Abstract

This paper is aimed at providing three versions to solve and characterize weak solutions for Dirichlet problems involving the -Laplacian and the -pseudo-Laplacian. In this way generalized versions for some results which use Ekeland variational principle, critical points for nondifferentiable functionals, and Ghoussoub-Maurey linear principle have been proposed. Three sequences of generalized statements have been developed starting from the most abstract assertions until their applications in characterizing weak solutions for some mathematical physics problems involving the abovementioned operators.

1. Introduction

Obtaining and/or characterization of weak solutions for problems of mathematical physics equations involving -Laplacian and -pseudo-Laplacian is a subject matter previously discussed by the author through several approach methods in [1–6]. New similar results can be found by other authors, for instance, Amiri and Zivari-Rezapour in [7], El Khalil and Ouanan in [8], Rhouma and Sayeb in [9], Yoshida in [10]. The importance of these operators also devolves from their involvement in actual modelings of natural phenomena as thermal transfer by Lanchon-Ducauquis, Tulita, and Meuris in [11] or glacier sliding or flow by Partridge in [12] and Pélissier in [13]. In this paper three methods are proposed following three sequences of results, starting from abstract statements and finishing with their application to find weak solutions of Dirichlet problems for -Laplacian and for -pseudo-Laplacian as well. In the first succession of assertions, two results of Ghoussoub from [14] have been generalized replacing the frame of Hilbert space by reflexive strictly convex Banach space and the condition imposed to the goal function to be of -Fréchet class by the weaker condition to have a lower semicontinuous and Gâteaux differentiable function. These two theoretical statements were used, together with other results concerning Dirichlet problems for both cited generalized operators, in finding weak solutions for these problems. The second proposition sequence involves critical points for nondifferentiable functional. In this context, two results of Chang from [15] have been generalized changing the space in and introducing and in some problems formulated for by Costa and Gonçalves in [16]. The last series of assertions starts from Ghoussoub-Maurey linear principle which is used here to characterize weak solutions for some mathematical physics equations. Moreover, the problems were discussed and solved using important findings for these operators obtained by the author in [5, 6] in connection with specific properties of the Sobolev spaces involved.

2. Critical Points and Weak Solutions for Elliptic Type Equations

2.1. Theoretical Support

In order to introduce the first result, a theoretical support will be given starting with the following.

Ekeland Principle (see [1, 17, 18]). Let be a complete metric space and bounded from below, lower semicontinuous, and proper. For any and of withand for any , there exists in such that

Definition 1. Let be a real normed space, be a bornology on , and let . Let be in and a nonempty subset of . verifies the Palais-Smale condition on the level around (or relative to ), , with respect to , when a sequence of points in for whichthis sequence has a convergent subsequence.
To clarify the above notation, let be a bornology on and let locally finite in the point . By definition is -differentiable in , if there exists in the dual such that for every in we have (uniform limit on for ). is the -derivative of in , and it is denoted by .

Through the minimization on of a functional (minimization with constraints) global critical points of this may be obtained.

As a preliminary, we generalize some results from [14] introducing Banach space instead of Hilbert space and the Gâteaux differentiability instead of -class Fréchet.

Proposition 2. Let be a real reflexive strictly convex Banach space, let be lower semicontinuous and Gâteaux differentiable and let be a closed subset of such that for every from with the metric gradient , for sufficiently small ,Then, if is lower bounded on , for every a minimizing sequence for on , there exists a sequence in such thatwhere and .

Remark 3. This result is reported in [14] as Lemma  9 in the frame of the Hilbert spaces having the function of -class Fréchet, but condition (5) is more complicated due to another condition imposed to the set .

Explanations. Let us introduce the definition of the metric gradient in order to provide other observations relative to this central notion for this statement. In a real normed space , consider the Gâteaux differentiable functional . The metric gradient of is the multiple-valued mapping , , where is the duality mapping on corresponding to the identity and the canonical injection of into , . Consequently, for any = . If is reflexive, for any , is nonempty. being strictly convex, is single-valued. So, if is reflexive and strictly convex, then , , and the following equalities hold:

Go now to the proof of Proposition 2.

Proof. Denote and let be from . For , hence , we have . Apply the enunciated Ekeland principle with , in with known properties. Thus we obtain the sequence satisfying (6), (7) , andVerify (5). It is sufficient to work under the assumption . Thus apply the hypothesis made in the statement with respect to with and denoting, for , replace in (9) and find,multiply this inequality by , , and take the limit for in order to keep the sense of the inequality. Remark that ; . Consider that the existence of the limit for implies the existence of the limit for together with their equality, = = , taking into account the definition of the Gâteaux derivative and the above considerations on the metric gradient, and (5) is also fulfilled.

Remark 4. The Gâteaux derivative from the above statement can be replaced by any -derivative and the result remains the same. In the case of the Fréchet derivative, it must remove the condition “ lower semicontinuous” from the statement.

Notation 1. -differentiable;

Proposition 5. Let be a real reflexive strictly convex Banach space and lower semicontinuous and Gâteaux differentiable and is a nonempty convex closed subset such that , the identity map. If is lower bounded on , then for every a minimizing sequence for on , there is a sequence in such thatMoreover, if satisfies , where , then

Proof. Applying Proposition 2, (4) is satisfied indeed: if and , then, being convex,Let be the sequence given by the statement. , hence . , hence , clearly , and consequently has a convergent subsequence , . This implies , and is a global critical point of contained in .

2.2. Weak Solutions

Open set of class in . Use the notations (the norm is that Euclidean from ): , , , . Let be an open nonempty set in , , and its boundary. By definition, is of class if from a neighbourhood of in and bijective such that , , , and . Weak solution. Let be an open bounded nonempty set in , , , and . Consider the problemsActually on means , where is the trace operator, a continuous linear mapping from in , . We have and . from is by definition a weak solution for and , respectively, if on and

Remark 6. Here is the weak gradient, and it is equal to the weak derivatives; . is endowed in the first case with the norm ; that is, , which is equivalent to the norm . For the second case , equip the same vector space with the norm , which is equivalent to . Nemytskii Operator. Let be , , the Lebesgue measure in , open nonempty Lebesgue measurable and . By definition is a Carathéodory function if is Lebesgue measurable and is continuous , . In this case, for every from one may consider the function , , Nemytskii operator. Suppose Then . Assume that satisfies the growth condition: with , where , , and , .

Then ; is continuous and bounded on . If is bounded and , then with continuous; moreover, with continuous (see, e.g., [2]), where , and , is of Fréchet class and [14], so it is also Gâteaux differentiable.

Theorem 7. Let be an open bounded nonempty set in and a Carathéodory function with the growth condition:where , when , and when , and where ,
Then the energy functional,where is Gâteaux differentiable on and

Proof. One may consider , in both cases, as the sum of two other functions. The second of these functions being Gâteaux differentiable (see the above consideration), it is sufficient to remark that also the maps and are Gâteaux differentiable on [2, 19] and then is Gâteaux differentiable on.

Corollary 8. Letand f be as in Theorem 7. Then the weak solutions of and , respectively, are precisely the critical points of the functional:

Proof. Indeed, if is a weak solution of and , respectively, then ((15) and (16), resp., Theorem 7); hence . The inverse assertion is obvious.

Weak Subsolutions and Weak Supersolutions of and . Let be an open bounded set of class in , , let be a Carathéodory function, and let . is a weak subsolution and a weak supersolution, respectively, of or if