. [3,5,6,13,14,15,20]), while at our best knowledge, for systems with such an operator the study was recently initiated in [18]. Also, results for systems having a radial structure in a ball were obtained in [4,16,17]. So, using a variational approach, in the case of one parameter systems of type    −M(u) = λF u (x, u, v), x ∈ Ω, −M(v) = λF v (x, u, v), x ∈ Ω, u| ∂Ω = 0 = v| ∂Ω , (2) it was proved in [18] that, under suitable assumptions on the potential F , there exists a positive number Λ such that (2) has at least two non-trivial and nonnegative solutions, for all λ > Λ. In [16], for systems involving Lane-Emden type perturbations of the operator M and having a variational structure, where B R = {x ∈ R N : |x| < R} (R > 0, N ≥ 2), the positive exponents p, q satisfy max{p, q} > 1 and the function µ : [0, R] → [0, ∞) is continuous and µ(r) > 0 for all r ∈ (0, R], it was shown that there exists Λ > 0 such that (3) has zero, at least one or at least two positive solutions according to λ ∈ (0, Λ), λ = Λ or λ > Λ. This result extends the corresponding one obtained in [7] in the case of a single equation. Then, in the recent paper [17] were considered non-potential radial systems having the form where λ 1 , λ 2 are two positive parameters, p 1 , p 2 , q 1 , q 2 are positive exponents with min{p 1 , q 2 } > 1 and the weight functions µ 1 , µ 2 : [0, R] → [0, ∞) are assumed to be continuous with µ 1 (r) > 0 < µ 2 (r) for all r ∈ (0, R]. Using some fixed point index estimations and lower and upper solutions method, it was proved the existence of a continuous curve Γ splitting the first quadrant into two disjoint open sets O 1 and O 2 such that the system (4) has zero, at least one or at least two positive (radial) solutions according to (λ 1 , λ 2 ) ∈ O 1 , (λ 1 , λ 2 ) ∈ Γ or (λ 1 , λ 2 ) ∈ O 2 , respectively. In view of the above it appears as being natural to consider also non-potential systems in a general bounded domain Ω ⊂ R N . Since the solvability of (1) is always guaranteed without any additional assumptions on the functions f 1 and f 2 (see [18]), a main interest concerns the multiplicity and possible localization of solutions.
It is the goal of this work to study non-existence, existence and multiplicity of strictly positive solutions (see Definition 2.2) for autonomous systems having the form where the parameters λ 1 , λ 2 are positive, under the following hypothesis: are continuous and quasi-monotone nondecreasing with respect to both u and v; (ii) there exist constants c > 0, p 1 , q 2 > 1 and q 1 , p 2 > 0 such that for all u, v > 0.
Notice that (H f ) is verified by Lane-Emden type nonlinearities, i.e. f 1 (u, v) = u p1 v q1 and f 2 (u, v) = v q2 u p2 . In our main result (Theorem 4.6) we obtain the existence of a continuous curve which separates the first quadrant in two disjoint sets, an open one O 0 and a closed one F, such that (5) has zero or at least one strictly positive solution, according to (λ 1 , λ 2 ) ∈ O 0 or (λ 1 , λ 2 ) ∈ F. The set O 0 is adjacent to the coordinates axes 0λ 1 and 0λ 2 and the curve approaches asymptotically to two lines parallel to the axes 0λ 1 and 0λ 2 . Moreover, we show that there exists (λ * 1 , λ * 2 ) ∈ F such that, for all λ 1 > λ * 1 and λ 2 > λ * 2 , problem (5) has at least two strictly positive solutions.
Our approach relies on a lower and upper solutions method, which we develop for the general system (1) under some hypotheses of quasi-monotonicity on the nonlinearities f 1 and f 2 . This allows us to depict multiplicity and localization informations about solutions. We note that our lower and upper solutions method -which is different from the one in [16,17] used for radial systems, as well as the multiplicity and localization results we obtain for problem (1) (Propositions 3.3 -3.5) are inspired by the corresponding ones proved for a single equation in [13,14].
