Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections

1 Introduction

In this article, we concentrate on the multivalued variational inequality given below and establish the corresponding existence and compactness results: find π ∈ K such that

where Ω is a bounded open set in R N ( N ≥ 2 ) with a Lipschitz boundary ∂ Ω , and we divide ∂ Ω into two relatively open subsets: Γ and its complement Γ 0 = ∂ Ω \ Γ such that ∂ Ω = Γ ∪ Γ 0 . Furthermore, consider K , a closed convex subset of V Γ 0 defined as

where the definition of space W 1 , A ( Ω ) will be given in Section 2. Moreover, G denotes a lower-order multivalued operator depending on the gradient of unknown functions (multivalued convection term), formulated by the related multivalued function g : Ω × R × R N → 2 R \ { ∅ } . However, F 1 and F 2 are two multivalued operators generated by multivalued function f 1 : Ω × R → 2 R \ { ∅ } and the boundary multivalued function f 2 : Γ × R → 2 R \ { ∅ } , respectively. Among (1.1), the nonlinear and nonhomogeneous operator J mapping from W 1 , A ( Ω ) to W 1 , A ( Ω ) * is given by

for all v ∈ W 1 , A ( Ω ) with A : Ω × [ 0 , + ∞ ) → [ 0 , + ∞ ) given by

for all x ∈ Ω and all t ∈ [ 0 , + ∞ ) , in which p , q ∈ C ( Ω ¯ ) fulfilling the assumptions 1 < p ( x ) < N , p ( x ) ≤ q ( x ) . In addition, 0 ≤ ϑ ( ⋅ ) ∈ L 1 ( Ω ) , e represents the Euler constant, and k is a positive constant. Note that to make sure the partial differential operator (1.3) is actually a double-phase operator, we assume that the domain where p ( x ) < q ( x ) is not a subset to the domain where ϑ equals zero, i.e., Ω < ≔ { x in Ω : p ( x ) < q ( x ) } ⫋ Ω 0 ≔ { x in Ω : ϑ ( x ) = 0 } .

The partial differential double-phase operator with k -logarithmic perturbation specified in (1.3) and (1.4) was introduced by Lu et al. [36], who studied some critical properties of operator (1.3) (including continuity, boundedness, coercivity, strict monotonicity, and S + -property), as well as some basic results of the Musielak-Orlicz (Sobolev) spaces driven by operator (1.3). One of the main characteristics of the multivalued variational inequality (1.1) is that it involves the so-called double-phase operator (which is fully nonlinear and has un-balance growth) and multivalued convection (this leads to the invalidity of variational method, so, our main method is based on topological approaches). As far as we know, the classical double-phase operator has the following form:

with the related energy functional:

Motivated by the study associating strongly anisotropic materials, Zhikov [59] first introduced the aforementioned energy functional in the 1980s, since it can well describe the fact that the energy density changes its ellipticity and growth characteristics in response to the specific point x within the domain Ω . Energy functional given by (1.5) and variants of it are called double-phase functional because when it comes to the domain where weighted function ϑ ( ⋅ ) equals zero, i.e., Ω 0 ≔ { x ∈ Ω : ϑ ( x ) = 0 } , it will show p -growth instead of q -growth. Nowadays, double-phase problems have gained increasing attention for its broad applications, concerning weighted anisotropic variant of the transonic flow problems, nonlinear Derrick’s problem, reaction diffusion systems, nonlinear theory of composite materials, image processing, and so on (see, for instance, [2,4,9,10,27,58]). As for new class of double-phase operators, we refer to Crespo-Blanco et al. [14], who studied the double-phase operator with variable exponents:

and showed certain properties of this type of operator and the corresponding Musielak-Orlicz Sobolev spaces. Then, Vetro and Zeng [54] considered a double-phase energy functional exhibiting log L -perturbed and p , q -growth formulated by

with

(see [54] for more details involving the properties of the associated Musielak Orlicz-Sobolev space as well as existence and uniqueness results for the solution of perturbed Dirichlet double-phase problems). Moreover, we can refer to Arora et al. [1] for the detailed study of the logarithmic double-phase-type operator

and existence and multiplicity results of the equations driven by the aforementioned logarithmic double-phase-type operator. For more impressive works dealing with multivalued double-phase problems without logarithmic perturbation, it could refer to Zeng et al. [55–57]. Particularly, double-phase problems with logarithmic perturbation can be well applied in the study of the plastic theory relating to logarithmic hardening, population dynamic problems as well as quantum mechanics (see Fuchs and Mingione [22], Fuchs and Seregin [23], Marcellini and Papi [37], Seregin and Frehse [48], and Vetro and Winkert [53]).

