The Existence of Solutions for Local Dirichlet ( r ( u ) , s ( u )) -Problems

: In this paper, we consider local Dirichlet problems driven by the ( r ( u ) , s ( u )) -Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r , s are real continuous functions and we have dependence on the solution u . The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a ﬁxed-point argument.


Introduction
Interest in general forms of differential problems, whose leading operator is of the (p, q)-Laplacian type, has greatly increased over the last few decades. The main reason is that this kind of nonlinear operator appears naturally in the study of nonlocal diffusion with special features (see Ružička [1]). Indeed, if the Laplacian operator (that is, p = q = 2) is recognized as a key mathematical prototype for the comprehensive study of linear elliptic equations in the context of physical phenomena, the (p, q)-Laplacian operator (in the case q = 2) extends the range of applications for nonlinear equations in the context of nonlinear physical phenomena as the viscosity analysis of materials with different hardening exponents in the growth rates, and the behavior of smart fluids with and without the influence of external fields (for example, an electromagnetic field). An interesting collection of monographies is available about the general theory of the Laplacian equation, p-Laplacian equation, (p, q)-Laplacian equation, (p(z), q(z))-Laplacian equation, and the suitable framework spaces (see, for example, [2][3][4][5]). Thus, the reader can find precise replies to any questions concerning the source of difficulties and extra complications in extending the regularity theory of the Laplacian equation up to (p(z), q(z))-Laplacian equations with variable exponent functions p and q (see also the book Rȃdulescu-Repovš [6]). Without being exhaustive, we briefly underline some facts which say that it is nontrivial to continue the theoretical study of (p(z), q(z))-Laplacian operators. Following the approach of the Calculus of Variations and Critical Point Theory, the natural starting point of the existing theory is qualitative works about the existence and regularity of solutions for variational integrals (total energy integrals) of the form I(u) = Ω g(z, ∇u(z))dz, where u : Ω → R N (Ω is an open bounded domain) and ∇u is the N × N matrix of the deformation gradient. The study is carried out imposing a growth assumption of the form c 0 |u| c 1 ≤ |g(z, u)| ≤ c 2 (1 + |u| c 3 ) for all (z, u) ∈ Ω × R, where c 0 , c 1 , c 2 , c 3 are positive constants and 1 ≤ c 1 ≤ c 3 ; we refer to [7][8][9][10][11] for a wide discussion on the topic. Moreover, we distinguish the case of constant exponents p, q (namely, isotropic equations) and the case of variable exponents p(z), q(z) (namely, anisotropic equations). The existing results were developed in the abstract settings of Lebesgue and Sobolev spaces with and without variable exponents, namely, L p (Ω), W 1,p (Ω), L p(z) (Ω), W 1,p(z) (Ω). Now, it is well-known that L p(z) (Ω) is not invariant with respect to translation (Kováčik-Rákosník [12]). This is a source of difficulties about convolutions and continuity of functions in the mean. Moreover, W 1,p(z) (Ω) presents difficulties about the density of smooth functions ), the Sobolev inequality, and embedding theorems (Edmunds-Rákosník [14], Kováčik-Rákosník [12]). This means that the passage from the constant exponent setting to the variable exponent setting needs attention to special cases, and thus, some challenging open problems remain (for further details, we refer to  and Cencelj-Rȃdulescu-Repovš [16], and the references therein).
In our paper, attention is focused on the following Dirichlet boundary value problem: where ∆ m(u) u := div(|∇u| m(u)−2 ∇u) is the m(u)-Laplacian, N is a given nonlinear reaction term, and Ω ⊂ R d is an open bounded domain with a smooth boundary. The main feature of problem (1) is that both r and s vary in respect of the solution u.
To the best of our knowledge, the study of qualitative behavior for anisotropic (r(u), s(u))-Laplacian equations with exponent functions depending on the solution u was not considered in previous works. Instead, there are two key papers dealing with the single r(u)-Laplacian equation, both with homogeneous Dirichlet boundary conditions (see Andreianov-Bendahmane-Ouaro [17] and Chipot-de Oliveira [18]). In these papers, the authors established existence and uniqueness results, assuming the nonlinearity N has an appropriate structure. However, due to the u-dependence, r and s continue to be strongly related to Ω. This leads us to keep in mind the natural position: p(z) = r(u(z)) and q(z) = s(u(z)). Consequently, in the sequel, we can work in the Lebesgue and Sobolev spaces with variable exponents according to Fan-Zhao [19] (see also Papageorgiou-Rǎdulescu-Repovš [20]).
The paper is organized as follows. In Section 2, we collect the preliminaries and give some auxiliary results. Section 3 is devoted to the proofs of existence theorems for regularized problems, in the case of a Carathéodory nonlinearity satisfying sign and polynomial growth conditions. Here, we use geometrical and compactness conditions to establish our results. Section 4 is devoted to the proofs of existence and uniqueness theorems in the case of a general nonlinearity with an appropriate structure of the Sobolev space. We work with regularized auxiliary problems and a priori estimates. Additionally, the Schauder fixed-point theorem is used in the proof of the second theorem.

