Existence of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian

Using the non-smooth critical point theory we investigate the existence and multiplicity of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian.

Mathematics Subject Classification 2000: 35D05; 35J20; 35J60; 35J70.

In this article, we study the existence and multiplicity of solutions for the differential inclusion problem with singular coefficients involving the p(x)-Laplacian of the form

where the following conditions are satisfied:

(P) Ω is a bounded open domain in ℝ^N, N ≥ 2, , 1 < p^- := inf_Ω p(x) ≤ p⁺ := sup_Ω p(x) < +∞, λ, μ ∈ ℝ.

(A) For i = 1, 2, for x ∈ Ω, G_i(x, u) is measurable with respect to x (for every u ∈ ℝ) and locally Lipschitz with respect to u (for a.e. x ∈ Ω), ∂G_i : Ω × ℝ → ℝ is the Clarke sub-differential of G_i and for x ∈ Ω, t ∈ ℝ and ξ_i ∈ ∂G_i, where c_i is a positive constant, , r_i(x) > q_i(x) for all x ∈ Ω, and

here

A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form

and in this case the assumption corresponding to (A) is the following

, for i = 1, 2, ∂G_i : Ω × ℝ → ℝ is the Clarke sub-differential of G_i and for x ∈ Ω, t ∈ ℝ and ξ_i ∈ ∂G_i, where c_i is a positive constant, , and

The operator -div(|∇u|^p(x)-2 ∇u) is said to be the p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electro-magnetic field [1,2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal baro-tropic gas through a porous medium [3,4]. Another field of application of equations with variable exponent growth conditions is image processing [5]. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [6-11] for an overview of and references on this subject, and to [12-21] for the study of the p(x)-Laplacian equations and the corresponding variational problems.

Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions for Dirichlet boundary value problems with discontinuous nonlinearities has been widely investigated in recent years. Chang [22] extended the variational methods to a class of non-differentiable functionals, and directly applied the variational methods for non-differentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Later Kourogenis and Papageorgiou [23] obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the p-Laplacian with discontinuous nonlinearities. In the celebrated work [24,25], Ricceri elaborated a Ricceri-type variational principle and a three critical points theorem for the Gâteaux differentiable functional, respectively. Later, Marano and Motreanu [26,27] extended Ricceri's results to a large class of non-differentiable functionals and gave some applications to differential inclusion problems involving the p-Laplacian with discontinuous nonlinearities.

In [21], by means of the critical point theory, Fan obtain the existence and multiplicity of solutions for (1.1) under the condition of satisfying the Carathéodory condition for i = 1, 2, x ∈ Ω. The aim of the present article is to generalize the main results of [21] to the case of the functional of problem (1.1) is nonsmooth.

This article is organized as follows: In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function; In Section 3, we give the variational principle which is needed in the sequel; In Section 4, using the critical point theory, we prove the existence and multiplicity results for problem (1.1).

2.1 Variable exponent Sobolev spaces

Let Ω be a bounded open subset of ℝ^N, denote .

For , denote

On the basic properties of the space W^1,p(x)(Ω) we refer to [7,28-30]. Here we display some facts which will be used later.

Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere. For , define the spaces L^p(x) (Ω) and W^1,p(x) (Ω) by

with the norm

and

with the norm

Denote by the closure of in W^1,p(x) (Ω) . Hereafter, we always assume that p^- > 1.

Proposition 2.1. [7,31]The spaces L^p(x) (Ω) , W^1,p(x) (Ω) and are separable and reflexive Banach spaces.

Proposition 2.2. [7,31]The conjugate space of L^p(x) (Ω) is , where . For any u ∈ L^p(x) (Ω) and v ∈ , .

Proposition 2.3. [7,31]In the Poincaré inequality holds, that is, there exists a positive constant c such that

So is an equivalent norm in .

Proposition 2.4. [7,28,29,31]Assume that the boundary of Ω possesses the cone property and . If and , then there is a compact embedding W^1,p(x)(Ω) → L^q(x) (Ω).

Let us now consider the weighted variable exponent Lebesgue space.

Let a ∈ S(Ω) and a(x) > 0 for x ∈ Ω. Define

with the norm

then is a Banach space. The following proposition follows easily from the definition of .

Proposition 2.5. (see [7,31]) Set ρ(u) = ∫_Ω a(x)|u(x)|^p(x) dx. For , we have

(1)

(2)

(3)

(4)

(5)

(6)

Proposition 2.6. (see [21]) Assume that the boundary of Ω possesses the cone property and . Suppose that a ∈ L^r(^x)(Ω), a(x) > 0 for x ∈ Ω, and r^- > 1. If and

then there is a compact embedding .

