Identification of discontinuous parameters in double phase obstacle problems

Shengda Zeng; Yunru Bai; Patrick Winkert; Jen-Chih Yao

doi:10.1515/anona-2022-0223

Open Access Published by De Gruyter August 19, 2022

Identification of discontinuous parameters in double phase obstacle problems

Shengda Zeng , Yunru Bai , Patrick Winkert and Jen-Chih Yao

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2022-0223

Abstract

In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.

Keywords: discontinuous parameter; double phase operator; elliptic obstacle problem; inverse problem; mixed boundary condition; multivalued convection; Steklov eigenvalue problem

MSC 2010: 35J20; 35J25; 35J60; 35R30; 49N45; 65J20

Special Issue: – Nonlinear analysis: perspectives and synergies

1 Introduction

The aim of this article is to study an inverse problem to an elliptic differential inclusion problem involving a double phase differential operator, a multivalued convection term (dependence on the gradient of the solution), a multivalued boundary condition and an obstacle constraint. To this end, let Ω be a bounded domain in R N ( N ≥ 2 ) with Lipschitz boundary Γ ≔ ∂ Ω such that Γ is divided into three mutually disjoint parts Γ 1 , Γ 2 and Γ 3 with Γ 1 having positive Lebesgue measure. We study the problem

(1.1) − div ( a ( x ) ∣ ∇ u ∣ p − 2 ∇ u + μ ( x ) ∣ ∇ u ∣ q − 2 ∇ u ) + g ( x , u ) + μ ( x ) ∣ u ∣ q − 2 u ∈ f ( x , u , ∇ u ) in Ω , u = 0 on Γ 1 , ∂ u ∂ ν a = h ( x ) on Γ 2 , ∂ u ∂ ν a ∈ U ( x , u ) on Γ 3 , u ( x ) ≤ Φ ( x ) in Ω ,

where 1 < p < N , p < q , μ : Ω ¯ → [ 0 , ∞ ) is a bounded function,

∂ u ∂ ν a ≔ ( a ( x ) ∣ ∇ u ∣ p − 2 ∇ u + μ ( x ) ∣ ∇ u ∣ q − 2 ∇ u ) ⋅ ν ,

with ν being the outward unit normal vector on Γ , f : Ω × R × R N → 2 R and U : Γ 3 × R → 2 R are two given multivalued functions, Φ : Ω → R is an obstacle function and a : Ω → ( 0 , + ∞ ) , h : Γ 2 → R are two possibly discontinuous parameters.

The main contribution of this article is twofold. The first intention of the article is to establish the nonemptiness, boundedness and closedness of the solution set to problem (1.1) (in the weak sense), in which our main methods are based on a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping, the theory of nonsmooth analysis and the properties of the Steklov eigenvalue problem for the p -Laplacian. The second contribution of the article is to develop a general framework for studying the inverse problem under consideration and to establish the solvability for such inverse problems. To the best of our knowledge, this is the first work studying the identification of discontinuous parameters for such general nonlinear elliptic equations. The problem under consideration combines several interesting phenomena such as double phase operators, multivalued right-hand sides, mixed boundary conditions and obstacle constraints.

First we point out that, motivated by several applications, the inverse problem of parameter identification in partial differential equations is an important field in mathematics and even though such problems in form of equations and inequalities have been studied a lot, there are still several open problems to be solved. Our work is motivated by the article of Migórski et al. [39], who studied the inverse problem of mixed quasi-variational inequalities of the form

⟨ T ( a , u ) , v − u ⟩ + φ ( v ) − φ ( u ) ≥ ⟨ m , v − u ⟩ for all v ∈ K ( u ) ,

where K : C → 2 C is a set-valued map, T : B × V → V ∗ is a nonlinear map, φ : V → R ∪ { + ∞ } is a functional and m ∈ V ∗ , while V is a real reflexive Banach space, B is another Banach space and C is a nonempty, closed, convex subset of V . Their abstract result applies to p -Laplacian inequalities, see also [38] for hemivariational inequalities. We also mention the works of Clason et al. [10] for noncoercive variational problems, Gwinner [27] for variational inequalities of second kind, Gwinner et al. [28] for an optimization setting and Migórski and Ochal [40] for nonlinear parabolic problems, see also references therein. In addition, we refer to the recent work of Papageorgiou and Vetro [45] about existence and relaxation theorems for different types of problems which can be applied to variational inequalities and control systems.

A second interesting phenomenon is the occurrence of the weighted double phase operator, namely,

(1.2) div ( a ( x ) ∣ ∇ u ∣ p − 2 ∇ u + μ ( x ) ∣ ∇ u ∣ q − 2 ∇ u ) for u ∈ W 1 , ℋ ( Ω ) .

For a ≡ 1 , this operator corresponds to the energy functional given by

(1.3) ∫ Ω ( ∣ ∇ u ∣ p + μ ( x ) ∣ ∇ u ∣ q ) d x .

Functionals of the form (1.3) have been initially introduced by Zhikov [56] in 1986 in order to describe models for strongly anisotropic materials and it also turned out its relevance in the study of duality theory as well as in the context of the Lavrentiev phenomenon, see Zhikov [57]. Observe that the energy density in (1.3) changes its ellipticity and growth properties according to the point in the domain. In general, double phase differential operators and corresponding energy functionals interpret various comprehensive natural phenomena and model several problems in Mechanics, Physics and Engineering Sciences. For example, in the elasticity theory, the modulating coefficient μ ( ⋅ ) dictates the geometry of composites made of two different materials with distinct power hardening exponents p and q , see Zhikov [58]. Functionals given in (1.3) have been intensively studied in the last few years concerning regularity of local minimizers. We mention the famous works of Baroni et al. [2,3], Byun and Oh [7], Colombo and Mingione [12,13], Marcellini [36,37] and Ragusa and Tachikawa [49].

