Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian

Lemma 2.1

[27] Assume that φ is a locally Lipschitz functional on a Banach space X and φ : X → ℝ satisfies:

1. φ is weakly lower semicontinuous;

2. φ is coercive.

Then there exists x^* ∈ X such that φ(x^*) = min_x∈X φ(x).

Lemma 2.2

[22] Let X be a Banach space and φ : X → ℝ a locally Lipschitz functional satisfying the (C) condition. If X = Y ⊕ V with Y a finite-dimensional subspace of X. φ satisfies the nonsmooth C-condition,

and there exists an r > 0 such that

where Γ = {γ ∈ C(E, X) : γ|_∂E = id}, E = {x ∈ Y : ║x║ ≤ r} and ∂E = {x ∈ Y : ║x║ = r}, then

and c is a critical value of φ. Moreover, if c = inf_V φ, then V ^ՈK_c ≠ ∅, where K_c = {x ∈ X : φ(x) = c, λ(x) = 0}.

Lemma 2.3

[22] Let X be a reflexive Banach space, ϕ : X → ℝ a locally Lipschitz functional satisfying the PS-condition. Assume that there exist x₀, x₁ ∈ X, c₀ ∈ ℝ and ϱ > 0 such that ║x₁ − x₀║ > ϱ and

Then, ϕ has a critical point x ∈ X with c = ϕ(x) ≥ c₀ given by

where Γ₁ = {γ ∈ C([0, 1], x) : γ(0) = x₀, γ(1) = x₁}.

In order to discuss (1.3), we recall some known results from critical point theory and the properties of space W^1,p(t) for the convenience of the readers.

Let p(t) ∈ C(0, T;ℝ) and 1 < p₋ := inf_t∈[0,T]p(t) ≤ sup_t∈[0,T]p(t) := p⁺ < ∞. Define

with the norm

For u ∈ L l o c 1 0 , T ; R N , let u^' denote the weak derivative of u, i.e., u ′ ∈ L l o c 1 0 , T ; R N and satisfy

Define

with the norm

We call the space L^p(t) a generalized Lebesgue space, it is a special kind of generalized Orlicz spaces. The space W^1,p(t) is called a generalized Sobolev space, it is a special kind of generalized Orlicz-Sobolev spaces. For the general theory of generalized Orlicz spaces and generalized Orlicz-Sobolev spaces, see [13,14].

Proposition 2.4

[14] Let

then

1. | u | p ( t ) < 1 ( = 1 ; > 1 ) ⟺ ρ ( u ) < 1 ( = 1 ; > 1 ) ;

2. |u|p(t)>1⇒|u|p(t)p−≤ρ(u)≤|u|p(t)p+,|u|p(t)<1⇒|u|p(t)p+≤ρ(u)≤|u|p(t)p−;

3. | u | p ( t ) → 0 ⟺ ρ ( u ) → 0 ; | u | p ( t ) → ∞ ⟺ ρ ( u ) → ∞ .

4. Let u ∈ L^p(t) \ {0}, then ║u║_p(t) = λ if and only if ρ u λ = 1.

Proposition 2.5

[13] L^p(t) and W^1,p(t) are Banach spaces with the norms defined above. When p⁻ > 1, they are reflexive.

Let C T ∞ = C T ∞ R , R N = u ∈ C ∞ R , R N : u is T-periodic}.

Definition 2.6

[13] Let u, v ∈ L¹(0, T;ℝ^N). If

then v is called a T-weak derivative of u and is denoted by u ˙ .

Definition 2.7

[24] Define

with the norm ∥ u ∥ W T 1 , p = | u | p p + | u ˙ | p p 1 / p .

Definition 2.8

[13, 14] Define

and H T 1 , p ( t ) 0 , T ; R N to be the closure of W T 1 , p ( t ) 0 , T ; R N .

From Definition 2.7 and 2.8 we see that, for u ∈ L¹(0, T;ℝ^N), the weak derivative u^' and the T-weak derivative are two different conceptions (see [13] for details). Although the two derivatives are distinct, we have

Proposition 2.9

[13, 14]

1. C T ∞ 0 , T ; R N is dense in W T 1 , p ( t ) 0 , T ; R N ;

2. W T 1 , p ( t ) 0 , T ; R N = H T 1 , p ( t ) 0 , T ; R N = u ∈ W T 1 , p ( t ) 0 , T ; R N : u ( 0 ) = u ( T ) ;

3. If u ∈ H T 1 , 1 , the weak derivative is also the T-weak derivative u ˙ , i . e . , u ′ = u ˙ .

Proposition 2.10

[13] Let u ∈ W T 1 , 1 , then

1. ∫ 0 T u ˙ d t = 0 ;

2. u has its continuous representation, which is still denoted by u,

   u ( t ) = ∫ 0 t u ˙ ( s ) d s + u ( 0 ) , u ( 0 ) = u ( T ) .

