Open Access Published by De Gruyter November 26, 2020

Convergence analysis for double phase obstacle problems with multivalued convection term

Shengda Zeng, Yunru Bai, Leszek Gasiński and Patrick Winkert

Abstract

In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by 𝓢 the solution set of the obstacle problem and by 𝓢n the solution sets of approximating problems, we prove the following convergence relation

w - lim sup n S n = s - lim sup n S n S ,

where w-lim supn→∞ 𝓢n and s-lim supn→∞ 𝓢n denote the weak and the strong Kuratowski upper limit of 𝓢n, respectively.

MSC 2010: 35J20; 35J25; 35J60

1 Introduction

Recently, based on a surjectivity result for pseudomonotone operators obtained by Le [25], the authors [44] have studied the nonemptyness, boundedness and closedness of the set of weak solutions to the following double phase problem with a multivalued convection term and obstacle effect

d i v | u | p 2 u + μ ( x ) | u | q 2 u f ( x , u , u ) in  Ω , u ( x ) Φ ( x ) in  Ω , u = 0 on  Ω , (1.1)

where Ω ⊆ ℝN is a bounded domain with Lipschitz boundary ∂Ω, 1 < p < q < N, μ: Ω → [0, ∞) is Lipschitz continuous, f: Ω × ℝ × ℝN → 2 is a multivalued function depending on the gradient of the solution and Φ: Ω → ℝ+ is a given function, see Section 3 for the precise assumptions.

As the obstacle effect leads to various difficulties in obtaining the exact and numerical solutions, it is reasonable to consider some appropriate approximating methods to overcome/avoid the obstacle effect. In the present paper, we are going to propose a family of approximating problems corresponding to (1.1) and deliver an important convergence theorem which indicates that the solution set of the obstacle problem can be approximated by the solutions of perturbation problems. More precisely, let {ρn} be a sequence of positive numbers such that ρn → 0 as n → ∞ and for each n ∈ ℕ, we consider the following problem

d i v | u | p 2 u + μ ( x ) | u | q 2 u + 1 ρ n u ( x ) Φ ( x ) + f ( x , u , u ) in  Ω , u = 0 on  Ω . (1.2)

Denoting by 𝓢 and 𝓢n the sets of solutions to problems (1.1) and (1.2), respectively, we shall establish the relations between the sets 𝓢, w-lim supn→∞ 𝓢n (being the weak Kuratowski upper limit of 𝓢n) and s-lim supn→∞ 𝓢n (being the strong Kuratowski upper limit of 𝓢n), see Definition 2.2.

The introduction of so-called double phase operators goes back to Zhikov [46] who described models of strongly anisotropic materials by studying the functional

u | u | p + μ ( x ) | u | q d x . (1.3)

The integral functional (1.3) is characterized by the fact that the energy density changes its ellipticity and growth properties according to the point in the domain. More precisely, its behavior depends on the values of the weight function μ(⋅). Indeed, on the set {xΩ : μ(x) = 0} it will be controlled by the gradient of order p and in the case {xΩ : μ(x) ≠ 0} it is the gradient of order q. This is the reason why it is called double phase.

Functionals of the expression (1.3) have been studied more intensively in the last five years. Concerning regularity results, we refer, for example, to the works of Baroni-Colombo-Mingione [4, 5, 6], Baroni-Kuusi-Mingione [7], Cupini-Marcellini-Mascolo [15], Colombo-Mingione [13], [14], Marcellini [28, 29] and the references therein.

Double phase differential operators and corresponding energy functionals appear in several physical applications. 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 q and p, see Zhikov [47]. We also refer to other applications which can be found in the works of Bahrouni-Rădulescu-Repovš [1] on transonic flows, Benci-D’Avenia-Fortunato-Pisani [8] on quantum physics and Cherfils-Il’yasov [9] on reaction diffusion systems.

Existence and uniqueness results have been recently obtained by several authors. In the case of single-valued equations with or without convection term, we refer to Colasuonno-Squassina [12], Gasiński-Papageorgiou [16, 17], Gasiński-Winkert [19, 20, 21], Liu-Dai [27], Perera-Squassina [39], Papageorgiou-Vetro-Vetro [34, 35] and the references therein.

