Convergence analysis for double phase obstacle problems with multivalued convection term

Proposition 2.1

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

Definition 2.2

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

and the τ-Kuratowski upper limit of the sets A_n

If

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

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.

Definition 3.2

1. We say that u ∈ K is a weak solution of problem (1.1) if there exists η ∈ Lq1q1−1 (Ω) such that η(x) ∈ f(x, u(x), ∇ u(x)) for a. a. x ∈ Ω and

   ∫Ω|∇u|p−2∇u⋅∇(v−u)+μ(x)|∇u|q−2∇u⋅∇(v−u)dx=∫Ωη(x)(v−u)dx

   for all v ∈ K, 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 ∈ Lq1q1−1 (Ω) such that η(x) ∈ f(x, u(x), ∇ u(x)) for a. a. x ∈ Ω and

   ∫Ω|∇u|p−2∇u+μ(x)|∇u|q−2∇u⋅∇vdx+1ρn∫Ωu(x)−Φ(x)+v(x)dx=∫Ωη(x)v(x)dx

   for all v ∈ W01,H (Ω).

Lemma 3.3

If hypothesis H(Φ) holds, then the function B: L^q₁(Ω) → Lq1′ (Ω) given by

is bounded, demicontinuous and monotone, where 〈⋅, ⋅, 〉_q1 denotes the duality pairing between L^q₁(Ω) and its dual space Lq1′ (Ω).

Theorem 3.4

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

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

2. it holds

   ∅≠w-lim supn→∞Sn=s-lim supn→∞Sn⊂S;

3. for each u ∈ s- lim supn→∞ 𝓢_n and any sequence {u͠_n} with

   u~n∈T(Sn,u) for each n∈N,

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

   T(Sn,u):={u~∈Sn∣∥u−u~∥1,H,0≤∥u−v∥1,H,0 for all v∈Sn}.

Proof

1. Let i: W01,H (Ω) → L^q₁(Ω) be the embedding operator from W01,H (Ω) to L^q₁(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 N͠_f: W01,H (Ω) ⊂ L^q₁(Ω) → 2Lq1′(Ω) associated to the multivalued mapping f given by

   N~f(u):=η∈Lq1′(Ω)|η(x)∈f(x,u(x),∇u(x))  for a.a. x∈Ω

   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 N͠_f has closed and convex values as well. Moreover, by hypothesis H(f)(iv) we have

   ∥η∥q1′q1′=∫Ω|η(x)|q1′dx≤∫Ωa1|∇u(x)|pq1′+a2|u(x)|q1−1+α(x)q1′dx≤M0∫Ω|∇u(x)|p+|u(x)|q1+α(x)q1′dx=M0∥∇u∥pp+∥u∥q1q1+∥α∥q1′q1′. (3.3)

   Notice that the embeddings W01,H (Ω) ⊂ W01,p (Ω) ⊂ L^q₁(Ω) are both continuous, so, N͠_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 η ∈ N͠_f(u) such that

   A(u)+1ρni∗B(u)−i∗N~f(u)∋0,

   where A: W01,H (Ω) → W01,H (Ω)^* and B: L^q₁(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 𝓢_n of solutions to problem (1.2) is nonempty, bounded and closed.

2. First, we prove that the set w- lim supn→∞ 𝓢_n is nonempty. Indeed, we have the following claims.

   1. The set ⋃n∈N 𝓢_n is uniformly bounded in W01,H (Ω).

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

      ∥un∥1,H,0→∞ as n→∞.

      Hence, for each n ∈ ℕ, we are able to find η_n ∈ N͠_f(u_n) such that

      ∫Ω|∇un|p−2∇un+μ(x)|∇un|q−2∇un⋅∇vdx+1ρn∫Ωun(x)−Φ(x)+v(x)dx=∫Ωηn(x)v(x)dx

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

      ∫Ω|∇un|p−2∇un+μ(x)|∇un|q−2∇un⋅∇undx−∫Ωηn(x)un(x)dx=−1ρn∫Ωun(x)−Φ(x)+un(x)dx.