The rest of the paper is organized as follows. In Section 2 we present the preliminary setup including some definitions and results later used in our approach. Section 3 is devoted to the lower and upper solutions method where the general results concerning the existence, multiplicity and localization of solutions for system (1) are provided. The main non-existence, existence and multiplicity result for the two parameters system (5) is proved in Section 4.

2.
Preliminaries. First, we list some definitions and notations which will be used throughout the paper. Let O be a bounded domain in R N with boundary ∂O of class C 2 . For two functions u, v : Then Concerning system (1), we adopt the following notion of solution.
It is worth pointing out that the notion of solution in the sense of Definition 2.2 fits with the above one in the case of a single equation, as well as with the definition of a solution for a system in the sense of paper [18]. Precisely, one has that if (u, v) is a solution of (1), then u ∈ W 2,r (Ω), v ∈ W 2,s (Ω), for all finite r, s ≥ 1, it satisfies the equations a.e. in Ω and vanishes on ∂Ω. This can be easily checked by following exactly the outline of the argument in [ By means of Lemma 2.1 (i) we can define the operator S : L ∞ (Ω) → W 2,r (Ω) ⊂ C 1 (Ω) (with some r > N fixed), which maps any h ∈ L ∞ (Ω) into the unique solution S(h) := u h ∈ W 2,r (Ω) of problem (7). According to [18,Remark 4.4] (also see [15,Lemma 2.3]) one has that S : L ∞ (Ω) → C 1 (Ω) is completely continuous. Also, from hypothesis (H f ) we have that the Nemytskii operators are continuous from the product space C 1 (Ω) × C 1 (Ω) to L 1 (Ω) and map bounded sets from C 1 (Ω) × C 1 (Ω) into bounded sets in L ∞ (Ω). From this, arguing as in the proof of [18,Theorem 4.5] we infer that the operator T : is completely continuous. A couple of functions (u, v) ∈ C 1 (Ω) × C 1 (Ω) is a solution of (1) iff it is a fixed point of the operator T . Then, since the convex set 3. Lower and upper solutions.
We say that a lower solution (α u , An upper solution is defined by reversing the above inequalities, i.e., • β u ≥ 0, β v ≥ 0 on ∂Ω.
We say that an upper solution (β u , Following the terminology in [16,19], a function f = f (x, s, t) : Ω × J 1 × J 2 → R is said to be quasi-monotone nondecreasing with respect to t (resp. s) if for a.a. x ∈ Ω, and all fixed s ∈ J 1 (resp. t ∈ J 2 ) one has where T is defined in (9) and Proof. The quasi-monotonicity of f 1 and f 2 will play a key role here. Adopting some ideas from [13,14], we divide the proof in three parts.
Step 1. Construction of a modified problem. We define the functions where γ u , γ v : Ω × R → R are given by Then we consider the modified problem which, being of type (1), we know that has a solution.
We prove that α u ≤ u; in a similar way one obtains that Summing up we get This and the strict monotonicity of the function y → y/ 1 − |y| 2 (y ∈ R N with |y| < 1) yield Then, either ∇(u − α u ) = 0 in {u < α u } or the N -dimensional measure of the set {u < α u } is zero. In both cases we get (u − α u ) − = 0 and hence u ≥ α u . Using (11) and similar arguments as above we infer that u ≤ β u and v ≤ β v .
Part 2. Existence of extremal solutions. We have that the fixed points of the operator T defined in (9) are precisely the solutions of problem (1). By the complete continuity of T the bounded set Step 1. We prove the existence of some (w 1 and as in Part 1, for all j = 0, 1, . . . , k, we define the functions and consider the modified problem As (14) is of type (1), we know that it has a solution (z 1 , z 2 ). We have to prove that for a.e. x ∈ Ω. Also, we have for every w ∈ W 1,1 0 (Ω) and, given any j = 1, . . . , k, we will show next that z 1 ≤ u j . Taking w = (z 1 − u j ) + ∈ W 1,1 0 (Ω) in (16) (i = 1) and in (8) (with (u, v) replaced by (u j , v j )) and using the first inequality in (15), we deduce that So, we infer that (z 1 − u j ) + = 0 and hence z 1 ≤ u j . The fact that z 2 ≤ v j results in a similar way. As the inequalities are valid for all j = 1, . . . , k we conclude that z 1 ≤ u 0 , resp. z 2 ≤ v 0 . This also imply that Γ 0 i (x, z 1 , z 2 ) = Γ j i (x, z 1 , z 2 ) for i = 1, 2 and all j = 1, . . . , k. Using this and proceeding exactly as in Step 2 from Part 1 by (16) and (10), it is not difficult to prove that α u ≤ z 1 (resp. α v ≤ z 2 ).
