Variational Inequalities with Multivalued Lower Order Terms and Convex Functionals in Orlicz-Sobolev Spaces

Dong, Ge; Fang, Xiaochun

doi:https://doi.org/10.1155/2015/321437

Journal of Function Spaces

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Research Article | Open Access

Volume 2015 | Article ID 321437 | https://doi.org/10.1155/2015/321437

Variational Inequalities with Multivalued Lower Order Terms and Convex Functionals in Orlicz-Sobolev Spaces

Ge Dong^1,2and Xiaochun Fang¹

Academic Editor: Dumitru Motreanu

Received12 Jan 2015

Accepted27 Feb 2015

Published25 Mar 2015

Abstract

We consider the existence of solutions of variational inequality form. Find , whose principal part is having a growth not necessarily of polynomial type, where is a second-order elliptic operator of Leray-Lions type, is a multivalued lower order term, and is a convex functional. We use subsupersolution methods to study the existence and enclosure of solutions in Orlicz-Sobolev spaces.

1. Introduction

Let be a bounded domain in with Lipschitz boundary, and let be a Leray-Lions operator defined on , .

Le [1] used subsupersolution methods to study the existence and enclosure of solutions of the variational inequality of the following form.

(P₀) Find such thatwhere is a multivalued lower order term and is a convex functional. Accordingly the function is supposed to satisfy polynomial growth conditions with respect to , where is the derivative of .

When trying to weaken this restriction on , one is led to replace by a Sobolev space built from an Orlicz space instead of . Here the -function which defines is related to the growth of the function .

It is well known that, in the study of differential equations, different classes of differential equations correspond to different function space settings. The classical Sobolev space is a special case of Orlicz-Sobolev spaces.

In this paper, it is our purpose to study the existence and enclosure of solutions to the problem (P₀) in the setting of the Orlicz-Sobolev spaces.

The paper is organized as follows: Section 2 contains some preliminaries and some technical lemmas which will be needed in Section 3. In Section 3, we first establish some basic properties of the operator in Orlicz-Sobolev spaces; next, following Le [1] in which a subsupersolution method for variational inequality of the form (P₀) in Sobolev spaces was established, we prove the existence and enclosure of solutions of the problem (P₀) in Orlicz-Sobolev spaces.

We refer to some results of a subsupersolution method for variational inequalities studied in variable exponent Sobolev spaces (cf., e.g., [2–4]) and partial differential equations in Musielak-Orlicz-Sobolev spaces (cf., e.g., [5]). For some classical results we also refer to [6–12].

2. Preliminaries

2.1. Function

Let be an -function; that is, is continuous and convex, with for , as , and as . Equivalently, admits the representation , where is a nondecreasing, right continuous function, with , for , and as .

The -function conjugated to is defined by , where is given by .

are called the right-hand derivatives of , respectively.

The -function is said to satisfy the condition near infinity (, for short), if for some and , , . It is readily seen that this will be the case if and only if for every there exists a positive constant and , such that , , and equivalently, there exists and , such that(cf., e.g., [13]).

Moreover, one has the following Young inequality: , .

We will extend these -functions into even functions on all .

Let be two -functions; we say that grows essentially less rapidly than near infinity, denoted as , if for every , as . This is the case if and only if (cf., e.g., [14, 15]).

For a measurable function on , its modular is defined by (cf., e.g., [16]).

2.2. Orlicz Spaces

Let be an open and bounded subset of and an -function. The Orlicz class (resp., the Orlicz space ) is defined as the set of (equivalence classes of) real valued measurable functions on such that is a Banach space under the (Luxemburg) normand is a convex subset of but not necessarily a linear space.

The closure in of the set of bounded measurable functions with compact support in is denoted by .

The equality holds if and only if ; moreover, is separable.

is reflexive if and only if and (cf., e.g., [16]).

Convergences in norm and in modular are equivalent if and only if (cf., e.g., [16]).

The dual space of can be identified with by means of the pairing , and the dual norm of is equivalent to (cf., e.g., [14, 16]).