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

      ∫Ω|∇un|p−2∇un+μ(x)|∇un|q−2∇un⋅∇undx−∫Ωηn(x)un(x)dx=−1ρn∫Ωun(x)−Φ(x)+−0−Φ(x)+un(x)dx≤0,

      thus

      ∥∇un∥pp+∥∇un∥q,μq−∫Ωηn(x)un(x)dx≤0. (3.4)

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

      ∫Ωηn(x)un(x)dx≤b1∥∇un∥pp+b2∥un∥pp+∥w∥1. (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

      ∥u∥pp≤λ1,p−1∥∇u∥ppfor all u∈W01,p(Ω),

      we get

      0≥∥∇un∥pp+∥∇un∥q,μq−∫Ωηn(x)un(x)dx≥∥∇un∥pp+∥∇un∥q,μq−b1∥∇un∥pp−b2∥un∥pp−∥w∥1≥1−b1−b2λ1,p−1∥∇un∥pp+∥∇un∥q,μq−∥w∥1≥1−b1−b2λ1,p−1∥∇un∥pp+∥∇un∥q,μq−∥w∥1≥1−b1−b2λ1,p−1min∥un∥1,H,0p,∥un∥1,H,0q−∥w∥1,

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

      1−b1−b2λ1,p−1minRp,Rq−∥w∥1>0.

      Therefore, we are able to find N₀ > 0 large enough such that ∥u_n∥_1,𝓗,0 > R₀ for all n ≥ N₀ and

      0≥1−b1−b2λ1,p−1min∥un∥1,H,0p,∥un∥1,H,0q−∥w∥1>0

      for all n ≥ N₀. This gives a contradiction, so Claim 1 is proved.

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

      un⇀u as n→∞ (3.6)

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

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

   2. u(x) ≤ Φ(x) for a.a. x ∈ Ω.

      For every n ∈ ℕ, we have

      1ρn∫Ωun(x)−Φ(x)+v(x)dx=〈Aun,−v〉H+∫Ωηn(x)v(x)dx. (3.7)

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

      ∫Ωηn(x)v(x)dx≤M01q1′∥∇un∥pp+∥un∥q1q1+∥α∥q1′q1′1q1′∥v∥q1. (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

      1ρn∫Ωun(x)−Φ(x)+v(x)dx≤∥Aun∥1,H,0∥v∥1,H,0+M01q1′∥∇un∥pp+∥un∥q1q1+∥α∥q1′q1′1q1′∥v∥q1≤M1∥v∥1,H,0

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

      ∫Ωun(x)−Φ(x)+v(x)dx≤ρnM1∥v∥1,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=∫Ωlimn→∞un(x)−Φ(x)+v(x)dx=limn→∞∫Ωun(x)−Φ(x)+v(x)dx≤limn→∞ρnM1∥v∥1,H,0=0

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

   3. u ∈ 𝓢.

      For each n ∈ ℕ, we have

      〈Aun,un−v〉H=1ρn∫Ωun(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 s ↦ s⁺ gives

      〈Aun,un−v〉H≤1ρn∫Ωv(x)−Φ(x)+(v(x)−un(x))dx+∫Ωηn(x)(un(x)−v(x))dx

      for all v ∈ W01,H (Ω). Hence,

      〈Aun,un−v〉H−∫Ωηn(x)(un(x)−v(x))dx≤0 (3.9)

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

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

      〈Aun,un−u〉H−∫Ωηn(x)(un(x)−u(x))dx≤0,

      that is,

      lim supn→∞〈Aun−i∗ηn,un−u〉H≤0.

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

      lim infn→∞〈Aun−i∗ηn,un−v〉H≥〈u∗(v),un−v〉.

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

      u∗(v)=Au−i∗η(v).

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

      〈Au,v−u〉H−∫Ωη(v)(x)(v(x)−u(x))dx≥0. (3.10)

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

      〈Au,v−u〉H−∫Ωη∗(x)(v(x)−u(x))dx≥0 (3.11)

      for all v ∈ K. Arguing by contradiction, suppose that for each η ∈ N͠_f(u), there is v ∈ K such that

      〈Au,v−u〉H−∫Ωη(x)(v(x)−u(x))dx<0.

      For any v ∈ K, let us consider the set R_v ⊂ N͠_f(u) defined by

      Rv:=η∈N~f(u)∣〈Au,v−u〉H−∫Ωη(x)(v(x)−u(x))dx<0

      for all v ∈ K. We now assert that for each v ∈ K, the set R_v is weakly open. Let {η_n} ⊂ Rvc be such that η_n ⇀ η for some η ∈ Lq1′ (Ω) as n → ∞, where Rvc denotes the complement of R_v. Hence,