Finally, papers or monographs dealing with certain types of double phase problems or multivalued problems can be found in Bahrouni-Rădulescu-Repovš [1], Bahrouni-Rădulescu-Winkert [2], [3], Carl-Le-Motreanu [10], Cencelj-Rădulescu-Repovš [11], Clarke [22], Gasiński-Papageorgiou [18], Marino-Winkert [30], Papageorgiou-Rădulescu-Repovš [32, 33], Papageorgiou-Vetro-Vetro [37], Rădulescu [40], Vetro [41], Vetro-Vetro [42], Zhang-Rădulescu [45], Zeng-Bai-Gasiński-Winkert [43] and the references therein.

The paper is organized as follows. In Section 2 we recall the definition of the Musielak-Orlicz spaces L𝓗(Ω) and its corresponding Sobolev spaces W1,𝓗(Ω) and we recall the definition of the Kuratowski lower and upper limit, respectively. In Section 3 we present the full assumptions on the data of problem (1.2), give the definition of weak solutions for (1.1) as well as (1.2) and state and prove our main result, see Theorem 3.4.

2 Preliminaries

Let Ω be a bounded domain in ℝN and let 1 ≤ r ≤ ∞. In what follows, we denote by Lr(Ω) : = Lr(Ω;ℝ) and Lr(Ω;ℝN) the usual Lebesgue spaces endowed with the norm ∥⋅∥r. Moreover, W1,r(Ω) and W 0 1 , r (Ω) stand for the Sobolev spaces endowed with the norms ∥⋅∥1,r and ∥⋅∥1,r,0, respectively. For any 1 < r < ∞ we denote by r′ the conjugate of r, that is, 1 r + 1 r = 1.

For the weight function μ and powers p, q we will assume that:

H(μ): μ : Ω → ℝ+ := [0, ∞) is Lipschitz continuous and 1 < p < q < N are chosen such that

q p < 1 + 1 N .

We consider the function 𝓗: Ω × ℝ+ → ℝ+ defined by

H ( x , t ) = t p + μ ( x ) t q for all  ( x , t ) Ω × R + .

Based on the definition of 𝓗 we are able to introduce the Musielak-Orlicz space L𝓗(Ω) given by

L H ( Ω ) = u | u : Ω R  is measurable and  ρ H ( u ) := Ω H ( x , | u | ) d x < + ,

endowed with the Luxemburg norm

u H = inf τ > 0 | ρ H u τ 1 .

We know that L𝓗(Ω) is uniformly convex and so a reflexive Banach space. In addition, we introduce the seminormed function space

L μ q ( Ω ) = u | u : Ω R  is measurable and  Ω μ ( x ) | u | q d x < + ,

which is equipped with the seminorm ∥⋅∥q,μ given by

u q , μ = Ω μ ( x ) | u | q d x 1 q .

It is known that the embeddings

L q ( Ω ) L H ( Ω ) L p ( Ω ) L μ q ( Ω )

are continuous, see Colasuonno-Squassina [12, Proposition 2.15 (i), (iv) and (v)]. Taking into account these embeddings we have the inequalities

min u H p , u H q u p p + u q , μ q max u H p , u H q (2.1)

for all uL𝓗(Ω).

By W1,𝓗(Ω) we denote the corresponding Sobolev space which is defined 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 ∥∇u𝓗 = ∥|∇u|∥𝓗.

By W 0 1 , H (Ω) we denote the completion of C 0 (Ω) in W1,𝓗(Ω), that is,

W 0 1 , H ( Ω ) = C 0 ( Ω ) ¯ W 1 , H ( Ω ) .

