Kirchhoff type via critical point theory

Opuscula Math. 36, no. 5 (2016), 631–649 http://dx.doi.org/10.7494/OpMath.2016.36.5.631

Opuscula Mathematica

MULTIPLICITY RESULTS FOR AN IMPULSIVE BOUNDARY VALUE PROBLEM OF p(t)-KIRCHHOFF TYPE VIA CRITICAL POINT THEORY A. Mokhtari, T. Moussaoui, D. O’Regan Communicated by Marek Galewski Abstract. In this paper we obtain existence results of k distinct pairs nontrivial solutions for an impulsive boundary value problem of p(t)-Kirchhoff type under certain conditions on the parameter λ. Keywords: genus theory, nonlocal problems, impulsive conditions, Kirchhoff equation, p(t)-Laplacian, variational methods, critical point theory. Mathematics Subject Classification: 35A15, 35B38, 34A37.

1. INTRODUCTION We study the multiplicity of nontrivial solutions for the problem   RT 1 0 p(t)  0 p(t)−2 0 0    dt |u (t)| · u (t) = λh(t, u(t)), − a + b p(t) |u (t)| 0

−∆u0 (tj ) = Ij (u(tj )),     u(0) = u(T ) = 0,

j = 1, 2, . . . , l,

t 6= tj , t ∈ [0, T ], (1.1)

where 0 = t0 < t1 < . . . < tl < tl+1 = T , λ > 0 is a numerical parameter, h is 0 − a Carathéodory function, Ij ∈ C(R, R), j = 1, 2, . . . , l, ∆u0 (tj ) = u0 (t+ j ) − u (tj ), 0 + 0 − u (tj ) and u (tj ) denote the right and left derivative of u at t = tj , j = 1, 2, . . . , l. Here p is a function in C([0, T ], R) with 1 < p− = inf p(t) ≤ p+ = sup p(t), t∈[0,T ]

t∈[0,T ]

and a, b are positive constants. c AGH University of Science and Technology Press, Krakow 2016

631

632

A. Mokhtari, T. Moussaoui, D. O’Regan

Impulsive problems for the p(t)-Laplacian were introduced in [12] and [13]. In [4] the authors considered  00  u (t) + λh(t, u(t)) = 0, t 6= tj , t ∈ [0, T ], −∆u0 (tj ) = Ij (u(tj )), j = 1, 2, . . . , l,   u(0) = u(T ) = 0, and the goal in this paper is to generalize the results so that (1.1) can be considered. The variable exponent Lebesgue space Lp(t) (0, T ) is defined by ZT

( p(t)

L

(0, T ) =

u : (0, T ) → R is mesurable,

) p(t)

|u(t)|

dt < +∞

0

endowed with the norm ( |u|p(t) = inf

) ZT u(t) p(t) λ>0: dt ≤ 1 . λ 0

The variable exponent Sobolev space W 1,p(t) (0, T ) is defined by W 1,p(t) (0, T ) = {u ∈ Lp(t) (0, T ) : u0 ∈ Lp(t) (0, T )} endowed with the norm kuk1,p(t) = |u|p(t) + |u0 |p(t) . Denote by C([0, T ]) the space of continuous functions on [0, T ] endowed with 1,p(t) the norm |u|∞ = supt∈[0,T ] |u(t)|. Now W0 (0, T ) denotes the closure of C0∞ (0, T ) 1,p(t) in W (0, T ). 1,p(t)

Proposition 1.1 ([11]). Lp(t) (0, T ), W 1,p(t) (0, T ) and W0 reflexive and uniformly convex Banach spaces.

(0, T ) are separable,

Proposition 1.2 ([11]). For any u ∈ Lp(t) (0, T ) and v ∈ Lq(t) (0, T ), where 1 1 p(t) + q(t) = 1, we have ZT  1 1  uv dt ≤ − + − |u|p(t) |v|q(t) . p q 0

Proposition 1.3 ([11]). Let ρ(u) = following assertions hold.   1. For u 6= 0, |u|p(t) = λ ⇔ ρ λu = 1.

