Gradient and Parameter Dependent Dirichlet ( p ( x ) , q ( x )) -Laplace Type Problem

: We analyze a Dirichlet ( p ( x ) , µ q ( x )) -Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter µ varies on the non-negative real axis. The results are obtained by applying the properties of pseudomonotone operators, jointly with certain a priori estimates. operator.


Introduction
We study a inhomogeneous equation with Dirichlet boundary condition of the form − ∆ p(x) u(x) − µ∆ q(x) u(x) = f (x, u(x), ∇u(x)) in Ω, u on a bounded domain Ω ⊆ R N , with smooth boundary ∂Ω. On the left-hand side, we find the sum of two m(x)-Laplace differential operators with m ∈ C(Ω), whose combined effects are related to the values of a non-negative real number µ. In details, we recall that the notation ∆ m(x) corresponds to the following largely investigated operator ∆ m(x) u = div(|∇u| m(x)−2 ∇u) for all u ∈ W 1,m(x) 0 (Ω), Indeed, this variational integral is related to the total energy of the equation and its manipulation leads to the proper definition of a weak solution to the same equation. This is done according to John Ball's total energy theorem, and a clear introduction to these arguments is the monography of Lindqvist [1]. Turning to the right-hand side of Equation (1), we find a gradient-dependent function, whose regularity and growth conditions will be given in Section 2 (see assumptions A 1 , A 2 ) and in Section 5 (see assumptions A 3 , A 4 , A 5 ).
We point out that the presence of the gradient-dependence is crucial in the choice of a working strategy, as it inhibits the use of variational methods. Consequently, we establish our results by using the properties of pseudomonotone operators.
Briefly, we give some comments over the existing literature. A special form of the m-Laplace equation in the case m(x) = m = constant was given attention by de Figueiredo-Girardi-Matzeu [2], Fragnelli-Papageorgiou-Mugnai [3] and Ruiz [4]. These papers deal respectively with mountain-pass techniques [2], the Leray-Schauder alternative principle [3], the blow-up argument and a Liouville-type theorem to obtain a priori estimates [4].
A feature of Equation (1) is the presence of a parameter µ acting on the q(x)-Laplace differential operator. In the case µ = 0, (1) reduces to the p(x)-Laplace equation, as it is given in Wang-Hou-Ge [9] (existence and uniqueness of weak solution). Similarly, Vetro [10] deals with the case µ = 0, but in the presence of a Kirchhoff term weighting the p(x)-Laplace differential operator (both degenerate and non-degenerate Kirchhoff type problems are considered). Dealing with the case µ = 0, we will analyze the asymptotic behavior of weak solutions to (1). The results are obtained working in the context of the variable exponent Lebesgue space L m(x) (Ω) and the variable exponent Sobolev spaces ). The required notions and notation are given in Section 2, but the readers can consult the books by Diening-Harjulehto-Hästö-Rȗzȋcka [11] and by Rȃdulescu-Repovš [12], for details. A discussion about the uniqueness of weak solution will conclude the work herein, using certain additional assumptions on the nonlinearity. For additional problems involving different p(x)-Laplace type differential operators, we suggest the works by Ekincioglu and co-workers [13][14][15][16][17]. Finally, we mention the recent work by Bahrouni-Repovš [18] dealing with the existence and the nonexistence of solutions for a new class of p(x)-curl systems arising in electromagnetism.

Functional Framework
We give some notions involving a reflexive Banach space (X, · ) with topological dual X * . By ·, · , we mean the duality brackets of (X * , X). According to Gasiński-Papageorgiou [19], we recall the following concept and lemmas of a generalized pseudomonotone operator. Definition 1. For a generalized pseudomonotone operator, we mean an operator T : X → X * such that, for every {x n } n∈N ⊆ X, with we obtain
We recall that an operator T : X → X * is strongly coercive if T(u), u u goes to +∞, as u goes to +∞ too. This property leads to the following surjectivity lemma.

