Multiplicity of Positive Solutions to a Singular (p 1 , P 2 )-laplacian System with Coupled Integral Boundary Conditions

In this work, we investigate the existence and multiplicity results for positive solutions to a singular (p 1 , p 2)-Laplacian system with coupled integral boundary conditions and a parameter (µ, λ) ∈ R 3 +. Using sub-super solutions method and fixed point index theorems, it is shown that there exists a continuous surface C which separates R 2 + × (0, ∞) into two regions O 1 and O 2 such that the problem under consideration has two positive solutions for (µ, λ) ∈ O 1 , at least one positive solution for (µ, λ) ∈ C, and no positive solutions for (µ, λ) ∈ O 2 .


Introduction
Nonlocal boundary conditions appear when the information on the boundary are connected to values inside the domain.Various types of boundary value problems involving nonlocal conditions have been extensively studied by various methods such as fixed point theorems on cones and the Leray-Schauder alternative, etc.We refer the reader to [11,14,15,24,[30][31][32][33]37] and the references therein.
Using fixed point index theory on cones of Banach spaces, they obtained several results about the existence, multiplicity, and nonexistence of positive solutions under various assumptions on the nonlinearity g 3 (t, s) which satisfies L 1 -Carathéodory condition.
The systems of differential equations equipped with a variety of boundary conditions have been extensively studied by many authors, see, e.g., [2-7, 10, 19, 21, 23, 25, 26, 34].For example, in [25], do Ó et al. considered a class of system of second-order differential equations of the form        −u = g 4 (t, u, v, a, b), in (0, 1), −v = g 5 (t, u, v, a, b), in (0, 1), where the nonlinearities g 4 and g 5 are superlinear at the origin as well as at infinity, and a, b ∈ R + .Using fixed point theorems of cone expansion/compression type, the upper-lower solutions method and degree argument, it was shown that there exists a continuous curve Γ which splits the positive quadrant of the (a-b)-plane into disjoint sets S 1 and S 2 such that (1.1) has at least two positive solutions in S 1 , has at least one positive solution on the boundary of S 1 , and has no positive solutions in S 2 .The result was applied to establish the existence and multiplicity of positive radial solutions for a certain class of semilinear elliptic systems in annular domains.We are concerned with the existence of positive solutions to the following singular (p 1 , p 2 )-Laplacian system with coupled integral boundary conditions (Φ(u (t))) + λh(t) • f (t, u(t)) = θ, t ∈ (0, 1), where Φ(s 1 , s 2 ) = (ϕ p 1 (s 1 ), ϕ p 2 (s 2 )), ϕ p i (s) := |s| p i −2 s with p i > 1 for i ∈ {1, 2}, θ is the origin of R 2 , α : + is a parameter, and • denotes the entrywise product, i.e., (a 1 , a 2 ) Throughout this paper, we assume the following hypotheses are satisfied unless otherwise stated: where s ĥ(τ)dτ ds + and I is the identity matrix of size 2.
For convenience, we identify (a, b) ∈ R 2 with the 1-by-2 matrix a b if necessary.Consequently The main purpose of this paper is to study the existence and multiplicity results for positive solutions to problem (P λ,µ ) using sub-super solutions method and fixed point index theorems.For sub-super solutions method concerning semilinear problems with nonlocal boundary conditions, we refer to [27][28][29].It seems not obvious that sub-super solutions method can be applicable to our problem with (p 1 , p 2 )-Laplacian due to the coupled integral boundary condition in (P λ,µ ).Thus we prove a theorem for sub-super solutions (see Theorem 2.12), and it is shown that there exists a continuous surface C which separates R 2  + × (0, ∞) into two regions O 1 and O 2 such that (P λ,µ ) has two positive solutions for (µ, λ) ∈ O 1 , at least one positive solution for (µ, λ) ∈ C, and no positive solutions for (µ, λ) ∈ O 2 (see Theorem 3.10).
As applications, we study existence results for positive radial solutions to p-Laplacian systems defined in an exterior domain as follows: subject to coupled integral boundary conditions where Using the main result (Theorem 3.10), we investigate the existence, multiplicity and nonexistence of positive solutions z , where u i and v i are i-th coordinates of u and v, respectively.For functions w 1 , . We also denote θ the zero function from [0, 1] to R 2 as well as the origin of R 2 .
This paper is organized as follows.In Section 2, well-known theorems such as generalized Picone identity and a fixed point index theorem are recalled, and a solution operator and a theorem for sub-super solutions related to problem (P λ,µ ) are also introduced.In Section 3, the main result in this paper is given (see Theorem 3.10).Finally, in Section 4, applications for problem (1.3)+(1.4)or (1.3)+(1.5)are given (see Corollary 4.1).