Besides, from condition H(μ) and Colasuonno-Squassina [12, Proposition 2.18] we can see that

u 1 , H , 0 = u H for all  u W 0 1 , H ( Ω )

is an equivalent norm on W 0 1 , H (Ω). Now we are able to adapt (2.1) in terms of W 0 1 , H (Ω)-norm as follows

min u 1 , H , 0 p , u 1 , H , 0 q u p p + u q , μ q max u 1 , H , 0 p , u 1 , H , 0 q (2.2)

for all u W 0 1 , H (Ω). Since both spaces W1,𝓗(Ω) and W 0 1 , H (Ω) are uniformly convex, we know that they are reflexive Banach spaces.

Furthermore, we have the following compact embedding

W 0 1 , H ( Ω ) L r ( Ω ) (2.3)

for each 1 < r < p*, where p* is the critical exponent to p given by

p := N p N p , (2.4)

see Colasuonno-Squassina [12, Proposition 2.15].

Let us now consider the eigenvalue problem for the negative r-Laplacian with homogeneous Dirichlet boundary condition and 1 < r < ∞ which is defined by

Δ r u = λ | u | r 2 u in  Ω , u = 0 on  Ω . (2.5)

From Lê [26] we know that the set σr being the set of all eigenvalues of (–Δr, W 0 1 , r (Ω)) has a smallest element λ1,r which is positive, isolated, simple and it can be variationally characterized through

λ 1 , r = inf u r r u r r : u W 0 1 , r ( Ω ) , u 0 .

Now, let A: W 0 1 , H (Ω) → W 0 1 , H (Ω)* be the operator defined by

A ( u ) , v H := Ω | u | p 2 u + μ ( x ) | u | q 2 u v d x , (2.6)

for u, v W 0 1 , H (Ω), where 〈 ⋅, ⋅〉𝓗 is the duality pairing between W 0 1 , H (Ω) and its dual space W 0 1 , H (Ω)*.

The properties of the operator A: W 0 1 , H (Ω) → W 0 1 , H (Ω)* can be summarized as follows, see Liu-Dai [27].

Proposition 2.1

The operator A defined by (2.6) is bounded, continuous, monotone (hence maximal monotone) and of type (S+).

Throughout the paper the symbols “⇀” and “→” stand for the weak and the strong convergence, respectively. Let (V, ∥⋅∥V) be a Banach space with its dual V* and denote by 〈⋅, ⋅〉 the duality pairing between V* and V. We end this section by recalling the following definition, see, for example, Papageorgiou-Winkert [38, Definition 6.7.4].

Definition 2.2

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

τ - lim inf 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 An

τ - lim sup n A n := x X x = τ - lim k x n k , x n k A n k , n 1 < n 2 < < n k < .

If

A = τ - lim inf n A n = τ - lim sup n A n ,

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

3 Main results

We assume the following hypotheses on the data of problem (1.2).

  1. H(f)

    The multivalued convection mapping f: Ω × ℝ × ℝN → 2 has nonempty, compact and convex values such that

    1. the multivalued mapping xf(x, s, ξ) has a measurable selection for all (s, ξ) ∈ ℝ × ℝN;

    2. the multivalued mapping (s, ξ) ↦ f(x, s, ξ) is upper semicontinuous for almost all (a. a.) xΩ;

    3. there exists α L q 1 q 1 1 (Ω) and a1, a2 ≥ 0 such that

      | η | a 1 | ξ | p q 1 1 q 1 + a 2 | s | q 1 1 + α ( x )

      for all ηf(x, s, ξ), for a. a. xΩ, all s ∈ ℝ and all ξ ∈ ℝN, where 1 < q1 < p* with the critical exponent p* given in (2.4);

    4. there exist w L + 1 (Ω) and b1, b2 ≥ 0 such that

      b 1 + b 2 λ 1 , p 1 < 1 ,

      and

      η s b 1 | ξ | p + b 2 | s | p + w ( x )

      for all ηf(x, s, ξ), for a. a. xΩ, all s ∈ ℝ and all ξ ∈ ℝN, where λ1,p is the first eigenvalue of the Dirichlet eigenvalue problem for the p-Laplacian, see (2.5).

  2. H(Φ)

    Φ: Ω → [0, ∞) is such that Φ L q 1 (Ω).

  3. H(0)

    {ρn} is a sequence with ρn > 0 for each n ∈ ℕ such that ρn → 0 as n → ∞.