From the above estimates we obtain that the solution (z 1 , z 2 ) of (14) satisfies (1). In particular, we have (z 1 , z 2 ) ∈ k j=1 K uj ,vj , which shows the finite intersection property for the family {K u,v : (u, v) ∈ S}. By the compactness of S, there exists is a minimal solution of (1) lying between the lower solution (α u , α v ) and the upper one (β u , β v ).
Step 2. In order to obtain the existence of (w 2 u , w 2 v ) with w 2 u ≥ u and w 2 v ≥ v for every (u, v) ∈ S, we use a symmetric procedure as the one employed above in Step 1. For each (u, v) ∈ S, we introduce the closed subset of S and we prove the finite intersection property for the family {K u,v : (u, v) ∈ S}. For k ∈ N * and (u 1 , v 1 ), . . . , (u k , v k ) ∈ S, we set u 0 = max{u 1 , . . . , u k }, resp. v 0 = max{v 1 , . . . , v k }. Then α u ≤ u 0 ≤ β u , α v ≤ v 0 ≤ β v and, for all j = 0, 1, . . . , k, we define now the functions and following the outline of the proof of the previous step, we show that any solution (u, v) ∈ S} has the finite intersection property. From this we easily deduce the existence of a maximal solution of (1) lying between (α u , α v ) and (β u , β v ).
Part 3. Degree computation. Let T : C 1 (Ω) × C 1 (Ω) → C 1 (Ω) × C 1 (Ω) be the fixed point operator associated with problem (13), defined as in (9) with N i (i = 1, 2) replaced by N i (u, v) = Γ i (·, u(·), v(·)) (i = 1, 2). We have that there exists a solution (u, v) of (1) with α u ≤ u ≤ β u , α v ≤ v ≤ β v and, since (α u , α v ) and (β u , β v ) are, respectively, a strict lower and a strict upper solution of (1), every such a solution satisfies α u u β u , resp. α v v β v , hence it belongs to U. Therefore, the set U is a non-empty open bounded set in  Assume (H f ) and suppose that f 1 (x, s, t) (resp. f 2 (x, s, t)) is quasi-monotone nondecreasing with respect to t (resp. s). If there exist a strict lower solution (α u , α v ) and a strict upper solution (β u , β v ) of (1) with α u ≤ β u or α v ≤ β v , then problem (1) has at least three solutions (u 1 , v 1 ), (u 2 , v 2 ) and (u 3 , v 3 ) such that and one of the following holds: Proof. The arguments are quite similar to those from the proof of Proposition 3.2 in [13] (also see [14,Proposition 2]). However, for the sake of completeness, we give a sketch of the proof below. Setting if R < s < R + 1, Notice that f R 1 , f R 2 satisfy hypothesis (H f ) and f R 1 (x, s, t), resp. f R 2 (x, s, t) are quasi-monotone nondecreasing with respect to t, resp. s. Now we consider the modified problem Due to the choice of R, Remark 2.4 implies that any solution of (19) is a solution of (1) and (α u , α v ), (β u , β v ) are strict lower and upper solutions of (19). Also, denotingᾱ = −R − 1 andβ = R + 1, clearly one has that (ᾱ,ᾱ), resp. (β,β) are strict lower resp. upper solutions of (19).
