The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions

In this paper we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sup- and supersolutions. Under very general assumptions on the data we then apply our result to get infinitely many solutions. Moreover, we also discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem.


Introduction
In this paper we consider the following variable exponent double phase system with nonlinear boundary conditions in Ω, in Ω, where Ω ⊆ R N , N ≥ 2, is a bounded domain with Lipschitz boundary ∂Ω, ν(x) denotes the unit normal of Ω at the point x ∈ ∂Ω, f i : Ω × R × R × R N × R N → R and g i : ∂Ω × R × R → R are Carathéodory functions for i = 1, 2 that satisfy local growth conditions (see hypotheses (H2)) and we suppose the following assumptions on the exponents and the weight functions: (H1) p i , q i ∈ C(Ω) such that 1 < p i (x) < N and p i (x) < q i (x) < p * i (x) for all x ∈ Ω, as well as 0 ≤ µ i (•) ∈ L ∞ (Ω), where p * i is given by for i = 1, 2. The operator in (1.1) is the so-called variable exponent double phase operator given by div |∇u i | pi(x)−2 ∇u i + µ i (x)|∇u i | qi(x)−2 ∇u i , u ∈ W 1,Hi (Ω), defined in a suitable Musielak-Orlicz Sobolev space W 1,Hi (Ω), i = 1, 2, which has been recently studied in Crespo-Blanco-Gasiński-Harjulehto-Winkert [8].The study of such operators goes back to Zhikov [39] who introduced for the first time energy functionals defined by ω → Ω |∇ω| p + a(x)|∇ω| q dx.Such functionals have been used to describe models for strongly anisotropic materials in the context of homogenization and elasticity.It also has several mathematical applications in the study of duality theory and of the Lavrentiev gap phenomenon; see Zhikov [40,41].
The main objective of our paper is to establish a method of sub-and supersolution in terms of trapping region of the system (1.1) under very general local structure conditions on the nonlinearities involved.As an application, we present some existence results to the system (1.1) under very mild and easily verifiable conditions on the data.In addition, we will also study the corresponding Dirichlet system and get a sub-supersolution approach including some existence results.The novelty of our paper is the combination of the variable exponent double phase operator with fully coupled convective right-hand sides along with coupled nonlinear boundary functions.To the best of our knowledge, such general systems have not been treated in the literature, even if we replace our operator with the p i -Laplacian, that is, µ i ≡ 0 for i = 1, 2.
Our paper is motivated by the work of Carl-Motreanu [5] who studied the elliptic system where they obtain extremal positive and negative solutions of the system by combining the theory of pseudomonotone operators, regularity results as well as a strong maximum principle.On the contrary, in the present paper we obtain existence and multiplicity results by using neither regularity theory nor strong maximum principle, which are not available in our setting.The method of sub-and supersolution is a very powerful tool and has been used in several works: here we mention, for example, the papers of Carl-Le-Winkert [4], Carl-Winkert [6], Motreanu-Sciammetta-Tornatore [32]; see also the monographs of Carl-Le [2] and Carl-Le-Motreanu [3].
The paper is organized as follows.In Section 2 we present the main preliminaries, including the properties of the Musielak-Orlicz Sobolev space, the double phase operator and the definition of trapping region (see Definition 2.5).Section 3 is devoted to our abstract existence result for given pairs of sub-supersolution (see Theorem 3.2), while in Section 4 we present several existence results with a construction of sub-supersolution (see Theorems 4.1 and 4.2).Finally, in Section 5 we consider the corresponding Dirichlet systems including the method of sub-supersolution and some existence results (see Theorems 5.3 and 5.4).
Let Ω be a bounded domain in R N with Lipschitz boundary ∂Ω and let Let M (Ω) be the space of all measurable functions u : Ω → R. For a given r ∈ C + (Ω), the variable exponent Lebesgue space L r(•) (Ω) is defined as equipped with the Luxemburg norm given by u r(•) = inf λ > 0 : We know that (L r(•) (Ω), • r(•) ) is a separable and reflexive Banach space.Similarly we introduce the variable exponent boundary Lebesgue space (L r(•) (∂Ω), • r(•),∂Ω ) by using the (N − 1)-dimensional Hausdorff surface measure σ.Let r ′ ∈ C + (Ω) be the conjugate variable exponent to r, that is, 1 r(x) = 1 for all x ∈ Ω.
Proposition 2.3.Let hypotheses (H1) be satisfied.Then, the operators A i defined in (2.1) are bounded, continuous, strictly monotone and of type (S + ), that is, Next, we define the product spaces equipped with the norms respectively.Based on Proposition 2.2 we have the compact embeddings and hold true for all (v 1 , v 2 ) ∈ W and all the integrals in (2.3) and (2.4) are finite.
Next, we introduce the notion of weak sub-and supersolution to (1.1).
Definition 2.5.We say that (u 1 , u 2 ), (u 1 , u 2 ) ∈ W form a pair of sub-and supersolution of problem and and with all integrals in (2.5) and (2.6) to be finite.
) is a pair of sub-and supersolution, then the order interval We now recall some definitions that we will use in the sequel (see Carl-Le-Motreanu [3, Definitions 2.95 and 2.96]).Definition 2.6.Let X be a reflexive Banach space, X * its dual space, and denote by • , • its duality pairing.Let A : X → X * .Then A is called (v) to satisfy the (S + )-property if Remark 2.7.In the context of Definition 2.6, any completely continuous operator is compact and any linear compact operator is completely continuous (see Zeidler [38,Proposition 26.2]).
Lemma 2.8.Let X be a reflexive Banach space and let A : X → X * be a demicontinuous operator satisfying the (S + )-property.Then A is pseudomonotone.
Lemma 2.9.Let X be a Banach space, A : X → X * be of type (S + ), and B : X → X * be compact.Then A + B is of type (S + ) as well.
Theorem 2.10.Let X be a real, reflexive Banach space, let A : X → X * be a pseudomonotone, bounded, and coercive operator, and b ∈ X * .Then, a solution of the equation Au = b exists.