Lemma 3.
Given two Banach spaces X and Y with X ⊆ Y, we have that: (a) if the embedding is continuous and X is dense in Y, then the embedding Y * ⊆ X * is continuous; (b) in addition to (a), if X is reflexive, then Y * is dense in X * . Now, we focus on the Lebesgue space L m(x) (Ω) and the Sobolev space W 1,m(x) (Ω), where the study of Equation (1) will be developed. Precisely, we consider About these norms, from [11], we know that This means that there is equivalence between u W 1,m(x) (Ω) and ∇u L m(x) (Ω) on holds true. Fan-Zhao [20] gives us that L m(x) (Ω), W 1,m(x) (Ω) and W 1,m(x) 0 (Ω), equipped with these norms, are separable, reflexive, and uniformly convex Banach spaces. In the same paper [20], some Sobolev embedding results are given. We recall them in the following lemma.

Lemma 4.
Let m 1 , m 2 ∈ C(Ω) be such that m 1 (x), m 2 (x) > 1 for all x ∈ Ω. Then, we have: Another significant result for our analysis is the following theorem of [20].
Then, the following relations hold: . Remark 1. The inequalities in Theorem 1 can be used to obtain some a priori estimates. For further use, starting from Precisely, we observe that which establishes (3). Following a similar argument, one can derive the inequality We will work with the integral operator (Ω) * and possessing the following features: (i) boundedness, that is, T m maps bounded sets to bounded sets; (ii) continuity; (iii) monotonicity, and hence maximal monotonicity; (Ω).
Since we know that there is absence of homogeneity in T m , we will impose the following assumptions for the exponents: Let α ∈ C(Ω) be such that and max Therefore, the inequality (5) leads to Observe that, for any x ∈ Ω, we have p( . Thus, we deduce that, for all u ∈ W 1,p(x) 0 (Ω), the following inequalities hold Integrating the above inequalities, we find (Ω).
By Sobolev embeddings, there exist positive constants C α + and C α − such that and Using again the fact that α − ≤ α + ≤ p(x) for any x ∈ Ω, we deduce that W 1,p(x) 0 (Ω) is continuously embedded in W 1,α +
Before stating the assumptions on the nonlinearity, we recall that a function f : Ω × R × R N → R is said to be "Carathéodory" provided that: (i) for all (z, y) ∈ R × R N , x → f (x, z, y) is measurable; (ii) for almost all x ∈ Ω, (z, y) → f (x, z, y) is continuous. Therefore, f is jointly measurable (see Hu-Papageorgiou [21], p. 142). We impose the following assumptions on the Carathéodory function f : Ω × R × R N → R: (5) and c > 0 such that α (x) ) for a.a. x ∈ Ω, all z ∈ R, all y ∈ R N ; We note that the interest for equations subject to m(x)-growth conditions (and hence the significance of assumptions as A 1 and A 2 ) is supported by their applications. For instance, there are fluids that start flowing only after a certain threshold/strength is overcome, but the same fluids freeze as soon as the forcing factor leaves (that is, the typical behavior of certain oil paints (Bingham fluids)). The study of these phenomena requires variable exponents spaces and variable exponents growth conditions (see again [11,12]).

Example 1.
A nonlinearity satisfying the assumptions A 1 and A 2 is obtained combining two power terms in the form for a.a. x ∈ Ω, all z ∈ R, all y ∈ R N .
hold for a.a. x ∈ Ω, all z ∈ R, all y ∈ R N .

Existence and Asymptotic Results
Before establishing the existence of a weak solution to (1), we define the Nemitsky (Ω).
Such map possesses some regularities. Indeed, referring to the work of Galewski [22], A 1 ensures the boundedness and continuity of N * f (·). With respect to the embedding i * : L α (x) (Ω) → W −1,p (x) (Ω), we deduce by Lemma 3 that i * is continuous. This fact leads to the boundedness and continuity of the operator f . Now, we say in which sense the solutions to (1) are considered here. By Lemma 4, a solution will be sought in the variable exponent space W These notions will be used to construct the following result, along with the theory of pseudomonotone operators.