Next, we introduce the following open bounded subsets of C 1 0 (Ω) × C 1 0 (Ω): and since (α u , α v ), (ᾱ,ᾱ) are strict lower solutions of (19), and (β u , β v ), (β,β) are strict upper solutions of (19), one has Then, from Proposition 3.3, we infer (20) and the additivity-excision property of the Leray-Schauder degree, imply (see [13,14]) Since U β α , Uβ α and V are pairwise disjoint, from the above degree calculations we infer that there exist three distinct fixed points (u 1 , v 1 ), This means that (17) is fulfilled and using again the fact that (α u , α v ) and (β u , β v ) are, respectively, a strict lower and a strict upper solution of (19), one of the following holds: Next, we observe that we can replace (u 1 , v 1 ) and (u 3 , v 3 ) with the minimal, resp. the maximal solution of (19) lying between (ᾱ,ᾱ) and (β,β) and (17) still remains valid. Additionally, we have that Then, it is easy to check that (i ) − (iv ) imply (i) − (iv). The conclusion follows from the fact that this solutions also solve problem (1).

Proposition 3.5.
Assume (H f ) and that f 1 (x, s, t) (resp. f 2 (x, s, t)) is quasimonotone nondecreasing with respect to t (resp. s). If there exists a lower solution (α u , α v ) of problem (1), then it has at least one solution (u, v) with u ≥ α u and v ≥ α v .
Proof. We consider problem (19) constructed in the proof of Proposition 3.4 with instead of R given by (18). From Remark 2.4 we see that any solution of (19) is a solution of the original problem (1). Settingβ = R + 1 one has that (α u , α v ), (β,β) are lower, respectively upper solutions for (19), with α u ≤β and α v ≤β. Then, Proposition 3.3 implies the existence of at least one solution of (19) lying between (α u , α v ) and (β,β), which also is a solution for (1).
4. Non-existence, existence, multiplicity and localization results. In this section, under hypothesis (H f ), we study the existence and multiplicity of strictly positive solutions for the autonomous system (5). In order to treat (5) we introduce g i (s, t) = f i (s + , t + ) (s, t ∈ R, i = 1, 2) and consider the modified system Remark 4.1. (i) Notice that on account of (H f ) (ii) one has that g 1 (ξ, 0) = g 1 (0, ξ) = 0 = g 2 (ξ, 0) = g 2 (0, ξ), ∀ ξ ≥ 0, hence, problem (21) always has the trivial solution. Also, each solution (u, v) of (21) satisfies u ≥ 0 ≤ v (see [18,Lemma 3.6]) and it solves (5). ) is a non-trivial solution of (21) with some λ 1 , λ 2 > 0, then (u, v) is strictly positive. Indeed, from (22) and since (u, v) is non-trivial we easily infer that uv > 0. This ensures that u(x) > 0 and v(x) > 0 for all x belonging to a set of positive measure. Then, on account of (6) one has −M(u) > 0 and −M(v) > 0 in Ω and Lemma 2.1 (ii) yields that u 0 and v 0. Proof. This can be done as in the proof of [12, Theorem 2.5] by using a time-map argument. Here, we give an alternative proof. With this aim, let us consider the Cauchy problem with r > 0. From the proof of Proposition 2 in [7], one has that (24) has an unique solution w which is the unique fixed point of the compact operator where φ −1 (y) = y/ 1 + y 2 , y ∈ R. Note that the solution of (24) is strictly decreasing on [a, b]. Next, let {r k } be a sequence of positive numbers such that r k 0, as k → +∞, and w k be the solution of (24) with r = r k . If we assume, by contradiction, that there are infinitely many w k which vanish in some R k ∈ [a, b] then, relabeling the sequence {w k } if necessary, we can write Using the mean value theorem, we get with some t k ∈ [a, R k ]. Then, as w + k ∞ = r k we infer a contradiction. Consequently, there exists a sequence {w k } in C 2 [a, b] of solutions of (23), such that, for every k, min [a,b] w k > 0 and lim k→+∞ w k ∞ = lim k→+∞ r k = 0.
Hereafter, for x ∈ R N and ρ > 0 we denote by B ρ (x) the open ball centered at x of radius ρ and by B ρ (x) its closure. Under hypothesis (H f ), there exist λ * 1 , λ * 2 > 0 such that problem (5) has at least two strictly positive solutions, for each λ 1 > λ * 1 and λ 2 > λ * 2 . Proof. We shall apply Proposition 3.4. With this aim we divide the proof in two steps.
To see that (ii) holds true, we apply Proposition 4.4 (ii).
Taking into account the definition of Λ(θ) and using Propositions 4.3 and 4.5 we get the conclusion.