Existence results for double phase implicit obstacle problems involving multivalued operators

In this paper we study implicit obstacle problems driven by a nonhomogenous differential operator, called double phase operator, and a multivalued term which is described by Clarke’s generalized gradient. Based on a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis, we prove the existence of at least one solution.

In this paper we prove the existence of at least one weak solution (see Definition (3.4)) of problem (1.1) by applying a surjectivity theorem for multivalued mappings, Kluge To the best of our knowledge, this is the first work which combines a double phase phenomena along with Clarke's generalized gradient and an implicit obstacle. A main tool in our treatment will be a surjectivity result of Le [24] for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping.
One difficulty in the study of (1.1) is the occurrence of the so-called double phase operator defined by −div |∇u| p−2 ∇u + μ(x)|∇u| q−2 ∇u , u ∈ W 1,H 0 ( ), where 1 < p < q < N . Although this operator looks like the ( p, q)-differential operator the difference is the weight function μ : → [0, ∞) which can have values in zero. That means we cannot search for weak solutions in the usual Sobolev space W 1, p 0 ( ), we need a certain type of a Sobolev-Musielak-Orlicz space equipped with the Luxemburg norm, see Sect. 2 for its definition. The idea to treat problems driven by the double phase operators goes back to the 1980s and the work of Zhikov [36] who introduced such classes of operators to describe models of strongly anisotropic materials by treating the functional see also Zhikov [37,38]  Works which are closely related to our paper dealing with certain types of double phase problems or multivalued problems can be found in Bahrouni-Rȃdulescu-Repovš [1], Bahrouni-Rȃdulescu-Winkert [2,3], Carl-Le-Motreanu [9], Cencelj-Rȃdulescu-Repovš [10], Clarke [12], Gasiński-Papageorgiou [18,19], Papageorgiou-Rȃdulescu-Repovš [30,31], Rȃdulescu [34], Zhang-Rȃdulescu [35] and the references therein.
The paper is organized as follows. In Sect. 2 we recall the definition of the used function spaces, some embedding results and we state the surjectivity results of Le [24] for multivalued mappings as well as Kluge's fixed point theorem. In Sect. 3 we present the full assumptions on the data of problem (1.1), give the definition of the weak solution and consider an auxiliary problem defined in (3.7). Next, we prove some properties of the solution set of (3.7) stated as Theorem 3.6 whose proof is mainly based on tools from nonsmooth analysis in terms of multivalued mappings. Taking these results into account we are able to prove our main result which says that the solution set of (1.1) is nonempty, bounded and weakly closed in W 1,H 0 ( ), see Theorem 3.5.

Preliminaries
In the whole paper we suppose that is a bounded domain in R N . Given 1 ≤ r ≤ ∞, L r ( ) and L r ( ; R N ) stand for the usual Lebesgue spaces equipped with the norm · p while W 1,r ( ) and W 1,r 0 ( ) denote the Sobolev spaces endowed with the norms · 1,r and · 1,r ,0 , respectively. By r , we denote the conjugate of r ∈ (1, ∞), that is, 1 r + 1 r = 1. For the weight function in (1.1) we suppose the following condition: is Lipschitz continuous and 1 < p < q < N are chosen such that Set R + := [0, ∞) and consider the modular function H : × R + → R + given by The Musielak-Orlicz space L H ( ) is defined by The space L H ( ) is uniformly convex and so a reflexive Banach space. Furthermore, we introduce the seminormed function space L q μ ( ) L q μ ( ) = u u : → R is measurable and μ(x)|u| q dx < +∞ endowed with the seminorm and is equipped with the norm From the assumption on μ : → R + in H(μ) combined with Colasuonno-Squassina [13, Proposition 2.18], it is known that is compact for each 1 < r < p * , where p * stands for the critical exponent to p given by Let us recall some properties of the eigenvalue problem for the r -Laplacian (1 < r < ∞) with homogeneous Dirichlet boundary condition given by It is known that the set σ r has a smallest element λ 1,r which is positive, isolated, simple and it can be variationally characterized through see Lê [25]. The notion of pseudomonotonicity for multivalued operators is recalled in the next definition.

Definition 2.2
Let X be a real reflexive Banach space. The operator A : X → 2 X * is called pseudomonotone if the following conditions hold: Let X be a real Banach space with its dual space X * . A function J : X → R is said to be locally Lipschitz at u ∈ X if there exist a neighborhood N (u) of u and a constant L u > 0 such that Since our results are based on fixed point results, so we now recall the fixed point theorem of Kluge [23]. Theorem 2.5 Let Z be a real reflexive Banach space and let C ⊂ Z be nonempty, closed and convex. Assume that : C → 2 C is a multivalued mapping such that for every u ∈ C, the set (u) is nonempty, closed, and convex and the graph of is sequentially weakly closed. If either C is bounded or (C) is bounded, then the map has at least one fixed point in C.
Finally, we end this section by recalling the following surjectivity theorem for multivalued mappings which was proved by Le [24, Theorem 2.2]. We use the notation B R (0) := {u ∈ X : u X < R}.
has a solution in D(F).

