Multiplicity results for system of Pucci’s extremal operator

Abstract

This article deals with the existence of multiple positive solutions to the following system of nonlinear equations involving Pucci’s extremal operators:

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2u_1)&=f_1(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2u_2)&=f_2(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ ~~~~~~~~\vdots&=~~~~~~~~~~~~ \vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2u_n)&=f_n(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ u_1=u_2=\dots =u_n&=0~~{} & {} \textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

where $\Omega$ is a smooth and bounded domain in $\mathbb {R}^N$ and $f_i:[0,\infty )\times [0,\infty )\dots \times [0,\infty )\rightarrow [0,\infty )$ are $C^{\alpha }$ functions for $i=1,2,\dots ,n$. The multiplicity result in this work is motivated by the work Amann (SIAM Rev 18(4):620–709, 1976), and Shivaji (Nonlinear analysis and applications (Arlington, Tex., 1986), Dekker, New York, 1987), where the three solutions theorem (multiplicity) has been proved for linear equations. Later on, it was extended for a system of equations involving the Laplace operator by Shivaji and Ali (Differ Integr Equ 19(6):669–680, 2006). Thus, the results here can be considered as a nonlinear analog of the results mentioned above. We also have applied the above results to show the existence of three positive solutions to a system of nonlinear elliptic equations having combined sublinear growth by explicitly constructing two ordered pairs of sub and supersolutions.

1 Introduction

This article is concerned with the existence of multiple positive solutions to the following system of nonlinear boundary value problems:

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2u_1)&=f_1(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2u_2)&=f_2(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ ~~~~~~~~\vdots&=~~~~~~~~~~~~ \vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2u_n)&=f_n(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ u_1&=u_2=\dots =u_n=0~~{} & {} \textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

(1.1)

where $\mathcal {M}_{\lambda ,\Lambda }^+$ is the Pucci’s extremal operator (see, Equation 2.1), $\Omega$ is a bounded domain with smooth boundary in $\mathbb {R}^{N}$ with $N \ge 1$ and $f_i$ are locally Hölder continuous function satisfying the following monotonicity type condition:

• (C1) For each fixed $1\le i\le n,$ there is a positive constant $\mu _i>0$ such that for any $r,s>0$ with $r>s$, we have

  $$\begin{aligned}{} & {} f_i(u_1,u_2,\dots ,u_{i-1},r,u_{i+1},\dots ,u_n)-f_i(u_1,u_2,\dots ,u_{i-1},s,u_{i+1},\dots ,u_n)\\{} & {} \quad >-\mu _i(r-s). \end{aligned}$$

We also impose the following cooperating coupling condition for the system (1.1), a.e.

• (C2) $u_{j} \rightarrow f_i(u_1,u_2,\dots ,u_n)$ is nondecreasing if $i\ne j,$ and $u_l$, for $l\ne j$, are fixed.

More precisely, under the assumptions $(C1)-(C2)$, we establish the analogous results of [1] in the context of system of fully nonlinear elliptic equations of the non-divergence form (1.1). As an application of it, we have shown the existence of three positive solutions to the system (1.1) when $f_i$ satisfies a combined sublinear growth at infinity. Here, the existence of three solutions has been accomplished by combining three solutions type theorem with the construction of two appropriate ordered pairs of sub and supersolutions of the considered problem. The order on the set of vector valued functions is induced by the order of their respective component functions for the precise description see the discussion after Definition 2.8.

The study of existence, uniqueness, multiplicity, and qualitative behavior of positive solutions to the system of equations has been attracting continuous attention for a long time. For the system involving linear and quasilinear equations, we refer to [1, 2, 11, 15, 16, 19, 21, 22, 34]. On the other hand for similar results in the setting of fully nonlinear elliptic equations of the non divergence form, we refer to [20, 24, 25, 32].

To put our problem in the right perspective, let us recall some already existing results in this direction. First of all, three solution type theorem in the setting of an abstract nonlinear operator from one Banach space to another appears in [4]. Subsequently, R. Shivaji successfully adopted the results of H. Aman in the context of semilinear elliptic equations to establish the existence of multiple(three) positive solutions to a class of semilinear elliptic equations [33]. Later on, it has been applied to show the existence of multiple positive solutions to the system of equations involving different types of elliptic operators. For instance, by following the same techniques J. Ali et al. have shown the existence of multiple positive solutions to a system of equations involving the Laplace operator [1]. For the existence of multiple positive solutions to the system of equations involving $p-q$ Laplace operator [34].

At the same time, the study of the existence of solutions to fully nonlinear elliptic operators also has dragged continuous attention. For the existence of solutions to equations involving fully nonlinear elliptic equations, we refer to [7, 9, 12,13,14, 28,29,30]. As far as the existence of multiple positive solutions to fully nonlinear elliptic equations is concerned we would first like to refer [8], where F. Charro et al. have proved the existence of multiple (at least two) positive solutions to equations having concave and convex nonlinearities. In [39], author proves the existence of two positive solutions to fully nonlinear elliptic equations having positive zeros along with superlinear growth at infinity. Recently, Sirakov et al. [26] studied the existence of multiple (at least two) positive solutions to fully nonlinear elliptic equations having superlinear growth in gradient. The authors, in [23], have proved the existence of three positive solutions to Pucci’s extremal equations having combined sublinear growth at infinity. For more results dealing with the existence of multiple positive solutions, we refer to [3, 12, 14, 36].

On the other hand during the last three decades there have been many attempts to extend the results available for scalar elliptic equations in non-divergence form to cooperative coupled elliptic systems of second order equations. We would like to provide some references in this direction. In [6], authors have proved the ABP Maximum principle and Harnack type inequality for system of fully nonlinear elliptic equations. For the results dealing with the generalization of Liouville type theorem to the system of fully nonlinear elliptic equations, we refer to [20, 24, 32]. For the existence of solution to a monotone system of fully nonlinear elliptic equations, we refer to [31]. In [32], authors also have established the existence of positive solutions to the system of equations having superlinear growth at infinity. In fact, the results of [32] can be seen as a direct extension of the results in [29] to the system of similar equations. Recently, Sirakov et al. [25] have established multiple positive solutions to a class of system of fully nonlinear elliptic equations having natural growth in gradient and linear growth in the dependent variable. Again this work can be seen as a generalization of [25] in the context of system of elliptic equations. By keeping the above works in mind here we would like to prove the existence of three positive solutions to a system of fully nonlinear elliptic equations. At this juncture, we would like to point out two facts. Firstly, this work can be thought of as a generalization of [23] to the system of equations. Secondly, on the one hand, our results assert the existence of multiple (three) positive solutions to system of equations having less general fully nonlinear operators than the operators considered in [32]. On the other hand, the nonlinearity in the dependent variable considered here is completely different from the one considered in [32]. Lastly, here we have established the existence of three positive solutions while [32] deals with the existence of two positive solutions.

This paper is divided into several sections. In Sect. 2, we present some basic definitions and preliminary results concerning Pucci’s extremal operator. Section 3 is devoted to the proof of three solution theorem (Theorem 3.7) for Problem 1.1. In Sect. 4, by using the results of Sect. 3 we state and prove our existence and multiplicity results (by explicitly constructing two ordered pairs of sub and supersolutions) for the problem of the type (1.1), where $f_i$ has a combined sublinear growth at infinity. The Last section contains an example of sublinear functions $f_i$ for which Problem 1.1 has at least three positive solutions.

2 Preliminaries

In this section we collect definitions and results which will be used throughout the article. For a given $0<\lambda <\Lambda$, Pucci’s extremal operators are defined as follows:

$$\begin{aligned} \mathcal {M}^{\pm }_{\lambda ,\Lambda }(M)=\Lambda \sum _{\pm e_{i}>0}e_{i}+\lambda \sum _{\pm e_{i}<0}e_{i}, \end{aligned}$$

(2.1)

where M is an $N\times N$ symmetric matrix and $e_i's$ are its eigenvalues. Many times in this article, we explicitly need the expression obtained by acting Pucci’s extremal operator on the Hessian of u. In general, it is very difficult to compute to eigenvalues of the Hessian of a function. However, if the function u is radial then the eigenvalues of $D^{2}u$ can easily be computed. In fact, we have the following lemma.

Lemma 2.1

(Lemma 3.1 [10]) Let $\tilde{u}:~[0, \infty )\longrightarrow \mathbb {R}$ be $C^{2}$ function such that $u(x)=\tilde{u}(|x|)$. Then for any $x\in \mathbb {R}^{N}\setminus \{0\},$ the eigenvalues of the Hessian $D^{2}u(x)$ are $\frac{\tilde{u}'(|x|)}{|x|}$ with multiplicity $N-1$ and $\tilde{u}''(|x|)$ with multiplicity 1.

