Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

In this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.


Introduction
Let ⊆ R N be a bounded domain with a Lipschitz-boundary ∂ . In this paper, we study the following nonlinear nonhomogeneous elliptic problem with a multivalued convection term and under obstacle condition where a : × R N → R N is continuous, monotone with respect to the second variable and satisfies particular other growth conditions to be described later, reaction term f : × R × R N → 2 R is multivalued and depends on the gradient of the solution (which makes the problem nonvariational) and obstacle : → [0, +∞] is a given function.
As the setting is general enough and hypotheses are mild and natural, we incorporate in our framework many differential operators, such as the p-Laplacian, the ( p, q)-Laplacian (i.e., the sum of a p-Laplacian and a q-Laplacian) and the generalized p-mean curvature differential operator. The precise conditions set on the data will be formulated in Sect. 3.
Moreover, for problems with double phase operators (which include the case for the ( p, q)-Laplacian) and multivalued terms (and also dealing with obstacle problems), we refer to Zeng-Gasiński-Winkert-Bai [23][24][25]. Finally, we mention that Mingione-Rȃdulescu [26] provided an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators.

Preliminaries
Let be a bounded domain in R N and let 1 ≤ r ≤ ∞. In what follows, we denote by L r ( ) and L r ( ; R N ) the usual Lebesgue spaces endowed with the norm · r . Moreover, W 1,r 0 ( ) stands for the Sobolev space endowed with the norm u := ∇u r for all u ∈ W 1,r 0 ( ).
Let us now consider the eigenvalue problem for the r -Laplacian with homogeneous Dirichlet boundary condition and 1 < r < ∞ which is defined by (2.1) A number λ ∈ R is an eigenvalue of − r , W 1,r 0 ( ) if problem (2.1) has a nontrivial solution u ∈ W 1,r 0 ( ) 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 ( ) . From Lê [27] 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 For s > 1 we denote by s = s s−1 its conjugate, the inner product in R N is denoted by · and the Euclidean norm of R N by |·|. Moreover, R + = [0, +∞) and the Lebesgue measure of a set A in R N is denoted by |A| N .
As for the data a of the problem (1.1) we assume that H (a) 0 : a : × R N → R N is a function such that a(x, ξ) = a 0 (x, |ξ |) ξ with a 0 ∈ C( × R + ) for all ξ ∈ R N and with a 0 (x, t) > 0 for all x ∈ , all t > 0.
For the regularity of a 0 as well as its behavior at zero, we will assume the following For the growth assumptions on a, we will exploit a function ϑ ∈ C 1 (0, ∞) satisfying for all t > 0, with some constants a 1 , a 2 , a 3 , a 4 > 0 and for 1 < q < p < ∞. Now we can state the growth of a as follows The above defined function ϑ will be also exploited in the next assumption guaranteeing coercivity-like behavior of the function a H (a) 3 Under the above hypotheses, we can state the following lemma in which we will summarize some properties of the function a : ×R N → R N (see e.g., Bai-Gasiński-Papageorgiou [3, Lemma 2.2]).

Proposition 2.2 If hypotheses H (a) 0 -H (a) 3 hold, then the operator A defined by (2.4) is bounded, monotone, continuous, hence maximal monotone and of type (S + ).
In the following example we indicate some operators fitting in our framework.

Example 2.3
In what follows for simplicity, we drop the dependence of the operators a on x. All the following maps satisfy hypotheses H (a) 0 -H (a) 3 : then the corresponding operator is the classical p-Laplacian and μ > 0 then the corresponding operator is the so called weighted ( p, q)-Laplacian defined by then the corresponding operator represents the generalized p-mean curvature differential operator defined by 1+|y| p with 2 < p < +∞, then the corresponding operator takes the form which arises in various problems of plasticity.
Next, let us recall the notions of pseudomonotonicity and generalized pseudomonotonicity for multivalued operators (see e.g., Gasiński-Papageorgiou [29, Definition 1.4.8]) which will be useful in the sequel.

Definition 2.4
Let X be a real reflexive Banach space. The operator A : X → 2 X * is called (a) pseudomonotone if the following conditions hold: (b) generalized pseudomonotone if the following holds: Let {u n } ⊂ X and {u * n } ⊂ X * with u * n ∈ A(u n ). If u n u in X and u * n u * in X * and lim sup n→∞ u * n , u n − u X * ×X ≤ 0, then the element u * lies in A(u) and u * n , u n X * ×X → u * , u X * ×X .
It is not difficult to see that every pseudomonotone operator is generalized pseudomonotone, see e.g., Carl-Le-Motreanu [ Proposition 2.5 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) is a nonempty, closed and convex subset of X * .
then u * ∈ A(u) and u * n , u n X * ×X → u * , u X * ×X .
Then the operator A : X → 2 X * is pseudomonotone.
Finally, we will state the following surjectivity theorem for multivalued mappings which are defined as the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping. The following theorem can be found in Le [20, Theorem 2.2]. We use the notation B R (0) := {u ∈ X : u X < R}. Theorem 2.6 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, all ξ ∈ F(u) and all η ∈ G(u). Then the inclusion F(u) + G(u) L has a solution in D(F).