A third interesting phenomenon is not only the multivalued right-hand side, which is motivated by several physical applications (see, e.g., Panagiotopoulos [42,43], Carl and Le [8] and references therein) but also its dependence on the gradient of the solutions often called convection term. The main difficulty in the study of gradient dependent right-hand sides is their nonvariational character, that is, the standard variational tools to the corresponding energy functionals are not applicable. In the past few years, several interesting works have been published with convection terms, we refer to the papers of El Manouni et al. [16], Faraci et al. [17], Faraci and Puglisi [18], Figueiredo and Madeira [20], Gasiński and Papageorgiou [23], Liu et al. [32], Liu and Papageorgiou [33], Marano and Winkert [35], Papageorgiou et al. [44] and Zeng and Papageorgiou [55].

Finally, we mention some existence results on the recent topic of double phase operators published within the last few years. We refer to Bahrouni et al. [1], Benslimane et al. [4], Biagi et al. [5], Colasuonno and Squassina [11], Fiscella [21], Farkas and Winkert [19], Gasiński and Papageorgiou [22], Gasiński and Winkert [24,25,26], Liu and Dai [31], Liu and Winkert [34], Papageorgiou et al. [46], Perera and Squassina [48], Stegliński [51] and Zeng et al. [52,53,54].

This article is organized as follows. Section 2 recalls preliminary material including Musielak-Orlicz Lebesgue and Musielak-Orlicz Sobolev spaces, the p -Laplacian eigenvalue problem with Steklov boundary condition, pseudomonotone operators and a surjectivity theorem for multivalued mappings. Under very general assumptions on the data, Section 3 proves the nonemptiness and compactness of the solution set to problem (1.1). In Section 4, we present a new existence result to the inverse problem of (1.1).

2 Preliminaries

This section is devoted to recall some basic definitions and preliminaries, which will be used in the next sections to derive the main results of the article. To this end, let Ω ⊂ R N be a bounded domain with Lipschitz boundary Γ ≔ ∂ Ω such that Γ is decomposed into three mutually disjoint parts Γ 1 , Γ 2 and Γ 3 with Γ 1 having positive Lebesgue measure. In what follows, we denote by M ( Ω ) the space of all measurable functions u : Ω → R . As usual, we identify two functions which differ on a Lebesgue-null set. Let r ∈ [ 1 , ∞ ) and D be a nonempty subset of Ω ¯ . We denote the usual Lebesgue spaces by L r ( D ) ≔ L r ( D ; R ) and L r ( D ; R N ) equipped with the standard r -norm ‖ ⋅ ‖ r , D and L r ( Γ ) stands for the boundary Lebesgue spaces with norm ‖ ⋅ ‖ r , Γ .

Let L r ( D ) + ≔ { u ∈ L r ( D ) : u ( x ) ≥ 0 for a.a. x ∈ Ω } . By W 1 , r ( Ω ) we define the corresponding Sobolev space endowed with the norm ‖ ⋅ ‖ 1 , r , Ω given by

‖ u ‖ 1 , r , Ω ≔ ‖ u ‖ r , Ω + ‖ ∇ u ‖ r , Ω for all u ∈ W 1 , r ( Ω ) .

For any fixed s > 1 , the conjugate of s is defined by s ′ > 1 such that 1 s + 1 s ′ = 1 . Moreover, we use the symbols s ∗ and s ∗ to represent the critical exponents to s in the domain and on the boundary, respectively, given by

(2.1) s ∗ = N s N − s if s < N , + ∞ if s ≥ N , and s ∗ = ( N − 1 ) s N − s if s < N , + ∞ if s ≥ N .

Let us comment on the r -Laplacian eigenvalue problem with Steklov boundary condition given by

(2.2) − Δ r u = − ∣ u ∣ r − 2 u in Ω , ∣ u ∣ r − 2 u ⋅ ν = λ ∣ u ∣ r − 2 u on Γ ,

for 1 < r < ∞ . From Lê [30] we know that (2.2) has a smallest eigenvalue λ 1 , r S > 0 , which is isolated and simple. Besides, we know that λ 1 , r S > 0 can be characterized by

(2.3) λ 1 , r S = inf u ∈ W 1 , r ( Ω ) ⧹ { 0 } ‖ ∇ u ‖ r , Ω r + ‖ u ‖ r , Ω r ‖ u ‖ r , Γ r .

The following assumptions are supposed in the entire article:

(2.4) 1 < p < N , p < q < p ∗ and 0 ≤ μ ( ⋅ ) ∈ L ∞ ( Ω ) .

Now we define the nonlinear function ℋ : Ω × [ 0 , ∞ ) → [ 0 , ∞ ) given by

ℋ ( x , t ) = t p + μ ( x ) t q for all ( x , t ) ∈ Ω × [ 0 , ∞ ) .

Then, the Musielak-Orlicz Lebesgue space L ℋ ( Ω ) driven by the function ℋ is given by

L ℋ ( Ω ) = { u ∈ M ( Ω ) : ρ ℋ ( u ) < + ∞ }

equipped with the Luxemburg norm

‖ u ‖ ℋ = inf τ > 0 : ρ ℋ u τ ≤ 1 .

Here, the modular function is given by

ρ ℋ ( u ) ≔ ∫ Ω ℋ ( x , ∣ u ∣ ) d x = ∫ Ω ( ∣ u ∣ p + μ ( x ) ∣ u ∣ q ) d x .

We know that L ℋ ( Ω ) is uniformly convex and so a reflexive Banach space. Moreover, we introduce the seminormed space L μ q ( Ω )

L μ q ( Ω ) = u ∈ M ( Ω ) : ∫ Ω μ ( x ) ∣ u ∣ q d x < + ∞

endowed with the seminorm

‖ u ‖ q , μ = ∫ Ω μ ( x ) ∣ u ∣ q d x 1 q .