Sub-Supersolution approach
In this section we are going to prove a sub-and supersolution existence result for the system (1.1) under very general structure conditions on the data.
Let u = (u 1 , u 2 ), u = (u 1 , u 2 ) be a pair of sub-and supersolution of problem (1.1) in the sense of Definition 2.5.We suppose the following assumptions.
Our main theorem in this section reads as follows.Proof.We split the proof into three steps.
Let λ = (λ 1 , λ 2 ) with λ k ≥ 0 and set Furthermore, we set where F k denote the Nemytskij operators related to f k , which are well defined for k = 1, 2 since the ranges of T 1 , T 2 lie within the trapping region [u, u].Therefore, due to the growth condition in (H2)(i) and the compact embedding W ֒→ L p1(•),p2(•) (Ω) (see (2.2)), we have that is bounded and compact.For the boundary term, we define where G k are the Nemytskij operators generated by g k .We know that is well defined, completely continuous, and bounded, due to (H2)(i), the compactness of the trace operator (see (2.2)), and Remark 2.7.Finally, let A(u) = (A 1 (u 1 ), A 2 (u 2 )) where A k are defined in (2.1).Because of Proposition 2.3, it is clear that A : W → W * is bounded, continuous, strictly monotone, and of type (S + ).We have the representations Using the notations above, u ∈ W ∩ [u, u] is a solution to (1.1) if and only if Step 2: , where T k are the truncation operators defined in (3.1).Now we consider the following auxiliary problem given in the form where λ = (λ 1 , λ 2 ) with λ k ≥ 0 to be specified later.Let Φ : W → W * be given by Φ(u) := A(u) + λB(u) − F (u) − G(u).
First, we know that Φ is bounded and continuous.Since A is of type (S + ) (see Proposition 2.3) and B, F , G are compact (and hence completely continuous; see Remark 2.7), we can apply Lemma 2.9 to get that Φ is of type (S + ) as well.Lemma 2.8 then implies that Φ is pseudomonotone.
Next, we are going to show that Φ : W → W * is coercive.To this end, using hypothesis (H2)(i) and Young's inequality we estimate for suitable C k depending on ϕ k and C ε,k > 0 depending on both ε and p k .
Next, we consider the operator G.To this end, for any u k ∈ L H k (Ω), we define and observe that η k > 1 for all u k ∈ L H k (Ω).Using the definition of η k along with (H2)(ii), Hölder's inequality, the embedding inequality u p k (•),∂Ω ≤ S k u 1,H k (cf.Proposition 2.2(iii)), and Young's inequality gives for suitable positive constants C k depending on ψ k 's, while C ε,k , Ĉε,k also depend on ε.On the other hand, we have for suitable ãk , bk > 0. Using this along with (3.4) we get Since Φ is bounded, continuous, pseudomonotone, and coercive, the main theorem on pseudomonotone operators (see Theorem 2.10) implies the existence of u ∈ W such that Φ(u) = 0.
Step 3: Comparison It remains to prove that u ∈ [u, u].We set (u − u) + = ((u 1 − u 1 ) + , (u 2 − u 2 ) + ).From Φ(u) = 0, besides recalling the definitions of F , G, we deduce 0 On the other hand, since u is a supersolution to (1.1), it turns out that Hence, from (3.9), (3.10) and (3.11) along with the monotonicity of A k (see Proposition 2.3), we obtain According to the definition of B k ( Thus, u k ≤ u k a. e. in Ω.Similarly, we show u k ≤ u k a. e. in Ω by applying the definition of subsolution.Therefore, we have shown that u ∈ [u, u] and so, by the definition of the truncations in (3.1) and the functions b k in (3.2), we see that u ∈ W turns out to be a weak solution of the system (1.1) lying within [u, u].