Main results
We impose the following assumptions for the data of problem (1.1). s) is measurable for all s ∈ R and there exists a function l ∈ L q 1 ( ) with for a. a. x ∈ and for all s ∈ R, where δ θ is defined by (iv) there exist c j ≥ 0 and γ j ∈ L q 1 q 1 −1 for a. a. x ∈ , for all ξ ∈ ∂ j(x, s) and for all s ∈ R, where ∂ j(x, s) stands for the generalized gradient of j with respect to the variable s and q 1 is given in (i); (v) there exists a constant m j ≥ 0 such that for a. a. x ∈ and for all s 1 , s 2 ∈ R whenever ξ 1 ∈ ∂ j(x, s 1 ) and ξ 2 ∈ ∂ j(x, s 2 ).

Remark 3.1 (a) Assumption H( j)(v)
is usually called the relaxed monotone condition for the locally Lipschitz function s → j(x, s), see for example, Migórski-Ochal-Sofonea [29]. It is equivalent to the inequality for a. a. x ∈ and for all s 1 , s 2 ∈ R.
(b) Positive homogeneity and subadditivity of T imply that T is also a convex function. On the other hand, it is not difficult to see that if T : W 1,H 0 ( ) → R is lower semicontinuous, then inequality (3.1) holds automatically.
Let {v n } ⊂ K (u) be a sequence such that v n → v in W 1,H 0 ( ) for some v ∈ W 1,H 0 ( ). Hence, for each n ∈ N, one has Passing to the upper limit as n → ∞ and taking inequality (3.1) into account, we obtain Hence, v ∈ K (u) which shows that K (u) is closed.
Let v 1 , v 2 ∈ K (u) and t ∈ (0, 1) be arbitrary. We set v t = tv 1 + (1 − t)v 2 . Then, we have T (v i ) ≤ U (u) for i = 1, 2. By the convexity of T , see Remark 3.1, it follows that Thus v t ∈ K (u). Therefore, we conclude that K (u) is a convex set in W 1,H 0 ( ).
Let us introduce the functional J : L q 1 ( ) → R defined by

Lemma 3.3 Under the assumptions H( j)(i) − (iv), the following hold:
with some c J > 0.

Moreover, if condition H( j)(v) holds, then the inequality
The weak solutions for problem (1.1) are understood in the following sense.

Definition 3.4
We say that u ∈ W 1,H 0 ( ) is a weak solution of problem (1.1) if u ∈ K (u) and where the multivalued function K is given by (3.2).
Our main results read as follows.
From Lemma 3.3(ii) we see that if u ∈ W 1,H 0 ( ) solves the following problem: Find u ∈ W 1,H 0 ( ) such that u ∈ K (u) and for all v ∈ K (u), then u is also a weak solution of problem (1.1). Based on this fact, in the sequel, we are going to explore the existence of solutions for problem (3.6).
To this end, we now introduce the following auxiliary problem: From the fact that f ∈ L p ( ) ⊂ W 1,H 0 ( ) * , problem (3.7) can be expressed as the variational-hemivariational inequality: Find u ∈ W 1,H 0 ( ) such that ( ) * is the double phase operator defined in (2.5). Employing the first separation theorem, see for example, Papageorgiou-Winkert [32, Theorem 3.1.57], it is not difficult to see that inequality problem (3.8) is equivalent to the following inclusion problem: Find u ∈ W 1,H 0 ( ) such that where the notation ∂ C I K (w) stands for the subdifferential of I K (w) in the sense of convex analysis.
First, we are going to apply the surjectivity theorem for multivalued mappings, see Theorem 2.6, in order to prove that problem (3.9) has at least one solution in W 1,H 0 ( ). In fact, we have the following claims. (3.10) Hence, for each n ∈ N, we are able to find ξ n ∈ ∂ J (u n ) such that The continuity of A, see Proposition 2.1, implies that We now prove that A + ∂ J is pseudomonotone. Let {u n } and {u * n } be sequences such that Our goal is to produce for each v ∈ W 1,H From (3.12), there is a sequence {ξ n } ⊂ W 1,H 0 ( ) * such that for each n ∈ N, ξ n ∈ ∂ J (u n ) and From By virtue of Theorem 2.2 of Chang [11], we know that which shows that ξ n , u n − u H = ξ n , u n − u L q 1 ( ) .

Claim 2 There exists R > 0 such that
for all u ∈ K (w) with u 1,H,0 = R, for all ξ ∈ ∂ J (u) and for all η ∈ ∂ C (I K (w) )(u).
Let u ∈ W 1,H 0 ( ) be fixed. Since 0 ∈ K (w) and f ∈ L p ( ) ⊂ W 1,H 0 ( ) * , for any ξ ∈ ∂ J (u) and η ∈ ∂ C (I K (w) )(u), we can find (3.17) Note that I K (w) : W 1,H 0 ( ) → R is a proper, convex and lower semicontinuous function. Hence, we can apply Proposition 1.10 of Brezis [8] to find a K (w) , b K (w) > 0 such that We consider now the two cases θ < p and θ = p. Suppose first θ < p and let c(θ ) > 0 be such that u θ ≤ c(θ ) u 1,H,0 for all u ∈ W 1,H 0 ( ) (3.20) due to the continuity of the embedding from W 1,H 0 ( ) to L r ( ) for all r ∈ (1, p * ). Applying (3.18) and (3.19) in (3.17) and using (3.20) we get where the last inequality is obtained via inequality (2.2). Since θ < p < q, it is clear that we can find a constant R 0 > 0 large enough such that R p 0 < R q 0 and Therefore, for each R ≥ R 0 fixed, the desired inequality (3.16) holds.