2.3. Orlicz-Sobolev Spaces

We now turn to the Orlicz-Sobolev space: (resp., ) is the space of all functions such that and its distributional partial derivatives lie in (resp., ). It is a Banach space under the norm Denote and . Clearly, is equivalent to .

Thus and can be identified with subspaces of the product of copies of . Denoting this product by , we will use the weak topologies and (cf. [15]).

If , then . If and , then are reflexive (cf., e.g., [14]); thus the weak topologies and are equivalent.

We recall the following notations and lemmas which will be used later.

Definition 1 (cf., e.g., [17, Definition 32.1]). Let be nonempty sets, and a multivalued mapping; that is, assigns to each point a subset of .(i)The set is called the effective domain of .(ii)The set is called the range of .(iii)The set is called the graph of .

Definition 2 (cf., e.g., [18, Definition 2.112]). Let be Banach spaces. The multivalued operator is called upper semicontinuous at , if for every open subset with , a neighborhood exists such that . is called upper semicontinuous in , if is upper semicontinuous at every .

The following definition can be referred to [19, Page 41], [18, Definition 2.120], [20, Definition 1], or [3, Definition 2.1(b)].

Definition 3. Let be a real reflexive Banach space. The operator is called pseudomonotone if the following conditions hold.(i)The set is nonempty, bounded, closed, and convex for all .(ii) is upper semicontinuous from each finite-dimensional subspace of to the weak topology on .(iii)If with and if is such that , then, to each element exists with

Lemma 4 (see [21, Lemma 2.1]). If , then and

Here , .

The following lemma can be referred to [18, Theorem 2.127] or [19, Theorem 2.11]. The proof of the lemma can be found in [19].

Lemma 5. Let be a real reflexive Banach space, a maximal monotone operator, and . Let be a pseudomonotone operator, and assume that either is quasibounded or is strongly quasibounded. Assume further that is -coercive; that is, a real-valued function exists with as such that, for all , we have . Then is surjective; that is, .

3. Main Results

Let be a bounded domain in with Lipschitz boundary, an -function, and the complementary function of . We say that satisfies (), if both and satisfy the condition near infinity, and satisfies the following coerciveness condition.

There exists a function such that as andLet be an -function such that , where is the Sobolev conjugate of . Assume that and satisfy the condition near infinity.

In what follows we denote by the set of all (equivalence classes of) Lebesgue measurable functions from to .

We consider the following variational inequality.

(P) Find such thatDetailed assumptions on together with a precise formulation of this inequality are presented in the following subsection.

3.1. Assumptions and Problem Settings

Let be a Carathéodory function satisfying the following conditions:(A1)For a.e. and all where , , and .(A2) is monotone in the following sense: for a.e. and all .

Define by

Example 6. (1) Let for , where . Then it can be verified that and are -functions satisfying condition near infinity and satisfies (8).
(2) Put , , . Then satisfies (A1)-(A2).

Remark 7. By (2), we have , , for some and .

Moreover, we have the following lemma.

Lemma 8. (1) Let and let satisfy (A1)-(A2). Then the operator is well-defined, bounded, and monotone.
(2) Let and and let satisfy (A1)-(A2). Then the operator is continuous. Moreover, is pseudomonotone.

Proof. (1) For every , by (10) and the convexity of , we havefor some , where . Therefore, . Then is well-defined.
From (14), one has that is bounded, for any bounded set .
The monotonicity of follows from (12).
(2) To prove the continuity of , letWe will prove that in , which is equivalent to that , orAssume that there exists such thatIn view of (15) and by [22, Chapter IV, Section 3, Theorem 3] there exists a subsequence of still denoted by and such that and for a.e. . Since is a Carathéodory function,By (10) and the convexity and -property of , we obtain that , for some . Therefore, where is a constant. By Lebesgue’s dominated convergence theorem, we getThis is a contradiction. Hence (16) holds. This shows that is continuous.
Since is monotone, hemicontinuous (in fact, continuous), and bounded on , it is pseudomonotone.