The following proposition plays an important role in the present article.

Proposition 2.7. Assume that the boundary of Ω possesses the cone property and . Suppose that a ∈ L^r(^x)(Ω), a(x) > 0 for x ∈ Ω, and r(x) > q(x) for all x ∈ Ω. If and

then there is a compact embedding .

Proof. Set , then and . Moreover, from (2.2) we can get

Using Proposition 2.6, we see that the embedding is compact.

■

2.2 Generalized gradient of the locally Lipschitz function

Let (X, || · ||) be a real Banach space and X* be its topological dual. A function f : X → ℝ is called locally Lipschitz if each point u ∈ X possesses a neighborhood Ω_u such that |f(u₁) - f(u₂)| ≤ L||u₁ - u₂|| for all u₁, u₂ ∈ Ω_u, for a constant L > 0 depending on Ω_u. The generalized directional derivative of f at the point u ∈ X in the direction v ∈ X is

The generalized gradient of f at u ∈ X is defined by

which is a non-empty, convex and w*-compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. We say that u ∈ X is a critical point of f if 0 ∈ ∂f(u). For further details, we refer the reader to Chang [22].

We list some fundamental properties of the generalized directional derivative and gradient that will be used throughout the article.

Proposition 2.8. (see [22,32]) (1) Let j : X → ℝ be a continuously differentiable function. Then ∂j(u) = {j'(u)}, j⁰(u; z) coincides with 〈j' (u), z〉_X and (f + j)⁰(u, z) = f⁰(u; z) + 〈j' (u), z〉_X for all u, z ∈ X.

(2) The set-valued mapping u → ∂f(u) is upper semi-continuous in the sense that for each u₀ ∈ X, ε > 0, v ∈ X, there is a δ > 0, such that for each w ∈ ∂f (u) with ||w - u₀|| < δ, there is w₀ ∈ ∂f (u₀)

(3) (Lebourg's mean value theorem) Let u and v be two points in X. Then there exists a point w in the open segment joining u and v and such that

(4) The function

exists, and is lower semi continuous; i.e., .

In the following we need the nonsmooth version of Palais-Smale condition.

Definition 2.1. We say that φ satisfies the (PS)_c-condition if any sequence {u_n} ⊂ X such that φ(u_n) → c and m(u_n) → 0, as n → +∞, has a strongly convergent subsequence, where m(u_n) = inf{||u*||_X* : u* ∈ ∂φ (u_n)}.

In what follows we write the (PS)_c-condition as simply the PS-condition if it holds for every level c ∈ ℝ for the Palais-Smale condition at level c.

In this section we assume that Ω and p(x) satisfy the assumption (P). For simplicity we write and ||u|| = |∇u|_p(_x) for u ∈ X. Denote by u_n ⇀ u and u_n → u the weak convergence and strong convergence of sequence {u_n} in X, respectively, denote by c and c_i the generic positive constants, B_ρ = {u ∈ X : ||u|| < ρ}, S_ρ = {u ∈ X : ||u|| = ρ}.

Set

where a_i and G_i (i = 1, 2) are as in (A).

Define the integral functional

We write

then it is easy to see that J ∈ C¹(X, ℝ) and φ = J - Ψ.

Below we give several propositions that will be used later.

Proposition 3.1. (see [19]) The functional J : X → ℝ is convex. The mapping J' : X → X* is a strictly monotone, bounded homeomorphism, and is of (S₊) type, namely

Proposition 3.2. Ψ is weakly-strongly continuous, i.e., u_n ⇀ u implies Ψ(u_n) → Ψ(u).

Proof. Define ϒ₁ = ∫_Ω G₁(x, u) dx and ϒ₂ = ∫_Ω G₂(x, u) dx. In order to prove Ψ is weakly-strongly continuous, we only need to prove ϒ₁ and ϒ₂ are weakly-strongly continuous. Since the proofs of ϒ₁ and ϒ₂ are identical, we will just prove ϒ₁.

We assume u_n ⇀ u in X. Then by Proposition 2.8.3, we have

where ξ_n ∈ ∂G₁(,τ_n(x)) for some τ_n(x) in the open segment joining u and u_n. From Chang [22] we know that . So using Proposition 2.5, we have

■

Proposition 3.3. Assume (A) holds and F satisfies the following condition:

(B) for a.e.x ∈ Ω, all u ∈ X and ξ₁ ∈ ∂G₁, ξ₂ ∈ ∂G₂, where θ is a constant, , .