      〈Au,v−u〉H−∫Ωηn(x)(v(x)−u(x))dx≥0

      for all n ∈ ℕ. Passing to the limit in the inequality above, we obtain that η ∈ Rvc . Therefore, for every v ∈ K, the set R_v is weakly open in Lq1′ (Ω). Besides, we observe that {R_v}_v∈K is an open covering of N͠_f(u). The latter coupled with the facts that L^q₁(Ω) is reflexive and N͠_f(u) is weakly compact and convex in Lq1′ (Ω), ensures that {R_v}_v∈K has a finite sub-covering of N͠_f(u), let us say {R_v1, R_v2, …, R_vn} for some points {v₁, v₂, …, v_n} ⊆ K. Let κ₁, κ₂, …, κ_n be a partition of unity for N͠_f(u), where for each i = 1, 2, …, n, κ_i : N͠_f(u) → [0, 1] is a weakly continuous function such that ∑i=1n κ_i(η) = 1 for all η ∈ N͠_f(u), see, for example, Granas-Dugundji [23, Lemma 7.3].

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

      M(η)=∑i=1nκi(η)vi for all η∈N~f(u).

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

      〈Au−i∗η,M(η)−u〉H=〈Au−i∗η,∑i=1nκi(η)vi−u〉H=∑i=1nκi(η)〈Au−i∗η,vi−u〉H<0 (3.12)

      for all η ∈ N͠_f(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 → 2^N͠_f(u) and Ψ: N͠_f(u) → 2^N͠_f(u) by

      Λ(v):=η∈N~f(u)∣〈Au,v−u〉H−∫Ωη(x)(v(x)−u(x))dx≥0

      for all v ∈ K, 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 N͠_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):=v∈K∣Λ(v)∩D≠∅

      is closed in W01,H (Ω). Let {v_n} ⊂ Λ^–(D) be a sequence such that v_n → v as n → ∞. Then, for each n ∈ ℕ, we are able to find η_n ∈ N͠_f(u) satisfying

      〈Au,vn−u〉H−∫Ωηn(x)(vn(x)−u(x))dx≥0. (3.13)

      From the weak compactness of N͠_f(u), without any loss of generality, we may suppose that η_n ⇀ η in Lq1′ (Ω), as n → ∞, for some η ∈ N͠_f(u). Passing to the upper limit as n → ∞ for (3.13), we have

      〈Au,v−u〉H−∫Ωη(x)(v(x)−u(x))dx≥0,

      that is, η ∈ Λ(v). But, the weak closedness of D implies that η ∈ D. Therefore, η ∈ Λ(v) ∩ D and so v ∈ Λ^–(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 η ∈ N͠_f(u) such that

      〈Au,M(η)−u〉H−∫Ωη(x)(M(η)(x)−u(x))dx≥0.

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

      Consequently, we conclude that ∅ ≠ w- lim supn→∞ 𝓢_n ⊂ 𝓢.

   4. It holds w- lim supn→∞ 𝓢_n = s- lim supn→∞ 𝓢_n.

      Since s- lim supn→∞ 𝓢_n ⊂ w- lim supn→∞ 𝓢_n, it is enough to verify the condition w- lim supn→∞ 𝓢_n ⊂ s- lim supn→∞ 𝓢_n. Let u ∈ w- lim supn→∞ 𝓢_n be arbitrary. Without any loss of generality, there exists a sequence, still denoted by {u_n} with u_n ∈ 𝓢_n such that u_n ⇀ u as n → ∞. We claim that u_n → u as n → ∞. For each n ∈ ℕ, it holds

      〈Aun,un−v〉H=−∫Ω(un(x)−Φ(x))+(un(x)−v(x))dx+∫Ωηn(x)(un(x)−v(x))dx

      for some η_n ∈ N͠_f(u_n) 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 supn→∞〈Aun,un−u〉H≤0.

      The latter combined with the convergence u_n ⇀ u as n → ∞ and the fact that A is of type (S₊) (see Proposition 2.1) implies that u_n → u as n → ∞. This means that u ∈ s- lim supn→∞ 𝓢_n. Therefore s- lim supn→∞ 𝓢_n = w- lim supn→∞ 𝓢_n.

3. Let u ∈ s- lim supn→∞ 𝓢_n be arbitrary. Since 𝓢_n is nonempty, bounded and closed, so, the set 𝓣(𝓢_n, u) is nonempty. Let {u͠_n} be any sequence such that

   u~n∈T(Sn,u) for each n∈N.

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

   u~n⇀u~ as n→∞

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

   〈Au~n,u~n−v〉H=1ρn∫Ωu~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 u͠ is a solution to problem (1.1) as well. Consequently, the desired conclusion is proved.□