Existence, Uniqueness and Asymptotic Behavior of Parametric Anisotropic (p, q)-Equations with Convection

In this paper we study anisotropic weighted (p, q)-equations with a parametric right-hand side depending on the gradient of the solution. Under very general assumptions on the data and by using a topological approach, we prove existence and uniqueness results and study the asymptotic behavior of the solutions when both the q(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q(\cdot )$$\end{document}-Laplacian on the left-hand side and the reaction term are modulated by a parameter. Moreover, we present some properties of the solution sets with respect to the parameters.


Introduction
Let ⊆ R N (N ≥ 2) be a bounded domain with smooth boundary ∂ . We consider the following nonlinear Dirichlet problem with parameter dependence in the leading term and with gradient and parameter dependence in the reaction term where μ ≥ 0 and λ > 0 are the parameters to be specified, the exponents p, q ∈ C( ) are such that 1 < q(x) < p(x) for all x ∈ and r (·) denotes the r (·)-Laplace differential operator defined by r (·) u = div |∇u| r (·)−2 ∇u for all u ∈ W 1,r (·) 0 ( ).
In the right-hand side of problem (1.1) we have a parametric reaction term in form of a Carathéodory function f : × R × R N → R which satisfies very general structure conditions, see hypotheses (H2) and (H3) in Sects. 2 and 3. Since the reaction term f : × R × R N → R also depends on the gradient ∇u of the solution u (that phenomenon is called convection), problem (1.1) does not have a variational structure and so we cannot apply tools from critical point theory. Instead we will use a topological approach based on the surjectivity of pseudomonotone operators.
We will not only present existence results under very general structure conditions but also sufficient conditions for the uniqueness of the solution of (1.1). Further, we study the asymptotic behavior of the solutions of (1.1) and prove some properties of the solution sets depending on the two parameters μ ≥ 0 and λ > 0 which are controlling the q(·)-Laplacian on the left-hand side and the reaction on the right-hand side. This leads to interesting results on certain ranges of μ and λ.
The novelty in our paper is the fact that we have an anisotropic nonhomogeneous differential operator and a parametric convection term on the right-hand side. If μ = 0 in (1.1), the operator becomes the anisotropic p-Laplacian and such equations have been studied for λ = 1 in the recent paper of Wang-Hou-Ge [25]. For constant exponents there exist several works but without parameter on the righthand side. Precisely, constant exponent p-Laplace problems with convection can be found in the papers of de Figueiredo-Girardi-Matzeu [4] for the Laplacian, Fragnelli-Papageorgiou-Mugnai [11] and Ruiz [24] both for the p-Laplacian. For ( p, q)-equation with constant exponents, convection term and λ = 1, we refer to the works of Averna-Motreanu-Tornatore [1] for weighted ( p, q)-equations, El Manouni-Marino-Winkert [6] for double phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian, Faria-Miyagaki-Motreanu [10] using a comparison principle and an approximation process, Gasiński-Winkert [13] for double phase problems, Liu-Papageorgiou [17] for resonant reaction terms using the frozen variable method together with the Leray-Schauder alternative principle, Marano-Winkert [18] with nonlinear boundary condition, Motreanu-Winkert [19] via sub-supersolution approach, Papageorgiou-Vetro-Vetro [20] for right-hand sides with a parametric singular term and a locally defined perturbation and [21] for semilinear Neumann problems, see also the references therein.
To the best of our knowledge, this is the first work dealing with an anisotropic differential operator and a parametric convection term. Such equations provide mathematical models of anisotropic materials. The parameter μ ≥ 0 modulates the effect of the q(·)-Laplace operator, and hence controls the geometry of the composite made of two different materials. In general, equations driven by the sum of two differential operators of different nature arise often in mathematical models of physical processes. We refer to the works of Bahrouni-Rȃdulescu-Repovš [2] for transonic flow prob-lems, Cherfils-Il'yasov [3] for reaction diffusion systems and Zhikov [26] for elasticity problems.
Finally, we mention that there are several relevant differences when dealing with anisotropic equations in contrast to constant exponent problems. We refer to the books of Diening-Harjulehto-Hästö-Rȗzȋcka [5], Harjulehto-Hästö [14] and Rȃdulescu-Repovš [23] for more information on the differences.
The paper is organized as follows. In Sect. 2 we collect some properties on variable exponent Sobolev spaces as well as on the p(·)-Laplacian and we present the hypotheses on the data of problem (1.1). Section 3 is devoted to the existence and uniqueness results as well as the asymptotic behavior when the parameter μ moves to 0 and +∞, respectively. We also show the boundedness of the set of solutions to problem (1.1). In Sect. 4 we complete the characterization of the set of solutions with respect to compactness and closedness.