RT 0

|u(t)|p(t) dt. For any u ∈ Lp(t) (0, T ), the

2. |u|p(t) < 1(= 1; > 1) ⇔ ρ(u) < 1(= 1; > 1). −

+

3. If |u|p(t) > 1, then |u|pp(t) ≤ ρ(u) ≤ |u|pp(t) . −

p 4. If |u|p(t) < 1, then |u|p+ p(t) ≤ ρ(u) ≤ |u|p(t) .

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 633

Proposition 1.4 ([11]). If u, uk ∈ Lp(t) (0, T ), k = 1, 2, . . ., then the following statements are equivalent: 1. 2.

lim |uk − u|p(t) = 0 (i.e. uk → u in Lp(t) (0, T )),

k→+∞

lim ρ(uk − u) = 0,

k→+∞

3. uk → u in measure in (0, T ) and lim ρ(uk ) = ρ(u). k→+∞

Proposition 1.5 ([9]). The Poincaré-type inequality holds, that is, there exists a positive constant c such that |u|p(t) ≤ c|u0 |p(t) ,

for all

1,p(t)

u ∈ W0

(0, T ).

1,p(t)

Thus |u0 |p(t) is an equivalent norm in W0 (0, T ). We will use this equivalent norm in the following discussion and write kuk = |u0 |p(t) for simplicity. We now recall the Krasnoselskii genus and information on this may be found in [1, 2, 14, 15]. Let E be a real Banach space. Let us denote by Σ the class of all closed subsets A ⊂ E\{0} that are symmetric with respect to the origin, that is, u ∈ A implies −u ∈ A. Definition 1.6. Let A ∈ Σ. The Krasnoselskii genus γ(A) is defined as being the least positive integer n such that there is an odd mapping ϕ ∈ C(A, Rn \ {0}). If such n does not exist, we set γ(A) = +∞. Furthermore, by definition, γ(∅) = 0. Theorem 1.7 ([14]). Let E = RN and ∂Ω be the boundary of an open, symmetric and bounded subset Ω ⊂ RN with 0 ∈ Ω. Then γ(∂Ω) = N . Note γ(S N −1 ) = N . If E is infinite dimension and separable and S is the unit sphere in E, then γ(S) = +∞. Proposition 1.8 ([14]). Let A, B ∈ Σ. Then, if there exists an odd map f ∈ C(A, B), then γ(A) ≤ γ(B). Consequently, if there exists an odd homeomorphism f : A → B, then γ(A) = γ(B). Definition 1.9. Let J ∈ C 1 (E, R). If any sequence (un ) ⊂ E for which (J(un )) is bounded and J 0 (un ) → 0 when n → +∞ in E 0 possesses a convergent subsequence, then we say that J satisfies the Palais-Smale condition (the (PS) condition). We now state a theorem due to Clarke. Theorem 1.10 ([5,17]). Let J ∈ C 1 (E, R) be a functional satisfying the Palais-Smale condition. Also suppose that: 1) J is bounded from below and even, 2) there is a compact set K ∈ Σ such that γ(K) = k and supx∈K J(x) < J(0). Then J possesses at least k pairs of distinct critical points and their corresponding critical values are less than J(0).

634

A. Mokhtari, T. Moussaoui, D. O’Regan 1,p(t)

Definition 1.11. We say that u ∈ W0 (1.1) if and only if 

ZT

(0, T ) = X is a weak solution of Problem

ZT  ZT 1 |u0 (t)|p(t) dt |u0 (t)|p(t)−2 · u0 (t)v 0 (t) dt = λ h(t, u(t))v(t) dt p(t)

a+b 0

0

0

+

l X

Ij (u(tj ))v(tj )

j=1

for all v ∈ X. In Section 2 we will use the following elementary inequalities (see [16]): for all x, y ∈ R, we have (|x|p(·)−2 x − |y|p(·)−2 y)(x − y) ≥

1 |x − y|p(·) 2p(·)

if p(·) ≥ 2,

(1.2)

and
(|x|p(·)−2 x − |y|p(·)−2 y)(x − y) ≥ p(·) − 1

|x − y|2 (|x| + |y|)2−p(·)

if 1 < p(·) < 2. (1.3)

Remark 1.12. Note (1.2) implies (|x|p(·)−2 x − |y|p(·)−2 y)(x − y) ≥

1 |x − y|p(·) 2p+

if p(·) ≥ 2.