3. u ˙ is the classical derivative of u if u ˙ ∈ C 0 , T ; R N .

Proposition 2.11

[13]

1. H T 1 0 , T ; R N is a reflexive Banach space if p⁻ > 1;

2. There is a continuous embedding W T 1 , p ( t ) H T 1 , p ( t ) ↪ C 0 , T ; R N . When p⁻ > 1, the embedding is compact.

Proposition 2.12

[14] If

then

1. (L^p(t))^* = L^q(t), where (L^p(t))^* is the dual space of L^p(t);

2. ∀ u ∈ L^p(t), v ∈ L^q(t), we have

Let

Obviously, W p e r 1 , p ( t ) 0 , T ; R N is a reflexive Banach space with the norm

Consider the following functional:

We know that (see [2]) J ∈ C 1 W p e r 1 , p ( t ) , R and p(t)-Laplacian operator (|u^'|^p(t)−2u^')^' is the derivative operator of J in the weak sense. Denote A = J ′ : W p e r 1 , p ( t ) 0 , T ; R N → W p e r 1 , p ( t ) 0 , T ; R N ∗ , then

Proposition 2.13

[14] J^' is a mapping of (S)₊, i.e., if

then u_n has a convergent subsequence in W p e r 1 , p ( t ) 0 , T ; R N .

For every u ∈ W p e r 1 , p ( t ) 0 , T ; R N , set

By virtue of [35], there exists a > 0 such that

The corresponding functional φ : W p e r 1 , p ( t ) 0 , T ; R N → R for (1.3) is defined by:

Theorem 3.1

If hypotheses (P), H(j)₁ hold, then problem (1.3) has at least one nontrivial periodic solution.

We can weaken hypothesis H(j)₁ (iv) at the expense of introducing an extra unilateral growth condition.

More precisely, the new hypotheses on j(t, u) are the following:

H(j)₂ j : [0, T] × ℝ^N is a function such that:

1. for all u ∈ ℝ^N , t ↦ j(t, u) is measurable;

2. for almost all t ∈ [0, T], u ↦ j(t, u) is locally Lipschitz;

3. for every w ∈ ∂j(t, u), there exists α ∈ L¹[0, T] such that for almost all t ∈ [0, T], we have |w| ≤ α(t);

4. j(t, u) → −∞ as |u| → ∞for almost all t ∈ E ⊂ [0, T] with measE > 0;

5. there exists β ∈ L¹[0, T] such that for almost all t ∈ [0, T], u ∈ ℝ^N, we have

   j ( t , u ) ≤ β ( t ) ;

6. there exists u₀ ∈ ℝ^N \ {0} such that ∫ 0 T j t , u 0 d t > 0 a n d ∫ 0 T j ( t , 0 ) d t ≤ 0.

Theorem 3.2

If hypotheses (P), H(j)₂ hold, then problem (1.3) has at least one nontrivial periodic solution.

In the previous existence theorems the energy functional φ defined in (2.2) was coercive and so the solution was obtained by an application of the least action principle. In the next existence theorem the energy functional φ is bounded below but not necessarily coercive. In this case the hypotheses on the nonsmooth potential j(t, u) are the following:

H(j)₃ j : [0, T] × ℝ^N is a function such that j(·, 0) ∈ L¹[0, T] and:

1. for all u ∈ R N , t ↦ j ( t , u ) is measurable;

2. for almost all t ∈ [0, T], u ↦ j(t, u) is locally Lipschitz;

3. for every r > 0, there exists α_r ∈ L¹[0, T] such that for almost all t ∈ [0, T], |u| ≤ r and all w ∈ ∂j(t, u), we have |w| ≤ α_r(t);

4. there exist 0 < μ < p⁻ and M > 0 such that for almost all t ∈ [0, T] and all |u| ≥ M, we have

   j 0 ( t , u ; u ) < μ j ( t , u ) ;

5. j(t, u) → +∞ uniformly for almost all t ∈ [0, T] as |u| → ∞.

Theorem 3.3

If hypotheses (P), H(j)₃ hold, then problem (1.3) has at least one nontrivial periodic solution.