The Musielak-Orlicz Sobolev space W 1 , ℋ ( Ω ) is given by

W 1 , ℋ ( Ω ) = { u ∈ L ℋ ( Ω ) : ∣ ∇ u ∣ ∈ L ℋ ( Ω ) }

equipped with the norm

‖ u ‖ 1 , ℋ = ‖ ∇ u ‖ ℋ + ‖ u ‖ ℋ ,

where ‖ ∇ u ‖ ℋ = ‖ ∣ ∇ u ∣ ‖ ℋ . As before, it is known that W 1 , ℋ ( Ω ) is a reflexive Banach space.

Next, we introduce a closed subspace V of W 1 , ℋ ( Ω ) given by

V ≔ { u ∈ W 1 , ℋ ( Ω ) : u = 0 on Γ 1 }

endowed with the norm ‖ u ‖ V = ‖ u ‖ 1 , ℋ for all u ∈ V . Of course, V is also a reflexive Banach space. In the following, we denote the norm of the dual space V ∗ of V by ‖ ⋅ ‖ V ∗ .

Let us recall some embedding results for the spaces L ℋ ( Ω ) and W 1 , ℋ ( Ω ) , see Gasiński and Winkert [26] or Crespo-Blanco et al. [14].

Proposition 2.1

Let (2.4) be satisfied and denoted by p ∗ , p ∗ the critical exponents to p as given in (2.1) for s = p .

L ℋ ( Ω ) ↪ L r ( Ω ) and W 1 , ℋ ( Ω ) ↪ W 1 , r ( Ω ) are continuous for all r ∈ [ 1 , p ] ;
W 1 , ℋ ( Ω ) ↪ L r ( Ω ) is continuous for all r ∈ [ 1 , p ∗ ] and compact for all r ∈ [ 1 , p ∗ ) ;
W 1 , ℋ ( Ω ) ↪ L r ( Γ ) is continuous for all r ∈ [ 1 , p ∗ ] and compact for all r ∈ [ 1 , p ∗ ) ;
L ℋ ( Ω ) ↪ L μ q ( Ω ) is continuous;
L q ( Γ ) ↪ L ℋ ( Ω ) is continuous.

We point out that if we replace the space W 1 , ℋ ( Ω ) by V in Proposition 2.1, then the embeddings (ii) and (iii) remain valid.

The following proposition is due to Liu and Dai [31, Proposition 2.1].

Proposition 2.2

Let (2.4) be satisfied and let y ∈ L ℋ ( Ω ) . Then the following hold:

if y ≠ 0 , then ‖ y ‖ ℋ = λ if and only if ρ ℋ y λ = 1 ;
‖ y ‖ ℋ < 1 (resp. > 1 and = 1 ) if and only if ρ ℋ ( y ) < 1 (resp. > 1 and = 1 );
if ‖ y ‖ ℋ < 1 , then ‖ y ‖ ℋ q ≤ ρ ℋ ( y ) ≤ ‖ y ‖ ℋ p ;
if ‖ y ‖ ℋ > 1 , then ‖ y ‖ ℋ p ≤ ρ ℋ ( y ) ≤ ‖ y ‖ ℋ q ;
‖ y ‖ ℋ → 0 if and only if ρ ℋ ( y ) → 0 ;
‖ y ‖ ℋ → + ∞ if and only if ρ ℋ ( y ) → + ∞ .

We suppose that

(2.5) a ∈ L ∞ ( Ω ) such that inf x ∈ Ω a ( x ) > 0 .

Next, we introduce the nonlinear operator F : V → V ∗ given by

(2.6) ⟨ F ( u ) , v ⟩ ≔ ∫ Ω ( a ( x ) ∣ ∇ u ∣ p − 2 ∇ u + μ ( x ) ∣ ∇ u ∣ q − 2 ∇ u ) ⋅ ∇ v d x + ∫ Ω ( ∣ u ∣ p − 2 u + μ ( x ) ∣ u ∣ q − 2 u ) v d x ,

for u , v ∈ V with ⟨ ⋅ , ⋅ ⟩ being the duality pairing between V and its dual space V ∗ . The following proposition states the main properties of F : V → V ∗ . We refer to Liu and Dai [31, Proposition 3.1] or Crespo-Blanco et al. [14, Proposition 3.4] for its proof.

Proposition 2.3

Let hypotheses (2.4) and (2.5) be satisfied. Then, the operator F defined by (2.6) is bounded, continuous, monotone (hence maximal monotone) and of type ( S + ) , that is,

u n ⟶ w u in V and limsup n → ∞ ⟨ F u n , u n − u ⟩ ≤ 0 ,

imply u n → u in V .

We now recall some notions and results concerning nonsmooth analysis and multivalued analysis. Throughout the article the symbols “ ⟶ w ” and “ → ” stand for the weak and the strong convergence, respectively, in various spaces. Moreover, let us recall the notions of pseudomonotonicity and generalized pseudomonotonicity in the sense of Brézis for multivalued operators (see, e.g., Migórski et al. [41, Definition 3.57]), which will be useful in the sequel.

Definition 2.4

Let X be a reflexive real Banach space. The operator A : X → 2 X ∗ is called

pseudomonotone (in the sense of Brézis) if the following conditions hold:
1. the set A ( u ) is nonempty, bounded, closed and convex for all u ∈ X ;
2. A is upper semicontinuous from each finite-dimensional subspace of X to the weak topology on X ∗ ;
3. if { u n } ⊂ X with u n ⟶ w u in X and u n ∗ ∈ A ( u n ) are such that
  limsup n → ∞ ⟨ u n ∗ , u n − u ⟩ X ∗ × X ≤ 0 ,
  then to each element v ∈ X , there exists u ∗ ( v ) ∈ A ( u ) with
  ⟨ u ∗ ( v ) , u − v ⟩ X ∗ × X ≤ liminf n → ∞ ⟨ u n ∗ , u n − v ⟩ X ∗ × X ;
generalized pseudomonotone (in the sense of Brézis) if the following holds: Let { u n } ⊂ X and { u n ∗ } ⊂ X ∗ with u n ∗ ∈ A ( u n ) . If u n ⟶ w u in X and u n ∗ ⟶ w u ∗ in X ∗ and
limsup n → ∞ ⟨ u n ∗ , u n − u ⟩ X ∗ × X ≤ 0 ,
then the element u ∗ lies in A ( u ) and
⟨ u n ∗ , u n ⟩ X ∗ × X → ⟨ u ∗ , u ⟩ X ∗ × X .