In particular, we can choose set K in (1.1) as the following bilateral constraint:

in which φ 1 , φ 2 ∈ W 1 , A ( Ω ) satisfying φ 1 ( x ) ≤ 0 along with φ 2 ( x ) ≥ 0 for a.a. x ∈ Γ 0 . In fact, the constraint set K defined earlier turns out to be a closed and convex subset of V Γ 0 . Moreover, we introduce the multivalued operators F 1 and F 2 by the Clarke’s generalized gradient corresponding to locally Lipschitz functions j 1 and j 2 respectively; then, the multivalued variational inequality (1.1) becomes equivalent to the next variational-hemivariational inequality:

for all v ∈ K , where ξ ∈ G ( π , ∇ π ) , j 1 o , and j 2 o denote the Clarke’s generalized directional derivative of j 1 and j 2 , respectively.

As we know, the initial study of variational inequalities originated from calculus of variations, and developed systematically by Fichera [20] and Stampacchia [50,51], in the 1960s to address challenges in mechanics, particularly obstacle problems in elasticity. Following the seminal contributions by Lions and Stampacchia [35], the research concerning variational inequalities developed strongly, for more details with respect to variational inequalities, readers can refer to monographs [29,47,52]. It is widely recognized that variational inequalities are intimately tied to the convexity of involved energy functionals; however, if these functionals are nonconvex but locally Lipschitz, there arises a generalization of variational inequalities, i.e., hemivariational inequalities. In fact, the study for hemivariational inequalities is fundamentally grounded in the theories involving Clarke’s generalized gradient of locally Lipschitz functionals, and we point out that the research for Clarke’s generalized gradient has its origin in Clarke and Chang [8], Clarke [11,12]. Based on the aforementioned fundamental works, a plenty of impressive works concentrating on variational or hemivariational inequalities appeared. For instance, Carl and Le [5] provided valuable insights into existence results for multivalued variational inequalities and inclusions, where the variational-hemivariational inequalities can be seen as particular cases of multivalued variational inequalities considered. Among multivalued variational inequality (1.1), the well-posedness result related to a type of variational-hemivariational inequality that incorporates constraints and history-dependent operators was studied by Liu et al. [34]. Recently, Carl [6] studied a quasilinear elliptic hemivariational inequality with bilateral constraints and obtained the existence results for weak solutions by appropriately designed penalty technique. For the study of variational problems with bilateral constraints, we refer to Kovalevsky [31,32], and more theoretical results can be found in [13,15,25,41,46]. For the aspect of practical applications, variational inequalities and hemivariational inequalities can be widely applied in various fields, for example, calculus of variations, economics, mathematical physics, and engineering, especially in contact mechanics as well as numerical analysis (see, e.g., previous works [3,26,38–40,49]).

Another intricate aspect of problem (1.1) lies in the presence of the multivalued convection term G , along with the multivalued term F 1 and boundary-defined multivalued term F 2 . In fact, equations with multivalued functions can be well applied in various problems of physics (see, Panagiotopoulos [42,43], Carl and Le [5], and their references). Moreover, the convection terms can be well used to model the convection effect of various fluid flows. In single or multiphase flows, convection arises naturally due to the coefficient of material heterogeneity and force influences (such as density and gravity). Generally, the standard variational tools and corresponding theory cannot be applied to problems involving convection terms due to its nonvariability. For more works dealing with problems with single-valued or multivalued convection terms, we refer to Dupaigne et al. [17], El Manouni et al. [18], Figueiredo and Madeira [21], Gasiński and Winkert [24], Liu et al. [34], and Papageorgiou et al. [44]. It is worth noting that, we do not assume any coercive condition throughout this article, but in general, some coercivity is required if we apply the surjective theorems. To overcome the lack of coercivity, usually, the sub- and supersolution method can be applied. However, instead of the sub-supersolution method, motivated by Carl [6], we will employ the penalty technique to construct an auxiliary penalty problem, which possesses the coercivity, and a properly structured penalty operator will be given in the sequel. As far as we know, this is the first work in addressing the multivalued variational inequality, incorporating a logarithmic perturbed double-phase operator and multivalued convection terms, utilizing a penalty-based approach.

