The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

Abstract

We prove the existence of extremal solutions of the following quasilinear elliptic problem under Dirichlet boundary condition in Orlicz-Sobolev spaces and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.

1. Introduction

The aim of this paper is to study some qualitative properties of solutions of the following quasilinear elliptic problem:on a bounded domain with a Lipschitz boundary in Orlicz-Sobolev spaces. The differential part is driven by a Leray-Lions operator, while the nonlinear term is a Carathéodory function satisfying a growth condition.

In [1, Chapter 3], the differential part of (1) is a Leray-Lions operator in Sobolev spaces and the nonlinearity satisfies the growth condition:with the constant and , for a.e. , all and all , where is the conjugate Hölder exponent to , i.e., . In [2], the nonlinearity satisfies the growth condition:with the constant and , , , , for a.e. , all and all , where is the Sobolev conjugate of . Faria [2] pointed that the condition (3) is more general than (2) because . However, if . Hence, the growth condition (3) is not more general than (2).

When trying to weaken the restriction on the Leray-Lions operator and the growth condition (2), one is led to replace with built from an Orlicz space instead of , where is an -function. The choice , leads to [1, Theorem 3.17]. A nonstandard example is (see, e.g., [2, 3]).

Many papers used the surjectivity result for pseudomonotone operators (see, e.g., [1, Theorem 2.99]) defined on reflexive spaces to prove the existence of the solution (see, e.g., [1, 2, 4, 5]). Our method does not need the reflexivity of the spaces. It is well known that the Orlicz space is reflex if and only if both and its complementary function satisfy -condition. However, there exist many spaces without reflexivity. For example, let ; then satisfies -condition, but its complementary function does not satisfy -condition; i.e., is not reflexive.

In this paper, we get rid of the restriction of the reflexivity of the spaces and get a weak solution for (1) in Orlicz-Sobolev spaces by using a linear functional analysis method. We also give the enclosure of solutions and prove the existence of extremal solutions.

This paper is organized as follows. Section 2 contains some preliminaries and some technical lemmas which will be needed. In Section 3, we use the linear functional analysis method to prove the existence of solutions for (1) in separable Orlicz-Sobolev spaces and the sub-supersolutions method to give the enclosure of solutions and the existence of extremal solutions between a subsolution and a supersolution. We also get the compactness and directness of the solutions set.

For some results, we also refer to [6–13].

2. Preliminaries

For quick reference, we recall some basic results of Orlicz spaces. Good references are Adams [14, Chapter 8], Krasnosel’skil [15], Chen [16], and Gossez [17].

2.1. N-Function

Let be an -function; i.e., is continuous, 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 , ,

Moreover, one has the following Young inequality: ,

For the -function one defines the Sobolev conjugate by .

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 .

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

For a measurable function on , its modular is defined by .

2.2. Orlicz Spaces

Let be an open and bounded subset of and be 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) norm:and 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 .

Convergences in norm and in modular are equivalent if and only if .

The dual space of can be identified with by means of the pairing , and the dual norm of is equivalent to .

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 spaces under the normDenote 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 .

If , then . If and , then are reflexive; thus the weak topologies and are equivalent.

Lemma 1 (See [18, Lemma 2.2]). For all , one haswhere is the diameter of .

Lemma 2 (See [19, Lemma 1]). Let be bounded and , , for . Then

Lemma 3 (See [20, Lemma 2.1]). If , then , and

Here , .

3. Main Results

Let be a bounded domain in with Lipschitz boundary, be two -functions, and be the complementary functions of , respectively. Assume that satisfies the condition near infinity and . By Theorem 2.2 and Proposition 2.1 in [21] the embeddings and are compact.

Let be the following quasilinear elliptic differential operator in divergence form:where the coefficients , , are assumed to satisfy the following: (H1)Each function is a Carathéodory function. Also there exists a positive constant and a nonnegative function such that for a.e. and for all , .(H2) for a.e. , all , and all with .(H3) for a.e. , all , and all , with some constant and a function .

The differential operator can be seen as a mapping from into its dual space given by

Example 4. (1) The p-Laplacian operator is form with the coefficients , , given by (see, e.g., [1, Example 2.110]).
(2) Let be a given positive and continuous function which increases from to and . Then , , satisfy the conditions (H1)-(H3).

Consider the following nonlinear elliptic equation:Here, is assumed to be a Carathéodory function.

Let denote the Nemytskij operator related to by

For , , we use the standard notations: , , for a.e. . A weak solution of (12) is called a solution for short.

By Lemma 3, is closed under and . In fact, since and , , , for any , .

The following lemma can be found in [5, Remark 3.1] as the setting of Musielak-Orlicz spaces. However, we give another proof.

Lemma 5. (a) (resp., ) is closed under “” and “”, i.e., if (resp., ), then (resp., ).
(b) The mappings and : (resp., ) are continuous, i.e., for any sequences in (resp., ), if in (resp., ), then , in (resp., ), as .