The operators considered here are highly nonlinear and therefore the associated equations in general do not have classical solutions. Moreover, these operators are of the nondivergence form. Therefore, the appropriate notion of weak solutions for studying the problem involving these operators is different from the notion of weak solutions emerging from the theory of integration by parts. The notion of solution is called viscosity solution, which is defined as follows:

Definition 2.2

(see [9, 18]) A vector valued function $u=(u_1,u_2,...,u_n)\in C(\overline{\Omega })\times C(\overline{\Omega })\times \dots \times C(\overline{\Omega })$ is called a viscosity subsolution (resp., supersolution) of (1.1) if for any $\phi \in C^{2}(\Omega )$ such that $u_{j}-\phi$ (for some $j=1,2...,n$) has a maximum (resp., minimum) at some point $x_{0}\in \Omega$ then we have

$$\begin{aligned} -\mathcal {M}_{\lambda _{j},\Lambda _{j}}^+ (D^2\phi (x_{0}))\le f(u_{j}(x_{0}))~~\Big (\text {resp.,}~~\ge f(u_{j}(x_{0}))\Big ). \end{aligned}$$

u is solution if it is both subsolution and supersolution at the same time. Moreover, u is called strict subsolution (resp., supersolution) of (1.1) if u is a subsolution (resp., supersolution) of

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2u_1)&=f_1(u_1,u_2,\dots ,u_n)+h_1~~~{} & {} \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2u_2)&=f_2(u_1,u_2,\dots ,u_n)+h_2~~~{} & {} \textrm{in}~~\Omega ,\\ ~~~~~~~~\vdots&=~~~~~~~~~~~~ \vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2u_n)&=f_n(u_1,u_2,\dots ,u_n)+h_{n}~~~{} & {} \textrm{in}~~\Omega , \end{aligned} \right. \end{aligned}$$

for some $h=(h_1,h_2,...,h_n)\in C(\overline{\Omega })\times \dots \times C(\overline{\Omega })$ such that $h_j(x)<0$ (resp., $h_{j}(x)>0$) on $\bar{\Omega }$ for all $j=1,2,...,n.$

As the system of equations considered here satisfied cooperative coupling condition. Therefore, we will quite frequently be borrowing results available for scalar equations. In order to prove the three solutions type theorem we convert the problem into another operator equation and show that the obtained operator is continuous, compact, etc. In order to prove the continuity of the operator we need the following fundamental result from the theory of viscosity solutions known as the Alexandroff-Bakelman-Pucci(ABP) maximum principle. For the scalar equation we denote the ellipticity constants by $\lambda , \Lambda$ instead of $\lambda _j,\Lambda _j.$

Theorem 2.3

(Theorem 2.12 [7]) Suppose $u\in C(\bar{\Omega })$ is a viscosity solution of

$$\begin{aligned}-\mathcal {M}^{+}_{\lambda ,\Lambda }(D^{2}u)+\mu u\le g(x)~\text {in}~~(resp.~-\mathcal {M}^{-}_{\lambda ,\Lambda }(D^{2}u)+\mu u\ge g(x)), \end{aligned}$$

in $\Omega ,$ where $g\in L^{N}(\Omega )\cap C(\Omega ).$ Then

$$\begin{aligned} \sup _{\Omega }u\le \sup _{\partial \Omega }u^{+}+\text {diam}(\Omega ). C_{1}\Vert g^{-}\Vert _{L^{N}(\Omega )} \end{aligned}$$

(2.2)

(resp., $\sup _{\Omega } u^{-}\le \sup _{\partial \Omega }u^{-}+\text {diam}(\Omega ). C_{1}\Vert g^{+}\Vert _{L^{N}(\Omega )}$), where $C_{1}$ is a positive constant which depends on $N,~\lambda , \Lambda , \text {diam}(\Omega )$.

As we have discussed above we have been converting the system (1.1) into the operator equations and then show that the operator is compact. In the process of proving the concerned operator is compact, we need the following theorem asserting the $C^{1,\alpha }-$estimate for solutions to fully nonlinear elliptic equations. On the one hand, In the Problem 1.1, the involving operators are convex in the Hessian so by Evans-Krylov theorem the solution would be classical as long as $g\in C^{\alpha }.$ On the other hand we have $g\in C(\Omega )$ only, so the solution is not classical but still we have the following $C^{1,\alpha }$ estimate.

Theorem 2.4

([37, 38]) There exist two positive constants $\alpha$ and C depending on $\lambda , \Lambda ,\mu ,\Omega$ such that for any solution of

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda ,\Lambda }^{+}(D^2u)+\mu u =&g~~\text{ in } ~~~\Omega ,\\ u=&0~\text{ on }~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

(2.3)

The following estimate holds

$$\begin{aligned}\Vert u\Vert _{C^{1,\alpha }(\overline{\Omega })}\le C(\Vert u\Vert _{L^{\infty }(\Omega )}+\Vert g\Vert _{L^{\infty }(\Omega )}). \end{aligned}$$

Moreover, since the operator is proper (that is, increasing in u), we have

$$\begin{aligned}\Vert u\Vert _{C^{1,\alpha }(\overline{\Omega })}\le C\Vert g\Vert _{L^{\infty }(\Omega )}. \end{aligned}$$

In the above $C^{1,\alpha }$ estimate we need that the function u to be a solution of (2.3). However, if u is not a solution but instead satisfies the following set of inequations

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda ,\Lambda }^+(D^2u)+\alpha u&\le |g(x)|~~ \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda ,\Lambda }^-(D^2u)+\alpha u&\ge -|g(x)|~~ \textrm{in}~~\Omega ,\\ u&=0~~\textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

(2.4)

then the Krylov–Safonov theorem implies that the solution is Hölder continuous see Theorem 1.10 [38] instead of $C^{1,\alpha }$. We also are using the following boundary Lipschitz estimate from [30]. It is essentially the same as in Proposition 4.9 [30] but we need the constant to be independent of f and u. The following theorem in the present form has been taken from [23].

Theorem 2.5

(Proposition 4.9 [30], Proposition 2.4 [35]) Suppose u satisfies (2.4). There exists a constant C which depends on $\lambda ,\Lambda , N, \text {diam}(\Omega ),$ such that for any $x\in \Omega$ and $x_{0}\in \partial \Omega$ we have

$$\begin{aligned} |u(x)|\le C(\Vert u\Vert _{L^{\infty }(\Omega )}+\Vert g\Vert _{L^{\infty }(\Omega )})|x-x_0|. \end{aligned}$$

(2.5)

Next, we state a result dealing with the existence of the eigenvalue and eigenfunction for Pucci’s extremal operator. We will be using this eigenfunction for the construction of the sub-solution of some appropriate equations.

Theorem 2.6

(Proposition 1.1 [5]) There exist a positive constant $\mu ^{+}_{1,i}$ and a function $\phi ^{+}_{1,i}\in C^{2}(\Omega )\cup C(\bar{\Omega })$ such that:

$$\begin{aligned} \left\{ \begin{aligned}{} -\mathcal {M}^{+}_{\lambda _i,\Lambda _i}(D^{2}\phi ^{+}_{1,i})&=\mu ^{+}_{1,i} \phi ^{+}_{1,i}~~\text {in}~\Omega \\ \phi ^{+}_{1,i}&=0~\text {on}~\partial \Omega . \end{aligned} \right. \end{aligned}$$

(2.6)

Moreover $\phi ^{+}_{1,i}>0$ in $\Omega .$

We also use the solution $e_i$ of (2.7) mentioned in the following theorem to define appropriate ordered Banach space as well as in the construction of a sub solution.

Theorem 2.7

(Theorem 17.18 [17]) Let $e_i\in C^2(\Omega )\cap C(\bar{\Omega })$ be the unique solution to the following problem

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2e_i)&=1~~\text{ in } ~~~\Omega ;\\ e_i&=0~\text{ on }~~\partial \Omega . \end{aligned} \right. \end{aligned}$$

(2.7)

It is clear that the solution $e_i\ge 0.$ Furthermore, by using the strong maximum principle and Höpf type lemma we can show that $e_i(x)\ge k_i d(x)$ for some $k_i>0$ and $d(x)=\text {dist}(x,~\partial \Omega ).$

For each $i=1,2,\dots ,n$ we denote by $C_{e_i}(\bar{\Omega })$ the set of functions in $u\in C_0(\bar{\Omega })$ (continuous functions which vanishes on $\partial \Omega$) such that $-te_i\le u\le te_i$ for some $t>0$. $C_{e_i}(\bar{\Omega })$ equipped with norm $\Vert u\Vert _{e_i}=\inf \{t>0:-te_i\le u\le te_i\}$ is an ordered Banach space with positive cone $P_{e_i}=\{u\in C_{e_i}(\bar{\Omega }):u(x)>0\}$ which is normal and has non empty interior. In fact, interior $\overset{o}{P}_{e_i}$ consists of all those functions $u\in C_0(\bar{\Omega })$ with $t_{1,i}e_i\le u\le t_{2,i}e_i$ for some $t_{1,i},t_{2,i}>0$. For each fixed i, we also have the following embedding

$$\begin{aligned} C_0^1(\bar{\Omega })\hookrightarrow C_{e_i}(\bar{\Omega })\hookrightarrow C_0(\bar{\Omega }). \end{aligned}$$

(2.8)

Next, we define the following operator. Our existence result will be the consequence of the properties of this operator. Indeed, we prove that this operator is completely continuous and strongly increasing in certain spaces.