It is not difficult to see that every pseudomonotone operator is generalized pseudomonotone, see, e.g., Carl et al. [9, Proposition 2.122]. Also, under an additional assumption of boundedness, we obtain the converse statement, see, e.g., Carl et al. [9, Proposition 2.123].

Proposition 2.5

Let X be a reflexive real Banach space and assume that A : X → 2 X ∗ satisfies the following conditions:

for each u ∈ X we have that A ( u ) is a nonempty, closed and convex subset of X ∗ .
A : X → 2 X ∗ is bounded.
A is generalized pseudomonotone, i.e., if u n ⟶ w u in X and u n ∗ ⟶ w u ∗ in X ∗ with u n ∗ ∈ A ( u n ) and
limsup n → ∞ ⟨ u n ∗ , u n − u ⟩ X ∗ × X ≤ 0 ,
then u ∗ ∈ A ( u ) and
⟨ u n ∗ , u n ⟩ X ∗ × X → ⟨ u ∗ , u ⟩ X ∗ × X .

Then the operator A : X → 2 X ∗ is pseudomonotone.

Let us now recall the definition of Kuratowski limits, see, e.g., Papageorgiou and Winkert [47, Definition 6.7.4].

Definition 2.6

Let ( X , τ ) be a Hausdorff topological space and let { A n } n ∈ N ⊂ 2 X be a sequence of sets. We define the τ -Kuratowski lower limit of the sets A n by

τ - liminf n → ∞ A n ≔ { x ∈ X : x = τ - lim n → ∞ x n , x n ∈ A n for all n ≥ 1 } ,

and the τ -Kuratowski upper limit of the sets A n

τ - limsup n → ∞ A n ≔ { x ∈ X : x = τ - lim k → ∞ x n k , x n k ∈ A n k , n 1 < n 2 < … < n k < … } .

A = τ - liminf n → ∞ A n = τ - limsup n → ∞ A n ,

then A is called τ -Kuratowski limit of the sets A n .

Finally, we recall the following surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping, see Le [29, Theorem 2.2].

Theorem 2.7

Let X be a real reflexive Banach space, let G : D ( G ) ⊂ X → 2 X ∗ be a maximal monotone operator, let ℱ : D ( ℱ ) = X → 2 X ∗ be a bounded multivalued pseudomonotone operator, let ℒ ∈ X ∗ and let B R ( 0 ) ≔ { u ∈ X : ‖ u ‖ X < R } . Assume that there exist u 0 ∈ X and R ≥ ‖ u 0 ‖ X such that D ( G ) ∩ B R ( 0 ) ≠ ∅ and

(2.7) ⟨ ξ + η − ℒ , u − u 0 ⟩ X ∗ × X > 0

for all u ∈ D ( G ) with ‖ u ‖ X = R , for all ξ ∈ G ( u ) and for all η ∈ ℱ ( u ) . Then the inclusion

ℱ ( u ) + G ( u ) ∋ ℒ

has a solution in D ( G ) .

Obviously, if

(2.8) lim ‖ u ‖ X → + ∞ u ∈ D ( G ) ⟨ ξ + η , u − u 0 ⟩ X ∗ × X ‖ u ‖ X = + ∞

is satisfied, then the estimate in (2.7) holds automatically for some R large enough.

3 Double phase elliptic obstacle inclusion problem

In this section, we are interested in the study of the existence of a solution to the double phase elliptic obstacle inclusion problem (1.1) and in deriving some relevant properties of the solution set to problem (1.1). More precisely, we are going to apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone operator, to examine the solvability of problem (1.1).

First, we formulate the hypotheses on the data of problem (1.1).

The multivalued convection mapping f : Ω × R × R N → 2 R has nonempty, bounded, closed and convex values and
1. the multivalued mapping x ↦ f ( x , s , ξ ) is measurable in Ω for all ( s , ξ ) ∈ R × R N ;
2. the multivalued mapping ( s , ξ ) ↦ f ( x , s , ξ ) is upper semicontinuous for a.a. x ∈ Ω ;
3. there exist α f ∈ L r r − 1 ( Ω ) + and a f , b f ≥ 0 such that
  ∣ η ∣ ≤ a f ∣ ξ ∣ p ( r − 1 ) r + b f ∣ s ∣ r − 1 + α f ( x ) ,
  for all η ∈ f ( x , s , ξ ) , for all s ∈ R , for all ξ ∈ R N and for a.a. x ∈ Ω , where 1 < r < p ∗ with the critical exponent p ∗ in the domain Ω given in (2.1) for s = p ;
4. there exist β f ∈ L 1 ( Ω ) + and constants c f , d f ≥ 0 such that
  η s ≤ c f ∣ ξ ∣ p + d f ∣ s ∣ p + β f ( x ) ,
  for all η ∈ f ( x , s , ξ ) , for all s ∈ R , for all ξ ∈ R N and for a.a. x ∈ Ω .

