Multiple Positive solutions of a $(p_1,p_2)$-Laplacian system with nonlinear BCs

Using the theory of fixed point index, we discuss existence, non-existence, localization and multiplicity of positive solutions for a $(p_1,p_2)$-Laplacian system with nonlinear Robin and/or Dirichlet type boundary conditions. We give an example to illustrate our theory.


Introduction
In the remarkable paper [39] Wang proved the existence of one positive solution of following one-dimensional p-Laplacian equation The results of [39] were extended by Karakostas [23] to the context of deviated arguments.
In both cases, the existence results are obtained via a careful study of an associated integral operator combined with the use of the Krasnosel'skiȋ-Guo Theorem on cone compressions and cone expansions.

The system of integral equations
We recall that a cone K in a Banach space X is a closed convex set such that λ x ∈ K for x ∈ K and λ ≥ 0 and K ∩ (−K) = {0}.
If Ω is a open bounded subset of a cone K (in the relative topology) we denote by Ω and ∂Ω the closure and the boundary relative to K. When Ω is an open bounded subset of X we write Ω K = Ω ∩ K, an open subset of K.
The following Lemma summarizes some classical results regarding the fixed point index, for more details see [2,13].
Let Ω be an open bounded set with 0 ∈ Ω K and Ω K = K. Assume that F : Ω K → K is a compact map such that x = F x for all x ∈ ∂Ω K . Then the fixed point index i K (F, Ω K ) has the following properties.
(3) If i K (F, Ω K ) = 0, then F has a fixed point in Ω K .
To the system (1.2)-(1.3) we associate the following system of integral equations, which is constructed in similar manner as in [39], where the case of a single equation is studied.
By a solution of (1.2)-(1.3), we mean a solution of the system (2.1).
In order to utilize the fixed point index theory we state the following assumptions on the terms that occur in the system (2.1).
that is, f i (·, u, v) is measurable for each fixed (u, v) and f i (t, ·, ·) is continuous for almost every (a.e.) t ∈ [0, 1], and for each r > 0 there exists φ i,r ∈ L ∞ [0, 1] such that Remark 2.2. The condition (2.2) is weaker than the condition In fact, for example, the function It is known (see e.g. [39]) that It follows that the functions in K i are strictly positive on the sub-interval [a i , b i ] and in particular we have In the following we make use of the notations: For a positive solution of the system (2.1) we mean a solution (u, v) ∈ K of (2.1) such that (u, v) > 0. We seek such solution as a fixed point of the following operator T .

Consider the integral operator
Under our assumptions, we can show that the integral operator T leaves the cone K invariant and is compact.
Lemma 2.4. The operator (2.4) maps K into K and is compact.
Firstly, we show that T sends bounded sets into bounded sets. Take (u, v) ∈ K such that (u, v) ≤ r. Then, for all t ∈ [0, 1] we have We prove now that T 1 sends bounded sets into equicontinuous sets. Let By the Ascoli-Arzelà Theorem we can conclude that T 1 is a compact map. In a similar manner we proceed for Moreover, the map T is compact since the components T i are compact maps.

Existence results
For our index calculations we use the following (relative) open bounded sets in K: and if ρ 1 = ρ 2 = ρ we write simply K ρ and V ρ . The set V ρ was introduced in [10] as an extension to the case of systems of a set given by Lan [27]. The use of different radii, in the spirit of the paper [21], allows more freedom in the growth of the nonlinearities.
The following Lemma is similar to the Lemma 5 of [10] and therefore its proof is omitted.
Lemma 3.1. The sets defined above have the following properties: We firstly prove that the fixed point index is 1 on the set K ρ 1 ,ρ 2 .
Lemma 3.2. Assume that In fact, if this does not happen, there exist λ ≥ 1 and Firstly we assume that u ∞ = ρ 1 and v ∞ ≤ ρ 2 .
Then we have Taking t = 0 gives ) .
Then we have Then, in both cases, we have Using the hypothesis (3.1) and the strictly monotonicity of ϕ −1 p 2 we obtain λρ 2 < ρ 2 . This contradicts the fact that λ ≥ 1 and proves the result.
We give a first Lemma that shows that the index is 0 on a set V ρ 1 ,ρ 2 .
Remark 3.4. We point out that a stronger, but easier to check, hypothesis than (3.2) is In the following Lemma we exploit an idea that was used in [19,21] and we provide a result of index 0 controlling the growth of just one nonlinearity f i , at the cost of having to deal with a larger domain. Nonlinearities with different growths were considered for examples in [35,36,45].
The proof of the next result regarding the existence of at least one, two or three positive solutions follows by the properties of fixed point index and is omitted. It is possible to state results for four or more positive solutions, in a similar way as in [26], by expanding the lists in conditions (S 5 ), (S 6 ).
Theorem 3.6. The system (2.1) has at least one positive solution in K if one of the following conditions holds.
, (I 0 r 1 ,r 2 ) hold. The system (2.1) has at least two positive solutions in K if one of the following conditions holds.
and (I 0 s 1 ,s 2 ) hold. (S 4 ) For i = 1, 2 there exist ρ i , r i , s i ∈ (0, ∞) with ρ i < c i r i and r i < s i such that (I 1 ρ 1 ,ρ 2 ), (I 0 r 1 ,r 2 ) and (I 1 s 1 ,s 2 ) hold. The system (2.1) has at least three positive solutions in K if one of the following conditions holds.

Non-existence results
We now provide some non-existence results for system (2.1).
Theorem 4.1. Assume that one of the following conditions holds.
(2) For i = 1, 2, Then there is no positive solution of the system (2.1) in K. Proof.
• Let be u ∞ = 0. Then we have Taking t = 0 gives a contradiction.
In the first case we have The proof is similar in the last case σ u,v > 1/2.
• Let be v ∞ = 0. We examine the case σ u,v ≥ b 2 . We have a contradiction. By similar proofs, the cases 0 < σ u,v ≤ a 2 and a 2 < σ u,v < b 2 can be examined.
(3) Assume, on the contrary, that there exists (u, v) ∈ K such that (u, v) = T (u, v) and (u, v) = (0, 0). If u ∞ = 0 then the function f 1 satisfies either (4.1) or (4.2) and the proof follows as in the previous cases. If v ∞ = 0 then the function f 2 satisfies either (4.1) or (4.2) and the proof follows as previous cases.

An example
We illustrate in the following example that all the constants that occur in the Theorem 3.6 can be computed.