EXISTENCE OF SOLUTIONS FOR DOUBLE PHASE OBSTACLE PROBLEMS WITH MULTIVALUED CONVECTION TERM

. The main goal of this paper is the study of an elliptic obstacle problem with a double phase phenomena and a multivalued reaction term which also depends on the gradient of the solution. Such term is called multivalued convection term. Under quite general assumptions on the data, we prove that the set of weak solutions to our problem is nonempty, bounded and closed. Our proof is based on a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping.


Introduction
Let Ω ⊆ R N be a bounded domain with Lipschitz boundary ∂Ω and let 1 < p < q < N . We study the following double phase problem with a multivalued convection term and obstacle effect − div |∇u| p−2 ∇u + µ(x)|∇u| q−2 ∇u ∈ f (x, u, ∇u) in Ω, in Ω, where µ : Ω → [0, ∞) is Lipschitz continuous, f : Ω × R × R N → 2 R is a multivalued function depending on the gradient of the solution and Φ : Ω → R is a given function. The precise conditions on the data will be presented in Section 3. The novelty of our work is the fact that we combine several different phenomena in one problem. To be more precise problem (1.1) contains (1) a double phase operator; (2) a multivalued convection term; (3) an obstacle restriction.
To the best of our knowledge, this is the first work which combines all these phenomena in one problem. We are going to prove that problem (1.1) has at least one solution. The proof is based on a surjectivity result of Le [20] for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping.

Preliminaries
Let Ω be a bounded domain in R N and let 1 ≤ r ≤ ∞. We denote by L r (Ω) := L r (Ω; R) and L r (Ω; R N ) the usual Lebesgue spaces endowed with the norms respectively. In what follows, for simplicity, the norms of L r (Ω; R) and L r (Ω; R N ) are both denoted · r , even if we do not mention it explicitly. Moreover, W 1,r (Ω) and W 1,r 0 (Ω) stand for the Sobolev spaces endowed with the norms · 1,r and · 1,r,0 , respectively. For any 1 < r < ∞ we denote by r the conjugate of r, that is, 1 r + 1 r = 1. In the entire paper we suppose the following condition: H(µ): µ : Ω → R + = [0, ∞) is Lipschitz continuous and 1 < p < q < N are chosen such that q p < 1 + 1 N .
We consider the function H : Ω × R + → R + defined by We know that L H (Ω) turns out to be uniformly convex and so it is a reflexive Banach space. In addition, we introduce the seminormed function space which is equipped with the seminorm · q,µ given by It is known that the embeddings Taking into account these embeddings we have the inequalities for all u ∈ L H (Ω). By W 1,H (Ω) we denote the corresponding Sobolev space which is defined by equipped with the norm (Ω). Now we are able to rewrite (2.1) for the space for all u ∈ W 1,H 0 (Ω). Since both spaces W 1,H (Ω) and W 1,H 0 (Ω) are uniformly convex, we know that they are reflexive Banach spaces.
Furthermore, we have the following compact embedding for each 1 < r < p * , where p * is the critical exponent to p given by Let us now consider the eigenvalue problem for the r-Laplacian with homogeneous Dirichlet boundary condition and 1 < r < ∞ which is defined by in Ω, has a nontrivial solution u ∈ W 1,r 0 (Ω) which is called an eigenfunction corresponding to the eigenvalue λ. We denote by σ r the set of eigenvalues of −∆ r , W 1,r 0 (Ω) . From Lê [21] we know that the set σ r has a smallest element λ 1,r which is positive, isolated, simple and it can be variationally characterized through (Ω) * be the operator defined by Definition 2.2. Let X be a real reflexive Banach space. The operator A : X → 2 X * is called (a) pseudomonotone if the following conditions hold: (i) The set A(u) is nonempty, bounded, closed and convex for all u ∈ X.
(ii) A is upper semicontinuous from each finite-dimensional subspace of X to the weak topology on X * .
then the element u * lies in A(u) and Proposition 2.3. Let X be a real reflexive Banach space and assume that A : X → 2 X * satisfies the following conditions: (i) For each u ∈ X we have that A(u) a is nonempty, closed and convex subset of X * . (ii) A : X → 2 X * is bounded. (iii) If u n u in X and u * n u * in X * with u * n ∈ A(u n ) and if lim sup n→∞ u * n , u n − u X * ×X ≤ 0, then u * ∈ A(u) and u * n , u n X * ×X → u * , u X * ×X . Then the operator A : X → 2 X * is pseudomonotone.