Definition 2.8

Define $T_i:C_0(\bar{\Omega })\times C_0(\bar{\Omega })\dots \times C_0(\bar{\Omega })\rightarrow C_0^1(\bar{\Omega }),$ where $T_i(u_1,u_2,\dots ,u_n)=w_i$ is the unique viscosity solution of

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2w_i)+\mu _{i} w_i&= f_i(u_1,u_2,\dots ,u_n)+\mu _{i} u_i ~\text {in}~~ \Omega ,\\ w_i&=0~\text {on}~~ \partial \Omega , \end{aligned} \right. \end{aligned}$$

and $\mu _i>0$ from (C1) for $i=1,2,\dots ,n$. Since the operator $-\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2\dot{)}+\mu _{i}$ is proper, the existence of unique $w_i$ is guaranteed by Theorem 4.9 [9]. Define $T:C_0(\bar{\Omega })\times C_0(\bar{\Omega })\dots \times C_0(\bar{\Omega })\rightarrow C_0^1(\bar{\Omega })\times C_0^1(\bar{\Omega })\dots \times C_0^1(\bar{\Omega })$ as $T(u_1,u_2,\dots ,u_n)= \big (T_1(u_1,u_2,\dots ,u_n),T_2(u_1,u_2,\dots ,u_n),\dots ,T_n(u_1,u_2,\dots ,u_n)\big ).$ In view of Theorem 2.4 and compact embedding $C^{1,\alpha }_{0}(\bar{\Omega })\hookrightarrow C^{1}_{0}(\bar{\Omega }),$ T is a compact operator. In fact, (2.8) implies that

$$\begin{aligned} T:C_{e_1}(\bar{\Omega })\times C_{e_2}(\bar{\Omega })\dots C_{e_n}(\bar{\Omega })\longrightarrow C_{e_1}(\bar{\Omega })\times C_{e_2}(\bar{\Omega })\dots C_{e_n}(\bar{\Omega }) \end{aligned}$$

(2.9)

is a compact operator.

After fixing $\lambda _1,\lambda _2,\dots ,\lambda _n$ and $\Lambda _1,\Lambda _2,\dots ,\Lambda _n$ we find that $e_1,e_2,\dots ,e_n$ are fixed. Therefore, from here onwards we denote $C_{e_1}(\bar{\Omega })\times C_{e_2}(\bar{\Omega })\dots C_{e_n}(\bar{\Omega })$ by $C_{E}(\overline{\Omega }).$ For any two $u=(u_1,u_2,\dots ,u_n), ~v=(v_1,v_2,\dots ,v_n)\in C_{E}(\overline{\Omega })$ we say that $u\le v$ if $u_i\le v_i$ for all $i=1,2,\dots ,n.$ We say that $T:C_{E}(\overline{\Omega })\longrightarrow C_{E}(\overline{\Omega })$ is increasing if $u\le v$ in $C_{E}(\overline{\Omega })$ implies $T(u)\le T(v),$ that is $T_i(u)\le T_{i}(v)$ for $i=1,2,\dots ,n.$

3 Main results

Lemma 3.1

A function u solves (1.1) iff u is a fixed point of the map T in $C_{E}(\overline{\Omega }).$

Proof

Suppose that $u=(u_1,u_2,\dots ,u_n)$ is a solution of (1.1). Then Theorem 2.4 and (2.8) implies that $u_{i}\in C_{e_i}(\overline{\Omega })$ for each $i=1,2,\dots ,n$ and $T(u)=u$ by definition of T. Conversely, suppose that $u=(u_1,u_2,\dots ,u_n)\in C_{E}(\overline{\Omega })$ is a fixed point of T. Again by Theorem 2.4, $u_i\in C^{1}_{0}(\bar{\Omega })$ for each $i=1,2,\dots ,n$ and so $g(u_i)=\mu _i u_i+f_i(u_1,u_2,\dots ,u_n)$ is Hölder continuous, consequently by Evans-Krylov theorem we get $u_i\in C^{2}(\Omega )$ for each $i=1,2,\dots ,n$ and $u=(u_1,u_2,\dots ,u_n)$ is a solution of (1.1).

In view of Lemma 3.1, proving the existence of three solutions to Problem 1.1 is equivalent to showing the existence of three fixed points of T. This will be achieved by applying the Lemma 3.6 stated below. In order to apply Lemma 3.6 we need to verify that T satisfies all the required assumptions.

Lemma 3.2

The map T is well defined and monotone operator from $C_{E}(\overline{\Omega })$ to $C_{E}(\overline{\Omega }).$

Proof

Let $u=(u_1,u_2,\dots ,u_n),v=(v_1,v_2,\dots ,v_n)\in C_{E}(\overline{\Omega })$ with $u\le v$ and $U=(U_1,U_2,\dots ,U_n)=T(u)$ and $V=(V_1, V_2,\dots ,V_n)=T(v)$. So we have

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}^{+}_{\lambda _i,\Lambda _i} (D^2U_i)+\mu _i U_i&= f_i(u)+\mu _i u_{i}~~\text {in}~~ \Omega ;\\ U_i&=0~\text {on}~~ \partial \Omega . \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}^{+}_{\lambda _i,\Lambda _i} (D^2V_i)+\mu _i V_i&= f_i(v)+\mu _i v_{i}~~\text {in}~~ \Omega ;\\ V_i&=0~\text {on}~~ \partial \Omega . \end{aligned} \right. \end{aligned}$$

for $i=1,2,\dots ,n$. Using (C1) and (C2), it is easy to observe that for each $i=1,2,\dots ,n,$

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^{+} (D^2(U_i-V_i)+\mu _i(U_i-V_i)&\le f_i(u)-f_i(v)+\mu _i(u_i-v_i)\le 0. \end{aligned}$$

By applying the ABP estimate we have

$$\displaystyle {\sup _{\Omega }}(U_i-V_i)\le \displaystyle {\sup _{\partial \Omega }}(U_i-V_i)^+=0~~~~(\text {since}~~U_i=V_i~~\text {on}~~\partial \Omega ).$$

Hence $U_i\le V_i$ for each $i=1,2,\dots ,n$, which is same as $T_i(u)\le T_i(v).$ The same argument works for other values of i,  which proves that, T is monotonically increasing. Moreover, the above calculation also asserts that T is well defined.

Our next aim is to show that T is continuous from $C_{E}(\overline{\Omega })$ to $C_{E}(\overline{\Omega })$. ABP maximum principle simply implies the continuity of T from $C_{0}(\bar{\Omega })\times C_{0}(\bar{\Omega })\times \dots \times C_{0}(\bar{\Omega })$ to itself. Here, we apply the boundary Lipschitz estimate Theorem 2.5 to show the continuity of T from $C_{E}(\overline{\Omega })$ to $C_{E}(\overline{\Omega }).$

Theorem 3.3

The map $T: C_{E}(\overline{\Omega })\rightarrow C_{E}(\overline{\Omega })$ is completely continuous.

Proof

We have already noticed that $T_i$ is compact for $i=1,2,\dots ,n$. Here we prove that $T_i$ is continuous which immediately implies that T is completely continuous. Let $u^{l}=(u^l_1,u^l_2,\dots ,u^l_n)\in C_{E}(\overline{\Omega })$ such that $u^l\rightarrow u$ in $C_{E}(\overline{\Omega })$ as $l\rightarrow \infty .$ Now, we want to show that $T_i(u^l)\rightarrow T_i(u)$ in $C_{e_i}(\bar{\Omega })$ for $i=1,2,\dots ,n.$

Claim: $T_i(u^{l})\rightarrow T_i(u)$ in $C_{0}(\bar{\Omega })$ for $i=1,2,\dots ,n.$

Observe that the following inequality holds in the viscosity sense

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+ \Big (D^2\big (T_i(u^l)-T_i(u)\big )\Big )+\mu _i \big (T_i(u^l)-T_i(u)\big )\le f_i(u^l)-f_i(u)+\mu _i (u^l_{i}-u_i). \end{aligned}$$

(3.1)

So by ABP estimate

$$\begin{aligned}\sup _{\Omega }(T_i(u^l)-T_i(u))\le C\big [\Vert f_i(u^l)-f_i(u)\Vert _{L^{N}(\Omega )}+\mu _i\Vert u_i^l-u_i\Vert _{L^{N}(\Omega )}\big ]. \end{aligned}$$

By changing the role of $T_i(u^l)$ and $T_i(u)$ we find

$$\begin{aligned} \Vert T_i(u^l)-T_i(u))\Vert _{L^{\infty }(\Omega )}\le C\big [\Vert f_i(u^l)-f_i(u)\Vert _{L^{N}(\Omega )}+\mu _i\Vert u_i^l-u_i\Vert _{L^{N}(\Omega )}\big ]. \end{aligned}$$

So the claim follows by observing that the right hand side goes to 0 as $l\rightarrow \infty .$

Claim: $T_{i}(u^l)\rightarrow T_i(u)$ in $C_{e_i}(\bar{\Omega })$ for $i=1,2,\dots ,n:$

By simple calculation we get

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^-\Big (D^2\big (T_i(u^l)-T_i(u)\big )\Big )+\mu _i \big (T_i(u^l)-T_i(u)\big )\ge f_i(u^l)-f_i(u)+\mu _i(u_i^l-u_i). \end{aligned}$$

(3.2)

In view of (3.1) and (3.2) it is clear that we can apply the boundary Lipschitz estimate Theorem 2.5. Let us take $x\in \Omega$ and find the corresponding point $x_0\in \partial \Omega$ such that $|x-x_0|=d(x)=\text {inf}\{|x-y|~|~y\in \partial \Omega \}.$ Now, by applying Theorem 2.5 we have

$$\begin{aligned} |T_i(u^l)(x)-T_i(u)(x)|&\le C\Big (\Vert T_i(u^l)-T_i(u)\Vert _{L^{\infty }(\Omega )}+\mu _i\Vert u_i^{l}-u_i\Vert _{L^{\infty }(\Omega )}\nonumber \\&\quad +\Vert f_i(u^{l})-f_i(u)\Vert _{L^{\infty }(\Omega )}\Big )|x-x_0|\nonumber \\&\le C\Big (\Vert T_i(u^l)-T_i(u)\Vert _{L^{\infty }(\Omega )}+\mu _i\Vert u_i^{l}-u_i\Vert _{L^{\infty }(\Omega )}\nonumber \\&\quad +\Vert f_i(u^{l})-f_i(u)\Vert _{L^{\infty }(\Omega )}\Big )d(x) \nonumber \\&\le \frac{C}{k_i}\Big (\Vert T_i(u^l)-T_i(u)\Vert _{L^{\infty }(\Omega )}\nonumber \\&\quad +\mu _i\Vert u_i^{l}-u_i\Vert _{L^{\infty }(\Omega )}+\Vert f_i(u^{l})-f_i(u)\Vert _{L^{\infty }(\Omega )}\Big )e_i(x), \end{aligned}$$