Mathematical Background
The aim of this section is to recall some results about the variable exponent Lebesgue and Sobolev spaces, the known embedding theorems, and Hölder-type inequalities. We fix the notation as follows. By ·, · we mean the duality brackets for the pair (X * , X), where X and X * are a Banach space and its topological dual, respectively.
To develop our results, we need some special features for the framework space. Thus, by M(Ω) we mean the space of Lebesgue-measurable functions m : Ω → [1, +∞), where m − := ess inf z∈Ω m(z), m + := ess sup z∈Ω m(z). (2) Fixed m ∈ M(Ω) ∩ L ∞ (Ω), then the space L m(z) (Ω) given as contains all Lebesgue-measurable functions u over the variable space Ω, with the bounded integral value ρ m (u). This space is Banach whenever we consider the usual Luxemburg norm defined by For further use, we recall the following properties of L m(z) (Ω).
To m(·), there corresponds m (·) related by the relation 1 m (·) + 1 m(·) = 1, and referred to as the Hölder conjugate exponent of m(·). Thus, we denote by L m (z) (Ω) the topological dual of L m(z) (Ω). We recall the crucial inequality Moreover, in the existing literature, some consolidated results link the norm with the integral in the definition of L m(z) (Ω). Precisely, we need the following theorems.
. Remark 1. By Theorem 4, we can easily deduce that These inequalities will be used to obtain certain a priori estimates in the sequel.
Moreover, if m − > 1, we recall the Hölder inequality in the form for u ∈ L m(z) (Ω), v ∈ L m (z) (Ω). This Hölder inequality leads to the proof of the following embedding theorem over a bounded domain Ω. Here, |Ω| denotes the Lebesgue measure of the set Ω.
On this basis, we recall the precise definition of the Lebesgue−Sobolev space W 1,m(z) (Ω) given by A further step toward the correct definition of the framework space leads to an introduction of the set Additionally, we have that W 1,m(z) 0 (Ω). Thus, we will use ∇u L m(z) (Ω) to replace u W 1,m(z) (Ω) and put As pointed out in the Introduction, the density of smooth functions is a source of (Ω). To solve this situation, by following the similar arguments in [18], we consider the set Indeed, ( [19], Theorem 2.6) gives us that (Ω), under the assumptions: • Ω is a bounded domain with ∂Ω being Lipschitz-continuous; • m(·) is log-Hölder continuous (that is, there exists c 5 > 0 such that Moreover, (7) implies that H 1,m(z) 0 (Ω). We point out that Finally, we recall the critical Sobolev exponent corresponding to m(·) given as Additionally, the classical Sobolev embedding theorem was generalized by Fan-Zhao ( [19], Theorem 2.3) in the following way.
In the fashion of inequality (7) We conclude this section with a result concerning the features of the functionals related to monotonicity (see Chipot [21]).

Lemma 1.
For all ξ, η ∈ R d , the following assertions hold true:

Carathéodory Nonlinearity
In this section, we assume r, s : R → (1, +∞) is continuous and N := g(z, u(z)), where g : Ω × R → R satisfies some standard sign and polynomial growth assumptions. Here, g(z, ξ) is a Carathéodory function (i.e., for each ξ ∈ R, z → g(z, ξ) is measurable and for a.e. z ∈ Ω, ξ → g(z, ξ) is continuous). The results are consistent with the theoretical analysis of Fan-Zhang [22], but we work on a regularized problem with a u-dependent (r(·), s(·))-Laplacian operator (instead of a problem with a z-dependent p(z)-Laplacian operator).
We recall that for a weak solution of the problem (P g ) (that is, (1) (Ω)). As usual, this means that we do not require more differentiability than it belongs to the first-order variable exponent Sobolev space. In order to perform an asymptotic analysis of the solutions to problem (1), we implemented an approximation strategy based on the following parametric auxiliary problem: where Ω ⊂ R d is an open bounded domain with a smooth boundary, g given as above has some regularities, and r and s are such that 1 < r − , s − and r + , s + < β, and ε > 0 is a parameter. We revisit the definition of a weak solution to (P g ) in the following form: that is, a function u ε ∈ W 1,β 0 (Ω) satisfying (10) for all v ∈ W 1,β 0 (Ω) is a weak solution to the problem (9). Now, we focus on the operator V ε : W for all u, v ∈ W 1,β 0 (Ω). Using the norm u = ∇u L β (Ω) , we discuss some properties of V ε (see also ([22], Theorem 3.1)).

