A Variational Method for Multivalued Boundary Value Problems

Abstract

In this paper, we investigate the existence of solution for differential systems involving a Laplacian operator which incorporates as a special case the well-known Laplacian operator. In this purpose, we use a variational method which relies on Szulkin’s critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive.

1. Introduction

This paper is devoted to the study of the following second-order differential systems:where is fixed, is a monotone homeomorphism, is a proper, convex, and lower semicontinuous function, is a Caratheodory mapping, continuously differentiable with respect to the second variable and satisfies some usual growth conditions, is the gradient of at and denotes the subdifferential of j in the sense of convex analysis.

Second-order differential problems with multivalued boundary value conditions have been studied by many authors. In this direction, we can cite the works of Bader and Papageorgioua [1], Béhi et al. [2], Gasinski and Papageorgiou [4], Jebelean [5], Jebelean and Morosanu [6], and references therein. In [1, 4–6], the authors investigate differential systems driven by a homogeneous Laplacian operator while in problem (1), we deal with a nonhomogeneous Laplacian operator which incorporates as a special case the Laplacian operator. As a consequence, problem (1) is a generalization of the following problem studied, in 2005, by Jebelean and Morosanu [6]:where denotes the homogeneous operator Laplacian. Indeed, in our work, the Laplacian operators are nonhomogeneous and are of the form with the Laplacian operator, , and a continuous map.

In order to obtain existence result for problem (1), we will use a variational method which relies on Szulkin’s critical point theory [7].

This paper is organized as follows: After introducing notations and preliminary results in Section 2, in Section 3 we give a variational approach to problem (1). In Section 4, using some results from Section 3, we prove our main result. In Section 4, we give an example to illustrate the applicability of our result. Finally, Section 5 is reserved for the conclusion.

2. Preliminaries

Let us recall some notions and results in the framework of Szulkin’s critical point theory [7] which are needed and also some notations which will be used in the sequel. At this end, is the Sobolev Banach space which will be endowed with the norm:where and is the norm on defined by

We denote , the norm on the set :

Let be a real Banach space and be a functional of typewhere and G is proper, convex, and lower semicontinuous (lsc). A point is said to be a critical point of B if it satisfies the inequality

A number such that contains a critical point is called a critical value of B.

To establish existence results for (1), we will need the following proposition due to Szulkin (see [7] Proposition 1.1).

Proposition 1. If B satisfies (7), each local minimum point of B is necessarily a critical point of B.

The functional B is said to satisfy the Palais-smale (in short, (PS)) condition if every sequence for which andwhere possesses a convergent subsequence.

3. A Variational Approach for Problem (1)

Before beginning the variational approach, we make the following hypotheses on the data of problem (1): : is a continuous map such that there exists such that : is a monotone homeomorphism such that  , , with the well-know Laplacian operator and with of class on , strictly convex (i.e., is a potential function corresponding to ϕ) and such that ;  there exist such that

 where denotes the Euclidean norm on .

Remark 1. Suppose that , , we have . Then this function satisfies hypotheses and . Other cases that satisfy Hypotheses and are when with and the function is a monotone homeomorphism on . This is the case, for example, when function a is equal to one of the following functions:with . Indeed, for these examples, the potentials functions corresponding to ϕ are respectively the functions and , . : is a Caratheodory mapping such that : F (t, .) is continuously differentiable, for a.e.  :  : for each there is some such thatOur notion of solution of problem (1) is defined as follows:

Definition 1. By a solution of the differential system (1), we will understand a function of class with absolutely continuous, which satisfies the equality in (1) a.e. on .
Now let us start the variational approach. In this purpose, for , let be defined by

Lemma 1. is proper, convex, and lower semicontinuous.

Proof. By hypothesis , it followsWhence is proper. Also since is convex, it follows that is convex. Finally, because of the lower semicontinuity of the functional norm on Banach space, is lower semicontinuous (in short, lsc).

Lemma 2. andwhere is the inner product in .