Let K be a subset of W 0 1 , H (Ω) defined by

K := u W 0 1 , H ( Ω ) | u ( x ) Φ ( x )  for a.a. x Ω . (3.1)

Remark 3.1

  1. The set K is a nonempty, closed and convex subset of W 0 1 , H (Ω).

  2. From assumption H(Φ) we see that 0 ∈ K.

The weak solutions for problems (1.1) and (1.2) are understood in the following way.

Definition 3.2

  1. We say that uK is a weak solution of problem (1.1) if there exists η L q 1 q 1 1 (Ω) such that η(x) ∈ f(x, u(x), ∇ u(x)) for a. a. xΩ and

    Ω | u | p 2 u ( v u ) + μ ( x ) | u | q 2 u ( v u ) d x = Ω η ( x ) ( v u ) d x

    for all vK, where K is given by (3.1).

  2. We say that u W 0 1 , H (Ω) is a weak solution of problem (1.2) if there exists η L q 1 q 1 1 (Ω) such that η(x) ∈ f(x, u(x), ∇ u(x)) for a. a. xΩ and

    Ω | u | p 2 u + μ ( x ) | u | q 2 u v d x + 1 ρ n Ω u ( x ) Φ ( x ) + v ( x ) d x = Ω η ( x ) v ( x ) d x

    for all v W 0 1 , H (Ω).

It is straightforward, to prove the following lemma.

Lemma 3.3

If hypothesis H(Φ) holds, then the function B: Lq1(Ω) → L q 1 (Ω) given by

B u , v q 1 = Ω u ( x ) Φ ( x ) + v ( x ) d x f o r a l l u , v L q 1 ( Ω ) , (3.2)

is bounded, demicontinuous and monotone, where 〈⋅, ⋅, 〉q1 denotes the duality pairing between Lq1(Ω) and its dual space L q 1 (Ω).

Now, we can state the main result of this paper.

Theorem 3.4

If hypotheses H(μ), H(f), H(Φ), and H(0) hold, then

  1. for each n ∈ ℕ, the set 𝓢n of solutions to problem (1.2) is nonempty, bounded and closed;

  2. it holds

    w - lim sup n S n = s - lim sup n S n S ;

  3. for each us- lim sup n 𝓢n and any sequence {n} with

    u ~ n T ( S n , u ) f o r e a c h n N ,

    there exists a subsequence of {n} converging strongly to u in W 0 1 , H (Ω), where the set 𝓣(𝓢n, u) is defined by

    T ( S n , u ) := { u ~ S n u u ~ 1 , H , 0 u v 1 , H , 0 f o r a l l v S n } .

Proof

  1. Let i: W 0 1 , H (Ω) → Lq1(Ω) be the embedding operator from W 0 1 , H (Ω) to Lq1(Ω) with its adjoint operator i*: L q 1 (Ω) → W 0 1 , H (Ω)*. Since 1 < q1 < p* the embedding operator i is compact and so i* as well. From hypotheses H(f)(i) and (iii), we see that the Nemytskij operator f: W 0 1 , H (Ω) ⊂ Lq1(Ω) → 2 L q 1 ( Ω ) associated to the multivalued mapping f given by

    N ~ f ( u ) := η L q 1 ( Ω ) | η ( x ) f ( x , u ( x ) , u ( x ) )  for a.a. x Ω

    for all u W 0 1 , H (Ω) is well-defined (see the proof of Proposition 3 in Papageorgiou-Vetro-Vetro [36]). The convexity and closedness of the values of f ensure that f has closed and convex values as well. Moreover, by hypothesis H(f)(iv) we have

    η q 1 q 1 = Ω | η ( x ) | q 1 d x Ω a 1 | u ( x ) | p q 1 + a 2 | u ( x ) | q 1 1 + α ( x ) q 1 d x M 0 Ω | u ( x ) | p + | u ( x ) | q 1 + α ( x ) q 1 d x = M 0 u p p + u q 1 q 1 + α q 1 q 1 . (3.3)

    Notice that the embeddings W 0 1 , H (Ω) ⊂ W 0 1 , p (Ω) ⊂ Lq1(Ω) are both continuous, so, f(u) is bounded in L q 1 (Ω) for each u W 0 1 , H (Ω).