(1.4)

i p(·) |x − y|p(·) p− − 1 2 (|x|p(·)−2 x − |y|p(·)−2 y)(x − y) ≥ √ 2−p(·) 2 (|x|p(·) + |y|p(·) ) 2

(1.5)

Also (1.3) implies h

if 1 < p(·) < 2. To see this note for any x, y ∈ R and 1 < p(·) < 2, from (1.3) we have h

(|x|p(·)−2 x − |y|p(·)−2 y)(x − y)

i p(·) 2

≥ (p− − 1)

|x − y|p(·) (|x| + |y|)p(·)

2−p(·) 2

and now using (|x| + |y|)p(·) ≤ 2p(·)−1 (|x|p(·) + |y|p(·) ) ≤ 2(|x|p(·) + |y|p(·) ), we obtain h

i p(·) 2 ≥ (p− − 1) (|x|p(·)−2 x − |y|p(·)−2 y)(x − y)

1 2

≥

2−p(·) 2

|x − y|p(·)
2−p(·) |x|p(·) + |y|p(·) 2

p− − 1 |x − y|p(·) √ . 2−p(·) 2 |x|p(·) + |y|p(·)
2

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 635

2. MAIN RESULTS Theorem 2.1. Assume the following are satisfied. (h1 ) There exist α, β ∈ L1 (0, T ) and a continuous function γ : [0, T ] → R such that 0 ≤ γ + = supt∈[0,T ] γ(t) < 2p− − 1 with |h(t, u)| ≤ α(t) + β(t)|u|γ(t)

for any (t, u) ∈ [0, T ] × R.

Ru (h2 ) h(t, u) is odd with respect to u and H(t, u) = 0 h(t, ξ) dξ > 0 for every (t, u) ∈ [0, T ] × R \ {0}. Ru (h3 ) Ij (u) (j = 1, 2, . . . , l) are odd and 0 Ij (s) ds ≤ 0 for any u ∈ R (j = 1, . . . , l). Then for any k ∈ N, there exists a λk such that when λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions. Proof. The corresponding functional to Problem (1.1) is defined as follows: ZT 2 1 1 b 0 p(t) |u (t)| dt + |u0 (t)|p(t) dt p(t) 2 p(t)

ZT ϕ(u) = a 0

−

0

l X

u(t Z j)

(2.1)

ZT

Ij (s) ds − λ

j=1 0

H(t, u(t)) dt. 0

From (h1 ) and the fact that Ij ∈ C(R, R) it is easy to see that ϕ ∈ C 1 (X, R) and for all u, v ∈ X ZT



0

ϕ (u) · v = a + b

 ZT 1 0 p(t) |u (t)| dt |u0 (t)|p(t)−2 u0 (t)v 0 (t) dt p(t)

0

−

l X

0

(2.2)

ZT Ij (u(tj ))v(tj ) − λ

j=1

h(t, u(t))v(t) dt. 0

Thus the critical points of ϕ are the weak solutions of (1.1). First we show that ϕ is bounded from below. From the continuous embedding of X in C([0, T ]), for any u ∈ X with kuk > 1, we have that ZT

γ(t)+1

|u(t)| 0

ZT dt ≤ 0

|u|γ(t)+1 dt ≤ cT kukγ ∞

+

+1

,

(2.3)

636

A. Mokhtari, T. Moussaoui, D. O’Regan γ(t)+1

where c = maxt∈[0,T ] c0 and c0 is the best constant of the continuous embedding. It follows from conditions (h1 ), (h2 ), (h3 ) and (2.3) that

ZT ϕ(u) = a

ZT 2 1 1 b |u0 (t)|p(t) dt + |u0 (t)|p(t) dt p(t) 2 p(t)

0

−

0

l X

u(t Z j)

ZT

Ij (s) ds − λ

j=1 0

a ≥ + p

ZT

0 0

p(t)

|u (t)| 0

ZT −λ

H(t, u(t)) dt

b  dt + 2(p+ )2

ZT

2 |u0 (t)|p(t) dt

(2.4)

0

α(t)|u(t)| + β(t)|u(t)|γ(t)+1 dt

0 − − + a b ≥ + kukp + kuk2p − λ|α|L1 c0 kuk − λ|β|L1 ckukγ +1 , + 2 p 2(p )

for any u ∈ X with kuk ≥ 1. Since γ + < 2p− − 1 , then limkuk→+∞ ϕ(u) = +∞ and consequently, ϕ is bounded from below. Next, we show that the functional ϕ satisfies the (PS) condition. Now for any u ∈ X with kuk ≤ 1, it is easy to see that