Next, we will establish some existence results by using nonsmooth Mountain Pass theorem. Our hypotheses on the nonsmooth potential function j(t, u) are the following:

H(j)₄ j : [0, T] × ℝ^N ↦ ℝ is a function such that j(t, 0) = 0 for every t ∈ [0, T];

1. for all u ∈ ℝ^N , t ↦ j(t, u) is measurable;

2. for almost all t ∈ [0, T], u ↦ j(t, u) is locally Lipschitz;

3. for every w ∈ ∂j(t, u), there exists α ∈ C([0, T], ℝ) such that for almost all t ∈ [0, T], we have

   | w | ≤ b ( t ) | u | α ( t ) − 1 ,

   where b ∈ L α ( t ) [ 0 , T ] ∩ L + ∞ [ 0 , T ] a n d p + < α − ≤ α ( t ) ≤ α + ;

4. there exists constants M > 0, α, β > 0 and ν < p⁻ such that for almost all t ∈ [0, T] and all u ∈ ℝ^N \ {0}, we have

   0 ≤ p + + 1 α + β | u | v j ( t , u ) ≤ − j 0 ( t , u ; − u ) ;

5. for every u ∈ ℝ^N , w ∈ ∂j(t, u) we have

   lim | u | → 0 w | u | p ( t ) − 1 = 0 ;

6. there exists a function q(t) > 0(p⁺ < q⁻ ≤ q(t) ≤ q⁺) such that for almost all t ∈ [0, T] and all u ∈ ℝ^N, we have

Theorem 3.4

If hypotheses (P), H(j)₄ hold, then problem (1.3) has at least one nontrivial periodic solution.

Theorem 3.5

If hypotheses (P), H(j)₄ (j), (i)-(iii), (v) and the following conditions hold:

(vi’) there exist μ > p⁺ and M > 0 such that

and ∫ 0 T j t , u 0 d t > 0  as  u 0 ∈ R N , u 0 ≥ M .

Then problem (1.3) has at least one nontrivial periodic solution.

Proof of Theorem 3.1

We consider the locally Lipschitz energy functional φ : W p e r 1 , p ( t ) 0 , T ; R N → R defined by

It is easy to verify that φ is locally Lipschitz by H(j)₁ (iii) and (2.1).

Because of hypothesis H(j)₁(iv) and Lemma 3 [35], for almost all t ∈ [0, T], u ∈ ℝ^N, we have

where γ ∈ L¹[0, T], G ∈ C(ℝ^N , ℝ) is subadditive, i.e., G(x + y) ≤ G(x) + G(y) for all x, y ∈ ℝ^N and is coercive, i.e., G(u) → +∞ as |u| → ∞.

Consider the direct sum decomposition

where

For every u ∈ W p e r 1 , p ( t ) 0 , T ; R N we have u = u ¯ + u ~ ,  where u ¯ ∈ R N , u ~ ∈ V . By (3.1), we have

From the properties of the continuous function G mentioned above, we have

Using (3.3) in (3.2), we obtain

where c₁, c₂ > 0.

Noting that ∥ u ∥ ≤ ∥ u ¯ ∥ + ∥ u ~ ∥ , by virtue of (2.1), (3.4) and the coercivity on G, we infer that φ is coercive.

Owing to the fact the compact embedding of W per  1 , p ( t ) 0 , T ; R N ↪ C 0 , T ; R N and the weak lower semi-continuity of the norm functional in a Banach space, we infer that φ is weakly lower semicontinuous. So by hypothesis H(j)₁(v) and the Weierstrass theorem (Lemma 2.1) we can find u₀ such that

Next, we will show that u the solution of (1.3).

In fact, let u ∈ W p e r 1 , p ( t ) 0 , T ; R N be such that 0 ∈ ∂φ(u). Define the nonlinear operator A : W p e r 1 , p ( t ) 0 , T ; R N → W p e r 1 , p ( t ) 0 , T ; R N ∗ as follows

Thus,

where w ∈ L^q(t)(0, T;ℝ^N) and w ∈ ∂j(t, u(t)). Here 1 p ( t ) + 1 q ( t ) = 1.