    It is easy to see that u W 0 1 , H (Ω) is a weak solution of problem (1.2) (see Definition 3.2(b)), if and only if u solves the following inclusion:

    Find u W 0 1 , H (Ω) and ηf(u) such that

    A ( u ) + 1 ρ n i B ( u ) i N ~ f ( u ) 0 ,

    where A: W 0 1 , H (Ω) → W 0 1 , H (Ω)* and B: Lq1(Ω) → L q 1 (Ω) are given by (2.6) and (3.2), respectively.

    Then, using the same arguments as in the proof of Zeng-Gasiński-Winkert-Bai [44, Theorem 3.3], we can conclude that for each n ∈ ℕ, the set 𝓢n of solutions to problem (1.2) is nonempty, bounded and closed.

  2. First, we prove that the set w- lim sup n 𝓢n is nonempty. Indeed, we have the following claims.

    1. Claim 1

      The set n N 𝓢n is uniformly bounded in W 0 1 , H (Ω).

      Arguing by contradiction, suppose that n N 𝓢n is unbounded. Without any loss of generality (passing to a subsequence if necessary), we may assume that there exists a sequence {un} ⊂ W 0 1 , H (Ω) with un ∈ 𝓢n for each n ∈ ℕ such that

      u n 1 , H , 0  as  n .

      Hence, for each n ∈ ℕ, we are able to find ηnf(un) such that

      Ω | u n | p 2 u n + μ ( x ) | u n | q 2 u n v d x + 1 ρ n Ω u n ( x ) Φ ( x ) + v ( x ) d x = Ω η n ( x ) v ( x ) d x

      for all v W 0 1 , H (Ω). Inserting v = un into the inequality above, we get

      Ω | u n | p 2 u n + μ ( x ) | u n | q 2 u n u n d x Ω η n ( x ) u n ( x ) d x = 1 ρ n Ω u n ( x ) Φ ( x ) + u n ( x ) d x .

      By the nonnegativity of Φ and the monotonicity of the function ss+, we have

      Ω | u n | p 2 u n + μ ( x ) | u n | q 2 u n u n d x Ω η n ( x ) u n ( x ) d x = 1 ρ n Ω u n ( x ) Φ ( x ) + 0 Φ ( x ) + u n ( x ) d x 0 ,

      thus

      u n p p + u n q , μ q Ω η n ( x ) u n ( x ) d x 0. (3.4)

      However, by hypothesis H(f)(iv), we have

      Ω η n ( x ) u n ( x ) d x b 1 u n p p + b 2 u n p p + w 1 . (3.5)

      Applying (3.5) in (3.4), using the continuity of the embedding W 0 1 , H (Ω) ⊆ W 0 1 , p (Ω) as well as the estimate

      u p p λ 1 , p 1 u p p for all  u W 0 1 , p ( Ω ) ,

      we get

      0 u n p p + u n q , μ q Ω η n ( x ) u n ( x ) d x u n p p + u n q , μ q b 1 u n p p b 2 u n p p w 1 1 b 1 b 2 λ 1 , p 1 u n p p + u n q , μ q w 1 1 b 1 b 2 λ 1 , p 1 u n p p + u n q , μ q w 1 1 b 1 b 2 λ 1 , p 1 min u n 1 , H , 0 p , u n 1 , H , 0 q w 1 ,

      where the last inequality is obtained by (2.2). Since 1 < p < q < N and b1 + b 2 λ 1 , p 1 < 1, we can take R0 > 0 large enough such that for all RR0 it holds

      1 b 1 b 2 λ 1 , p 1 min R p , R q w 1 > 0.

      Therefore, we are able to find N0 > 0 large enough such that ∥un1,𝓗,0 > R0 for all nN0 and

      0 1 b 1 b 2 λ 1 , p 1 min u n 1 , H , 0 p , u n 1 , H , 0 q w 1 > 0

      for all nN0. This gives a contradiction, so Claim 1 is proved.