Preliminaries and Hypotheses
In this section we give a brief overview about variable exponent Lebesgue and Sobolev spaces, see the books of Diening-Harjulehto-Hästö-Růžička [5], Harjulehto-Hästö [14] and the papers of Fan-Zhao [7], Kováčik-Rákosník [16]. Moreover, we recall some facts about pseudomonotone operators and we state the hypotheses on the data of problem (1.1).
To this end, let be a bounded domain in R N (N ≥ 2) with smooth boundary ∂ . For r ∈ C + ( ), where C + ( ) is given by we denote Moreover, denoting by M( ) the space of all measurable functions u: → R, the variable exponent Lebesgue space L r (·) ( ) for a given r ∈ C + ( ) is defined as equipped with the Luxemburg norm given by Here the corresponding modular ρ r : L r (·) ( ) → R is given by We know that (L r (·) ( ), · r (·) ) is a separable, reflexive and uniformly convex Banach space.
The following proposition gives the relation between the norm · r (·) and the modular ρ r (·). Proposition 2.1 For all u ∈ L r (·) ( ) we have the following assertions:

Remark 2.2 A direct consequence of Proposition 2.1 is the following relation
Let r ∈ C + ( ) be the conjugate variable exponent to r , that is, We know that L r (·) ( ) * = L r (·) ( ) and Hölder's inequality holds, that is, for all u ∈ L r (·) ( ) and for all v ∈ L r (·) ( ).
If r 1 , r 2 ∈ C + ( ) and r 1 (x) ≤ r 2 (x) for all x ∈ , then we have the continuous embedding For r ∈ C + ( ) we define the variable exponent Sobolev space W 1,r (·) ( ) by where ∇u r (·) = |∇u| r (·) . Furthermore, we define The spaces W 1,r (·) ( ) and W 1,r (·) 0 ( ) are both separable and reflexive Banach spaces, in fact uniformly convex Banach spaces. In the space W 1,r (·) 0 ( ), we have Poincaré's inequality, that is, For r ∈ C + ( ) we introduce the critical variable Sobolev exponent r * defined by The following proposition states the Sobolev embedding theorem for variable exponent Sobolev spaces.
Let us now recall some definitions which are used in the sequel.

Definition 2.4
Let X be a reflexive Banach space, X * its dual space and denote by · , · its duality pairing. Let A: X → X * , then A is called Remark 2. 5 We point out that if the operator A: X → X * is bounded, then the definition of pseudomonotonicity in Definition 2.4 (ii) is equivalent to u n u in X and In the following we are going to use this definition since our operators involved are bounded.
Theorem 2.6 Let X be a real, reflexive Banach space, let A: X → X * be a pseudomonotone, bounded, and coercive operator, and b ∈ X * . Then, a solution of the equation Au = b exists.

Next, we introduce the nonlinear operator
This operator has the following properties, see Fan-Zhang [9, Theorem 3.1].

Proposition 2.7
The operator A r (·) (·) is bounded (that is, it maps bounded sets to bounded sets), continuous, monotone (thus maximal monotone) and of type (S + ).
Now we can formulate the hypotheses on the data of problem (1.1).
(H2) f : × R × R N → R is a Carathéodory function such that (i) there exist σ ∈ L α (·) ( ) with 1 < α(x) < p * (x) for all x ∈ and c > 0 such that for a. a. x ∈ , for all s ∈ R and for all ξ ∈ R N , where p * is the critical exponent to p given in (2.2) for r = p; for a. a. x ∈ , for all s ∈ R and for all ξ ∈ R N .