Theorem 2.
Assume that A 1 , A 2 and (6) are satisfied, then Equation (1) has at least one weak solution for all µ ≥ 0.
Proof. Let µ ≥ 0 be fixed. We consider the operator T : W 1,p(x) 0 (Ω) → W −1,p (x) (Ω) given as This operator possesses some regularities. Indeed, boundedness and continuity can be deduced easily by definition. Thus, we focus on the pseudo-monotonicity of T(·). We observe that T(·) is everywhere defined and bounded, and hence, with respect to ([19], Proposition 3.2.49), we remain to prove that T(·) is generalized pseudomonotone. Thus, we assume it satisfies the hypotheses u n From (14), we have lim sup Now, assumption A 1 leads to the following estimate (by Hölder inequality) (by (3) and (4)).
The importance of this estimate lays in the fact that, along with the boundedness of {u n } n∈N in W 1,p(x) 0 (Ω) and the convergence u n → u in L α(x) (Ω), we obtain Ω f (x, u n , ∇u n )(u n − u)dx → 0 as n → +∞.
(Ω), together with the fact that T(·) is continuous, give us T(u n ), u n → T(u), u .
The proof of general pseudo-monotonicity of T(·) is completed, and hence also the pseudo-monotonicity of T(·) is established.
Next, we show the strong coercivity of T(·), using assumption A 2 . Precisely, we have As q + < p − by (6), we deduce the strong coercivity of T(·). By Lemma 2, every pseudomonotone strongly coercive operator is surjective. Consequently, there exists u ∈ W 1,p(x) 0 (Ω) such that T( u) = 0. We conclude that Equation (1) has at least one weak solution for all µ ≥ 0.
Next, we will analyze the asymptotic behavior of weak solutions to (1). We indicate some of the notations used throughout this section. Let S µ = set of solutions to Equation (1), fixed µ ≥ 0, S = ∪ µ≥0 S µ = set of solutions to Equation (1).
We observe that these two sets are bounded in W 1,p(x) 0 (Ω). We give the proof in the following lemma.

Lemma 5.
Assume that A 1 , A 2 , and (6) are satisfied, then S µ is a bounded set in W 1,p(x) 0 Proof. We first establish the boundedness of S µ in W 1,p(x) 0 (Ω) for a fixed µ ≥ 0. Thus, without loss of generality, we consider a solution to (1), namely u ∈ W 1,p(x) 0 (Ω), such that u > 1. From the definition of weak solution (see (13)), choosing the test function h = u, we deduce that (2)).
We remain to prove that S = ∪ µ≥0 S µ is bounded in W 1,p(x) 0 (Ω) too. Observe that (17) is independent from µ, and hence holds for each u ∈ S. Consequently, S is bounded in W 1,p(x) 0 (Ω).
Before stating our next lemma, we remark that, throughout this paper, given a sequence {u n } n∈N , we denote every relabeled subsequence again with {u n } n∈N .
The first lemma concerns the behavior of (1) in the case µ → 0 + .

Lemma 6.
Assume that A 1 , A 2 and (6) are satisfied. Given a sequence of parameters {µ n } n∈N converging to 0 + , and a sequence {u n } n∈N of solutions to Equation (1) such that u n ∈ S µ n for all n ∈ N, then there is a relabeled subsequence of {u n } n∈N such that u n → u in W 1,p(x) 0 (Ω) with u ∈ W 1,p(x) 0 (Ω) solution to (1).
Proof. Let u n ∈ S µ n for all n ∈ N. The proof of the boundedness of S = ∪ µ S µ in W (Ω) and u n → u in L α(x) (Ω), for some (Ω). By (16), we derive that Ω f (x, u n , ∇u n )(u n − u)dx → 0 as n → +∞, whenever u n → u in L α(x) (Ω) (by assumption A 1 ). From u n ∈ S µ n for all n ∈ N, we obtain (Ω).
If we take the limit in (21) for n → +∞, we obtain that (Ω) be the multivalued mapping defined by This mapping represents the solution mapping of Equation (1). We show that F possesses some regularities.

Proposition 3.
Assume that A 1 , A 2 , and (6) are satisfied; then, the multivalued mapping F defined by (24) is upper semicontinuous.
Proof. Observe that the upper semicontinuity of (24) means that, for every closed subset C Consider {µ n } n∈N ⊂ F − (C) satisfying µ n → µ in [0, +∞). Clearly, for every n ∈ N, there exists u n ∈ F(µ n ) ∩ C. From the last sentence in the proof of Lemma 5 (boundedness of S), we know that {u n } n∈N is a bounded sequence. Moreover, from the proof of Lemma 6, we know that u n → u in W 1,p(x) 0 (Ω). Using the similar arguments as in the proof of Proposition 2 (recall u n ∈ S µ n ), we obtain u ∈ S µ = F(µ). Since we know that u ∈ C as C is closed, then µ ∈ F − (C).
We conclude that u 1 = u 2 , which contradicts the assumption u 1 = u 2 . We deduce that S µ is singleton, and hence the solution to Equation (1)