Main Results
Let us start this section with the assumption of the multivalued convection term f : × R × R N → 2 R which will be needed in the existence result for problem (1.1). First of them provides general information on the regularity of f .
has nonempty, compact and convex values; for all (s, ξ) ∈ R × R N , the multivalued mapping x → f (x, s, ξ) has a measurable selection; for almost all x ∈ , the multivalued mapping (s, ξ) → f (x, s, ξ) is upper semicontinuous.
Next two assumptions provide the growth conditions on f . In what follows by p * we denote the critical exponent corresponding to p, namely for all η ∈ f (x, s, ξ), for a.a. x ∈ , all s ∈ R and all ξ ∈ R N . H ( f ) 2 : there exist w ∈ L 1 + ( ) and b 1 , b 2 ≥ 0 are such that [see (2.3) for the definition of a 3 and (2.2) for the definition of λ 1, p ] and for all η ∈ f (x, s, ξ), for a.a. x ∈ , all s ∈ R and all ξ ∈ R N .
We can provide an explicit example of a function f satisfying the above hypotheses.
It is obvious that the set K is a nonempty, closed and convex subset of W The weak solutions for problem (1.1) are understood in the following sense.

Definition 3.3 We say that u ∈ K is a weak solution of problem (1.1) if there exists
where K is given by (3.1).
The main result of this paper, providing existence of solutions as well as the properties of the solution set, is stated as the next theorem. Proof The proof of the theorem is divided into three steps.
Consider the embedding operator i : W 1, p 0 ( ) → L q 1 ( ) and denote by i * : L q 1 ( ) → W 1, p 0 ( ) * its adjoint operator. As 1 < q 1 < p * , the embedding operator i is compact and so is i * . Moreover, by virtue of hypotheses H ( f ) 0 and H ( f ) 1 , the Nemytskij operator N f : W 1, p 0 ( ) ⊂ L q 1 ( ) → 2 L q 1 ( ) associated to the multivalued mapping f : 0 ( ) * and introduce the indicator function It is easy to see that u ∈ K is a weak solution of problem (1.1) (see Definition 3.3), if and only if u solves the following inequality: given by (2.4) and ·, · stands for the duality pairing between W 1, p 0 ( ) * and W 1, p 0 ( ). Next, let us consider the multivalued operator A : W Now we can reformulate problem (3.3) in the following equivalently way: 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.
In order to prove that problem (3.4) has at least one weak solution, we will apply the surjectivity result for multivalued pseudomonotone operators (see Theorem 2.6). Let u ∈ W 1, p 0 ( ) and η ∈ N f (u) be arbitrary. By condition H ( f ) 1 , we have Remembering that 1 < q 1 < p * and using Proposition 2.2 (or Lemma 2.1(ii)) we get that A : W 1, p 0 ( ) → 2 W 1, p 0 ( ) * is a bounded mapping. Next, using Proposition 2.5, we will prove that A is a pseudomonotone operator.
By hypotheses on f , it is clear that A(u) is nonempty, closed and convex subset of W 1, p ( ) * for all u ∈ W 1, p 0 ( ). Moreover, as we just showed, A is a bounded mapping. So, it is enough to show that A is a generalized pseudomonotone operator (see Proposition 2.5). Let 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 . Because the embedding W 1, p 0 ( ) → L q 1 ( ) is compact, from (3.6), we get that u n → u in L q 1 ( ). On the other hand, by virtue of (3.5), we have that the sequence {ξ n } is bounded in L q 1 ( ). Therefore, by (3.8) we get This fact together with (3.6) and the (S + )-property of A (see Proposition 2.2), imply that u n → u in W As ξ n ∈ N f (u n ), we have Estimate (3.5) and convergence (3.6) imply that the sequence {ξ n } is bounded in L q 1 ( ). Passing to a subsequence if necessary, we may assume that for some ξ ∈ L q 1 ( ). Recall that u n → u in W 1, p 0 ( ), so, passing to a subsequence if necessary, we have u n (x) → u(x) and ∇u n (x) → ∇u(x) as n → ∞ for a.e. x ∈ .
, we can find R 0 > 0 large enough such that for all R ≥ R 0 we have Therefore, inequality (3.12) holds. Because ∂ I K : W 1, p 0 ( ) → 2 W 1, p 0 ( ) * is a maximal monotone operator, we can apply Theorem 2.6 with F = ∂ I K , G = A, L = 0, and conclude that inclusion (3.4) has at least one solution u ∈ K which is a solution of (3.3) and so also a solution of (1.1) in the sense of Definition 3.3. Thus, S = ∅.
Step 2. The set S is closed in W 1, p 0 ( ). Let {u n } ⊂ S be a sequence such that u n → u in W 1, p 0 ( ), (3.16) for some u ∈ W 1, p 0 ( ). For each n ∈ N, we can find ξ n ∈ N f (u n ) such that for all v ∈ W 1, p 0 ( ). By hypothesis H ( f ) 1 and (3.16) we know that {ξ n } is bounded in L q 1 ( ). So, passing to a subsequence if necessary, we may assume that ξ n ξ in L q 1 ( ).
As before, using Theorem 7.2.2 of Aubin and Frankowska [31, p. 273], we obtain that ξ(x) ∈ f (x, u(x), ∇u(x)) for a.a. x ∈ , i.e., ξ ∈ N f (u). Passing to the upper limit in (3.17) as n → ∞ and using the lower semicontinuity of I K we conclude that u ∈ K is a solution of problem (1.1). Thus, the set S is closed.
Step 3. The set S is bounded.
If the set K is bounded the result is clearly true. So, let us assume that K be unbounded. Proceeding by contradiction, assume that the set S is unbounded, so we can find a sequence {u n } ⊆ S such that u n → +∞. (3.18) Similarly as in (3.13), we show that for some ξ n ∈ N f (u n ). This inequality together with (3.18) give a contradiction. Therefore, the set S is bounded.