Furthermore, we will state the following surjectivity theorem for multivalued mappings which is formulated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping. The following theorem was proved in Le [20, Theorem 2.2]. We use the notation B R (0) := {u ∈ X : u X < R}.
Theorem 2.4. Let X be a real reflexive Banach space, let F : D(F ) ⊂ X → 2 X * be a maximal monotone operator, let G : D(G) = X → 2 X * be a bounded multivalued pseudomonotone operator and let L ∈ X * . Assume that there exist u 0 ∈ X and R ≥ u 0 X such that D(F ) ∩ B R (0) = ∅ and ξ + η − L, u − u 0 X * ×X > 0 for all u ∈ D(F ) with u X = R, for all ξ ∈ F (u) and for all η ∈ G(u). Then the inclusion F (u) + G(u) L has a solution in D(F ).

Main results
We assume the following hypotheses on the multivalued nonlinearity f : The multivalued convection mapping f : Ω × R × R N → 2 R has nonempty, compact and convex values such that (i) the multivalued mapping x → f (x, s, ξ) has a measurable selection for all (s, ξ) ∈ R × R N ; (ii) the multivalued mapping (s, ξ) → f (x, s, ξ) is upper semicontinuous; (iii) there exists α ∈ L q 1 q 1 −1 (Ω) and a 1 , a 2 ≥ 0 such that for all η ∈ f (x, s, ξ), for a. a. x ∈ Ω, for all s ∈ R and for all ξ ∈ R N , where 1 < q 1 < p * with the critical exponent p * given in (2.4); (iv) there exist w ∈ L 1 + (Ω) and b 1 , b 2 ≥ 0 is such that b 1 + b 2 λ −1 1,p < 1, and for all η ∈ f (x, s, ξ), for a. a. x ∈ Ω, for all s ∈ R and for all ξ ∈ R N , where λ 1,p is the first eigenvalue of the Dirichlet eigenvalue problem for the p-Laplacian, see (2.5). It is obvious that the set K is a nonempty, closed and convex subset of W 1,H 0 (Ω).
The weak solutions for problem (1.1) are understood in the following sense.
Definition 3.2. We say that u ∈ K is a weak solution of problem (1.1) if there exists η ∈ L q 1 The main result of this paper is stated as the next theorem. (Ω) to L q1 (Ω) with its adjoint operator i * : L q 1 (Ω) → W 1,H 0 (Ω) * . Since 1 < q 1 < p * the embedding operator i is compact and so i * as well. However, from hypotheses H(f )(i) and (iii), we can use the same process as the proof of Papageorgiou-Vetro-Vetro [28,Proposition 3] to see that the Nemytskij operator N f : W 1,H 0 (Ω) ⊂ L q1 (Ω) → 2 L q 1 (Ω) associated to the multivalued mapping f given by (Ω) * . Also, let us consider the indicator Under the definitions above, it is not difficult to see that u ∈ K is a weak solution of problem (1.1), see Definition 3.2, if and only if u solves the following inequality: Find u ∈ K and η ∈ N f (u) such that (Ω) * is given in (2.6). Consider the multivalued operator A : Then, using a standard procedure, we can reformulate problem (3.3) to the following inclusion problem: Find u ∈ K such that A(u) + ∂I K (u) 0, (3.4) where the notation ∂I K stands for the subdifferential of I K in the sense of convex analysis.