(3.3)

where we have used $k_id(x)\le e_i(x)$ for any $x\in \Omega .$ Since (3.3) is true for any $x\in \Omega ,$ so by the definition of $\Vert \cdot \Vert _{e_i}$ we have

$$\begin{aligned} \Vert T_i&(u^l)-T_i(u)\Vert _{C_{e_i}(\overline{\Omega })}\nonumber \\&\le \frac{C}{k_i}\Big (\Vert T_i(u^{l})-T_i(u)\Vert _{L^{\infty }(\Omega )}+\mu _i\Vert u^{l}_{i}-u_i\Vert _{L^{\infty }(\Omega )}+\Vert f_i(u^{l})-f_i(u)\Vert _{L^{\infty }(\Omega )}\Big ). \end{aligned}$$

(3.4)

Again, since the right hand side goes to zero as $l\rightarrow \infty ,$ the claim follows.

Theorem 3.4

The map $T:C_{E}(\overline{\Omega }) \rightarrow C_{E}(\overline{\Omega })$ is strongly increasing.

Proof

Let $u,v\in C_{E}(\overline{\Omega })$ with $u\le v$ but $u\not = v.$ This implies that $u_i\le v_i$ for all $i=1,2,\dots ,n$ but $u_j\not = v_j$ for some j. We show that $T(u)\le T(v)$ and $T_j(u)\not =T_j(v).$ Since T is increasing so $T(u)\le T(v).$ To prove the other claim let us denote by $U_j=T_j(u)$ and $V_j=T_j(v)$ and $W_j=V_j-U_j.$ By similar calculation as in Equation 3.2 and using the condition (C1) and (C2) we have $-\mathcal {M}_{\lambda _j,\Lambda _j}^-(D^2 W_j)+\mu _j W_j\ge 0$ in $\Omega$ and $W_j=0$ in $\partial \Omega .$ Hence by ABP maximum principle we have $W_j\ge 0$ in $\Omega ,$ consequently, by Höpf lemma $\frac{\partial W_j}{\partial \eta }<0$ on $\partial \Omega$ (here it is important to note that $U_j,~V_j\in C^{1,\alpha }(\bar{\Omega })$ so is $W_j$) and $W_j>0$ on $\Omega$.

The above fact implies that there is a $t>0$ such that $W_j>te_j.$ If not, then for each $m\in \textbf{N}$ there is $x_m\in \bar{\Omega }$ such that $W_j(x_m)<\frac{1}{m}e_j(x_m),$ consequently, $W_j(x_m)\le \frac{1}{m}\Vert e_j\Vert _{L^{\infty }(\Omega )}.$ Now since $W_j>0$ in $\Omega ,$ so $x_{m}\rightarrow x$ for some $x\in \partial \Omega .$ Let us consider

$$\begin{aligned} \displaystyle {\lim _{m\rightarrow \infty }}\Big |\frac{W_j(x_m)-W_j(x)}{x_m-x}\Big |&\le \displaystyle {\lim _{m\rightarrow \infty }}\big |\frac{\frac{1}{m}e_j(x_n)-\frac{1}{m}e_j(x)}{x_m-x}\big |~ (\text {since}~W_j(x)=e_j(x)=0)\\&\le \displaystyle {\lim _{m\rightarrow \infty }}\frac{1}{m}\frac{|\nabla e_i(z_m)||(x_m-x)|}{|x_m-x|}\\&\le \displaystyle {\lim _{m\rightarrow \infty }}\frac{1}{m}\Vert \nabla e_i\Vert _{L^{\infty }(\Omega )}\rightarrow 0 \end{aligned}$$

Which is a contradiction to the fact that $\frac{\partial W_j }{\partial \eta }<0$ on $\partial \Omega$. Hence there exist $t>0$ such that $W_j>te_j.$ Hence T is a strongly increasing.

Before stating and proving the next result. Suppose that $u,v\in C_{E}(\overline{\Omega })$ with $u\le v.$ Then we say that $w\in [u,v]$ if $u_i\le w_i\le v_i$ for $i=1,2,\dots ,n.$

Theorem 3.5

(Minimal and maximal solution) Let $\psi =(\psi _1,\psi _2,\dots ,\psi _n),\phi =(\phi _1,\phi _2,\dots ,\phi _n)$ be positive sub and supersolution of (1.1) satisfying $\psi \le \phi .$ Then there exists a minimal as well as a maximal solution for (1.1) in the order interval $[\psi ,\phi ]$.

Proof

Since $\psi$ is a subsolution of (1.1) so for each $i=1,2,\dots n,$ we have

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+(D^2(\psi _i))+\mu _i \psi _i \le f_i(\psi )+\mu _i \psi _i=-\mathcal {M}_{\lambda _i,\Lambda _i}^+(D^2 T_i(\psi ))+\mu _i T_i(\psi ). \end{aligned}$$

So by ABP maximum principle we have $\psi _i\le T_i(\psi )$ and similarly $T_i(\phi )\le \phi _i$ for $i=1,2,\dots ,n$ and also $T_i(\psi )\le T_i(\phi )$ (by monotonicity of T). Let P be the positive cone of $C_{E}(\overline{\Omega }).$ Then $(C_{E}(\overline{\Omega }),~P)$ is an ordered Banach space. Theorem 3.2 and 3.3 implies that $T:[\psi ,\phi ] \rightarrow C_{E}(\overline{\Omega })$ is an increasing and completely continuous map. Also $\psi \le T(\psi )\le T(\phi )\le \phi$. Thus by Corollary 6.2 [4] there exists a minimal and a maximal fixed points of the map T in the interval $[\psi ,\phi ]$. In view of Lemma 3.1, minimal and maximal fixed points of the map T turn out to be the minimal and maximal solutions of (1.1) in the interval $[\psi ,\phi ].$

We have taken the following lemma from [4]. Our multiplicity results for the problem (1.1) is based on the following Lemma 3.6 applied to the map T.

Lemma 3.6

([4]) Let X be a retract of some Banach space and $F:X \rightarrow X$ be a completely continuous map. Suppose that $X_1$ and $X_2$ are disjoint retracts of X and let $U_k, k=1,2$ be open subsets of X such that $U_k\subset X_k,$ $k=1,2$. Moreover, suppose that $F(X_k)\subset X_k$ and that F has no fixed points on $X_k\setminus U_k, k=1,2$. Then F has at least three distinct fixed points, $x,x_1,x_2$ with $x_k\in X_k, k=1,2$ and $x\in X\setminus (X_1\cup X_2)$.

Theorem 3.7

[Three solution theorem] Suppose there exist two pairs of ordered sub and supersolutions $(\psi ^1,\phi ^1)$ and $(\psi ^2,\phi ^2)$ of (1.1) with the properties that $\psi ^1\le \psi ^2\le \phi ^1,$ $\psi ^1\le \phi ^2\le \phi ^1$ and $\psi ^2\nleq \phi ^2.$ In addition assume that $\psi ^2,\phi ^2$ are not solutions of (1.1). Then there exist at least three solutions $u^i~(i=1,2,3)$ to (1.1), where $u^1\in [\psi ^1,\phi ^2],~u^2\in [\psi ^2,\phi ^1]$ and $u^3\in [\psi ^1,\phi ^1$ $setminus([\psi ^1,\phi ^2]cup[\psi ^2,\phi ^1]).$

Proof

In view of Lemma 3.1, finding a solutions to (1.1) is equivalent to finding a fixed point of T. Let $X=[\psi ^1,\phi ^1],$ $X_1=[\psi ^1,\phi ^2]$ and $X_2=[\psi ^2,\phi ^1].$ Then from the inequalities properties satisfied by $\phi ^i$ and $\psi ^i,$ it is clear that $X_1,X_2\subset X$ and $X_1,X_2$ are disjoint. Also T is increasing and $\psi ^1, \phi ^1$ are ordered sub and supersolutions of (1.1) we have $\psi ^1\le T(\psi ^1)\le T(\phi ^1)\le \phi ^1$ i.e. $T(X)\subset X.$ Similarly, $T(X_k)\subset X_k$ for $k=1,2.$ It is also clear that T is a retract of X and $X_k$, $k=1,2$ in the Banach space $C_e(\bar{\Omega }).$ Moreover, the map T restricted to X has the following properties:

Claim $T:X\rightarrow X$ is completely continuous and strongly increasing.

By the Theorem 3.3 and 3.4, we know that T is strongly increasing and completely continuous when we restrict to the above spaces $X, X_1$ and $X_2$.

