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BY 4.0 license Open Access Published by De Gruyter November 26, 2020

Convergence analysis for double phase obstacle problems with multivalued convection term

  • Shengda Zeng EMAIL logo , 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 supnSn=s-lim supnSnS,

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

div|u|p2u+μ(x)|u|q2uf(x,u,u)in Ω,u(x)Φ(x)in Ω,u=0on 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

div|u|p2u+μ(x)|u|q2u+1ρnu(x)Φ(x)+f(x,u,u)in Ω,u=0on 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|qdx. (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 W01,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, 1r+1r=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

qp<1+1N.

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

H(x,t)=tp+μ(x)tqfor all (x,t)Ω×R+.

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

LH(Ω)=u|u:ΩR is measurable and ρH(u):=ΩH(x,|u|)dx<+,

endowed with the Luxemburg norm

uH=infτ>0|ρHuτ1.

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

Lμq(Ω)=u|u:ΩR is measurable and Ωμ(x)|u|qdx<+,

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

uq,μ=Ωμ(x)|u|qdx1q.

It is known that the embeddings

Lq(Ω)LH(Ω)Lp(Ω)Lμq(Ω)

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

minuHp,uHqupp+uq,μqmaxuHp,uHq (2.1)

for all uL𝓗(Ω).

By W1,𝓗(Ω) we denote the corresponding Sobolev space which is defined by

W1,H(Ω)=uLH(Ω):|u|LH(Ω)

equipped with the norm

u1,H=uH+uH,

where ∥∇u𝓗 = ∥|∇u|∥𝓗.

By W01,H (Ω) we denote the completion of C0 (Ω) in W1,𝓗(Ω), that is,

W01,H(Ω)=C0(Ω)¯W1,H(Ω).

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

u1,H,0=uHfor all uW01,H(Ω)

is an equivalent norm on W01,H (Ω). Now we are able to adapt (2.1) in terms of W01,H (Ω)-norm as follows

minu1,H,0p,u1,H,0qupp+uq,μqmaxu1,H,0p,u1,H,0q (2.2)

for all u W01,H (Ω). Since both spaces W1,𝓗(Ω) and W01,H (Ω) are uniformly convex, we know that they are reflexive Banach spaces.

Furthermore, we have the following compact embedding

W01,H(Ω)Lr(Ω) (2.3)

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

p:=NpNp, (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

Δru=λ|u|r2uin Ω,u=0on Ω. (2.5)

From Lê [26] we know that the set σr being the set of all eigenvalues of (–Δr, W01,r (Ω)) has a smallest element λ1,r which is positive, isolated, simple and it can be variationally characterized through

λ1,r=infurrurr:uW01,r(Ω),u0.

Now, let A: W01,H (Ω) → W01,H (Ω)* be the operator defined by

A(u),vH:=Ω|u|p2u+μ(x)|u|q2uvdx, (2.6)

for u, v W01,H (Ω), where 〈 ⋅, ⋅〉𝓗 is the duality pairing between W01,H (Ω) and its dual space W01,H (Ω)*.

The properties of the operator A: W01,H (Ω) → W01,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 infnAn:=xXx=τ-limnxn,xnAnfor all n1,

and the τ-Kuratowski upper limit of the sets An

τ-lim supnAn:=xXx=τ-limkxnk,xnkAnk,n1<n2<<nk<.

If

A=τ-lim infnAn=τ-lim supnAn,

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. 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 Lq1q11 (Ω) and a1, a2 ≥ 0 such that

      |η|a1|ξ|pq11q1+a2|s|q11+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

      b1+b2λ1,p1<1,

      and

      ηsb1|ξ|p+b2|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. Φ: Ω → [0, ∞) is such that Φ Lq1 (Ω).

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

Let K be a subset of W01,H (Ω) defined by

K:=uW01,H(Ω)|u(x)Φ(x) for a.a.xΩ. (3.1)

Remark 3.1

  1. The set K is a nonempty, closed and convex subset of W01,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 Lq1q11 (Ω) such that η(x) ∈ f(x, u(x), ∇ u(x)) for a. a. xΩ and

    Ω|u|p2u(vu)+μ(x)|u|q2u(vu)dx=Ωη(x)(vu)dx

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

  2. We say that u W01,H (Ω) is a weak solution of problem (1.2) if there exists η Lq1q11 (Ω) such that η(x) ∈ f(x, u(x), ∇ u(x)) for a. a. xΩ and

    Ω|u|p2u+μ(x)|u|q2uvdx+1ρnΩu(x)Φ(x)+v(x)dx=Ωη(x)v(x)dx

    for all v W01,H (Ω).

It is straightforward, to prove the following lemma.

