Multiple solutions for a system involving an anisotropic variable exponent operator

In this paper, the existence of a solution for an anisotropic variable exponent system is obtained and proved under general hypotheses. By considering additional conditions, it is proved a multiplicity result. The proofs are based on an application of appropriated L ∞ estimates, a sub-supersolution argument, and the Mountain Pass Theorem.

It will be considered that a weak solution for the system (S) is a pair (u, v) ∈ Z, where for all (ϕ, ψ) ∈ Z. Denote by · ∞ the norm in the space L ∞ ( ). Through minimization and subsupersolutions arguments, it is obtained the existence result below. Theorem 1.1 Suppose that the hypotheses (H), (F 1 )-(F 2 ) hold. Then, there exists ν > 0 such that the system (S) has a solution for max{ a ∞ , b ∞ } < ν.
In the last decades, Partial Differential Equations with variable exponents have been attracting the attention of several scientists due to their applicability in several relevant models. The main application of this kind of equation is in the study of electrorheological fluids. As mentioned in [1], the study of such fluids arose when fluids that stop spontaneously were discovered, also known as Bingham fluids. In the classical reference [2], due to W. Winslow, it was presented one of the main properties of electrorheological fluids. Parallel and string-like formations arise in this kind of fluid when it is considered the presence of an electrical field. This pattern is known as theWinslow effect. Moreover, the electrical field can raise the viscosity of the fluid by five orders of magnitude, see reference [1]. As pointed out in the interesting work [3], several studies with electrorheological fluids have been considered in NASA laboratories.
On the other hand, Anisotropic Partial Differential Equations can also be applied in several models. For example, in the classical reference [4], a model was presented that was applied for both image enhancement and denoising in terms of anisotropic problems as well as allowed the preservation of significant image features. We also quote the applicability in the study of the spread of epidemic disease in heterogeneous environments. In Physics, such an equation can be applied to consider the dynamics of fluids with different conductivities in different directions. We point out the references [4][5][6][7] for more details.
An important fact is that there is increasing interest in anisotropic problems with variable exponents. In the paper [8], the regularity of solutions of a stationary system is obtained, which is motivated by the theory of electrorheological fluids. In [9], a strong maximum principle is gained in the variable exponent setting, generalizing the classical principal of the Laplacian operator. The paper [10] presents the mathematical theory, which allows considering problems involving anisotropic operators with variable exponents. Moreover, several applications were considered. We also point out the interesting references [11][12][13][14][15][16][17][18][19][20] and the paper [21], which provides an overview concerning elliptic variational problems with nonstandard growth conditions and refers to different kinds of nonuniformly elliptic operators. See also [1,22] for a complete presentation of the theory of the Sobolev spaces with variable exponents and its applications.
The study of the system (S) is motivated by the problem considered in the reference [23], where it was proved, in an anisotropic setting, versions of Theorems 1.1 and 1.2 with α, β ≡ 2, and [24], where it was considered a scalar version of the system.
Regarding the remainder of the paper, we mention that in Sect. 2, it is considered some preliminary facts regarding the theory of the anisotropic variable spaces. The proofs of Theorems 1.1 and 1.2 are provided in Sects. 3 and 4, respectively.

Preliminaries
The Lebesgue space with a variable exponent is defined by with the Luxemburg's norm In what follows, we point out some results that can be found, for example, in [25].

Theorem 2.2 Consider functions p, q ∈ C + ( ). The statements below hold.
( Some results on anisotropic variable exponents [10] will be presented below. Consider The anisotropic variable exponent Sobolev space is defined by which is a Banach space with the norm Define the functions p( . Under the condition p(x) < p * (x) for all x ∈ , it holds the Poincaré type inequality below where C is a positive constant that does not depend on u ∈ W 1, − − → p(x) 0 ( ). Thus, it holds that the norm defined by is equivalent to the one given in (2.2).
An important fact is that it holds the compact embedding The results below, which will play an important role in our arguments, can be found in [24].

Proof of Theorem 1.1
The proof of Theorem 1.1 will be split into some steps. The first one consists of obtaining appropriated sub-supersolutions for the system (S). After this, the existence of solutions for an auxiliary system will be proved, which solves (S).
In what follows, it will be considered the definition of sub-supersolution for the system (S) and an auxiliary lemma.
It will be considered that (u, v) In the next result, it is obtained appropriated sub-supersolutions for (S).
Proof The lemmas 2.3 and 2.5 imply that there are unique nonnegative solutions u, u ∈ where Considering, if necessary, ρ > 0 smaller such that max{C 1 a ∞ , C 2 b ∞ } ≤ 1, if max{ a ∞ , b ∞ } < ρ, it will follow that the right-hand sides in (3.2) and (3.3) are nonnegative, providing the result.
Proof of Theorem 1.1 Consider the sub-supersolution pair provided in the proof of Lemma 3.1. Define the operators T : , and the auxiliary system , which is a Banach space. The solutions of (S ) coincide with the critical points of the C 1 functional defined by We have that J is a coercive and sequentially weakly lower semicontinuous. Consider the set which is closed and convex and hence weakly closed in W . Thus, it follows that J| A attains its infimum at some function u 0 ∈ A. Similar reasoning with respect to the proof of [26,Theorem 2.4] provides that J (u 0 ) = 0, which proves the result.

Proof of Theorem 1.2
Let u ∈ W whose solutions are given by the critical points of the C 1 functional where W was defined in the proof of Theorem 1.1 and

Lemma 4.1 The Palais-Smale condition is satisfied by the functional L.
Proof Consider (u n , v n ) ⊂ W a sequence such that L (u n , v n ) → 0 and L(u n , v n ) → c for some c ∈ R. With respect to the first part of (i), note that (F 3 ) holds with θ , ξ > 0 such that max{ 1 α -, θ } < θ < 1 which provide that (u n , v n ) is bounded in W . With respect to the second case of (i), that is β + < q --, we have constants C i > 0, i = 1, . . . , 5 with where θ > 0 was provided in the first part of the proof (i). Thus, the continuous embedding which is given by (2.4), implies that , for constants C i > 0, i = 1, . . . , 5. Since β + < q -, we obtain that the sequence (u n , v n ) is bounded in W .
Thus, for a subsequence still denoted by (u n , v n ), we obtain that for all h, k ∈ C( ) with 1 < h -≤ h + < (p ) -, 1 < k -≤ k + < (q )and some pair (u, v) ∈ W . From Lebesgue's Dominated Convergence Theorem and (4.1), it follows that Since p --, q --≥ 2, we have the result by the inequality (see, for instance, [27, page 97]) for all x, y ∈ R N and m ≥ 2, where ·, · denotes the usual Euclidean inner product in R N .
The next result provides the Mountain Pass Geometry for the functional L.