It is given that $\phi ^2=(\phi ^2_1,\phi ^2_2,\dots ,\phi ^2_n)$ is a strict supersolution of (1.1), that is,

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2\phi ^2_1)&\ge f_1(\phi ^2)~+h_1~~{} & {} \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2\phi ^2_2)&\ge f_2(\phi ^2)~+h_2~~{} & {} \textrm{in}~~\Omega ,\\ ~~~~~~~~\vdots&=~~~~~~~~~~~~ \vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2\phi ^2_n)&\ge f_n(\phi ^2)~+h_n~~{} & {} \textrm{in}~~\Omega ,\\ \phi ^2_1=\phi ^2_2=\dots =&\phi ^2_n=0~~{} & {} \textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

(3.5)

for some $h_1,h_2,\dots ,h_n\in C(\bar{\Omega })$ with $h_i>0$ on $\bar{\Omega },$ for all $i=1,2,\dots ,n.$ Also from the definition of T we have

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2T_1(\phi ^2))+\mu _1 T_1(\phi ^2)&= f_1(\phi ^2)~+\mu _1\phi ^2_1~~{} & {} \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2T_2(\phi ^2))+\mu _2T_2(\phi ^2)&=f_2(\phi ^2)+\mu _2\phi ^2_2~~{} & {} \textrm{in}~~\Omega ,\\ ~~~~~~~~\vdots&=~~~~~~~~~~~~ \vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2T_n(\phi ^2))+\mu _n T_n(\phi ^2)&= f_n(\phi ^2)~+\mu _n\phi ^2_n~~{} & {} \textrm{in}~~\Omega ,\\ T_1(\phi ^2)=T_2(\phi ^2)=\dots =&T_n(\phi ^2)=0~~{} & {} \textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

(3.6)

For each $i=1,2,\dots n,$ in view of (3.5) and (3.6) we have the following inequality in viscosity sense

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+\big (D^2(T_i(\phi ^2)-\phi ^2_i)\big )+\mu _i(T_i(\phi ^2)-\phi ^2_i)\le -h_i<0. \end{aligned}$$

(3.7)

By ABP maximum principle $T_i(\phi ^2)-\phi ^2_i\le 0$ in $\Omega$ for $i=1,2,\dots n.$ Again by strong maximum principle, for each $i=1,2,\dots ,n,$ we have $T_i(\phi ^2)<\phi ^2_i$ because $\phi ^2$ is strict supersolution. Now, by Corollary 6.2 [4] T has a maximal fixed point $u^1\in X_1$ and $\psi ^1\le u^1<\phi ^2.$ Since $\psi ^2$ is a strict subsolution of (1.1). By similar argument as above T has a minimal fixed point $u^2\in X_2$ with $\psi ^2<u^2\le \phi ^1.$ Since $u^1$ is a solution and $\phi ^2$ is a strict supersolution, we can easily check that for each $i=1,2,\dots ,n$ we have

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^-(D^2(\phi _i^2-u_i^2))+\mu _i(\phi _i^2-u_i^2)\ge f_i(\phi ^2)-f_i(u^1)+\mu _i(\phi ^2_i-u_i^1)+g_i.\nonumber \\ \end{aligned}$$

(3.8)

From the hypothesis (C1) and (C2) we have

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^-(D^2(\phi ^2_i-u^1_i))+\mu _i(\phi ^2_i-u^1_i)\ge g_i>0. \end{aligned}$$

(3.9)

Now repeating the argument as in Theorem 3.4, we can find a constant $t_{1,i}>0$ such that $\phi ^2_i-u^1_i>t_{1,i}e_i.$ Similarly there exists a constant $t_{2,i}>0$ such $u^2_i-\psi ^2_i>t_{2,i}e_i.$ Define

$$\begin{aligned} B_k= & {} X\cup \{z\in C_{e_1}(\bar{\Omega })\times C_{e_2}(\bar{\Omega })\times \dots \times C_{e_n}(\bar{\Omega }):\Vert z_i-u^k_i\Vert \nonumber \\{} & {} <t_{k,i}~\textrm{for}~i=1,2,\dots ,n\};~~~\textrm{for}~~k=1,2. \end{aligned}$$

(3.10)

Then $B_k\subset X_k$ and $B_k$ is open in X, $k=1,2$. Hence $X_k$ has non empty interior and let $U_k$ be the largest open set in $X_k$ containing $u^k$ such that T has no fixed point in $X_k\in U_k$ for $(k=1,2).$ Now the existence of third fixed point for T follows from Lemma 3.7 and hence the problem (1.1) has at least three solutions $u^1\in [\psi ^1,\phi ^2]$, $u^2\in [\psi ^2,\phi ^1]$ and $u^3\in [\psi ^1,\phi ^1]\setminus ([\psi ^1, \phi ^2]\cap [\psi ^2,\phi ^1])$.

4 Results applied to sublinear case

This section contains a result asserting the existence of multiple(three) positive solutions to a system of equations of the type (1.1). These existence and multiplicity results have been obtained by using the three solution theorem(Theorem 3.7) established in the previous section. Let us consider the following system of equations

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2u_1)&=\mu \left( u_1^{1-\alpha _1}+f_1(u_2)\right) ~~~{} & {} \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2u_2)&=\mu \left( u_2^{1-\alpha _2}+f_2(u_3)\right) ~~~{} & {} \textrm{in}~~\Omega ,\\ ~~~~~~~~\vdots&=~~~~~~~~~~~~ \vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2u_n)&=\mu \left( u_n^{1-\alpha _n}+f_n(u_1)\right) ~~~{} & {} \textrm{in}~~\Omega ,\\ u_1=u_2&=\dots =u_n=0~~{} & {} \textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

(4.1)

where $\mu$ is a positive parameter, $\alpha _{i}\in (0,1)$ and $f_{i}\in C^1[0,\infty )$ for $i=1,2,\dots ,n.$ In addition to the above assumption let us also assume that $f_i's$ satisfies the following hypotheses:

• (C3) $f_{i}$’s are nondecreasing with $f_{i}(0)=0$ for $i=1,2,\dots ,n.$

• (C4) $\displaystyle {\lim _{s \rightarrow \infty } \frac{f_{1}^{[m]}\circ f_{2}^{[m]}\circ \dots f_{n-1}^{[m]}\circ f_{n}^{[m]}(s)}{s}=0}$ for every $m>0$, where $f_{i}^{[m]}(s)=f_{i}(ms).$

From the right hand side structure of equation (4.1) and condition (C3) it is clear that condition (C1) and (C2) follows for the system (4.1).

First of all, we establish bifurcation result at $(0,\underline{0})$ from the trivial branch $(\mu ,\underline{0}).$

Theorem 4.1

Let (C3) hold. Then there exists $\mu _{0}>0$ such that for every $0<\mu <\mu _0,$ (4.1) has a positive classical solution $u=(u_1,u_2,\dots ,u_n).$ Moreover, $\displaystyle {\sum _{i=1}^n \Vert u_i\Vert _{L^{\infty }(\Omega )}\rightarrow 0}$ as $\mu \rightarrow 0.$

Proof

Let $e_i\in C^2(\Omega )\cap C(\overline{\Omega })$ be from Theorem 2.7. Since $f_i(0)=0$ we can choose sufficiently small $\mu _{0}>0$ such that for $i=1,2,\dots ,n-1$

$$\begin{aligned} \left( \mu _{0}\Vert e_i\Vert _{\infty }\right) ^{1-\alpha _i}+f_i(\mu _{0}\Vert e_{i+1}\Vert _{\infty })<1 \end{aligned}$$

and

$$\begin{aligned} \left( \mu _{0}\Vert e_n\Vert _{\infty }\right) ^{1-\alpha _n}+f_n(\mu _{0}\Vert e_1\Vert _{\infty })<1. \end{aligned}$$

For a fixed $\mu <\mu _{0},$ we define $\phi _i=\mu e_i.$ Then,

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2\phi _i)=-\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2 (\mu e_i))=\mu \big [\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2e_i)\big ]=\mu . \end{aligned}$$

Now, by using the fact that $f_i's$ are non decreasing( from (C3)) we have

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2\phi _i)=\mu .1>&\mu \left\{ (\mu _{0}\Vert e_i\Vert _{\infty })^{1-\alpha _i}+f_i(\mu _0\Vert e_{i+1}\Vert _{\infty })\right\} \\ >&\mu \left\{ (\mu \Vert e_i\Vert _{\infty })^{1-\alpha _i}+ f_i(\mu \Vert e_{i+1}\Vert _{\infty })\right\} \\ \ge&\mu \left\{ \left( \mu e_i\right) ^{1-\alpha _i}+f_i\left( \mu e_{i+1}\right) \right\} .\\ =&\mu \left\{ \phi _i^{1-\alpha _i}+f_i\left( \phi _{i+1}\right) \right\} , \text {for} i=1,2,\dots ,n-1. \end{aligned}$$

Similarly, we can also show that

$$\begin{aligned} -\mathcal {M}_{\lambda _n,\Lambda _n}^+ (D^2\phi _n)\ge \mu \left\{ \phi _n^{1-\alpha _n}+f_n\left( \phi _{1}\right) \right\} . \end{aligned}$$