Then φ satisfies the nonsmooth (PS) condition on X.

Proof. Let {u_n} be a nonsmooth (PS) sequence, then by (B) we have

and consequently {u_n} is bounded.

Thus by passing to a subsequence if necessary, we may assume that u_n ⇀ u in X as n → ∞. We have

with ε_n ↓ 0, where ξ_in(x) ∈ ∂G_i(x, u_n) for a.e. x ∈ Ω, i = 1, 2. From Chang [22] or Theorem 1.3.10 of [33], we know that . Since X is embedded compactly in , we have that u_n → u as n → ∞ in . So using Proposition 2.2, we have

Therefore we obtain . But we know that J' is a mapping of type (S₊). Thus we have

Remark 3.1. Note that our condition (1.2) is stronger than (1.2) of [21]. Because Ψ' is weakly-strongly continuous in [21], to verify that φ satisfies (PS) condition on X, it is enough to verify that any (PS) sequence is bounded. However, in this paper we do not know whether ξ(u) is weakly-strongly continuous, where ξ(u) ∈ ⇀Ψ. Therefore, it will be very useful to consider this problem.

Below we denote

We shall use the following conditions.

(B₁) ∃ c₀ > 0 such that G₂(x, t) ≥ - c₀ for x ∈ Ω and t ∈ ℝ.

(B₂) and M > 0 such that 0 < G₂(x, u) ≤ θ 〈u, ξ₂〉 for x∈ Ω, u ∈ X and |u| ≥ M, ξ₂ ∈ ⇀G₂.

Corollary 3.1. Assume (P), (A) and (A₁) hold. Then φ satisfies nonsmooth (PS) condition on X provided either one of the following conditions is satisfied.

(1). λ ∈ ℝ and μ = 0.

(2). λ ∈ ℝ, μ = 0 and (B₁) holds.

(3). λ ∈ ℝ, μ ∈ ℝ and (B₂) holds.

Proof. In case (1) or (2), we have, for x ∈ Ω and t ∈ ℝ,

which shows that the condition (B) with θ = 0 is satisfied.

In case (3), noting that (B₂) and (A) imply (B₁), by the conclusion (1) and (2) we know φ satisfies (PS) condition if μ ≤ 0. Below assume μ > 0. The conditions (B₂) and (A) imply that, for x ∈ Ω and u ∈ X,

so we have

which shows (B) holds. The proof is complete. ■

As X is a separable and reflexive Banach space, there exist (see [[34], Section 17]) such that

For k = 1, 2, . . . , denote

Proposition 3.5. [35]Assume that Ψ : X → ℝ is weakly-strongly continuous and Ψ (0) = 0. Let γ > 0 be given. Set

Then β_k → 0 as k → ∞.

Proposition 3.6. (Nonsmooth Mountain pass theorem, see [23,33]) If X is a reflexive Banach space, φ : X → ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)_c-condition, and for some r > 0 and e₁ ∈ X with ||e₁|| > r, max{φ(0), φ(e₁)} ≤·inf{φ(u) : ||u|| = r}. Then φ has a nontrivial critical u ∈ X such that the critical value c = φ(u) is characterized by the following minimax principle

where Γ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e₁}.

Proposition 3.7. (Nonsmooth Fountain theorem, see [36]) Assume (F₁) X is a Banach space, φ : X → ℝ be an invariant locally Lipschitz functional, the subspaces X_k, Y_k and Z_k are defined by (3.3).

If, for every k ∈ ℕ, there exist ρ_k > r_k > 0 such that

(F₂)

(F₃)

(F₄) φ satisfies the nonsmooth (PS)_c condition for every c > 0, then φ has an unbounded sequence of critical values.

Proposition 3.8. (Nonsmooth dual Fountain theorem, see [37]) Assume (F₁) is satisfied and there is a k₀ > 0 such that, for each k ≥ k₀, there exists ρ_k > γ_k > 0 such that

(D₁)

(D₂)

(D₃)

(D₄) φ satisfies the nonsmooth condition for every , then φ has a sequence of negative critical values converging to 0.

Remark 3.2. We say φ that satisfies the nonsmooth condition at level c ∈ ℝ (with respect to (Y_n)) if any sequence {u_n} ⊂ X such that

contains a subsequence converging to a critical point of φ.

In this section, using the critical point theory, we give the existence and multiplicity results for problem (1.1). We shall use the following assumptions:

such that

such that

(S) For i = 1, 2, G_i(x, -t) = G_i(x, t), ∀x ∈ Ω, ∀t ∈ ℝ.