      Let {un} ⊂ W 0 1 , H (Ω) with un ∈ 𝓢n for each n ∈ ℕ be an arbitrary sequence. Claim 1 indicates that {un} is bounded in W 0 1 , H (Ω). Then, we may assume that along a relabeled subsequence we have

      u n u  as n (3.6)

      for some u W 0 1 , H (Ω). This guarantees that the set w- lim sup n 𝓢n is nonempty.

      Next, we are going to demonstrate that w- lim sup n 𝓢n is a subset of 𝓢. Let uw- lim sup n 𝓢n be arbitrary. Without loss of generality, we may suppose that there exists a subsequence {un} ⊂ W 0 1 , H (Ω) with un ∈ 𝓢n for all n ∈ ℕ, satisfying (3.6). Our goal is to prove that u ∈ 𝓢.

    2. Claim 2

      u(x) ≤ Φ(x) for a.a. xΩ.

      For every n ∈ ℕ, we have

      1 ρ n Ω u n ( x ) Φ ( x ) + v ( x ) d x = A u n , v H + Ω η n ( x ) v ( x ) d x . (3.7)

      It follows from Hölder’s inequality and (3.3) that

      Ω η n ( x ) v ( x ) d x M 0 1 q 1 u n p p + u n q 1 q 1 + α q 1 q 1 1 q 1 v q 1 . (3.8)

      Putting (3.8) into (3.7), employing the boundedness of A (see Proposition 2.1), the convergence (3.6), and the embedding (2.3), we have

      1 ρ n Ω u n ( x ) Φ ( x ) + v ( x ) d x A u n 1 , H , 0 v 1 , H , 0 + M 0 1 q 1 u n p p + u n q 1 q 1 + α q 1 q 1 1 q 1 v q 1 M 1 v 1 , H , 0

      for some M1 > 0, where M1 > 0 is independent of n, that is

      Ω u n ( x ) Φ ( x ) + v ( x ) d x ρ n M 1 v 1 , H , 0

      for all v W 0 1 , H (Ω). Passing to the limit in the inequality above, using convergence (3.6), the compact embedding (2.3), and the Lebesgue Dominated Convergence Theorem, we conclude that

      Ω u ( x ) Φ ( x ) + v ( x ) d x = Ω lim n u n ( x ) Φ ( x ) + v ( x ) d x = lim n Ω u n ( x ) Φ ( x ) + v ( x ) d x lim n ρ n M 1 v 1 , H , 0 = 0

      for all v W 0 1 , H (Ω). Therefore, we have (u(x) – Φ(x))+ = 0 for a.a. xΩ, thus, u(x) ≤ Φ(x) for a.a. xΩ.

    3. Claim 3

      u ∈ 𝓢.

      For each n ∈ ℕ, we have

      A u n , u n v H = 1 ρ n Ω u n ( x ) Φ ( x ) + ( v ( x ) u n ( x ) ) d x + Ω η n ( x ) ( u n ( x ) v ( x ) ) d x

      for all v W 0 1 , H (Ω). The latter combined with the monotonicity of ss+ gives

      A u n , u n v H 1 ρ n Ω v ( x ) Φ ( x ) + ( v ( x ) u n ( x ) ) d x + Ω η n ( x ) ( u n ( x ) v ( x ) ) d x

      for all v W 0 1 , H (Ω). Hence,

      A u n , u n v H Ω η n ( x ) ( u n ( x ) v ( x ) ) d x 0 (3.9)

      for all vK, where K is defined in (3.1).

      Claim 2 indicates that uK, so, we put v = u in (3.9) to obtain

      A u n , u n u H Ω η n ( x ) ( u n ( x ) u ( x ) ) d x 0 ,

      that is,

      lim sup n A u n i η n , u n u H 0.

      It follows from the proof of Theorem 3.3 in Zeng-Gasiński-Winkert-Bai [44] that the multivalued mapping 𝓐 = Ai*f is pseudomonotone. So, for each vK, there exists u* ∈ 𝓐u such that

      lim inf n A u n i η n , u n v H u ( v ) , u n v .