Now, let us look at the main assumptions of data throughout this article:

1. Let p , q ∈ C ( Ω ¯ ) be such that 1 < p ( x ) < N and p ( x ) ≤ q ( x ) < p * ( x ) for all x ∈ Ω ¯ with Ω < ≔ { x ∈ Ω : p ( x ) < q ( x ) } ⫋ Ω 0 ≔ { x ∈ Ω : ϑ ( x ) = 0 } ; also, ϑ is a non-negative function belonging to L ∞ ( Ω ) .

2. Suppose that g : Ω × R × R N → 2 R \ { ∅ } , f 1 : Ω × R → 2 R \ { ∅ } , and f 2 : Γ × R → 2 R \ { ∅ } are graph measurable functions with g ( x , ⋅ , ⋅ ) : R × R N → 2 R \ { ∅ } , and f 1 ( x , ⋅ ) : R → 2 R \ { ∅ } being upper semicontinuous for a.a. x ∈ Ω and f 2 ( x , ⋅ ) : R → 2 R \ { ∅ } being upper semicontinuous for a.a. x ∈ Γ .

3. There exist σ ∈ C ( Ω ¯ ) , σ 1 ∈ C ( Ω ¯ ) , and σ 2 ∈ C ( Γ ) fulfilling 1 < σ ( x ) < p * ( x ) , 1 < σ 1 ( x ) < p * ( x ) , 1 < σ 2 ( x ) < p * ( x ) , β , γ , β 1 , and β 2 being positive constants and 0 < α ∈ L σ ′ ( Ω ) , α 1 ∈ L σ 1 ′ ( Ω ) , α 2 ∈ L σ 2 ′ ( Γ ) such that for all t ∈ R , y ∈ R N and for a.a. x ∈ Ω :

   sup { ∣ ξ ∣ : ξ ∈ g ( x , t , y ) } ≤ α ( x ) + β ∣ t ∣ σ ( x ) − 1 + γ ∣ y ∣ p ( x ) σ ′ ( x ) , sup { ∣ η ∣ : η ∈ f 1 ( x , t ) } ≤ α 1 ( x ) + β 1 ∣ t ∣ σ 1 ( x ) − 1 ,

   also for all t ∈ R and a.a. x ∈ Γ :

   sup { ∣ ζ ∣ : ζ ∈ f 2 ( x , t ) } ≤ α 2 ( x ) + β 2 ∣ t ∣ σ 2 ( x ) − 1 .

In the sequel, we will use the embedding operator i σ ( ⋅ ) mapping from W 1 , A ( Ω ) to L σ ( ⋅ ) ( Ω ) and i σ 1 ( ⋅ ) mapping from W 1 , A ( Ω ) to L σ 1 ( ⋅ ) ( Ω ) , as well as the trace operator i σ 2 ( ⋅ ) mapping from W 1 , A ( Ω ) to L σ 2 ( ⋅ ) ( Γ ) . According to (H2) together with Proposition 2.2(ii) and (iii) (Section 2), one can demonstrate that operators i σ ( ⋅ ) , i σ 1 ( ⋅ ) , and i σ 2 ( ⋅ ) are compact. Let i σ ( ⋅ ) * : L σ ′ ( ⋅ ) ( Ω ) → W 1 , A ( Ω ) * , i σ 1 ( ⋅ ) * : L σ 1 ′ ( ⋅ ) ( Ω ) → W 1 , A ( Ω ) * , and i σ 2 ( ⋅ ) * : L σ 2 ′ ( ⋅ ) ( Γ ) → W 1 , A ( Ω ) * be the adjoint operators of i σ ( ⋅ ) , i σ 1 ( ⋅ ) , and i σ 2 ( ⋅ ) , respectively, with W 1 , A ( Ω ) * being the dual space of W 1 , A ( Ω ) . Let M ( Ω ) denote the space of measurable functions from Ω to R . For any π ∈ M ( Ω ) and y ∈ [ M ( Ω ) ] N , we define the measurable selections of g ( ⋅ , π , y ) as

invoking hypothesis (H1), we see that the aforementioned set is nonempty. For the same reason, for any π ∈ M ( Ω ) (resp., π ∈ M ( Γ ) ), define measurable selections of f 1 ( ⋅ , π ) as

and of f 2 ( ⋅ , π ) :

which are nonempty.