Therefore, $\phi =(\phi _1,\phi _2,\dots ,\phi _n)$ is a supersolution of (4.1). Next, we will construct a subsolution of (4.1). Let $\phi _{1,i}^+>0,$ be the eigen function from Theorem 2.6 with $\Vert \phi _{1,i}^+\Vert _\infty =1.$ For a fix $\mu >0,$ we can find $m_{\mu }>0$ such that $\mu ^{+}_{1,i} m_{\mu }^{\alpha _i}\le \mu$ ($\mu ^{+}_{1,i}$ is the principal half eigenvalue corresponding to $\phi _{1,i}^+,$ see Theorem 2.6). Let us set $\psi _i=m_{\mu }\phi _{1,i}^+.$ Then for $i=1,2,\dots ,n-1$

$$\begin{aligned}&-\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2\psi _i)=\mu _{1,i}^+(m_{\mu }\phi _{1,i}^+)\le \mu ^+_{1,i}m_{\mu }^{\alpha _i} (m_{\mu }\phi _{1,i}^+)^{1-\alpha _i}\le \mu \psi _i^{1-\alpha _i}\\&\le \mu \left[ \psi _i^{1-\alpha _i}+f_i(\psi _{i+1})\right] , \end{aligned}$$

and by following the calculation we also have

$$\begin{aligned} -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2\psi _n)\le \mu \left[ \psi _n^{1-\alpha _n}+f_n(\psi _{1})\right] . \end{aligned}$$

Thus, $\psi =(\psi _1,\psi _2,\dots ,\psi _n)$ is a subsolution to (4.1). Since $\dfrac{\partial e_i}{\partial n}<0$ on $\partial \Omega$ we can choose $m_{\mu }$ sufficiently small such that $\psi _i\le \phi _i$ for each $i=1,2,\dots ,n$. So by Theorem 3.5 there exist a solution $(u_1,u_2,\dots ,u_n)$ to (4.1) such that $\psi \le (u_1,u_2,\dots ,u_n)\le \phi .$ Moreover,

$$\begin{aligned} \displaystyle {\sum _{i=1}^n}\Vert u_i\Vert _{\infty }\le \displaystyle {\sum _{i=1}^n}\Vert \phi _i\Vert _{\infty }=\mu \displaystyle {\sum _{i=1}^n}\Vert e_i\Vert _{\infty }\rightarrow 0 \end{aligned}$$

as $\mu \rightarrow 0.$

Following Theorem 4.2 gives the existence of a positive solution to the Problem 4.1 for all $\mu >0.$ Here, in addition to the above conditions (C3) we also assume that $f_i's$ satisfy the combined sublinear at $\infty ,$ that is, the hypothesis (C4).

Theorem 4.2

Let (C3)–(C4) hold, then (4.1) has a positive solution u for all $\mu >0.$

Proof

Let $(\psi _1,\psi _2,\dots ,\psi _n)$ be the subsolution constructed in Theorem 4.1. Next, we construct a supersolution $\tilde{\phi }$ of (4.1). Using the fact that $f_i's$ are nondecreasing, for a given $\mu >0$ we can find $\tilde{m}_{\mu }\gg 1$ such that $\tilde{m}_{\mu } f_{i}^{[\tilde{\beta }_i]}\circ f_{i+1}^{\tilde{\beta }_{i+1}]}\circ \dots \circ f_{n}^{[\tilde{\beta }_{n}]}\left( \tilde{m}_{\mu }\Vert e_1\Vert _{\infty }\right) >1$ and

$$\begin{aligned} \left[ \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }_i]}\circ f_{i+1}^{\tilde{\beta }_{i+1}]}\circ \dots \circ f_{n}^{[\tilde{\beta }_{n}]}\left( \tilde{m}_{\mu }\Vert e_1\Vert _{\infty }\right) \right] ^{\alpha _{i}}\ge \left( 2\mu \Vert e_i\Vert _{\infty }\right) ^{1-\alpha _{i}} \end{aligned}$$

(4.2)

where

$$\begin{aligned} \tilde{\beta }_i= {\left\{ \begin{array}{ll} 2\mu \tilde{m}_{\mu }\Vert e_{i+1}\Vert _{\infty };~~~i=1,2,\dots ,n-1,\\ 1;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i=n, \end{array}\right. } \end{aligned}$$

(4.3)

and $e_i's$ are the solution of (2.7) for $i=2,\dots ,n.$ Let us set $\tilde{\beta }=\displaystyle {\max \nolimits _{i=1,2,\dots ,n}\{\tilde{\beta }_i\}}.$ Then in view of the condition (C4),  we can choose $m_{\mu }>\tilde{m}_{\mu }$ such that

$$\begin{aligned} \dfrac{1}{2\mu \Vert e_1\Vert _{\infty }}\ge \dfrac{f_{1}^{[\tilde{\beta }]}\circ f_{2}^{[\tilde{\beta }]}\circ \dots \circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) }{m_{\mu }\Vert e_1\Vert _{\infty }} \end{aligned}$$

(4.4)

and

$$\begin{aligned} \dfrac{m_{\mu }^{\alpha _{1}}}{2}\ge \mu \Vert e_1\Vert _{\infty }^{1-\alpha _{1}}. \end{aligned}$$

(4.5)

Now, we define $\tilde{\phi }=(\tilde{\phi }_1,\tilde{\phi }_2,\dots ,\tilde{\phi }_n),$ where

$$\begin{aligned} \tilde{\phi }_{i}= {\left\{ \begin{array}{ll} m_{\mu }e_1; ~~i=1,\\ \left( 2\mu \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \right) e_i; ~~ i=2,\dots ,n. \end{array}\right. } \end{aligned}$$

(4.6)

In view of the fact that $f_i's$ are non-decreasing and (4.2) we have

$$\begin{aligned}&\left[ \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{\tilde{\beta }]}\circ \dots \circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \right] ^{\alpha _{i}}\nonumber \\&\quad \ge \left[ \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }_i]}\circ f_{i+1}^{\tilde{\beta }_{i+1}]}\circ \dots \circ f_{n}^{[\tilde{\beta }_{n}]}\left( \tilde{m}_{\mu }\Vert e_1\Vert _{\infty }\right) \right] ^{\alpha _{i}}\nonumber \\&\quad \ge \left( 2\mu \Vert e_i\Vert _{\infty }\right) ^{1-\alpha _{i}}, \end{aligned}$$

(4.7)

for $i=2,\dots ,n.$

Let us consider

$$\begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+ (D^2\tilde{\phi }_1)=&\frac{m_{\mu }}{2}+\frac{m_{\mu }}{2}\\ \ge&m_{\mu }^{1-\alpha _1}\frac{m^{\alpha _1}_{\mu }}{2}\\&\quad +\mu f_{1}^{[\tilde{\beta }]}\circ f_{2}^{[\tilde{\beta }]}\circ \dots \circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) ~~~~~(\text {from}(4.4))\\ \ge&\mu \left( m_{\mu }\Vert e_1\Vert _{\infty }\right) ^{1-\alpha _1}\\&\quad +\mu f_{1}^{[\tilde{\beta _1}]}\circ f_{2}^{[\tilde{\beta }]}\circ \dots \circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) ~~~\\&\quad (\text {since}\, f_1's\, \text {are non-decreasing})\\ \ge&\mu \left[ \left( m_{\mu }\Vert e_1\Vert _{\infty }\right) ^{1-\alpha _1}+f_{1}\left( 2\mu \tilde{m}_{\mu }\Vert e_2\Vert _{\infty }f_{2}^{[\tilde{\beta }]}\circ \dots \circ \right. \right. \\&\left. \left. \quad f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \right) \right] \\ \ge&\mu \left[ \left( m_{\mu }e_1\right) ^{1-\alpha _1}+f_{1}\left( 2\mu \tilde{m}_{\mu } f_{2}^{[\tilde{\beta }]}\circ \dots \circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) e_2\right) \right] \\ =&\mu \left[ \left( \tilde{\phi }_1\right) ^{1-\alpha _1}+f_1\left( \tilde{\phi }_2\right) \right] . \end{aligned}$$

Next, will consider the case when $i=2,3,\dots ,n-1$

$$\begin{aligned}&-\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2\phi _i)=2\mu \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \\&\quad =\mu \left[ \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \right] ^{\alpha _i}\\&\quad \quad \times \left[ \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \right] ^{1-\alpha _i}\\&\quad \quad +\mu \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \\&\quad \ge \mu \left[ 2\mu \Vert e_i\Vert _{\infty }\right] ^{1-\alpha _i}\left[ \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \right] ^{1-\alpha _i}~\\&\qquad ~(\text {in view of}~ (4.2))\\&\quad \quad +\mu f_{i}\left( 2\mu \tilde{m}_{\mu }\Vert e_{i+1}\Vert _{\infty } f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) \right) \\&\quad \quad ~(\tilde{m} \gg 1~\text {and monotonicity of}~f_i)\\&\quad \ge \mu \left[ 2\mu \tilde{m}_{\mu } f_{i}^{[\tilde{\beta }]}\circ f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) e_i\right] ^{1-\alpha _i}\\&\quad \quad +\mu f_{i}\left( 2\mu \tilde{m}_{\mu } f_{i+1}^{[\tilde{\beta }]}\circ \dots \circ f_{n-1}^{[\tilde{\beta }]}\circ f_{n}^{[\tilde{\beta }]}\left( m_{\mu }\Vert e_1\Vert _{\infty }\right) e_{i+1}\right) \\&\quad =\mu \left( \tilde{\phi }_i^{1-\alpha _i}+f_i\left( \tilde{\phi }_{i+1}\right) \right) \end{aligned}$$