Proof. (a) By Lemma 3, (resp., ) is closed under and . In fact, since and , (resp., ), for any (resp., ).
(b) Let in (resp., ), as . Suppose that there exists such that for any , then .
Therefore, we have , and , as . By passing to a subsequence if necessary, , , a.e. in , as , and there exist such that , and , which yields that , , for a.e. .
Hence, a.e. in , as , and for a.e. .
By Lebesgue’s theorem, we get , as ; this is a contradiction. Consequently, , as . Similarly, we can deduce that , and , as ; that is, the mappings and are continuous.

A function is called a (weak) solution of (12) if , and satisfies the following:

A function is called a subsolution (resp. supersolution) of (12) if , , and (16) holds with “” replaced with “” (resp. “”) for every nonnegative functions in .

By Young inequality and , there exist and , such that for all . Hence,

Theorem 6. Let and be a subsolution and a supersolution of problem (12), respectively, such that . Assume (H1)-(H3) and the following local growth condition for the nonlinearity :for a.e. , all , and all , with , , . Then there exists at least one solution of problem (12) with .

Proof. Denote . For , , we putThen . By Lemma 5, is continuous. It is easy to see that is bounded.
We define the cutoff function given byfor , . Then satisfies the following condition:for and all .
Since is convex and , there exist and such that whenever , and whenever for , . For all , we have where and the constants .
Define ,, where is a parameter to be specified later. Then is well defined.
Since , there exists a sequence such that dense in . Let and consider . and are two norms of equivalent to the usual norm of finite dimensional vector spaces.
Similar to the proof of Proposition 3.1 in [22], we can deduce that the mapping is continuous.
By (H3), (18), and (22), for every , where , such that and the constants . Let . Then we can deduce thatBy Lemma 1, we getwhere the constant . By Lemma 2, we immediately haveCombining (25) and (27), we obtainBy Remark 2.1 in [22], for every , there is a Galerkin solution such thatBy the density of , we get As the same proof in [22], we can deduce that the sequence is bounded in and there exists and a subsequence of , such thatas .
From (21), is bounded in . By Lemma 4.4 of [17],as .
On the other hand, thanks to (32) and (33), we haveas . Thus we obtain thatSimilar to the proof of Proposition 3.1 in [22], we can construct a subsequence still denoted by such thatHence,as .
Following the lines of Theorem 1 in [2], we can deduce that for every . By (34), .
Denote and . Then , for every . It follows from (39) that, passing to a subsequence if necessary, Therefore,Since and are bounded in , is bounded in and is bounded in . By Lemma 4.4 of [17], weakly in for . Thanks to (35), one has , for any . Therefore, we obtain that is a solution of (12).

Under the assumptions of Theorem 6, we define

Theorem 7. Under the assumptions of Theorem 6, the set is compact in .

Proof. Let be a sequence in . It follows from the coerciveness of that is bounded in . As the same proof of Theorem 6, there exists that is a solution of (12) and , i.e., .

To show that the set is directed with respect to the usual pointwise order, the following additional assumption on the coefficients is required.(H4)Let a nonnegative function and a continuous function exist such that  holds for a.e. , for all and for all , where satisfies , that is, for every , .

Similar to the proof of [1, Theorem 3.20], we can deduce the following result.

Theorem 8. Assume hypotheses (H1)-(H4), and let and be subsolutions of (12) such that the Nemytskii operatoris well defined. Then is a subsolution of (12). Analogously, if and are supersolutions of (12) with the same assumption on the Nemytskii operator , then is a supersolution.

Theorem 9. Let the assumptions of Theorem 6 and (H4) hold. Then the following assertions about are true. (a) is a direct set in both directions; that is, if then there exist such that and .(b) possesses extremal elements; i.e., there are such that , for all .

Proof. (a) Let . Then and are both subsolutions and supersolutions of (12). It follows, from Theorem 8, is a subsolution and is a supersolution of (12). The claim in (a) is now a straightforward consequence of Theorem 6.
(b) Since is separable, there exists a countable, dense subset of . Let . By (a), we can select such that . Thus, we get a bounded increasing sequence . Consequently, , for a.e. , and there exists a subsequence such that weakly in for as . Similar to the proof of Theorem 6, we can deduce that . From the density of , we can get that is the greatest element of . The existence of the smallest element of can be deduced in the same way.

Remark 10. A special case in Theorem 6 is that . In this case, choice leads to Theorem 3.17 in [1].

Remark 11. The above results can be extended to the more general situation of Musielak-Orlicz-Sobolev spaces following our method developed in this paper.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first author is supported by ’Chen Guang’ Project (supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation) (10CGB25) and the Teaching Reform Project of Jianqiao University (JGXM201608). The second author is supported by the National Natural Science Foundation of China (11371279) and the Fundamental Research Funds for the Central Universities.