      This means that for each vK, there is an element η(v) ∈ f(u) satisfying

      u ( v ) = A u i η ( v ) .

      For each vK, passing to the lower limit as n → ∞ in inequality (3.9), we are able to find an element η(v) ∈ f(u) such that

      A u , v u H Ω η ( v ) ( x ) ( v ( x ) u ( x ) ) d x 0. (3.10)

      We shall prove that uK is a weak solution to problem (1.1), namely, there exists an element η*f(u), which is independent of v, such that

      A u , v u H Ω η ( x ) ( v ( x ) u ( x ) ) d x 0 (3.11)

      for all vK. Arguing by contradiction, suppose that for each ηf(u), there is vK such that

      A u , v u H Ω η ( x ) ( v ( x ) u ( x ) ) d x < 0.

      For any vK, let us consider the set Rvf(u) defined by

      R v := η N ~ f ( u ) A u , v u H Ω η ( x ) ( v ( x ) u ( x ) ) d x < 0

      for all vK. We now assert that for each vK, the set Rv is weakly open. Let {ηn} ⊂ R v c be such that ηnη for some η L q 1 (Ω) as n → ∞, where R v c denotes the complement of Rv. Hence,

      A u , v u H Ω η n ( x ) ( v ( x ) u ( x ) ) d x 0

      for all n ∈ ℕ. Passing to the limit in the inequality above, we obtain that η R v c . Therefore, for every vK, the set Rv is weakly open in L q 1 (Ω). Besides, we observe that {Rv}vK is an open covering of f(u). The latter coupled with the facts that Lq1(Ω) is reflexive and f(u) is weakly compact and convex in L q 1 (Ω), ensures that {Rv}vK has a finite sub-covering of f(u), let us say {Rv1, Rv2, …, Rvn} for some points {v1, v2, …, vn} ⊆ K. Let κ1, κ2, …, κn be a partition of unity for f(u), where for each i = 1, 2, …, n, κi : f(u) → [0, 1] is a weakly continuous function such that i = 1 n κi(η) = 1 for all ηf(u), see, for example, Granas-Dugundji [23, Lemma 7.3].

      Also, we introduce a function 𝓜: f(u) → W 0 1 , H (Ω) defined by

      M ( η ) = i = 1 n κ i ( η ) v i  for all  η N ~ f ( u ) .

      Obviously, the function 𝓜 is also weakly continuous due to the weak continuity of κi for i = 1, 2, …, n. For any ηf(u), we have

      A u i η , M ( η ) u H = A u i η , i = 1 n κ i ( η ) v i u H = i = 1 n κ i ( η ) A u i η , v i u H < 0 (3.12)

      for all ηf(u), where the last inequality is obtained by the use of Lemma 7.3(ii) of Granas-Dugundji [23].

      Let us define two multivalued functions Λ: K → 2f(u) and Ψ: f(u) → 2f(u) by

      Λ ( v ) := η N ~ f ( u ) A u , v u H Ω η ( x ) ( v ( x ) u ( x ) ) d x 0

      for all vK, and

      Ψ ( η ) := Λ ( M ( η ) )  for all  η N ~ f ( u ) .

      Then, Ψ has nonempty, weakly compact and convex values (by (3.10) and because f(u) is bounded closed and convex in L q 1 (Ω)) and Λ is upper semicontinuous from the normal topology of K to weak topology of L q 1 (Ω). From Migórski-Ochal-Sofonea [31, Proposition 3.8], it is enough to verify that for each weakly closed set D in L q 1 (Ω), the set

      Λ ( D ) := v K Λ ( v ) D

      is closed in W 0 1 , H (Ω). Let {vn} ⊂ Λ(D) be a sequence such that vnv as n → ∞. Then, for each n ∈ ℕ, we are able to find ηnf(u) satisfying

      A u , v n u H Ω η n ( x ) ( v n ( x ) u ( x ) ) d x 0. (3.13)