Sub-and supersolutions
This section is devoted to the construction of pairs of sub-and supersolution for the system (1.1).Following ideas of Guarnotta-Marano [24] (see also the papers of D'Aguì-Sciammetta [9] and Motreanu-Sciammetta-Tornatore [32] for a single equation), we prove the existence of infinitely many solutions to (1.1) under suitable sign conditions on the nonlinearities, exhibiting an oscillatory behavior.We suppose the following assumptions on the Carathéodory functionsf (H3) There exist h i , k i ∈ R such that h i ≤ k i and and for all ξ i ∈ R N .We have the following existence result.Theorem 4.1.Let hypotheses (H1) and (H3) be satisfied.Suppose that (H2) is fulfilled for a. a. x ∈ Ω, for all s i ∈ [h i , k i ], and for all ξ i ∈ R N , i = 1, 2. Then there exists a weak solution (u 1 , u 2 ) ∈ W of system (1.1) Proof.We set u i := h i and u i := k i .By (H3) we have u i ≤ u i .For all (v 1 , v 2 ) ∈ W with v 1 , v 2 ≥ 0 a. e. in Ω and for all (w 1 , w 2 ) ∈ W such that u i ≤ w i ≤ u i , we get Analogous computations concerning u 1 , u 2 prove that (u 1 , u 2 ) and (u 1 , u 2 ) form a pair of sub-and supersolution of problem (1.1).Then, Theorem 3.2 implies the existence of a weak solution (u 1 , u 2 ) ∈ W of (1.1) satisfying u i ≤ u i ≤ u i for i = 1, 2.
If we strengthen our assumptions, we can obtain more solutions.For this purpose, we assume the following hypothesis.
(H4) For all n ∈ N, there exist h for a. a. x ∈ Ω, 1 , s 2 for a. a. x ∈ ∂Ω, 2 , ξ 1 , 0 for a. a. x ∈ Ω, for a. a. x ∈ ∂Ω, 2 ], for all ξ i ∈ R N , and for all n ∈ N. Theorem 4.2.Let hypotheses (H1) and (H4) be satisfied.Suppose that, for all n ∈ N, (H2) is fulfilled for a. a. x ∈ Ω, for all s i ∈ [h and for all ξ i ∈ R N .Then there exists a sequence {(u

Proof. It suffices to apply Theorem 4.1 for all n ∈ N, with h
(the other case works similarly).

The Dirichlet problem
In this section we want to discuss the situation when we have a Dirichlet boundary condition instead of a nonhomogeneous Neumann one.We consider the system where p i , q i , µ i , i = 1, 2 satisfy hypotheses (H1).Instead of W, we consider its subspace W 0 = W 1,H1 0 × W 1,H2 0 equipped with the norm induced by the one of W.
Definition 5.1.We say that (u 1 , u 2 ) ∈ W 0 is a weak solution to (5.1) if and hold true for all (v 1 , v 2 ) ∈ W 0 and all the integrals in (5.2) and (5.3) are finite.
The definition of a sub-and a supersolution of problem (5.1) reads as follows.
Definition 5.2.We say that (u 1 , u 2 ), (u 1 , u 2 ) ∈ W form a pair of sub-and supersolution of problem (5.1) if u i ≤ 0 ≤ u i a. e. in Ω for i = 1, 2 and in Ω and for all (w 1 , w 2 ) ∈ W such that u i ≤ w i ≤ u i for i = 1, 2, with all integrals above to be finite.
are continuous and bounded.Next, we introduce the cut-off functions b