Remark 4.1.

(1) It follows from (A), (A₂) and (O₂) that

(2)It follows from (A) and (B₂) that (see [33, p. 298])

The following is the main result of this article.

Theorem 4.1. Assume (P), (A), (A₁) hold.

(1) If (B₁) holds, then for every λ ∈ ℝ and μ ≤ 0, problem (1.1) has a solution which is a minimizer of the corresponding functional φ.

(2) If (B₁), (A₂), (O₁), (O₂) hold, then for every λ > 0 and μ ≤ 0, problem (1.1) has a nontrivial solution v₁ such that v₁ is a minimizer of φ and φ(v₁) < 0.

(3) If (A₂), (B₂), (O₂) hold, then for every μ > 0, there exists λ₀(μ) > 0 such that when |λ| ≤ λ₀(μ), problem (1.1) has a nontrivial solution u₁ such that φ(u₁) > 0.

(4) If (A₂), (B₂), (O₁), (O₂) holds, then for every μ > 0, there exists λ₀(μ) > 0 such that when 0 < λ ≤ λ₀(μ), problem (1.1) has two nontrivial solutions u₁ and v₁ such that φ(u₁) > 0 and φ(v₁) < 0.

(5) If (A₂), (B₂), (O₁), (O₂) and (S) holds, then for every μ > 0 and λ ∈ ℝ, problem (1.1) has a sequence of solutions {±u_k} such that φ(±u_k) → ∞ as k → ∞.

(6) If (A₂), (B₂), (O₁), (O₂) and (S) holds, then for every λ > 0 and μ ∈ ℝ, problem (1.1) has a sequence of solutions {±v_k} such that φ(±v_k) < 0 and φ(±v_k) → 0 as k → ∞.

Proof. We will use c, c' and c_i as a generic positive constant. By Corollary 3.1, under the assumptions of Theorem 4.1, φ satisfies nonsmooth (PS) condition. We write

then Ψ = Ψ₁ + Ψ₂, φ(u) = J(u) - Ψ (u) = J(u) - Ψ₁(u) - Ψ₂(u). Firstly, we use to denote its extension to , where i = 1, 2. From (A) and Theorem 1.3.10 of [33] (or Chang [22]), we see that (u) is locally Lipschitz on and for a.e. x ∈ Ω and i = 1, 2. In view of Proposition 2.4 and Theorem 2.2 of [22], we have that is also locally Lipschitz, and ∂Ψ₁(u) ⊆ λ ∫_Ω a₁(x) ∂G₁(x, u) dx, ∂Ψ₂(u) ⊆ μ ∫_Ω a₂(x) ∂G₁(x, u) dx, (see [38]), where stands for the restriction of to X for i = 1, 2. Therefore, φ is a locally Lipschitz functional on X.

(1) Let λ ∈ ℝ and μ ≤ 0. By (A),

By (B₁), Ψ₂(u) ≤ - μc₀ ∫_Ω a₂(x) dx = c₅. Hence . By (A₁), , so φ is coercive, that is, φ(u) → ∞ as ||u|| → ∞. Thus φ has a minimize which is a solution of (1.1).

(2) Let λ > 0, μ ≤ 0 and the assumptions of (2) hold. By the above conclusion (1), φ has a minimize v₁. Take such that 0 ≤ v₀(x) ≤ min{δ₁, δ₂}, and . By (O₁) and (O₂) we have, for t ∈ (0, 1) small enough,

Since , we can find t₀ ∈ (0, 1) such that φ(t₀v₀) < 0, and this shows φ(v₁) = inf_u∈X φ(u) < 0. So v₁ ≠ 0 because φ(0) = 0. The conclusion (2) is proved.

(3) Let μ > 0 and the assumptions of (3) hold. By Remark 4.1.(1), for sufficiently small ||u||

Since and , there exists γ > 0 and α > 0 such that J(u) - Ψ₂(u) ≥ α for u ∈ S_γ. We can find λ₀(μ) > 0 such that when |λ| ≤ λ₀(μ), Ψ₁(u) ≤ α/2 for u ∈ S_γ. So when |λ| ≤ λ₀(μ), φ(u) ≥ α/2 > 0 for u ∈ S_γ. By Remark 4.1.(2), noting that , we can find a u₀ ∈ X such that ||u₀|| > γ and φ(u₀) < 0. By Proposition 3.6 problem (1.1) has a nontrivial solution u₁ such that φ(u₁) > 0.