Example 2.9
Let d 1 , d 2 > 0 and consider the function defined by for a. a. x ∈ , for all s ∈ R and for all ξ ∈ R N with 0 = σ ∈ L p (·) ( ). It is easy to see that f fulfills hypotheses (H2).

( ) is a weak solution to (1.1) if
is satisfied for all h ∈ W 1, p(·) 0 ( ). We also recall the following result, see Gasiński-Papageorgiou [ Lemma 2.10 If X,Y are two Banach spaces such that X ⊆ Y , the embedding is continuous and X is dense in Y , then the embedding Y * ⊆ X * is continuous. Moreover, if X is reflexive, then Y * is dense in X * .

Existence and Uniqueness Results and Asymptotic Behavior
Now we state and prove the following existence result for problem (1.1). In the sequel we use the abbreviation Hypothesis (H2)(i) implies that N * f (·) is well-defined, bounded and continuous, see Fan-Zhao [7] and Kováčik-Rákosník [16]. By Lemma 2.10, the embedding i * : L α (·) ( ) → W −1, p (·) ( ) is continuous and hence the operator f is bounded and continuous. We fix μ ≥ 0 as well as λ ∈ ]0, λ * [ and consider the operator V : W u − u n α(·) → 0 as n → ∞.
Therefore, if we pass to the limit in the weak formulation in (2.4) replacing u by u n and h by u n − u and using (3.2), it follows that Therefore, by the weak convergence of {u n } n∈N , lim sup n→+∞ A p(·) (u n ), u n − u ≤ 0.
Taking the (S + )-property of A p(·) (·) into account (see Proposition 2.7) along with (3.1) gives u n → u in W 1, p(·) 0 ( ). From the strong convergence and the continuity of V , Applying (H2)(ii) and (3.3) along with Proposition 2.1(ii), we obtain for u ∈ W 1, p(·) 0 Let us now consider equation (1.1) under stronger assumptions in order to prove a uniqueness result. We suppose the additional assumptions.
(H3) (i) There exists a constant a 1 > 0 such that for a. a. x ∈ , for all s, t ∈ R and for all ξ ∈ R N . (ii) There exist a function ψ ∈ L r (·) ( ) with r ∈ C + ( ) such that r (x) < p * (x) for all x ∈ and a constant a 2 > 0 such that the function ξ → f (x, s, ξ) − ψ(x) is linear for a. a. x ∈ , for all s ∈ R and | f (x, s, ξ) − ψ(x)| ≤ a 2 |ξ | for a. a. x ∈ , for all s ∈ R and for all ξ ∈ R N .

Example 3.2
The following function satisfies hypotheses (H1)-(H3), where we drop the s-dependence: a. x ∈ and for all ξ ∈ R N , with p − = 2 , 0 = ψ ∈ L 2 ( ) and β = (β 1 , . . . , β N with λ 1 > 0 being the first eigenvalue of the Laplacian with Dirichlet boundary condition given by Our uniqueness result reads as follows. ( ) of (1.1). We test the corresponding weak formulations given in (2.4) with h = u − v and subtract these equations. This leads to First, it is easy to see that the left-hand side of (3.5) can be estimated via Now we apply the conditions in (H3) along with Hölder's inequality and (3.4) to the right-hand side of (3.5) in order to obtain (3.7) From (3.5), (3.6) and (3.7) we conclude that Now, we study the asymptotic behavior of problem (1.1) as the parameters μ and λ vary in an appropriate range. We introduce the following two sets S μ (λ) = u : uis a solution of problem (1.1) for fixed μ ≥ 0 and λ ∈ 0, λ * , First, we show the boundedness of S μ (λ) and S(λ) in W 1, p(·) 0 ( ).
Therefore, taking the limit in (3.10) as n → +∞, we conclude that u ∈ W 1, p(·) 0 ( ) is a weak solution of (1.1) with μ = 0, that is, a weak solution of the following problem − p(·) u = λ f (x, u, ∇u) in , Let us now study the case when μ → +∞.