Let be a convex, proper (cf. [19]), lower semicontinuous function. The effective domain of is , and let be the subdifferential of .

For any topological vector space , we use the notation , is closed and convex. Let be a function from to such that(F1) is superpositionally measurable (cf. [1, 23] or [24]);(F2)for a.e. , the function is upper semicontinuous.

The lower order term is defined by the (multivalued) integral For a precise definition of and an interpretation of this integral, we need some further notations. For any , let be the set of all measurable selections of . We know that since is measurable.

3.2. A Subsupersolution Method

For , we use the standard notation , .

By Lemma 4, is closed under and . In fact, since and , , for any .

Let be the set of all proper (not necessarily convex) functionals from to .

We introduce the following definitions as in [1].

Definition 9. Let . We say that (or equivalently ) if and only if , .

From Definition 9, if , then (cf. e.g. [1]).

Definition 10. (1) is called a solution of (9) if and there exists and such that , for a.e. , and(2) is called a subsolution of (9) if there exist , , and such that , , , for a.e. , andfor all .
(3) is called a supersolution of (9) if there exists , , and such that , , , for a.e. , andfor all .

From Definition 10, we see that is a solution of (9) if and only if is a solution of the following inclusion: find such that

We will study the existence and some properties of solutions of (9). The proof of the following theorem is based on the ideas and arguments in the proof of Proposition 2.2 in [4], Theorem 2.11 in [1], and Theorem 4.2 in [2] and is divided into several steps. Le [1] points that if is graph measurable then is superpositionally measurable. It is only needed in our proof is superpositionally measurable as [1]. However, we do not need the condition (F3) in [1].

Theorem 11. Let , , and satisfy (), (A1)-(A2), and (F1)-(F2), respectively. Assume that there are subsolutions and supersolutions of (9) such thatand there exists such thatfor a.e. , all .
Then, there exists a solution of (9) such that