According to the growth conditions given in (H2), we deduce that for any π ∈ L σ ( ⋅ ) ( Ω ) , and ∇ π ∈ L p ( ⋅ ) (resp., π ∈ L σ 1 ( ⋅ ) ( Ω ) , π ∈ L σ 2 ( ⋅ ) ( Γ ) ), and g ˜ ( π ) ⊂ L σ ′ ( ⋅ ) ( Ω ) (resp., f ˜ 1 ( π ) ⊂ L σ 1 ′ ( ⋅ ) ( Ω ) , f ˜ 2 ( π ) ⊂ L σ 2 ′ ( ⋅ ) ( Γ ) ); thus, the mapping g ˜ is from L σ ( ⋅ ) ( Ω ) × [ L p ( ⋅ ) ( Ω ) ] N to L σ ′ ( ⋅ ) ( Ω ) with π ↦ g ˜ ( π ) (resp., f ˜ 1 is from L σ 1 ( ⋅ ) ( Ω ) to L σ 1 ′ ( ⋅ ) ( Ω ) with π ↦ f ˜ 1 ( π ) , f ˜ 2 from L σ 2 ( ⋅ ) ( Γ ) to L σ 2 ′ ( ⋅ ) ( Γ ) with π ↦ f ˜ 2 ( π ) ). On this basis, we define the multivalued operator as G = i σ ( ⋅ ) * g ˜ : W 1 , A ( Ω ) → 2 W 1 , A ( Ω ) * (resp., F 1 = i σ 1 ( ⋅ ) * f ˜ 1 i σ 1 ( ⋅ ) : W 1 , A ( Ω ) → 2 W 1 , A ( Ω ) * , F 2 = i σ 2 ( ⋅ ) * f ˜ 2 i σ 2 ( ⋅ ) : W 1 , A ( Ω ) → 2 W 1 , A ( Ω ) * ), i.e., G ( π , ∇ π ) = { ξ ˜ ∈ W 1 , A ( Ω ) * : ξ ˜ ∈ g ˜ ( π , ∇ π ) } (resp., F 1 ( π ) = { η ˜ ∈ W 1 , A ( Ω ) * : η ˜ ∈ f ˜ 1 ( π ) } , F 2 ( π ) = { ζ ˜ ∈ W 1 , A ( Ω ) * : ζ ˜ ∈ f ˜ 2 ( π ) } ).

Now, we are ready to introduce the concept of weak solution for the multivalued variational inequality (1.1).

Among (1.7), the boundary integral ∫ Γ ζ ( v − π ) d ς means that

with i τ 2 ( ⋅ ) : W 1 , A ( Ω ) → L τ 2 ( ⋅ ) ( ∂ Ω ) representing the trace operator, where i τ 2 ( ⋅ ) v ∣ Γ denotes the restriction of i τ 2 ( ⋅ ) v on the boundary Γ .

The structure of this article is outlined as follows. In Section 2, some critical preliminaries will be given, including essential findings related to variable exponent Sobolev spaces and Musielak-Orlicz spaces in the context of the functional A given in (1.4), as well as the definition of penalty operators related to set K . Section 3 is aimed at showing our main existence results with the aid of penalty technique and a surjective theorem related to multivalued pseudomonotone operators. Then, in Section 4, we establish the compactness of the solution set pertaining to (1.1) under suitable assumptions. Finally, in Section 5, we consider the special instances to multivalued variational inequality (1.1), i.e., the set K is defined as a bilateral constraint set, and we choose the multivalued terms F 1 and F 2 defined, respectively, in the domain and its boundary as the Clarke’s subdifferential of locally Lipschitz functions.

2 Preliminaries

In this section, let us introduce basic and useful notations and prove conclusions involving the variable exponent Lebesgue space and Musielak-Orlicz spaces formulated by (1.4) (i.e., L A ( Ω ) and W 1 , A ( Ω ) ), which are pivotal to our proof of the primary findings, most of them taken from Diening et al. [16], Fan and Zhao [19], Harjulehto and Hästö [28], Kováčik and Rákosník [30], and Lu et al. [36].

In the sequel, we choose Ω a bounded domain in R N ( N ≥ 2 ) with Lipschitz boundary ∂ Ω , for a given continuous function ℓ such that 1 < ℓ ( x ) for all x ∈ Ω ¯ , we define ℓ − and ℓ + by

Also, let ℓ ′ ∈ C ( Ω ¯ ) be the conjugate of ℓ , i.e., 1 ℓ ( x ) + 1 ℓ ′ ( x ) = 1 for all x ∈ Ω ¯ . Given ℓ ∈ C ( Ω ¯ ) with 1 < ℓ ( x ) for all x ∈ Ω ¯ , we introduce the variable exponent Lebesgue space related to ℓ :

where ρ ℓ ( ⋅ ) represents the corresponding modular function:

Generally, the variable exponent Lebesgue space L ℓ ( x ) ( Ω ) is endowed with the Luxemburg norm, which is defined as follows:

and ( L ℓ ( ⋅ ) ( Ω ) , ‖ π ‖ ℓ ( ⋅ ) ) is shown to be a separable and reflexive Banach space. Similarly, one can give the definition of space ( L ℓ ( ⋅ ) ( ∂ Ω ) , ‖ π ‖ ℓ ( ⋅ ) , ∂ Ω ) . The next Hölder-type inequality with respect to L ℓ ( ⋅ ) ( Ω ) and its dual space L ℓ ′ ( ⋅ ) ( Ω ) is a useful tool for inequality estimation:

for all π ∈ L ℓ ( x ) ( Ω ) , and all v ∈ L ℓ ′ ( x ) ( Ω ) . In addition, if ℓ 1 , ℓ 2 ∈ C ( Ω ¯ ) fulfilling 1 < ℓ 1 ( x ) ≤ ℓ 2 ( x ) for all x ∈ Ω ¯ , then there holds the continuous embedding:

Furthermore, we give the definition of variable exponent Sobolev space

equipped with the norm

with ‖ ∇ π ‖ ℓ ( ⋅ ) ≔ ‖ ∣ ∇ π ∣ ‖ ℓ ( ⋅ ) . Hereafter, the completion of C 0 ∞ ( Ω ) under the norm ‖ ⋅ ‖ 1 , ℓ ( ⋅ ) is given as

which is a subspace of W 1 , ℓ ( ⋅ ) ( Ω ) . Both W 1 , ℓ ( ⋅ ) ( Ω ) and W 0 1 , ℓ ( ⋅ ) ( Ω ) are separable, reflexive, and uniformly convex Banach spaces. Moreover, for all π ∈ W 0 1 , ℓ ( ⋅ ) ( Ω ) and c 0 > 0 , we have the next Poincaré inequality:

Thus, we can replace the norm of space W 0 1 , ℓ ( ⋅ ) ( Ω ) with the equivalent norm ‖ ∇ π ‖ ℓ ( ⋅ ) , i.e.,

Now, let us recall the definitions and some crucial properties to the Musielak-Orlicz Lebesgue space L A ( Ω ) , as well as its associated Musielak-Orlicz Sobolev spaces W 1 , A ( Ω ) . To start with, we give the definition for L A ( Ω ) as follows:

We point out that A is a generalized N -function (see [36, Definition 2.7]); thus, according to Harjulehto-Hästö [28], the related modular function is defined as

If L A ( Ω ) is equipped with the following Luxemburg norm

then ( L A ( Ω ) , ‖ π ‖ A ) is a reflexive and separable Banach space (the precise proof can be found in [36]).

Now, we are ready to examine the relationship with respect to modular ρ A and the Luxemburg norm ‖ ⋅ ‖ A , referring to [36, Proposition 2.20].

Analogously, the definition of Musielak-Orlicz Sobolev space W 1 , A ( Ω ) is as follows:

with the norm given by

where ‖ ∇ π ‖ A = ‖ ∣ ∇ π ∣ ‖ A . For the convenience, we use the notation ‖ π ‖ representing the norm given in (2.1). In addition, the space W 0 1 , A ( Ω ) denotes the completion of C 0 ∞ ( Ω ) within W 1 , A ( Ω ) , and W 0 1 , A ( Ω ) can be endowed with the equivalent norm

due to the Poincaré inequality given by [36, Proposition 2.23]. Note that W 1 , A ( Ω ) and its subspace W 0 1 , A ( Ω ) are reflexive and separable Banach spaces (proved by [36]). According to [36, Proposition 2.21], if we take the equivalent norm in space W 1 , A ( Ω ) , i.e.,

where

for π ∈ W 1 , A ( Ω ) ; then, the conclusions in Proposition 2.1 still hold true with ρ A replaced by ρ ˆ A and L A ( Ω ) replaced with W 1 , A ( Ω ) .

The next proposition gives some embedding results with respect to the variable exponent Lebesgue space, L A ( Ω ) as well as W 1 , A ( Ω ) , that can be found in [36, Propositions 2.22 and 2.23]. Recall that X ↪ ↪ Y means space X is compactly embedded into space Y .

In the following parts, we define t + ≔ max { t , 0 } for any t ∈ R (in the same time, define t − ≔ − min { t , 0 } for any t ∈ R ), and let π ± ( ⋅ ) ≔ [ π ( ⋅ ) ] ± for any function π : Ω → R .