The function g : Ω × R → R is such that
1. for all s ∈ R , the function x ↦ g ( x , s ) is measurable;
2. for a.a. x ∈ Ω , the function s ↦ g ( x , s ) is continuous;
3. there exist a g > 0 and b g ∈ L 1 ( Ω ) such that
  g ( x , s ) s ≥ a g ∣ s ∣ ς − b g ( x ) ,
  for all s ∈ R and for a.a. x ∈ Ω , where p < ς < p ∗ ;
4. for any u , v ∈ L p ∗ ( Ω ) , the function x ↦ g ( x , u ( x ) ) v ( x ) belongs to L 1 ( Ω ) .

The function Φ : Ω → [ 0 , ∞ ) is measurable, that is, Φ ∈ M ( Ω ) .

U : Γ 3 × R → 2 R satisfies the following conditions:
1. U ( x , s ) is a nonempty, bounded, closed and convex set in R for a.a. x ∈ Γ 3 and for all s ∈ R ;
2. x ↦ U ( x , s ) is measurable on Γ 3 for all s ∈ R ;
3. s ↦ U ( x , s ) is u.s.c.;
4. there exist α U ∈ L δ ′ ( Γ 3 ) + and a U ≥ 0 such that
  ∣ U ( x , s ) ∣ ≤ α U ( x ) + a U ∣ s ∣ δ − 1
  for a.a. x ∈ Γ 3 and for all s ∈ R , where 1 < δ < p ∗ with the critical exponent p ∗ on the boundary Γ given in (2.1);
5. there exist β U ∈ L 1 ( Γ 3 ) + and b U ≥ 0 such that
  ξ s ≤ b U ∣ s ∣ p + β U ( x )
  for all ξ ∈ U ( x , s ) , for all s ∈ R and for a.a. x ∈ Γ 3 .

a ∈ L ∞ ( Ω ) is such that inf x ∈ Ω a ( x ) ≥ c Λ > 0 and h ∈ L p ′ ( Γ 2 ) .

The inequality holds
c Λ − c f − b U ( λ 1 , p S ) − 1 > 0 ,
where λ 1 , p S is the first eigenvalue of the p -Laplacian with Steklov boundary condition, see (2.2) and (2.3).

Remark 3.1

It should be mentioned that if hypotheses H( f )(iv) and H( U )(v) are replaced by the following conditions:

there exist β f ∈ L 1 ( Ω ) + and constants c f , d f ≥ 0 such that
η s ≤ c f ∣ ξ ∣ ϱ 1 + d f ∣ s ∣ p + β f ( x )
for all η ∈ f ( x , s , ξ ) , for all s ∈ R , for all ξ ∈ R N and for a.a. x ∈ Ω , where 1 < ϱ 1 < p ;
there exist β U ∈ L 1 ( Γ 3 ) + and b U ≥ 0 such that
ξ s ≤ b U ∣ s ∣ ϱ 2 + β U ( x )
for all ξ ∈ U ( x , s ) , for all s ∈ R and for a.a. x ∈ Γ 3 , where 1 < ϱ 2 < p ,

then hypothesis H(1) can be removed. Indeed, it follows from Young’s inequality with ε > 0 that

η s ≤ c f ∣ ξ ∣ ϱ 1 + d f ∣ s ∣ p + β f ( y ) ≤ ε ∣ ξ ∣ p + c 1 ( ε ) + d f ∣ s ∣ p + β f ( y ) ξ s ≤ b U ∣ s ∣ ϱ 2 + β U ( x ) ≤ ε ∣ s ∣ p + c 2 ( ε ) + β U ( x )

for all η ∈ f ( y , s , ξ ) , for all ξ ∈ U ( x , s ) , for all s ∈ R , for all ξ ∈ R N , for a.a. y ∈ Ω and for a.a. x ∈ Γ 3 with some c 1 ( ε ) , c 2 ( ε ) > 0 . If we choose ε ∈ 0 , c Λ 1 + ( λ 1 , p S ) − 1 , then the inequality in H(1) holds automatically.

Let K be a subset of V given by

(3.1) K ≔ { v ∈ V : v ≤ Φ in Ω } .

Under H( Φ ) we see that the set K is a nonempty, closed and convex subset of V . In fact, from H( Φ ) (i.e., Φ ( x ) ≥ 0 for a.a. x ∈ Ω ), we know that 0 ∈ K , i.e., K ≠ ∅ . Furthermore, it is clear that K is convex. For the closedness, let { u n } n ∈ N ⊂ K be a sequence such that u n → u in V for some u ∈ V . The continuity of V into L p ( Ω ) implies that u n → u in L p ( Ω ) . Passing to a subsequence if necessary, we may suppose that u n ( x ) → u ( x ) for a.a. x ∈ Ω . Therefore,

Φ ( x ) ≥ lim n → ∞ u n ( x ) = u ( x ) for a.a. x ∈ Ω .

Hence, u ∈ K and so K is closed.

Next, we state the definition of a weak solution to problem (1.1).

Definition 3.2

A function u ∈ K is said to be a weak solution of problem (1.1), if there exist functions η ∈ L r ′ ( Ω ) and ξ ∈ L δ ′ ( Γ 3 ) with η ( x ) ∈ f ( x , u ( x ) , ∇ u ( x ) ) for a.a. x ∈ Ω , ξ ( x ) ∈ U ( x , u ( x ) ) for a.a. x ∈ Γ 3 and the equality

∫ Ω ( a ( x ) ∣ ∇ u ∣ p − 2 ∇ u + μ ( x ) ∣ ∇ u ∣ q − 2 ∇ u ) ⋅ ∇ ( v − u ) d x + ∫ Ω g ( x , u ) ( v − u ) d x + ∫ Ω μ ( x ) ∣ u ∣ q − 2 u ( v − u ) d x ≥ ∫ Ω η ( x ) ( v − u ) d x + ∫ Γ 2 h ( x ) ( v − u ) d Γ + ∫ Γ 3 ξ ( x ) ( v − u ) d Γ

is satisfied for all v ∈ K , where the set K is defined by (3.1).