Proof. Denote . Since , is compact ([25, Theorem 2.2]). By Proposition 2.1 in [25], . Hence, is compact. We will use and for the embeddings from into and , respectively. Let and be the adjoints of and , respectively. Thus , , , and are compact. Note that is the usual identity embedding, for ; that is, for a.e. . Similarly, for , .
Step 1. Let and satisfy the conditions as in Definition 10 of sub- and supersolutions. Let , and for . Similarly, let , and for . Then . Define and , where is the characteristic function of . It is clear that and for a.e. .
For , , putThen, as in [1, 26], we can check that satisfies (F1) and (F2). From (31) and (27), we see thatWe define given byfor , . Then is a Carathéodory function andwith . By Young inequality and the convexity of , the operator defined by is well defined. From (34), we see that is a bounded operator. Moreover, the mapping is continuous from to . In fact, let in , as . If is not continuous, then there exists such that , . By passing to a subsequence, if necessary, we have a.e. and there is such that for a.e. . Then a.e. , as . Since satisfies condition and is convex, where and the function in the right-hand side belongs to . Using Lebesgue’s dominated convergence theorem, , as . This is a contradiction. Due to the compact embedding , is weakly-strongly continuous from into its dual space . It follows that is a (single-valued) pseudomonotone operator from into .
For and , define and , whereSince is compact, the mapping and mapping are weakly-strongly continuous and bounded. Consequently, and are (single-valued) pseudomonotone operators from into .
Next, we will find and such thatStep 2. We will prove that is a pseudomonotone and bounded mapping from to .
(i) We prove that is a nonempty, bounded, closed, and convex subset of for all . Moreover, is a bounded mapping from to .
Clearly, for any , is a nonempty, bounded, closed, and convex subset of ; in particular, . Moreover, is a bounded operator from to .
For any , from the boundedness of and the above arguments, we get that is a nonempty, bounded, closed, and convex subset of . Moreover, since , for some constant , it follows from the boundedness of that is also a bounded mapping. Next, we prove that is closed in . In fact, assume , with , , andBecause , is a bounded sequence in . By passing to a subsequence if necessary, we can assume without loss of generality thatSince is weakly convex and closed in and is reflexive, , and thus . On the other hand, since is continuous from to with both weak topologies, we have from (40) that in which combined with (39) yields . Hence, is closed in .
(ii) Let be a finite-dimensional subspace of . We will show that the restriction of on is upper semicontinuous from into with respect to the weak topology on .
In fact, assume . To prove the upper semicontinuity of at , we assume by contradiction that there is a weakly open neighborhood of in and a sequence such thatand there exists a sequence such that , . We see that is a weakly open neighborhood of in . Moreover, since , there exists such thatWe have , . As is a bounded sequence in , it follows from (i) that is a bounded sequence in . Also, as mentioned above, by passing to a subsequence, we can assume thatfor some . By (41), in . Hence, by passing to a subsequence if necessary, we can assume thatFor a.e. and for any , denote , where . Then, is open set in and . By (F2), there exists such that if then . In view of (44), there exists , such that , . Hence, , . It follows that, for any , , . Since , for a.e. ; thus, there is a sequence such that as for a.e. , which implies On the other hand, since , we have , for some . By (32), , for a.e. , all . Using Lebesgue’s dominated convergence theorem, we obtain thatSince ,In view of (47) and (43), we can deduce thatas . Therefore, by the convexity and closeness of and the reflexivity of . Consequently, . By (43) and the reflexivity of , there exists , such that , for any . From (42), ; this is a contradiction.
(iii) Referring to [19, Proposition 2.2] we can get that if with and if such that , then to each element , exists withBy (i)–(iii) and using Definition 3, we get that is a pseudomonotone and bounded mapping from to .
Step 3. By Step 1 and Step 2, is a multivalued bounded pseudomonotone mapping on with domain .
From the definition of (38) is equivalent to . By [17, Proposition 32.17] or [27, Theorem 4], is maximal monotone.
Step 4. For , we check that is -coercive.
In fact, for any , by (A1), Young inequality, and the convexity of , we havewhere and is a positive constant independent of .
From (32), for any ,for some constant .
Since is convex and , there exists such that whenever , and whenever for , . Note that , . In the sequel, we use the set notation , , and . Then we have where , . Taking and , we can deduce thatFor , we have and thusfor some . Similarly,for some .
Define, for , and . Then, it is easy to see that .
Let . Then . Combining the estimates from (50)–(56) and using (8), we obtain that By the arbitrary of , we have Hence, where . Therefore, is -coercive.
We have verified that satisfy the conditions of Lemma 5. By Lemma 5, is surjective, that is, . Hence, (38) has a solution .
Step 5. Let be any solution of (38). Following the lines in [1], we can deduce (28).
From (28) and the definitions of and , we have a.e in , for all , . Also, we get that for a.e. . In view of these observations, (38) reduces to (25). Our proof of Theorem 11 is complete.

The choice , in Theorem 11 leads to Theorem 2.11 in [1].

Under the assumptions of Theorem 11, we define and then the set is nonempty. As consequences of Theorem 11, some further properties of are given in the following corollary. The proofs of these properties are omitted, since they do not require substantial modifications as in Theorem 11.

Corollary 12. Under the assumptions of Theorem 11, the following assertions are true.(i)Any is both a subsolution and a supersolution of (9).(ii) is directed downward and upward; that is, for all , there exist such that and .

Remark 13. If in problem , then the condition (8) can be omitted. In fact, by Lemma 3.14 and Lemma 5.7 in [15] the coerciveness holds in Step 4 of the proof of Theorem 11.
Moreover, if is given by an integral functional such as , is a single-valued term, then problem implies problem (1.1) in [28].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are highly grateful for the referee’s careful reading and comments on this paper. The first author was supported by “Chen Guang” Project (supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation) (10CGB25) and Shanghai Colleges for Outstanding Young Teachers’ Scientific Research Selection and Training Special Fund (sjq08011). The second author was supported by the National Natural Science Foundation of China (11371279).

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Copyright © 2015 Ge Dong and Xiaochun Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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