Lemma 3.3

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

Bu,vq1=Ωu(x)Φ(x)+v(x)dxforallu,vLq1(Ω), (3.2)

is bounded, demicontinuous and monotone, where 〈⋅, ⋅, 〉q1 denotes the duality pairing between Lq1(Ω) and its dual space Lq1 (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 supnSn=s-lim supnSnS;
  3. for each us- lim supn 𝓢n and any sequence {n} with

    u~nT(Sn,u)foreachnN,

    there exists a subsequence of {n} converging strongly to u in W01,H (Ω), where the set 𝓣(𝓢n, u) is defined by

    T(Sn,u):={u~Snuu~1,H,0uv1,H,0forallvSn}.

Proof

  1. Let i: W01,H (Ω) → Lq1(Ω) be the embedding operator from W01,H (Ω) to Lq1(Ω) with its adjoint operator i*: Lq1 (Ω) → W01,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: W01,H (Ω) ⊂ Lq1(Ω) → 2Lq1(Ω) associated to the multivalued mapping f given by

    N~f(u):=ηLq1(Ω)|η(x)f(x,u(x),u(x)) for a.a.xΩ

    for all u W01,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

    ηq1q1=Ω|η(x)|q1dxΩa1|u(x)|pq1+a2|u(x)|q11+α(x)q1dxM0Ω|u(x)|p+|u(x)|q1+α(x)q1dx=M0upp+uq1q1+αq1q1. (3.3)

    Notice that the embeddings W01,H (Ω) ⊂ W01,p (Ω) ⊂ Lq1(Ω) are both continuous, so, f(u) is bounded in Lq1 (Ω) for each u W01,H (Ω).

    It is easy to see that u W01,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 W01,H (Ω) and ηf(u) such that

    A(u)+1ρniB(u)iN~f(u)0,

    where A: W01,H (Ω) → W01,H (Ω)* and B: Lq1(Ω) → Lq1 (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 supn 𝓢n is nonempty. Indeed, we have the following claims.

    1. The set nN 𝓢n is uniformly bounded in W01,H (Ω).

      Arguing by contradiction, suppose that nN 𝓢n is unbounded. Without any loss of generality (passing to a subsequence if necessary), we may assume that there exists a sequence {un} ⊂ W01,H (Ω) with un ∈ 𝓢n for each n ∈ ℕ such that

      un1,H,0 as n.

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

      Ω|un|p2un+μ(x)|un|q2unvdx+1ρnΩun(x)Φ(x)+v(x)dx=Ωηn(x)v(x)dx

      for all v W01,H (Ω). Inserting v = un into the inequality above, we get

      Ω|un|p2un+μ(x)|un|q2unundxΩηn(x)un(x)dx=1ρnΩun(x)Φ(x)+un(x)dx.

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

      Ω|un|p2un+μ(x)|un|q2unundxΩηn(x)un(x)dx=1ρnΩun(x)Φ(x)+0Φ(x)+un(x)dx0,

      thus

      unpp+unq,μqΩηn(x)un(x)dx0. (3.4)

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

      Ωηn(x)un(x)dxb1unpp+b2unpp+w1. (3.5)

      Applying (3.5) in (3.4), using the continuity of the embedding W01,H (Ω) ⊆ W01,p (Ω) as well as the estimate

      uppλ1,p1uppfor all uW01,p(Ω),

      we get

      0unpp+unq,μqΩηn(x)un(x)dxunpp+unq,μqb1unppb2unppw11b1b2λ1,p1unpp+unq,μqw11b1b2λ1,p1unpp+unq,μqw11b1b2λ1,p1minun1,H,0p,un1,H,0qw1,

      where the last inequality is obtained by (2.2). Since 1 < p < q < N and b1 + b2λ1,p1 < 1, we can take R0 > 0 large enough such that for all RR0 it holds

      1b1b2λ1,p1minRp,Rqw1>0.

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

      01b1b2λ1,p1minun1,H,0p,un1,H,0qw1>0

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

      Let {un} ⊂ W01,H (Ω) with un ∈ 𝓢n for each n ∈ ℕ be an arbitrary sequence. Claim 1 indicates that {un} is bounded in W01,H (Ω). Then, we may assume that along a relabeled subsequence we have

      unu asn (3.6)

      for some u W01,H (Ω). This guarantees that the set w- lim supn 𝓢n is nonempty.

      Next, we are going to demonstrate that w- lim supn 𝓢n is a subset of 𝓢. Let uw- lim supn 𝓢n be arbitrary. Without loss of generality, we may suppose that there exists a subsequence {un} ⊂ W01,H (Ω) with un ∈ 𝓢n for all n ∈ ℕ, satisfying (3.6). Our goal is to prove that u ∈ 𝓢.

    2. u(x) ≤ Φ(x) for a.a. xΩ.