For every x, y ∈ ℝ^N, the following inequalities hold [23]:

and

An argument similar to the one used in [37] shows that for every v ∈ W p e r 1 , p ( t ) 0 , T ; R N , thus

which implies that (|u^'(t)|^p(t)−2u^'(t))^' has T-weak derivative and satisfy

It follows from C [ 0 , T ] , R N ⊆ W p e r 1 , p ( t ) [ 0 , T ] , R N  and  W p e r 1 , p ( t ) [ 0 , T ] , R N ↪ C [ 0 , T ] , R N that u(t) is continuous for every t ∈ [0, T]. It is obviously that (|u^'(t)|^p(t)−2u^'(t))^' ∈ L^q(t)([0, T], ℝ^N). Since L q ( t ) ↪ L 1 , then we have (|u^'(t)|^p(t)−2u^'(t))^' ∈ L¹([0, T], ℝ^N), which implies that u ′ ( t ) p ( t ) − 2 u ′ ( t ) ′ ∈ W p e r 1 , 1 [ 0 , T ] , R N . By Proposition 2.4 (2), we have

Noting that p(0) = p(T), we can obtain u^'(0) = u^'(T). Since u ∈ W p e r 1 , p ( t ) 0 , T ; R N , then u(0) = u(T). So u is the solution of (1.3).

Proof of Theorem 3.2

Given δ > 0, by Lemma 2 [35] and H(j)₂ (iv), we can find E_δ ⊆ E such that meas(E\E_δ)< δ as |u| → ∞, j(t, u) → −∞ uniformly for all t ∈ E_δ , u ∈ ℝ^N, so for almost all t ∈ E_δ , u ∈ ℝ^N we have

with G ∈ C(ℝ^N , ℝ), γ ∈ L¹(E_δ) as in the proof of Theorem 3.1. Then for every u ∈ W p e r 1 , p ( t ) 0 , T ; R N , by (3.4), (3.7) and H(j)₂ (v) we have

where c₃, c₄ > 0. Therefore φ is coercive. The rest of the proof goes as that of Theorem 3.1.

Proof of Theorem 3.3

Consider the locally Lipschitz energy functional φ : W p e r 1 , p ( t ) 0 , T ; R N → R defined by

Step 1: φ satisfies the nonsmooth C-condition.

Let {u_n}_n≥1 be a (C)-sequence of φ, i.e., there exists M₁ > 0 such that

Since ∂ φ u n ⊆ W p e r 1 , p ( t ) 0 , T ; R N ⋆ is weakly compact and the norm functional in a Banach space is weakly compact, from the Weierstrass theorem (Lemma 2.1), we can find u n ⋆ ∈ ∂ φ u n such that m u n = u n ⋆ .

Define the nonlinear operator A : W p e r 1 , p ( t ) 0 , T ; R N → W p e r 1 , p ( t ) 0 , T ; R N ⋆

Then

where w_n ∈ ∂j(t, u_n).

From the choice of the sequence u n n ≥ 1 ⊆ W p e r 1 , p ( t ) 0 , T ; R N we have

Thus,

and also

It follows from (3.8) and (3.9) that

By H(j)₃ (iv), we have

where c₅ > 0 is independent of n.

Therefore, from (3.10) we have

By the Poincare-Wirtinger inequality (2.1), u ~ n is bounded in W p e r 1 , p ( t ) 0 , T ; R N .

It follows from H(j)₃ and Lemma 3 [35] that for almost all t ∈ [0, T] and all u ∈ ℝ^N, we have

with γ ∈ L¹[0, T], G ∈ C(ℝ^N , ℝ) is subadditive, coercive and satisfies G(u) ≤ |u| + 4 for all u ∈ ℝ^N.

From the choice of the sequence u n n ≥ 1 ⊆ W p e r 1 , p ( t ) 0 , T ; R N we have

Since u ~ n is bounded, there exists M₂ > 0, n ≥ 1 such that

By (3.11), we get

thus, there exists M₃ > 0 such that G u ¯ n b ≤ M 3 , n ≥ 1.

Due to the coercivity of G, we infer that u ¯ n n ≥ 1 ⊆ R N is bounded. Therefore u n n ≥ 1 ⊆ W p e r 1 , p ( t ) 0 , T ; R N is bounded and so by passing to a subsequence if necessary, we may assume that

Next, we will prove that u n → u  in  W per  1 , p ( t ) 0 , T ; R N . By Proposition 2.13, it suffices to prove that the following inequality hold:

In fact, from the choice of the sequence {u_n}_n≥1 we have

Recall that u n ⋆ = A u n − w n , then we have

By H(j)₃(iii), {w_n}⊆ L¹[0, T] is bounded and

Then

So lim ¯ n → ∞ < A u n − A ( u ) , u n − u >≤ 0 , ε n ↓ 0.

Step 2: Similar to the proof in [30, 31], we have j(t, sx) ≤ s^μj(t, x) for every s ≥ 1, we omit its proof course.

Step 3: φ|_V is coercive, i.e., for every v ∈ V, φ(v) → +∞ as ║v║ → ∞.

Let v ∈ V be such that |{t ∈ T : ║v(t) > M║}|₁ > 0, we have