Finally, for $i=n$

$$\begin{aligned} -\mathcal {M}_{\lambda _n,\Lambda _n}^{+} (D^2\tilde{\phi }_n)=&2\mu \tilde{m}_{\mu }f_n^{[\tilde{\beta }]}(m_{\mu }\Vert e_1\Vert _{\infty })\\ =&\mu \left[ \left( \tilde{m}_{\mu }f_n^{[\tilde{\beta }]}(m_{\mu }\Vert e_1\Vert _{\infty })\right) ^{\alpha _n}\left( \tilde{m}_{\mu }f_n^{[\tilde{\beta }]}(m_{\mu }\Vert e_1\Vert _{\infty })\right) ^{1-\alpha _n}\right. \\&\quad \left. +\tilde{m}_{\mu }f_n^{[\tilde{\beta }]}m_{\mu }\Vert e_1\Vert _{\infty })\right] \\&\ge \mu \left[ \left( 2\mu \tilde{m}_{\mu }\Vert e_n\Vert _{\infty }\right) ^{1-\alpha _n}\times \left( m_{\mu }f_n^{[\tilde{\beta }]}(m_{\mu }\Vert e_1\Vert _{\infty })\right) ^{1-\alpha _n}\right. \\&\quad \left. +\tilde{m}_{\mu }f_n(\tilde{\beta }m_{\mu }\Vert e_1\Vert _{\infty })\right] \\&\ge \mu \left[ \left( 2\mu \tilde{m}_{\mu }f_n^{[\beta _n]}\left( m_\mu \Vert e_1\Vert _\infty \right) e_n\right) ^{1-\alpha _n}+f_n\left( m_{\mu }e_1\right) \right] \\&=\mu \left[ \phi _n^{1-\alpha _n}+f_n\left( \phi _1\right) \right] , \end{aligned}$$

where we have used the fact that $f_n$ is non-decreasing, $\tilde{m}_{\mu }\gg 1$ and $\tilde{\beta }>1.$ Thus, $\tilde{\phi }=(\tilde{\phi }_1,\tilde{\phi }_2,\dots ,\tilde{\phi }_n)$ is a supersolution of (4.1) for all $\mu >0.$ Moreover, in view of $\dfrac{\partial e_i}{\partial n}<0,$ and $\tilde{m}_\mu \gg 1,$ we can choose $m_{\mu }$ sufficiently small such that $\psi _i\le \phi _i,$ for all $i=1,2,\dots ,n.$ Therefore, for each $\mu >0,$ Problem 4.1 has a solution $u=(u_1,u_2,\dots ,u_n)$ satisfying $\psi \le (u_1,u_2,\dots ,u_n)\le \tilde{\phi }.$

Our next result(Theorem (4.3)) deals with the existence of multiple(three) solutions to our problem (4.1) for certain range of values of $\mu .$ In order to prove this we require the following additional condition of $f_i$’s.

• (C5) Suppose there exist two positive constants $0<a<b$ with $f_i(a)\ne 0$ and $f_i(b)\ne 0$ for $i=1,2,\dots n$ and

  $$\displaystyle {\min _{i=1,2,\dots ,n}\left\{ \dfrac{1}{2\Vert e_i\Vert _{\infty }}\min \left\{ a^{\alpha _i},\dfrac{a}{f_i(a)}\right\} \right\} }>\displaystyle {\max _{i=1,2\dots ,n}\left\{ A_i \dfrac{b}{f_i(b)}\right\} },$$

where $A_i=\displaystyle {\inf _{\epsilon }}\dfrac{N_i^{-}R^{N_i^+-1}}{\epsilon ^{N_i^{-}}(R-\epsilon )}$, $N_i^-=\frac{\Lambda _i}{\lambda ^i}(N-1)+1$, $N_i^+=\frac{\lambda _i}{\Lambda _i}(N-1)+1$ and R is the radius of the largest inscribed ball $B_R$ in $\Omega .$ We have given an example in Sect. 5 verifying the above assumptions.

Theorem 4.3

Let $f_i'$s satisfy (C3) and (C5). Then there exist $0\le \mu _*<\mu ^*$ such that for any $\mu \in [\mu _{*},\mu ^{*}],$ Problem 4.1 has at least three solutions.

Proof

In view of Theorem 3.7, the existence of three solutions follow once we construct two pairs of sub and supersolutions for (4.1) ordered in an appropriate sense. Here, we assume $\Omega =B_{R}.$ Let $\psi$ and $\tilde{\phi }$ are same as in Theorem 4.2. Next, we construct a strict supersolution $\Phi$ to (4.1). Let us set $\Phi =(\Phi _1,\Phi _2,\dots ,\Phi _n)=\left( \dfrac{ae_1}{\Vert e_1\Vert _\infty },\dfrac{ae_2}{\Vert e_2\Vert _\infty },\dots ,\dfrac{ae_n}{\Vert e_n\Vert _\infty }\right)$ where $e_i's$ are as above. Thus, for $\mu <\mu ^*=\displaystyle {\min _{i=1,2,\dots ,n}\left\{ \frac{1}{2\Vert e_i\Vert _{\infty }}\min \left\{ a^{\alpha _i},\frac{a}{f_i(a)}\right\} \right\} },$ and $i=1,2,\dots ,n-1,$ we have

$$\begin{aligned} -\mathcal {M}_{\lambda _i,\Lambda _i}^+ (D^2\Phi _i)&=\frac{a}{2\Vert e_i\Vert _\infty }+\dfrac{a}{2\Vert e_i\Vert _\infty }>\mu a^{1-\alpha _i}+\mu f_i(a)\\&\ge \mu \left[ \left( \dfrac{a}{\Vert e_i\Vert _\infty }e_i\right) ^{1-\alpha _i}+f_i\left( \dfrac{a}{\Vert e_{i+1}\Vert _\infty }e_{i+1}\right) \right] \\&=\mu \left( \Phi _i^{1-\alpha _i}+f_i(\Phi _{i+1})\right) .\\ \end{aligned}$$

Similarly, for $i=n$

$$\begin{aligned} -\mathcal {M}_{\lambda _n,\Lambda _n}^+ (D^2\Phi _n)&\ge \mu \left( \Phi _n^{1-\alpha _n}+f_n(\Phi _1)\right) . \end{aligned}$$

Thus, $\Phi$ is a strict supersolution of (4.1) for $\mu <\mu ^{*}.$ Next, we construct a positive strict subsolution $\Psi$ of (4.1) for $\mu >\mu _{*}=\displaystyle {\max _{i=1,2\dots ,n}\left\{ A_i \dfrac{b}{f_i(b)}\right\} }$. For $0<\epsilon <R;~l,m>1,$ define $\rho :[0,R]\rightarrow [0,1]$ as

$$\begin{aligned} \rho (r)=\left\{ \begin{array}{l l} 1 &{} \quad 0 \le r \le \epsilon ,\\ 1-\left( 1-\left( \dfrac{R-r}{R-\epsilon }\right) ^m\right) ^l &{}\quad \epsilon < r \le R. \end{array}\right. \end{aligned}$$

and $d(r)=b\rho (r).$ Then

$$\begin{aligned} \rho ^{\prime }(r)=\left\{ \begin{array}{ll} 0 &{} \quad 0 \le r \le \epsilon , \\ -\dfrac{lm}{R-\epsilon }\left( 1-\left( \dfrac{R-r}{R-\epsilon }\right) ^m\right) ^{l-1}\left( \dfrac{R-r}{R-\epsilon }\right) ^{m-1} &{}\quad \epsilon < r \le R. \end{array}\right. \end{aligned}$$

Consequently, we have

$$\begin{aligned} |\rho ^{\prime }(r)| \le \dfrac{lm}{R-\epsilon }~~\text {and}~|d^{\prime }(r)| \le \dfrac{b lm}{R-\epsilon } \end{aligned}$$

(4.8)

Let us consider the solution of the following system:

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+ (D^2\Psi _1)&=\mu f_1(d(r))~~{} & {} \text {in}~~ B_{R},\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+ (D^2\Psi _2)&=\mu f_2(d(r))~~{} & {} \text {in}~~ B_{R},\\&\vdots =\vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+ (D^2\Psi _n)&=\mu f_n(d(r))~~{} & {} \text {in}~~ B_{R},\\ \Psi _1=\Psi _2=\Psi _n&=0~~{} & {} \text {on}~~\partial B_{R}, \end{aligned} \right. \end{aligned}$$

(4.9)

and set $\Psi =(\Psi _1,\Psi _2,\dots ,\Psi _n).$ By well known regularity results the function $\Psi _i\in C^{2}(\overline{B_{R}}),$ for $i=1,2,\dots , n$. Therefore by Theorem 1.1 [24], we find that $\Psi _i$ is radially symmetric. Consequently, we have

$$\begin{aligned} \left\{ \begin{aligned} -&\theta _1(\Psi _1''(r))\Psi _1''(r)-\frac{N-1}{r}\theta _1(\Psi _1'(r))\Psi _1'(r)= \mu f_1(d(r)) ~{} & {} r\in (0,R);\\ -&\theta _2(\Psi _2''(r))\Psi _2''(r)-\frac{N-1}{r}\theta _2(\Psi _2'(r))\Psi _2'(r)= \mu f_2(d(r)) ~{} & {} r\in (0,R);\\&\vdots{} & {} \vdots \\ -&\theta _n(\Psi _n''(r))\Psi _n''(r)-\frac{N-1}{r}\theta _n(\Psi _n'(r))\Psi _n'(r)= \mu f_n(d(r)) ~{} & {} r\in (0,R);\\&\Psi _1'(0)=\Psi _2'(0)=\Psi _n'(0)=0;\\&\Psi _1(R)=\Psi _2(R)=\Psi _n(R)=0. \end{aligned} \right. \end{aligned}$$

(4.10)

where

$$\begin{aligned} \theta _i(s)= {\left\{ \begin{array}{ll} \Lambda _i,&{} \text {if}~s\ge 0\\ \lambda _i,&{} \text {if}~s<0. \end{array}\right. } \end{aligned}$$