Proposition 2.
The following statements hold: Proof. (i) Clearly, V ε is continuous and bounded (by definition). On the other hand, strict monotonicity follows by Lemma 1.
(ii) With respect to the third term in Equation (11), we note that B : W is an operator of type (S + ), and hence, V ε is an operator of type (S + ) too.
(iii) We know that and hence, V ε is coercive.
(iv) According to (i) and (iii), the operator V ε is continuous, strictly monotone (hence, maximal monotone too), and coercive. It follows that V ε is surjective (see Corollary 2.8.7, p. 135, Papageorgiou-Rǎdulescu-Repovš [20]). Consequently, V ε admits an inverse operator, is sufficient enough to say that V ε is a homeomorphism. Moreover, for h n , h ∈ W −1,β (Ω) with h n convergent to h as n goes to infinity, we can set u n = V −1 ε (h n ) and u = V ε (h) so that V ε (u n ) = h n and V ε (u) = h. It follows that the sequence {u n } is bounded in W 1,β 0 (Ω). With no loss of generality, suppose that u n w − → u 0 as n → +∞.
By the convergence of h n to h as n goes to infinity, we deduce that ⇒ u n → u 0 as n → +∞ (recall that V ε is of type (S + )), This fact leads to u n → u as n → +∞, and hence, V −1 ε is a continuous operator. This completes the proof.
Proof. With respect to the right-hand side of Equation (10), Proposition 1 permits us to say that is a continuous linear operator on W 1,β 0 (Ω). By the proof of Proposition 2 (iv), V ε is a strictly monotone surjective operator, and hence, (9) admits a unique solution.
Notice that the inequality in the statement of Proposition 3 implies that β > γ (·). On the other hand, using an appropriate growth condition, we establish the following existence result (this time we restore the u-dependence of g). Theorem 6. Let g : Ω × R → R be a Carathéodory function satisfying the growth condition |g(z, ξ)| ≤ c 7 + c 8 |ξ| α(z)−1 for all z ∈ Ω, all ξ ∈ R, some c 7 , c 8 ≥ 0 and α ∈ C(Ω).
As an illustrative example of the growth condition (12) (and hence, to apply Theorem 6), we construct a nonlinear reaction term starting from a locally defined Carathéodory function. This choice is motivated by the fact that boundary value problems of the form (P g ) (or more generally, of the form (1)) are considered in the context of the regularization theory in image processing for various image restoration problems (see also the comments in the Introduction of [17,21] and the references cited therein). In particular, we recall that regularization techniques often combine local smoothing effects and estimates by suitable cut-off functions. Thus, let f : Ω × [0, b] → R (b > 0) be a Carathéodory function satisfying f (z, 0) = 0 for a.e. z ∈ Ω and (i) 0 ≤ f (z, ξ) ≤ a b (z) for a.e. z ∈ Ω, all 0 ≤ ξ ≤ b with a b ∈ L ∞ (Ω); (ii) for α ∈ C(Ω) such that 1 ≤ α(z) ≤ α + < β for all z ∈ Ω, there exists ν ∈ (0, b) sufficiently small enough to have f (z, ξ) ≤ ξ α + −1 , ξ β * ≤ ξ β for a.e. z ∈ Ω, all 0 ≤ ξ ≤ ν. Next, we introduce a cut-off function ψ ∈ C c (R) satisfying and define the function We note that g : Ω × R → R is a Carathéodory function, also involving a critical growth term ξ → ξ β * −1 , satisfying the condition As already mentioned in the Introduction, our strategy of proof for the existence of at least a weak nontrivial solution to problem (9) also uses geometrical and compactness conditions, and it applies to the functional I. In detail, to use the Critical Point Theory, we impose a Palais-Smale condition (i.e., a compactness-type condition) in the sense of the following definition. We recall that the main source of difficulty in the construction of a Palais-Smale sequence (that is, the sequence given in Definition 1) is in establishing the boundedness of the sequence. To this aim, we impose the classical Ambrosetti-Rabinowitz condition to the nonlinearity g.
Proof. We start with a sequence {u n } ⊂ W 1,β 0 (Ω) such that {I(u n )} is bounded and I (u n ) converges to zero in W −1,β (Ω) as n goes to infinity. Thus, we can find a constant c 10 > 0 such that (for some c 11 > 0; using the position p(z) = r(u n (z)) and q(z) = s(u n (z))) Thus, we deduce that {u n } is a bounded sequence. Now, there is no loss of generality in assuming that u n w − → u as n → +∞. Consequently, we have that H (u n ) converges to H (u) (recall the definition of H at the beginning of the proof of Theorem 6). Next, we use Proposition 2. We have and hence, u n → u as n → +∞, since V ε is a mapping of type (S + ). We conclude that the functional I admits the Palais-Smale property.
The Ambrosetti-Rabinowitz condition (AR) is a strong condition in the sense that it determines the supercritical behavior of the nonlinearity g, but it gives us the so-called mountain pass geometry of Ambrosetti and Rabinowitz [23] (that is, a key result to establish the existence of critical points of the Euler-Lagrange functional associated to the problem under investigation). Of course, the common (without z-dependence) superlinear nonlinearity g(ξ) = ξ θ−1 if ξ ≥ 0 satisfies the condition (AR). On this basis, we consider the local infinitesimal condition with respect to u: g(z, u) = o(|u| β−1 ) as u goes to 0 uniformly for z ∈ Ω.
Then, we give the following result about the nontriviality of solutions to (9). (13) and (AR) hold, then (9) admits at least one nontrivial weak solution.
Notice that I(0) = 0, and hence, the functional I admits mountain pass geometry. Consequently, I admits at least one non-zero critical point that is an exact nontrivial solution to (9).

