Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian

Proposition 2.1

L a p ( t ) , W a 1 , p ( t ) , W 0 1 , p ( t ) are reflexive Banach spaces with norms defined above when p − > 1 .

Proposition 2.2

Let ρ ( u ) = ∫ Ω a ( t ) ∣ u ( t ) ∣ p ( t ) d t for any u , v ∈ L a p ( t ) , then the following properties hold:

1. ρ ( u ) = 0 ⇔ u = 0 ;

2. ρ ( u ) = ρ ( − u ) ;

3. ρ ( α u + β v ) ≤ α ρ ( u ) + β ρ ( v ) for any α , β ≥ 0 , α + β = 1 ;

4. ρ ( u + v ) ≤ 2 p + ( ρ ( u ) + ρ ( v ) ) ;

5. If λ > 1 , then

   λ p + ρ ( u ) ≤ ρ ( λ u ) ≤ λ p − ρ ( u ) ≤ λ ρ ( u ) ≤ ρ ( u ) ;

6. ‖ u ‖ p ( t ) , a = 1 if and only if ρ u λ = 1 , for any u ∈ L a p ( t ) ⧹ { 0 } .

Proposition 2.3

For any u ∈ L a p ( t ) , the following properties hold:

1. ∣ u ∣ p ( t ) , a < 1 ( = 1 ; > 1 ) ⇔ ρ ( u ) < 1 ( = 1 ; > 1 ) ;

2. If ∣ u ∣ p ( t ) , a > 1 , then ∣ u ∣ p ( t ) , a p − ≤ ρ ( u ) ≤ ∣ u ∣ p ( t ) , a p + ;

3. If ∣ u ∣ p ( t ) , a < 1 , then ∣ u ∣ p ( t ) , a p + ≤ ρ ( u ) ≤ ∣ u ∣ p ( t ) , a p − ;

4. ∣ u ∣ p ( t ) , a → 0 ⇔ ρ ( u ) → 0 ;

5. ∣ u ∣ p ( t ) , a → ∞ ⇔ ρ ( u ) → ∞ .

Proposition 2.4

Let ϕ ( u ) = ∫ Ω ( ∣ u ˙ ∣ p ( t ) + a ( t ) ∣ u ∣ p ( t ) ) d t for any u ∈ W a 1 , p ( t ) , then the following properties hold:

1. ‖ u ‖ < 1 ( = 1 ; > 1 ) ⇔ ϕ ( u ) < 1 ( = 1 ; > 1 ) ;

2. If ‖ u ‖ > 1 , then ‖ u ‖ p − ≤ ϕ ( u ) ≤ ‖ u ‖ p + ;

3. If ‖ u ‖ < 1 , then ‖ u ‖ p + ≤ ϕ ( u ) ≤ ‖ u ‖ p − ;

4. ‖ u ‖ → 0 ⇔ ϕ ( u ) → 0 ;

5. ‖ u ‖ → ∞ ⇔ ϕ ( u ) → ∞ .

Proposition 2.5

Let ρ ( u ) = ∫ Ω a ( t ) ∣ u ∣ p ( t ) d t for any u ∈ L a p ( t ) , { u n } ⊂ L a p ( t ) , then the following properties are equivalent:

1. lim n → ∞ ∣ u n − u ∣ p ( t ) , a = 0 ;

2. lim n → ∞ ρ ( u n − u ) = 0 ;

3. u n → u a.e. t ∈ Ω and lim n → ∞ ρ ( u n ) = ρ ( u ) .

Proposition 2.6

( L p ( t ) ) ∗ = L q ( t ) with 1 / p ( t ) + 1 / q ( t ) = 1 and

where ( L p ( t ) ) ∗ is the dual space of L p ( t ) .

Proposition 2.7

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

Proposition 2.8

Let u ∈ W a 1 , p ( t ) , then

1. u ∈ C ( R , R N ) and u ( t ) → 0 as ∣ t ∣ → ∞ . Moreover, the embedding W a 1 , p ( t ) ↪ L ∞ ( R , R N ) is continuous, and there exists a constant κ > 0 such that

   ‖ u ‖ L ∞ ≤ κ ‖ u ‖ , ∀ u ∈ W a 1 , p ( t ) ;

2. If H(p),H(a) hold and a ( t ) → + ∞ as ∣ t ∣ → ∞ , then the embedding W a 1 , p ( t ) ↪ L ∞ ( R , R N ) is compact.

Proposition 2.9

I ′ is a mapping of type ( S ) + , i.e., if

then u n has a convergent subsequence in W a 1 , p ( t ) .

Proposition 2.10

The mapping A is a strictly monotone, bounded homeomorphism and is of type ( S ) + in W a 1 , p ( t ) .

Definition 2.11

Let X be a Banach space and let X ∗ be its topological dual. By ⟨ ⋅ ⟩ we denote the duality brackets for the pair ( X , X ∗ ) . A function ϕ : X → R is said to be locally Lipschitz, if for every x ∈ X there exist U ∈ N ( x ) and a constant k U > 0 , such that

Definition 2.12

For a given locally Lipschitz function ϕ : X → R , the generalized directional derivative of φ at x ∈ X in the direction h ∈ X is defined by

Definition 2.13

Let ϕ : X → R be a locally Lipschitz function. Then generalized subdifferential of ϕ at x ∈ X is the nonempty set ∂ ϕ ( x ) ⊆ X ∗ defined by

Definition 2.14

Let ϕ : X → R be a locally Lipschitz function. A point x ∈ X is said to be a critical point of ϕ if 0 ∈ ∂ ϕ ( x ) .

Definition 2.15

The locally Lipschitz function ϕ : X → R satisfies the nonsmooth Palais-Smale (PS) condition, if any sequence { x n } n ≥ 1 ⊆ X such that

has a strongly convergent subsequence, where m ( x n ) = min [ ‖ x ∗ ‖ : x ∗ ∈ ∂ ϕ ( x n ) ] .

Lemma 2.16

(Lebourg’s mean value Theorem [39]). Given the points x and y in X and a real-valued function ϕ which is Lipschitz continuous on an open set containing the segment [ x , y ] = { ( 1 − t ) x + t y : t ∈ [ 0 , 1 ] } , there exist z = x + t 0 ( y − x ) , with 0 < t 0 < 1 , and x ∗ ∈ ∂ ϕ ( z ) such that

Definition 2.17

The locally Lipschitz function ϕ : X → R satisfies the nonsmooth Cerami condition (C-condition), if any sequence { x n } n ≥ 1 ⊆ X such that

has a strongly convergent subsequence.

Lemma 2.18

(Weierstrass theorem [43]). 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.

Lemma 2.19

(Nonsmooth mountain pass theorem [35]). Let X be a reflexive Banach space, ϕ : X → R a locally Lipschitz functional satisfying the PS-condition. Assume that there exist x 0 , x 1 ∈ X , c 0 ∈ R and ϱ > 0 such that ‖ x 1 − x 0 ‖ > ϱ and

Then ϕ has a critical point x ∈ X with c = ϕ ( x ) ≥ c 0 , where c is given by