For a fixed $1\le i\le n,$ let us rewrite the ith equation in (4.10) as follows:

$$\begin{aligned} \Psi _i''(r)+\frac{(N-1)\theta _{i}(\Psi _i'(r))}{r\theta _i(\Psi _i''(r))}\Psi _i'(r)=-\frac{\mu }{\theta _i(\Psi _i''(r))}f_i(d(r)). \end{aligned}$$

(4.11)

Moreover, by setting $v_i(r)=\frac{\theta _i(\Psi _i'(r))(N-1)}{\theta _i(\Psi _i''(r))r}$, $\tau _i(r)=\exp {{\int _1^rv_i(s)ds}}$ and $\tilde{\tau _i}(r)=\frac{\tau _i(r)}{\theta _i(\Psi _i''(r))},$ Equation 4.11 can further be rewritten as follows:

$$\begin{aligned} \begin{aligned}{} \tau _i(r)\Psi _i''(r)+\tau _{i}(r)\frac{\theta _{i}(\Psi _i'(r))}{\theta _{i}(\Psi _i''(r))}\frac{N-1}{r}\Psi _i'(r)&=-\mu \tilde{\tau _i}(r)f_i(d(r)),\\ \left( \tau _i(r)\Psi _i'(r)\right) '&=-\mu \tilde{\tau _{i}}(r)f_{i}(d(r)), \end{aligned} \end{aligned}$$

(4.12)

Integrating (4.12) from 0 to r we have

$$\begin{aligned} \int _0^r\left( \tau _i(s)\Psi _i'(s)\right) 'ds=-\mu \int _0^r\tilde{\tau _i}(s)f_i(d(s))ds \end{aligned}$$

(4.13)

which implies

$$\begin{aligned} \Psi _i'(r)=-\frac{\mu }{\tau _i(r)}\int _0^r\tilde{\tau _i}(r)f_i(d(s))ds~~~(\text {since}~~\Psi '_i(0)=0). \end{aligned}$$

For each fixed $1\le i\le n,$ we claim that for $\mu >\mu _{*},$ $\Psi _i(t)>d(t)$ for all $t\in [0,R).$ Suppose the claim is true then

$$-\mathcal {M}_{\lambda _i,\Lambda _i}^+(D^2\Psi _i)=\mu f_i(d)<\mu \left( \Psi _i^{1-\alpha _i}+f_i(\Psi _{i+1})\right) ~~~\textrm{in}~~B_R,$$

that is, $(\Psi _1,\Psi _2,\dots ,\Psi _n)$ is a strict supersolution of (4.1).

Proof of claim: In order to show $d(t)<\Psi _i(t),$ it is enough to show $\Psi _i'(t)<d'(t)$ on (0, R],  since $\Psi _i(R)=d(R)=0.$ For any $r\in (0,\epsilon )$, $f_i(d(r))=f_i(b)$ and $\tilde{\tau _i}(r)>0,$ hence $\psi _i'(r)<0=d'(r)$ and the claim follows. In order to prove the claim for $r\in (\epsilon ,R).$ We observe that

• (i)  $N_{i}^+-1\le v_i(r)r\le N_{i}^--1,$

• (ii)  $r^{N_{i}^--1}\le \tau _i(r)\le r^{N_{i}^+-1}$ and $\frac{\tau _i(r)}{\Lambda _i}\le \tilde{\tau _i}(r)\le \frac{\tau _i(r)}{\lambda _i}.$

Now consider

$$\begin{aligned} -\Psi _i'(r)=&\frac{\mu }{\tau _i(r)}\int _0^r\tilde{\tau _i}(s)f_i(d(s))ds\nonumber \\&\ge \frac{\mu }{\tau _i(r)}\int _0^{\epsilon }\tilde{\tau }_i(s)f_i(d(s))ds~~~~(\text {since}~r>\epsilon ) \nonumber \\&\ge \frac{\mu }{r^{N_{i}^+-1}}\int _0^\epsilon \tilde{\tau _i}(s)f_i(d(s))ds\nonumber \\&\ge \frac{\mu }{\Lambda _i r^{N_{i}^+-1}}\int _0^{\epsilon }s^{N_{i}^--1}f_i(b)ds\nonumber \\&\ge \frac{\mu f_i(b)}{\Lambda _i R^{N_{i}^+-1}}\int _0^{\epsilon }s^{N_{i}^--1}ds\nonumber \\&=\frac{\mu f_i(b)\epsilon ^{N_{+}^-}}{\beta R^{N_{i}^+-1}N_{i}^-}. \end{aligned}$$

(4.14)

Thus, in order to show $-\Psi _i'(r)< -d'(r),$ in view of (4.14) and (4.8), it is sufficient to show that

$$\begin{aligned} \frac{\mu g_i(b)\epsilon ^{N^{-}_i}}{\Lambda _i R^{N_{+}^i-1}N^{-}_i}>\frac{lmb}{R-\epsilon }. \end{aligned}$$

(4.15)

Let us take $\epsilon _i=\frac{N_{i}^-R}{N_{i}^-+1}$ at which the function $\frac{1}{(R-\epsilon )\epsilon ^{N_{i}^-}}$ has minimum for each $i=1,2,\dots ,n$. From (C5) we have $\mu >\displaystyle {\max \nolimits _{i=1,2,\dots ,n}\left\{ \frac{b}{f_i(b)}\frac{\Lambda _i N_{i}^-R^{N_{i}^+-1}}{(R-\epsilon _i)\epsilon _i{^{N_{i}^-}}}\right\} }=\mu _*$. Hence we can choose $l,m>1$ such that

$$\begin{aligned} \mu >\frac{lmb\Lambda _i N_{i}^-R^{N_{i}^+-1}}{f_i(b)(R-\epsilon _i)\epsilon _i^{N_{i}^-}}. \end{aligned}$$

(4.16)

is satisfied for $i=1,2,\dots ,n$. Hence $\Psi$ is a strict subsolution for $\mu >\mu _*$. Since $\Psi _i(0)>d(0)=b\rho (0)=b>a=\Vert \Phi _i\Vert _{\infty }$, hence we have $\Psi _i\nleq \Phi _i$. From Theorem 4.2 we can choose sufficiently small subsolution $\psi$ and sufficiently large supersolution $\phi$ such that $\psi \le \Psi \le \tilde{\phi }$, $\psi \le \Phi \le \tilde{\phi }$. Hence by Theorem 3.7 we have at least three positive solutions for $\mu \in (\mu _*,\mu ^*)$.

Next for $\Omega$ is a general bounded domain, let $B_R$ be the largest inscribed ball in $\Omega$. To show the existence of three positive solution we will use these sub and supersolutions in $B_R$ and in $\Omega \setminus B_R$ extend by zero(regularity). It is easy to check that we can get at least three positive solutions for $\mu \in (\mu _*,\mu ^*)$.

5 Example

Here, we consider an example that satisfies the hypothesis of Theorems 4.1-4.2. In the case of the operator being Laplacian, the below problem arises in combustion theory and studied by many authors e.g. Parks [27]. Consider

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2u_1)&=\mu \left( u_1^{1-\alpha _1}+e^{\frac{\tau u_2}{\tau +u_2}}-1\right) ;{} & {} ~~~\textrm{in}~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2u_2)&=\mu \left( u_2^{1-\alpha _2}+u_3^{\beta _1}\right) ;{} & {} ~~~\textrm{in}~\Omega ,\\ \vdots&\vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2u_n)&=\mu \left( u_n^{1-\alpha _n}+u_1^{\beta _{n-1}}\right) ;{} & {} ~~~\textrm{in}~\Omega ,\\ u_1=u_2=\dots&=0;{} & {} ~~~\textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

(5.1)

where $\alpha _i\in (0,1)$ for $i=1,2,\dots ,n$, $\tau >0$. Here $f_1(s)=e^{\frac{\tau u}{\tau +u}}-1$ and $f_i(s)=s^{\beta _i}, \beta _i>0$ for $i=1,2,\dots ,n-1$, it is clear that $f_i(0)=0,$ $f_i(s)$ are increasing and has combined sublinear growth at infinity. So, it satisfies the hypothesis (C3)-(C4). Hence Theorem 4.1-4.2 Moreover, for $\tau \gg 1,$ by choosing $a=1$, $b=\tau$, we have $\mu ^*=\displaystyle {\min _{i=1,2,\dots ,n}\left\{ \frac{1}{2\Vert e_i\Vert _{\infty }}\min \left\{ a^{\alpha _i},\frac{a}{f_i(a)}\right\} \right\} }>\frac{1}{4}\displaystyle {\min _{1,2,\dots ,n}\left\{ \frac{1}{\Vert e_i\Vert _{\infty }}\right\} }$, and

$\mu _*=\displaystyle {\max _{1,2,\dots ,n}\left\{ A_i\frac{\tau }{f(\tau )}\right\} =A^*\max \left\{ \frac{\tau }{e^{.5\tau }}, \tau ^{1-\beta _1},\dots ,\tau ^{1-\beta _{n-1}}\right\} }\rightarrow 0$, where $A^*=\max \left\{ A_i\right\}$. Consequently, $\mu _*<\mu ^*$ for $\tau \gg 1$. Therefore, $f_i$’s in (5.1) also satisfies (C5). Hence for $\mu \in (\mu _*,\mu ^*),$ Theorem 4.3 implies that (5.1) has at least three positive solutions.