      From the weak compactness of f(u), without any loss of generality, we may suppose that ηnη in L q 1 (Ω), as n → ∞, for some ηf(u). Passing to the upper limit as n → ∞ for (3.13), we have

      A u , v u H Ω η ( x ) ( v ( x ) u ( x ) ) d x 0 ,

      that is, ηΛ(v). But, the weak closedness of D implies that ηD. Therefore, ηΛ(v) ∩ D and so vΛ(D). Applying Migórski-Ochal-Sofonea [31, Proposition 3.8] derives that Λ is strongly-weakly upper semicontinuous. On the other hand, the continuity of 𝓜 and Theorem 1.2.8 of Kamenskii-Obukhovskii-Zecca [24] imply that Ψ is also strongly-weakly upper semicontinuous.

      We are now in a position to employ Tychonov fixed point principle, (see, for example, Granas-Dugundji [23, Theorem 8.6]) for function Ψ, to conclude that there exists ηf(u) such that

      A u , M ( η ) u H Ω η ( x ) ( M ( η ) ( x ) u ( x ) ) d x 0.

      This leads to a contraction with (3.12). Consequently, we infer that uK solves problem (1.1) as well, that means, there exists ηf(u), which is independent of v, such that (3.11) holds.

      Consequently, we conclude that ∅ ≠ w- lim sup n 𝓢n ⊂ 𝓢.

    4. Claim 4

      It holds w- lim sup n 𝓢n = s- lim sup n 𝓢n.

      Since s- lim sup n 𝓢nw- lim sup n 𝓢n, it is enough to verify the condition w- lim sup n 𝓢ns- lim sup n 𝓢n. Let uw- lim sup n 𝓢n be arbitrary. Without any loss of generality, there exists a sequence, still denoted by {un} with un ∈ 𝓢n such that unu as n → ∞. We claim that unu as n → ∞. For each n ∈ ℕ, it holds

      A u n , u n v H = Ω ( u n ( x ) Φ ( x ) ) + ( u n ( x ) v ( x ) ) d x + Ω η n ( x ) ( u n ( x ) v ( x ) ) d x

      for some ηnf(un) and for all v W 0 1 , H (Ω). Inserting v = u into the above inequality and passing to the upper limit as n → ∞ for the resulting inequality, we can use the compact embedding (2.3) to get

      lim sup n A u n , u n u H 0.

      The latter combined with the convergence unu as n → ∞ and the fact that A is of type (S+) (see Proposition 2.1) implies that unu as n → ∞. This means that us- lim sup n 𝓢n. Therefore s- lim sup n 𝓢n = w- lim sup n 𝓢n.

  3. Let us- lim sup n 𝓢n be arbitrary. Since 𝓢n is nonempty, bounded and closed, so, the set 𝓣(𝓢n, u) is nonempty. Let {n} be any sequence such that

    u ~ n T ( S n , u )  for each n N .

    It follows from Claim 1 that the sequence {n} is bounded. So, passing to a subsequence, we may assume, that

    u ~ n u ~  as n

    for some W 0 1 , H (Ω). Thus, using the same argument as the proof of Claim 2, we get that K. Then, for each n ∈ ℕ, we have

    A u ~ n , u ~ n v H = 1 ρ n Ω u ~ n ( x ) Φ ( x ) + ( v ( x ) u ~ n ( x ) ) d x + Ω η n ( x ) ( u ~ n ( x ) v ( x ) ) d x

    for all v W 0 1 , H (Ω). Proceeding in the same way as in the proof of Claim 3, we conclude that is a solution to problem (1.1) as well. Consequently, the desired conclusion is proved.□

Acknowledgment

The authors wish to thank the three knowledgeable referees for their useful remarks in order to improve the paper.

Project supported by the NNSF of China Grant No. 12001478, H2020-MSCA-RISE-2018 Research and Innovation Staff Exchange Scheme Fellowship within the Project No. 823731 CONMECH, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2020ZK07, and International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0.

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Received: 2019-06-10
Accepted: 2020-09-27
Published Online: 2020-11-26

© 2021 S. Zeng et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.