Proof. Let us consider the product space , , equipped with the normWe define such thatLet us show that is bounded.
We haveIt followsSo is bounded.
Let us show that is continuous.
Let be a sequence such that in . Then in and in . Whence in and in . We setBy the previous arguments, the sequence in . We infer that the sequence in H. So is continuous.
We consider the functional:Let us show that the linear operator Q is continuous on .
Using Hölder’s inequality, we obtainSo Q is continuous.
Let us show that is Frechet differentiable in and in the sense of (15).
Using Fubini’s inequality, for arbitrary , we obtainwith and ; and .
Arguing as in the proof of the continuity of and the fact it is bounded on , we show that L and are continuous and bounded on . Moreover, using Lebesgue’s dominated comvergence theorem, we haveTherefore, (15) is proved.
We know that where denotes the dual of .
Let us show that is continuous.
We haveUsing Hölder’s inequality, we haveWhenceThus, since is continuous, we obtain the continuity of .
We introduce also the functional: defined byRecall that j is proper, convex, and lsc. Then, J is also proper, convex, and lsc. Let us setSince and J are proper, convex and lsc, it follows that is proper, convex and lsc on . Assuming that hypotheses and on the Caratheodory function F hold, for each , we obtain:with . Equation (30) comes from inequality (12) and the estimation:equation (30) and the embedding allow us to introduce the functional: defined by

Lemma 3. If the hypotheses , and hold, then

Proof. The proof is similar to the one of Theorem 2.6 of [2].
Considering the functional framework in Section 2, we set in (32), in (29) and :

Proposition 2. Let the Caratheodory function satisfies , and and let . If u is a critical point of the functional defined by (35), in the sense (7), i.e.,then u is a solution of problem (1). The converse implication is also true.

Proof. We suppose that u is a critical point of . In the inequality (36), we take . Then dividing by s and letting , we obtainwhere is the directional derivative of the convex function J at u in the direction . From (28) and (37), it followsSince , using Hahn–Banach’s theorem (see Theorem I.1 of Brezis [3]), (38) impliesUsing (15), (34), and (39), we obtainFrom hypothesis and , it follows thatAlso, impliesEquations (40) and (42) implyWith ϕ being a homeomorphism, (43) ensures that . This together with (44) shows that u is the solution of the differential system (1). Furthermore, (38) and (44) yieldThus,which, by a standard result from convex analysis (Theorem 23.2 of Rockafellar [8]), means that the boundary conditions in (1) are true.
Now let us show the converse implication.
By multiplying (1) with and integrating by parts on , we obtainUsing inequalities (15) and (34) in (47), it followsBy using Theorem 23.2 of Rockafellar [8], we obtainUsing some previous arguments, we infer that

4. Existence Results for Problem (1)

At first, let us introduce the constantfor . If , then and if , then . To obtain the existence result, we make the following hypothesis:  .

Proposition 3. If is true, thenwhere J is defined by (28).

Proof. Since , by (51) we haveWhence,ThenIt followsWhenceFurthermore, we haveThenSoInequalities (57) and (60) yield the result.
If the nonlinearity F lies asymptotically on the left of , then problem (1) is solvable.

Theorem 1. Assume , , and . Ifthe problem (1) has at least a solution.

Proof. Let us show that the functional in (35) is sequentially weakly lsc and coercive on the space .
Let us show that in (32) is sequentially weakly continuous.
Let be such that , with some . By , there is an such thatWe estimateand by (62), it followsBy the compactness of the embedding and (64), it follows that is sequentially weakly continuous on . Then, by the weak lower semicontinuity of in (29), is sequentially weakly lower semicontinuous.
Furthermore, from (61), there are constant and such thatThen, (30) and (65) yieldwhich, by (32), givewith . Using (29), (67), and (53) and the Proposition 3, we estimate on as follows:Since j is convex and lsc, it is bounded from below by an affine functional. Therefore, by (28) there are constants such thatFrom (69) and the continuity embedding of , we obtainwith some . Consequently,meaning that is coercive on .
Since the functional is sequentially weakly lsc and coercive, by a well-known result from calculus of variations, is bounded from below and attains its infimum at some . Then, by Proposition 1 and 3, u is a solution of problem (1).
Remarque 2. The problem (1) incorporates classical problems of Dirichlet, Neumann, Periodic, antiperiodic, and Sturm–Louiville (see Remark 3.9 of Jebelean [5]).

5. Example

We consider the following problem:where , is the well-known Laplacian operator. Let us set . Whence . When, , we see that ϕ is a monotone homeomorphism such that . Moreover, satisfies hypothesis . Therefore, by Theorem 1, problem (72) has at least a solution.

6. Conclusion

In this article, using variational method which relies on Szulkin theory, we have established existence for second-order problems with multivalued boundary conditions. We give an example but more examples and applications can be given.

Data Availability

No data were used to support this study.

Conflicts of Interest

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