The following theorem which is the main result in this section shows that for each pair ( a , h ) ∈ L ∞ ( Ω ) + × L p ′ ( Γ 2 ) satisfying H(0), the solution set to problem (1.1), denoted by S ( a , h ) , is nonempty, bounded and weakly closed.

Theorem 3.3

Let hypotheses (2.4), H( f ), H( g ), H( Φ ), H( U ), H(0) and H(1) be satisfied. Then, the solution set of problem (1.1) is nonempty, bounded and weakly closed (hence, weakly compact).

Proof

We divide the proof into three parts.

I Existence:

First, we consider the following nonlinear functions F : V → V ∗ , G : V ⊂ L ς ( Ω ) → L ς ′ ( Ω ) and L : L p ( Ω ) → L p ′ ( Ω ) defined by

⟨ F u , v ⟩ ≔ ∫ Ω ( a ( x ) ∣ ∇ u ∣ p − 2 ∇ u + μ ( x ) ∣ ∇ u ∣ q − 2 ∇ u ) ⋅ ∇ v d x + ∫ Ω ( ∣ u ∣ p − 2 u + μ ( x ) ∣ u ∣ q − 2 u ) v d x , ⟨ G u , w ⟩ L ς ′ ( Ω ) × L ς ( Ω ) ≔ ∫ Ω g ( x , u ) w d x ⟨ L y , z ⟩ L p ′ ( Ω ) × L p ( Ω ) ≔ ∫ Ω ∣ y ∣ p − 2 y z d x

for all u , v ∈ V , for all w ∈ L ς ( Ω ) and for all y , z ∈ L p ( Ω ) .

Let u ∈ V be fixed. By the Yankov-von Neumann-Aumann selection theorem (see e.g., Papageorgiou and Winkert [47, Theorem 2.7.25]) and assumptions H( f )(i) and (ii), we know that the multivalued function x ↦ f ( x , u ( x ) , ∇ u ( x ) ) admits a measurable selection. Let η : Ω → R be a measurable selection of x ↦ f ( x , u ( x ) , ∇ u ( x ) ) , that is, η ( x ) ∈ f ( x , u ( x ) , ∇ u ( x ) ) for a.a. x ∈ Ω . From H( f )(iii) and the inequality

( ∣ r 1 ∣ + ∣ r 2 ∣ ) s ≤ 2 s − 1 ( ∣ r 1 ∣ s + ∣ r 2 ∣ s ) for all r 1 , r 2 ∈ R with s ≥ 1 ,

it follows that there exist constants M 1 , M 2 > 0 satisfying

(3.2) ∫ Ω ∣ η ( x ) ∣ r ′ d x ≤ ∫ Ω a f ∣ ∇ u ∣ p r ′ + b f ∣ u ∣ r − 1 + α f ( x ) r ′ d x ≤ M 1 ∫ Ω ( ∣ ∇ u ∣ p + ∣ u ∣ r + α f ( x ) r ′ ) d x = M 1 ( ‖ ∇ u ‖ p , Ω p + ‖ u ‖ r , Ω r + ‖ α f ‖ r ′ , Ω r ′ ) ≤ M 2 ( ‖ u ‖ V p + ‖ u ‖ V r + ‖ α f ‖ r ′ , Ω r ′ ) ,

where we have used the fact that the embeddings of V into W 1 , p ( Ω ) and of V into L r ( Ω ) are continuous. Hence, η ∈ L r ′ ( Ω ) . This permits us to consider the Nemytskij operator N f : V ⊂ L r ( Ω ) → 2 L r ′ ( Ω ) associated with the multivalued mapping f defined by

N f ( u ) ≔ { η ∈ L r ′ ( Ω ) : η ( x ) ∈ f ( x , u ( x ) , ∇ u ( x ) ) for a.a. x ∈ Ω }

for all u ∈ V . Similarly, because of hypotheses H ( U ) (i), (ii) and (iii), for each u ∈ L δ ( Γ 3 ) fixed, we are able to find a measurable function ξ : Γ 3 → R satisfying ξ ( x ) ∈ U ( x , u ( x ) ) for a.a. x ∈ Γ 3 and

(3.3) ‖ ξ ‖ δ ′ , Γ 3 δ ′ = ∫ Γ 3 ∣ ξ ( x ) ∣ δ ′ d Γ ≤ ∫ Γ 3 ( α U ( x ) + a U ∣ u ∣ δ − 1 ) δ ′ d Γ ≤ M 3 ∫ Γ 3 ( α U ( x ) δ ′ + ∣ u ∣ δ ) d Γ = M 3 ( ‖ α U ‖ δ ′ , Γ 3 δ ′ + ‖ u ‖ δ , Γ 3 δ )

for some M 3 > 0 . Therefore, in what follows, we denote by N U : L δ ( Γ 3 ) → 2 L δ ′ ( Γ 3 ) the Nemytskij operator corresponding to the multivalued mapping U defined by

N U ( u ) ≔ { η ∈ L δ ′ ( Γ 3 ) : η ( x ) ∈ U ( x , u ( x ) ) for a.a. x ∈ Γ 3 }

for all u ∈ L δ ( Γ 3 ) .

Let ι : V → L r ( Ω ) , ω : V → L ς ( Ω ) and β : V → L p ( Ω ) be the embedding operators of V to L r ( Ω ) , V to L ς ( Ω ) and V to L p ( Ω ) , respectively, with its adjoint operators ι ∗ : L r ′ ( Ω ) → V ∗ , ω ∗ : L ς ′ ( Ω ) → V ∗ and β ∗ : L p ′ ( Ω ) → V ∗ , respectively. Also, we denote by γ : V → L δ ( Γ 3 ) the trace operator of V into L δ ( Γ 3 ) with its adjoint operator γ ∗ : L δ ′ ( Γ 3 ) → V ∗ . Consider the indicator function of the set K formulated as

I K ( u ) ≔ 0 if u ∈ K , + ∞ if u ∉ K .