(4) Let μ > 0 and the assumptions of (4) hold. By the conclusion (3), we know that, there exists λ₀(μ) > 0 such that when 0 < λ ≤ λ₀(μ), problem (1.1) has a nontrivial solution u₁ such that φ(u₁) > 0. Let γ and α be as in the proof of (3), that is, φ(u) ≥ α/2 > 0 for u ∈ S_γ. By (O₁), (O₂) and the proof of (2), there exists w ∈ X such that ||w|| < γ and φ(w) < 0. It is clear that there is v₁ ∈ B_γ, a minimizer of φ on B_γ. Thus v₁ is a nontrivial solution of (1.1) and φ(v₁) < 0.

(5) Let μ > 0, λ ∈ ℝ and the assumptions of (5) hold. By (S), we can use the nonsmooth version Fountain theorem with the antipodal action of ℤ₂ to prove (5) (see Proposition 3.7). Denote

Let β_k(γ) be as in Proposition 3.5. By Proposition 3.5, for each positive integer n, there exists a positive integer k₀(n) such that β_k(n) ≤ 1 for all k ≥ k₀(n). We may assume k₀(n) < k₀(n + 1) for each n. We define {γ_k : k = 1, 2, . . . , } by

Note that γ_k → ∞ as k → ∞. Then for u ∈ Z_k with ||u|| = γ_k we have

and consequently

i.e., the condition (F₂) of Proposition 3.7 is satisfied.

By (A), (A₁), (B₂) and Remark 4.1.(2), we have

Noting that and all norms on a finite dimensional vector space are equivalent each other, we can see that, for each Y_k, φ(u) → - ∞ as u ∈ Y_k and ||u|| → ∞. Thus for each k there exists ρ_k > γ_k such that φ(u) < 0 for u ∈ Y_k ∩ S_ρk, so the condition (F₃) of Proposition 3.7 is satisfied. As was noted earlier, φ satisfies nonsmooth (PS) condition. By Proposition 3.7 the conclusion (5) is true.

(6) Let λ > 0, μ ∈ ℝ and the assumptions of (5) hold. Let us verify the conditions of the Nonsmooth dual Fountain theorem (see Proposition 3.8). By (S), φ is invariant on the antipodal action of ℤ₂. For Ψ(u) = ∫_Ω F(x, u)dx = Ψ₁(u)+ Ψ₂(u) let β_k(1) be as in Proposition 3.5, that is

By Proposition 3.5, there exists a positive integer k₀ such that for all k ≥ k₀. Setting ρ_k = 1, then for k ≥ k₀ and u ∈ Z_k ∩ S₁, we have

which shows that the condition (D₁) of Proposition 3.8 is satisfied.

Since is the closure of in , we may choose {Y_k : k = 1, 2, . . . , }, a sequence of finite dimensional vector subspaces of X defined by (3.5), such that for all k. For each Y_k, because all norms on Y_k are equivalent each other, there is ε ∈ (0, 1) such that for every and By (O₁) and (O₂), for u ∈ Y_k ∩ B_ε we have

Because there exists γ_k ∈ (0, ε) such that

thus the condition (D₂) of Proposition 3.8 is satisfied.

Because Y_k ∩ Z_k ≠ ∅ and γ_k < ρ_k, we have

On the other hand, for any u ∈ Z_k with ||u|| ≤ 1 = ρ_k, we have φ(u) = J(u) - Ψ(u) ≥ -Ψ(u) ≥ -β_k(1). Noting that β_k → 0 as k → ∞, we obtain d_k → 0, i.e., (D₃) of Proposition 3.8 is satisfied.

Finally let us prove that φ satisfies nonsmooth condition for every c ∈ R. Suppose . Similar to the process of verifying the (PS) condition in the proof of Proposition 3.3, we can get in X. Let us prove 0 ∈ ∂φ(u) below. Notice that

Using Proposition 2.8.4, Going to limit in the right side of above equation, we have

so m(u) ≡ 0, i.e., 0 ∈ ∂φ(u), this shows that φ satisfies the nonsmooth condition for every c ∈ ℝ. So all conditions of Proposition 3.8 are satisfied and the conclusion (6) follows from Proposition 3.8. The proof of Theorem 4.1 is complete.    ■

Remark 4.2

Theorem 4.1 includes several important special cases. In particular, in the case of the problem (1.4), i.e., in the case that

all conditions of Theorem 4.1 are satisfied provided (P), (A*), (A₁), and (A₂) hold.

The authors declare that they have no competing interests.

GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. RM participated in the design of the study. All authors read and approved the final manuscript.

The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030, 10971087), 1107RJZA223 and the Fundamental Research Funds for the Gansu Universities.

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