We are going to apply the surjectivity result for multivalued pseudomonotone operators, see Theorem 2.4. To this end, for any u ∈ W 1,H 0 (Ω) and η ∈ N f (u), by condition H(f )(iii), we obtain (Ω) ⊂ L q1 (Ω), 1 < q 1 < p * and Proposition 2.1 implies that A : W 1,H 0 (Ω) → 2 W 1,H 0 (Ω) * is a bounded mapping. We claim that A is pseudomonotone. In order to prove this, we are going to apply Proposition 2.3. Indeed, by hypotheses H(f ) we know that A has nonempty, closed and convex values. Moreover, as we just showed, A is a bounded mapping. So, it is enough to verify that A is a generalized pseudomonotone operator. (Ω) be such that So, for each n ∈ N, we are able to find an element ξ n ∈ N f (u n ) such that u * n = A(u n ) − i * ξ n . From the fact that the embedding from W 1,H 0 (Ω) to L q1 (Ω) is compact, see (2.3), we have u n → u in L q1 (Ω). Moreover, from (3.5), we see that the sequence {ξ n } is bounded in L q 1 (Ω). So, (3.7) leads to This fact along with (3.6) and the (S + )-property of A, see Proposition 2.1, implies that u n → u in W 1,H 0 (Ω). This yields due to the continuity of A, see Proposition 2.1. Since ξ n ∈ N f (u n ) we have ξ n (x) ∈ f (x, u n (x), ∇u n (x)) for a. a. x ∈ Ω. However, (3.5) and (3.6) imply that the sequence {ξ n } is bounded in L q 1 (Ω). Passing to a subsequence if necessary, we may suppose that ξ n ξ in L q 1 (Ω) for some ξ ∈ L q 1 (Ω). Employing Mazur's theorem, we are able to find a sequence {η n } of convex combinations of {ξ n } such that η n → ξ in L q 1 (Ω). Therefore, we can say that  (3.6) and condition H(f )(iii) we see that the sequence {ξ n (x)} is bounded for a. a. x ∈ Ω. So, by (3.8), we find a subsequence {ξ n (x)} for a. a. x ∈ Ω, still denoted by {ξ n (x)}, such that Keeping in mind that u n → u in W 1,H 0 (Ω) and W 1,H 0 (Ω) ⊂ W 1,p 0 (Ω) leads to u n (x) → u(x) and ∇u n (x) → ∇u(x) as n → ∞.
Because A is a bounded operator with nonempty, closed and convex values, we are now in the position to apply Proposition 2.3 in order to conclude that A is a pseudomonotone operator. Furthermore, we are going to prove that there exists a constant R > 0 such that for all u * ∈ A(u), for all η ∈ ∂I K (u) and for all u ∈ W 1,H 0 (Ω) with u 1,H,0 = R. For any u * ∈ A(u), we can find ξ ∈ N f (u) such that u * = A(u) − i * ξ. Recall that 0 ∈ K, one has (3.10) Note that I K : W 1,H 0 → R is a proper, convex and lower semicontinuous function. Hence, we can apply Proposition 1.3.1 in Gasiński-Papageorgiou [17] to find a K , b K > 0 such that (3.11) (Ω) ⊆ W 1,p 0 (Ω) as well as u p p ≤ λ −1 1,p ∇u p p for all u ∈ W 1,p 0 (Ω), into account, we get where the last inequality is obtained by using inequality (2.2). Since 1 < p < q < N and b 1 + b 2 λ −1 1,p < 1, we can take R 0 > 0 large enough such that for all R ≥ R 0 it holds (Ω) * is a maximal monotone operator. Therefore, we can apply Theorem 2.4 for F = ∂I K , G = A and L = 0. This shows that inclusion (3.4) has at least one solution u ∈ K which is a solution of (3.3) and so, a solution from (1.1) in the sense of Definition 3.2. Thus, S = ∅.
Next, we are going to show that the set of solutions of problem (1.1) is closed in W 1,H 0 (Ω). Let {u n } ⊂ S be a sequence such that u n → u in W 1,H 0 (Ω) (3.13) for some u ∈ W 1,H 0 (Ω). So, for each n ∈ N, there exists ξ n ∈ N f (u n ) such that A(u n ), v − u n H + ξ n , v − u n L q 1 (Ω) + I K (v) − I K (u n ) ≥ 0 (3.14) for all v ∈ W 1,H 0 (Ω). Hypothesis H(f )(iii) and the convergence in (3.13) ensure that {ξ n } is bounded in L q 1 (Ω). So, we may assume that ξ n ξ in L q 1 (Ω).
As before, from Mazur's theorem and the upper semicontinuity of (s, η) → f (x, s, η), we can show that ξ(x) ∈ f (x, u(x), ∇u(x)) for a. a. x ∈ Ω, that is, ξ ∈ N f (u). Passing to the upper limit in (3.14) as n → ∞ and taking the lower semicontinuity of I K into account it follows that u ∈ K is a solution of problem (1.1). Hence, S is closed.
In the last part of the proof we need to show that S is bounded. If K is bounded, the desired conclusion holds automatically. Let us suppose that K is unbounded and in addition, let us assume that S is unbounded. Then, there exists a sequence {u n } ⊆ S such that u n 1,H,0 → +∞. (3.15) As before, see (3.10), we can show via a simple calculation that 0 ≥ A(u n ) − i * ξ n , u n H ≥ 1 − b 1 − b 2 λ −1 1,p min u n p 1,H,0 , u n q 1,H,0 − w 1 − a K u n 1,H,0 − b K for some ξ n ∈ N f (u n ) where we have used the fact that 0 ∈ K. Combining the inequality above and (3.15) yields a contradiction. Therefore, S is bounded.