Under the aforementioned definitions, we could use a standard procedure for variation calculus to obtain that u ∈ K is a weak solution of problem (1.1) if and only if it solves the following nonlinear inclusion problem:

F u + ω ∗ G u − β ∗ L u − ι ∗ N f ( u ) − γ ∗ N U ( u ) + ∂ c I K ( u ) ∋ h in V ∗ ,

where ∂ c I K is the convex subdifferential operator of I K .

Observe that the functions F , G and L are bounded. The latter combined with (3.2), (3.3) and hypotheses H ( f ) and H ( U ) implies that for each u ∈ V the set

H ( u ) ≔ F u + ω ∗ G u − β ∗ L u − ι ∗ N f ( u ) − γ ∗ N U ( u )

is nonempty, bounded, closed and convex. We show that H is a pseudomonotone operator. Let { u n } n ∈ N ⊂ V , { ζ n } n ∈ N ⊂ V ∗ be sequences and let ( u , ζ ) ∈ V × V ∗ be such that

(3.4) ζ n ∈ H ( u n ) for each n ∈ N , ζ n ⟶ w ζ and limsup n → ∞ ⟨ ζ n , u n − u ⟩ ≤ 0 .

Then, for every n ∈ N , there are η n ∈ N f ( u n ) and ξ n ∈ N U ( u n ) such that

ζ n = F u n + ω ∗ G u n − β ∗ L u n − ι ∗ η n − γ ∗ ξ n for all n ∈ N .

Taking (3.2) and (3.3) into account, we can see that the sequences { η n } n ∈ N ⊂ L r ′ ( Ω ) and { ξ n } n ∈ N ⊂ L δ ′ ( Γ 3 ) are both bounded. Without any loss of generality, we may assume that there exist functions ( η , ξ ) ∈ L r ′ ( Ω ) × L δ ′ ( Γ 3 ) such that

η n ⟶ w η in L r ′ ( Ω ) and ξ n ⟶ w ξ in L δ ′ ( Γ 3 ) .

Recall that V is embedded compactly into L ς ( Ω ) , L r ( Ω ) and L p ( Ω ) , respectively, and γ : V → L δ ( Γ 3 ) is compact. Using this we have

(3.5) lim n → ∞ ⟨ ω ∗ G u n , u n − u ⟩ = lim n → ∞ ⟨ G u n , ω ( u n − u ) ⟩ L ς ′ ( Ω ) × L ς ( Ω ) = 0 , lim n → ∞ ⟨ β ∗ L u n , u n − u ⟩ = lim n → ∞ ⟨ L u n , β ( u n − u ) ⟩ L p ′ ( Ω ) × L p ( Ω ) = 0 , lim n → ∞ ⟨ ι ∗ η n , u n − u ⟩ = lim n → ∞ ⟨ η n , ι ( u n − u ) ⟩ L r ′ ( Ω ) × L r ( Ω ) = 0 , lim n → ∞ ⟨ γ ∗ ξ n , u n − u ⟩ = lim n → ∞ ⟨ ξ n , γ ( u n − u ) ⟩ L δ ′ ( Γ 3 ) × L δ ( Γ 3 ) = 0 .

Inserting (3.5) into the inequality in (3.4) yields

0 ≥ limsup n → ∞ ⟨ ζ n , u n − u ⟩ ≥ limsup n → ∞ ⟨ F u n , u n − u ⟩ + liminf n → ∞ ⟨ ω ∗ G u n , u n − u ⟩ − limsup n → ∞ ⟨ β ∗ L u n , u − u n ⟩ + liminf n → ∞ ⟨ ι ∗ η n , u − u n ⟩ + liminf n → ∞ ⟨ γ ∗ ξ n , u − u n ⟩ ≥ limsup n → ∞ ⟨ F u n , u n − u ⟩ .

From Proposition 2.3 we know that F is of type ( S + ) . Therefore,

u n → u in V .

Passing to a subsequence if necessary, we may assume that

(3.6) u n ( x ) → u ( x ) and ∇ u n ( x ) → ∇ u ( x ) for a.a. x ∈ Ω .

Applying Mazur’s theorem there exists a sequence { χ n } n ∈ N of convex combinations to { η n } n ∈ N satisfying

χ n → η in L r ′ ( Ω ) .

Therefore, we can suppose that χ n ( x ) → η ( x ) for a.a. x ∈ Ω . Due to the convexity of f we see that

χ n ( x ) ∈ f ( x , u n ( x ) , ∇ u n ( x ) ) for a.a. x ∈ Ω .

Recall that f is u.s.c. and has nonempty, bounded, closed and convex values (see hypotheses H( f )(i) and (ii)). So, we can use Proposition 4.1.9 of Denkowski et al. [15] to infer that the graph of ( s , ξ ) ↦ f ( x , s , ξ ) is closed for a.a. x ∈ Ω . Taking the convergence properties in (3.6) and χ n ( x ) → η ( x ) for a.a. x ∈ Ω into account, we obtain

η ( x ) ∈ f ( x , u ( x ) , ∇ u ( x ) ) for a.a. x ∈ Ω .

This shows that η ∈ N f ( u ) . Applying the same arguments as we did before, we conclude that ξ ∈ N U ( u ) . Recall that F , G and L are continuous. So we can use the convergence (3.4) in order to obtain

ζ n = F u n + ω ∗ G u n − β ∗ L u n − ι ∗ η n − γ ∗ ξ n ⟶ w F u + ω ∗ G u − β ∗ L u − ι ∗ η − γ ∗ ξ = ζ in V ∗ .