Preliminaries
For semilinear problems, we usually use integration by parts twice in order to obtain the useful information for solutions such as a block for parameters λ and a priori estimates for solutions.However, it is not effective for the p-Laplacian problem.The following generalized Picone identity can be used to overcome the difficulty (see Lemma 3.1 and Lemma 3.3).The identity can be verified by straightforward differentiation, but for completeness, we give the proof of it.Theorem 2.1 (generalized Picone identity, see, e.g., [13,20]).Let us define l p [y] = (ϕ p (y )) + b 1 (t)ϕ p (y), where ϕ p (s) = |s| p−2 s, s ∈ R, p > 1 and b 1 , b 2 are continuous functions on an interval I. Let y and z be functions such that y, z, ϕ p (y ), ϕ p (z ) are differentiable on I and z(t) = 0 for t ∈ I.
Now we recall a well-known theorem for the existence of a global continuum of solutions by Leray and Schauder [22] and a fixed point index theorem: Theorem 2.3 (see, e.g., [35,Corollary 14.12]).Let X be a Banach space with X = {0} and let P be an order cone in X.Consider x = H(λ, x), where λ ∈ R + and x ∈ P. If H : R + × P → P is completely continuous and H(0, x) = 0 for all x ∈ P, then C + (P ), the component of the solution set of (2.5) containing (0, 0), is unbounded.
Theorem 2.4 (see, e.g., [12]).Let X be a Banach space, P be a cone in X and O be a bounded open set containing θ in X, where θ is the origin of X.Let A : P ∩ O → P be completely continuous.Suppose that Ax = νx for all x ∈ P ∩ ∂O and all ν ≥ 1.Then i(A, O ∩ P, P ) = 1.

Solution operator
In this subsection, we define an operator related to problem (P λ,µ ) and prove the complete continuity of it. Denote , and By (H 3 ), det K > 0 and Then all entries of K −1 are nonnegative by (H 2 ) and nonnegativity of k ij for i, j ∈ {1, 2}. Define where and With the above transformations (2.7) and (2.8), we have the following lemma.
To show the complete continuity of T, we first prove the following lemma.
Proof.We only prove the case M n i → 0, since the other case can be proved in a similar manner.
where γ q = max{1, 2 q−1 } for q > 0. It follows from and thus the proof is complete.

Sub-super solutions theorem
In this subsection, we give a theorem for sub-super solutions to problem (P λ,µ ).
To get a theorem for sub-super solutions to problem (P λ,µ ), we make the following hypotheses: ) is quasi-monotone nondecreasing with respect to u.
Note that (H 2 ) implies (H 2 ).Now, a theorem for sub-super solutions for the problem (P λ,µ ) is given as follows.
where, for i ∈ {1, 2}, Let λ > 0 and µ ∈ R 2 + be fixed, and consider the following modified problem where Thus, for given v ∈ X, there is a unique solution β γ [v] ∈ R 2 of the equation x = g v (x), in other words, it is the unique element of R 2 which satisfies that From this fact, it follows that under the transformations (2.11) Thus (2.10) can be equivalently rewritten as follows: where Consequently, v is a solution to problem (2.12) if and only if u is a solution to problem (2.10) under the transformations (2.11), respectively.Now, define T γ = (T γ 1 , T γ 2 ) : X → P by, for each i = 1, 2 and v ∈ X, where Then v is a fixed point of T γ in X if and only if v is a solution to problem (2.12).It follows that T γ is completely continuous on X and T γ (X) is bounded in X.

Main results
First, we give a hypothesis which will be used in this section: (F ∞ ) For each i ∈ {1, 2}, there exists an interval Now we give a priori estimates for solutions as follows.
By similar arguments as in the proof of Lemma 3.1, we have the following lemma.We omit the proof of it.
. Then, by Proposition 3.9, C is a continuous surface in R 3 + , and it separates R 3  + into two regions Moreover, by Theorem 3. Proof.For fixed µ ∈ R 2 + , we will show that (P λ,µ ) has at least two positive solutions for λ ∈ (0, λ * (µ)), and thus the proof is complete by Theorem 3.5.
and α i ∈ (0, ∞).Using the Leray-Schauder alternative, the existence of at least one solution is obtained for two cases:

1
γ (u)(t) := u(t) − ((1 − t)α γ [u] + tµ).Then T γ is completely continuous on P, and T γ has a fixed point in P if and only if u is a non-negative solution to problem (3.8).Moreover, there exists a positive constant R such that T γ (u) < R for all u ∈ P and Γ ⊂ B R , where B R is an open ball with center θ and radius R in X. Applying Theorem 2.4 with O = B R , i(T γ , B R ∩ P, P ) = 1.Since all fixed points of T γ are contained in Γ, by the excision property, i(T γ , Γ ∩ P, P ) = i(T γ , B R ∩ P, P ) = 1.Since problem (P λ,µ ) with λ = λ 0 is equivalent to problem (3.8) on Γ ∩ P, (P λ,µ ) has a positive solution in Γ ∩ P for λ = λ 0 ∈ (0, λ * (µ)).Assume