      For every n ∈ ℕ, we have

      1ρnΩun(x)Φ(x)+v(x)dx=Aun,vH+Ωηn(x)v(x)dx. (3.7)

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

      Ωηn(x)v(x)dxM01q1unpp+unq1q1+αq1q11q1vq1. (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

      1ρnΩun(x)Φ(x)+v(x)dxAun1,H,0v1,H,0+M01q1unpp+unq1q1+αq1q11q1vq1M1v1,H,0

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

      Ωun(x)Φ(x)+v(x)dxρnM1v1,H,0

      for all v W01,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

      Ωu(x)Φ(x)+v(x)dx=Ωlimnun(x)Φ(x)+v(x)dx=limnΩun(x)Φ(x)+v(x)dxlimnρnM1v1,H,0=0

      for all v W01,H (Ω). Therefore, we have (u(x) – Φ(x))+ = 0 for a.a. xΩ, thus, u(x) ≤ Φ(x) for a.a. xΩ.

    3. u ∈ 𝓢.

      For each n ∈ ℕ, we have

      Aun,unvH=1ρnΩun(x)Φ(x)+(v(x)un(x))dx+Ωηn(x)(un(x)v(x))dx

      for all v W01,H (Ω). The latter combined with the monotonicity of ss+ gives

      Aun,unvH1ρnΩv(x)Φ(x)+(v(x)un(x))dx+Ωηn(x)(un(x)v(x))dx

      for all v W01,H (Ω). Hence,

      Aun,unvHΩηn(x)(un(x)v(x))dx0 (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

      Aun,unuHΩηn(x)(un(x)u(x))dx0,

      that is,

      lim supnAuniηn,unuH0.

      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 infnAuniηn,unvHu(v),unv.

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

      u(v)=Auiη(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

      Au,vuHΩη(v)(x)(v(x)u(x))dx0. (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

      Au,vuHΩη(x)(v(x)u(x))dx0 (3.11)

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

      Au,vuHΩη(x)(v(x)u(x))dx<0.

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

      Rv:=ηN~f(u)Au,vuHΩη(x)(v(x)u(x))dx<0

      for all vK. We now assert that for each vK, the set Rv is weakly open. Let {ηn} ⊂ Rvc be such that ηnη for some η Lq1 (Ω) as n → ∞, where Rvc denotes the complement of Rv. Hence,

      Au,vuHΩηn(x)(v(x)u(x))dx0

      for all n ∈ ℕ. Passing to the limit in the inequality above, we obtain that η Rvc . Therefore, for every vK, the set Rv is weakly open in Lq1 (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 Lq1 (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=1n κi(η) = 1 for all ηf(u), see, for example, Granas-Dugundji [23, Lemma 7.3].

      Also, we introduce a function 𝓜: f(u) → W01,H (Ω) defined by

      M(η)=i=1nκi(η)vi for all ηN~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

      Auiη,M(η)uH=Auiη,i=1nκi(η)viuH=i=1nκi(η)Auiη,viuH<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):=ηN~f(u)Au,vuHΩη(x)(v(x)u(x))dx0

      for all vK, and

      Ψ(η):=Λ(M(η)) for all ηN~f(u).

      Then, Ψ has nonempty, weakly compact and convex values (by (3.10) and because f(u) is bounded closed and convex in Lq1 (Ω)) and Λ is upper semicontinuous from the normal topology of K to weak topology of Lq1 (Ω). From Migórski-Ochal-Sofonea [31, Proposition 3.8], it is enough to verify that for each weakly closed set D in Lq1 (Ω), the set

      Λ(D):=vKΛ(v)D

      is closed in W01,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

      Au,vnuHΩηn(x)(vn(x)u(x))dx0. (3.13)

      From the weak compactness of f(u), without any loss of generality, we may suppose that ηnη in Lq1 (Ω), as n → ∞, for some ηf(u). Passing to the upper limit as n → ∞ for (3.13), we have

      Au,vuHΩη(x)(v(x)u(x))dx0,

      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

      Au,M(η)uHΩη(x)(M(η)(x)u(x))dx0.

      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 supn 𝓢n ⊂ 𝓢.

    4. It holds w- lim supn 𝓢n = s- lim supn 𝓢n.

      Since s- lim supn 𝓢nw- lim supn 𝓢n, it is enough to verify the condition w- lim supn 𝓢ns- lim supn 𝓢n. Let uw- lim supn 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

      Aun,unvH=Ω(un(x)Φ(x))+(un(x)v(x))dx+Ωηn(x)(un(x)v(x))dx

      for some ηnf(un) and for all v W01,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 supnAun,unuH0.

      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 supn 𝓢n. Therefore s- lim supn 𝓢n = w- lim supn 𝓢n.

  3. Let us- lim supn 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~nT(Sn,u) for eachnN.

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

    u~nu~ asn

    for some W01,H (Ω). Thus, using the same argument as the proof of Claim 2, we get that K. Then, for each n ∈ ℕ, we have

    Au~n,u~nvH=1ρnΩu~n(x)Φ(x)+(v(x)u~n(x))dx+Ωηn(x)(u~n(x)v(x))dx

    for all v W01,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.

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