Sobolev-Type Nonlinearity
This section is based on the ideas and results of Chipot and de Oliveira [18]. We mention that in the Ref. [18], the authors appealed to the pioneering work of Zhikov [24] about the appropriate way to take the limit in a sequence of nonlinear elliptic equations. In more detail, the Ref. [18] pointed out that to get how |∇u| r(u) is in the L 1 space over Ω, one can make use of the following lemma (see [18], p. 289).
Proof. Hypothesis (i) and Lemma 3 (separately for (p, p n ) and (q, q n )) give us that Then, summing the two obtained inequalities, we trivially deduce that (14) holds. Now, we focus on the existence problem of solutions to (1) in the case N := g ∈ W −1,α (Ω). According to the existing theory for (p(z), q(z))-Laplacian equations, we work on the same solution set used in the Ref. [18], that is, With respect to the previously defined norm (see (6)), we know that W 1,r(u) 0 (Ω) is a Banach space, whenever the exponent is finite and greater than 1 (i.e., r(u) ∈ (1, +∞) for all u ∈ R). Moreover, classical results say that u (Ω) ⊆ W 1,α 0 (Ω) is closed, whenever the inequality 1 < α ≤ r(·) holds for a r continuous function and some constant α. Consequently, W 1,r(u) 0 (Ω) is a separable and reflexive Banach space. Moreover, we also work on the set W 1,s(u) 0 (Ω), which of course has similar properties given above for (Ω). Now, for each ε > 0, we consider the auxiliary problem (namely, the regularized problem) in the following form −div ((|∇u| r(u)−2 + |∇u| s(u)−2 )∇u) − εdiv (|∇u| β−2 ∇u) = g in Ω, where the exponent functions r, s : R → (1, +∞) are continuous and satisfy the range condition separately d < α ≤ s(u), r(u) ≤ β < +∞ (that is, r and s are not related to each other). Different from problem (9), here we impose that the nonlinearity is g ∈ W −1,α (Ω). Consequently, (15) admits a weak solution u ∈ W 1,β for each v ∈ W 1,β 0 (Ω), and ·, · are the duality brackets for the pair (W −1,α (Ω), W 1,α 0 (Ω)). Now, we have all the ingredients to produce a weak solution of (15). Our strategy to obtain this solution is the following. Using technical hypotheses on r, s and g, we first create the setting for an application of Proposition 3, which gives us the existence of a particular solution. Then, this solution is used to define a self-map over a set with precise bounds in a norm. On account of the Schauder fixed-point theorem and Lebesgue theorem (of dominated convergence), we establish the continuity of the above map and hence, the existence of a fixed point, which gives us the required solution.
Passing to a subsequence if necessary (namely again {u n }), for a certain u ∈ W 1,β We return to (17), so that considering (u n , x n ) instead of (u, x), we have Ω |∇u n | r(x n )−2 + |∇u n | s(x n )−2 ∇u n ∇vdz Since the operator on the left-hand side of (21) is monotone, then we deduce that for each v ∈ W 1,β 0 (Ω). Considering (21) with u n − v instead of v, we use (22) to deduce that for each v ∈ W 1,β 0 (Ω). The convergence in (19), passing eventually to a subsequence, implies x n → x a.e. in Ω.
It follows that h is continuous, and this establishes the existence of the fixed point which is the exact weak solution to (15).
The next theorem needs the following revised definition of a weak solution.
Note that r(u), s(u) ∈ M(Ω) and the essential infimum and the essential supremum of r(u), s(u) satisfy the condition α ≤ s − (u) ≤ r + (u) ≤ β for all u ∈ W 1,r(u) 0 (Ω). For the strategy to work, we need to slightly strengthen hypotheses on the framework structure. Thus, the new conditions on Ω and (r, s) are the following: (Ω).