This implies that ζ ∈ H ( u ) . Hence, we have

lim n → ∞ ⟨ ζ n , u n ⟩ = lim n → ∞ ⟨ F u n + ω ∗ G u n − β ∗ L u n − ι ∗ η n − γ ∗ ξ n , u n ⟩ = lim n → ∞ ⟨ F u n + ω ∗ G u n − β ∗ L u n , u n ⟩ − lim n → ∞ ⟨ η n , ι u n ⟩ L r ′ ( Ω ) × L r ( Ω ) − lim n → ∞ ⟨ ξ n , γ u n ⟩ L δ ′ ( Γ 3 ) × L δ ( Γ 3 ) = ⟨ F u + ω ∗ G u − β ∗ L u − ι ∗ η − γ ∗ ξ , u ⟩ = ⟨ ζ , u ⟩ .

This shows that H is a generalized pseudomonotone operator. Employing Proposition 2.5, we conclude that H is pseudomonotone.

Next, we show the coercivity of H . To this end, we introduce a subspace W of W 1 , p ( Ω ) defined by

(3.7) W ≔ { u ∈ W 1 , p ( Ω ) : u = 0 on Γ 1 } .

Because Γ 1 has positive measure, it is not difficult to prove that W endowed with the norm

‖ u ‖ W ≔ ‖ ∇ u ‖ p , Ω for all u ∈ W

is a reflexive and separable Banach space. Moreover, since the embedding of V into W is continuous, there exists a constant C V W > 0 such that

‖ u ‖ W ≤ C V W ‖ u ‖ V for all u ∈ V .

Let u ∈ V and ζ ∈ H ( u ) be arbitrary. Then, we can find functions η ∈ N f ( u ) and ξ ∈ N U ( u ) such that ζ = F u + ω ∗ G u − β ∗ L u − ι ∗ η − γ ∗ ξ and

(3.8) ⟨ ζ , u ⟩ = ⟨ F u , u ⟩ + ⟨ ω ∗ G u − β ∗ L u , u ⟩ − ⟨ η , u ⟩ L r ′ ( Ω ) × L r ( Ω ) − ⟨ ξ , u ⟩ L δ ′ ( Γ 3 ) × L δ ( Γ 3 ) ≥ c Λ ‖ ∇ u ‖ p , Ω p + ‖ ∇ u ‖ q , μ q + ‖ u ‖ q , μ q − ∫ Ω c f ∣ ∇ u ∣ p + d f ∣ u ∣ p + β f ( x ) d x − ∫ Γ 3 b U ∣ u ∣ p + β U ( x ) d Γ + ∫ Ω a g ∣ u ∣ ς − b g ( x ) d x ≥ ( c Λ − c f ) ‖ ∇ u ‖ p , Ω p + ‖ ∇ u ‖ q , μ q + a g ‖ u ‖ ς , Ω ς + ‖ u ‖ q , μ q − ‖ b g ‖ 1 , Ω − d f ‖ u ‖ p , Ω p − ‖ β f ‖ 1 , Ω − b U ‖ u ‖ p , Γ 3 p − ‖ β U ‖ 1 , Γ 3 .

We set

ε = a g 2 ( ( λ 1 , p S ) − 1 b U + d f + 1 ) .

Keeping in mind that ς > p , it follows from Young’s inequality and the eigenvalue problem of the p -Laplacian with Steklov boundary condition (see (2.2) and (2.3)) that the following inequalities hold

(3.9) b U ‖ u ‖ p , Γ 3 p ≤ b U ( λ 1 , p S ) − 1 ( ‖ ∇ u ‖ p , Ω p + ‖ u ‖ p , Ω p )

and

(3.10) ‖ u ‖ p , Ω p = ∫ Ω ∣ u ∣ p d x ≤ ε ∫ Ω ∣ u ∣ ς d x + c ( ε ) = ε ‖ u ‖ ς , Ω ς + c ( ε )

with some c ( ε ) > 0 . Using (3.9) and (3.10) in (3.8), we obtain

(3.11) ⟨ ζ , u ⟩ ≥ ( c Λ − c f − b U ( λ 1 , p S ) − 1 ) ‖ ∇ u ‖ p , Ω p + ‖ ∇ u ‖ q , μ q + a g 2 ‖ u ‖ ς , Ω ς + ‖ u ‖ p , Ω p + ‖ u ‖ q , μ q − ‖ b g ‖ 1 , Ω − ‖ β f ‖ 1 , Ω − ‖ β U ‖ 1 , Γ 3 − c ( ε ) ≥ M ˆ 0 ( ‖ ∇ u ‖ p , Ω p + ‖ ∇ u ‖ q , μ q + ‖ u ‖ p , Ω p + ‖ u ‖ q , μ q ) + a g 2 ‖ u ‖ ς , Ω ς − ‖ b g ‖ 1 , Ω − ‖ β f ‖ 1 , Ω − ‖ β U ‖ 1 , Γ 3 − c ( ε ) = M ˆ 0 ϱ ℋ ( u ) + a g 2 ‖ u ‖ ς , Ω ς − ‖ b g ‖ 1 , Ω − ‖ β f ‖ 1 , Ω − ‖ β U ‖ 1 , Γ 3 − c ( ε ) ≥ M ˆ 0 min { ‖ u ‖ V p , ‖ u ‖ V q } + a g 2 ‖ u ‖ ς , Ω ς − ‖ b g ‖ 1 , Ω − ‖ β f ‖ 1 , Ω − ‖ β U ‖ 1 , Γ 3 − c ( ε ) ,

where M ˆ 0 > 0 is defined by

M ˆ 0 ≔ min { c Λ − c f − b U ( λ 1 , p S ) − 1 , 1 } .

Since c Λ − c f − b U ( λ 1 , p S ) − 1 > 0 , we deduce that H is coercive.

It is well-known that I K is a proper, convex and l.s.c. function. Note that (see, e.g., Proposition 1.10 of Brézis [6])