ϕ(u) ≥

+ + − a b kukp + kuk2p − λ|α|L1 c0 kuk − λ|β|L1 ckukγ +1 . + + 2 p 2(p )

(2.5)

Let (un ) ⊂ X be a Palais-Smale sequence for ϕ, i.e. (ϕ(un )) is a bounded sequence and limn→+∞ ϕ0 (un ) = 0. Thus there exists a positive constant B such that

ϕ(un ) ≤ B.

(2.6)

From (2.4), (2.5), (2.6) and since γ + < 2p− − 1, in all cases we deduce that the sequence (un ) is bounded in X. Thus, passing to a subsequence if necessary, there

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 637

exists u ∈ X such that un * u weakly in X. Moreover, by (2.1) and (2.2), we have

ϕ0 (un ) − ϕ0 (u) · un − u ZT



= a+b

 ZT 1 0 p(t) |u (t)| dt |u0n (t)|p(t)−2 u0n (t)(u0n (t) − u0 (t)) dt p(t) n

0

−

l X

0

ZT Ij (un (tj ))(un (tj ) − u(tj )) − λ

j=1

ZT



− a+b

 ZT 1 0 p(t) |u (t)| dt |u0 (t)|p(t)−2 u0 (t)(u0n (t) − u0 (t)) dt p(t) 0

0

+

l X

ZT Ij (u(tj ))(un (tj ) − u(tj )) + λ

j=1

h(t, u(t))(un (t) − u(t)) dt 0

 = a+b

ZT

 ZT 1 0 p(t) |u (t)| dt |u0n (t)|p(t)−2 u0n (t)(u0n (t) − u0 (t)) dt p(t) n

0

0

ZT



− a+b 0

−

h(t, un (t))(un (t) − u(t)) dt 0

 ZT 1 0 p(t) |u (t)| dt |u0 (t)|p(t)−2 u0 (t)(u0n (t) − u0 (t)) dt p(t) 0

l X

(Ij (un (tj ) − Ij (u(tj ))(un (tj ) − u(tj ))

j=1

ZT −λ

(h(t, un (t)) − h(t, u(t)))(un (t) − u(t)) dt. 0

Since the embedding of X in C([0, T ]) is compact, then (un ) uniformly converges to u in C([0, T ]), by using (h1 ) and the Lebesgue Dominated Convergence Theorem, we have that  ZT        λ h(t, u (t)) − h(t, u(t)) u (t) − u(t) dt → 0,  n n 

(2.7)

0

l     X    I (u (t ) − I (u(t ) u (t ) − u(t ) → 0,  j n j j j n j j 

as n → ∞.

j=1



Since ϕ0 (un ) → 0 and un * u in X, then we have ϕ0 (un ) − ϕ0 (u) · un − u → 0 as n → ∞.

638

A. Mokhtari, T. Moussaoui, D. O’Regan

Consequently, from (2.7) we have Sn → 0 as n → ∞, where ZT



Sn = a + b

 ZT 1 0 p(t) |u (t)| dt |u0n (t)|p(t)−2 u0n (t)(u0n (t) − u0 (t)) dt p(t) n

0

0

ZT



− a+b

 ZT 1 0 p(t) |u (t)| dt |u0 (t)|p(t)−2 u0 (t)(u0n (t) − u0 (t)) dt. p(t)

0

0

We can rewrite Sn as ZT   1 |u0n (t)|p(t) dt Sn = a + b p(t) 0

ZT ×


|u0n (t)|p(t)−2 u0n (t) − |u0 (t)|p(t)−2 u0 (t) (u0n (t) − u0 (t)) dt

0



ZT



ZT

+ a+b

 ZT 1 0 p(t) |u (t)| dt |u0 (t)|p(t)−2 u0 (t)(u0n (t) − u0 (t)) dt p(t) n

0

− a+b

0

 ZT 1 0 p(t) |u (t)| dt |u0 (t)|p(t)−2 u0 (t)(u0n (t) − u0 (t)) dt. p(t)

0

0 1,q(t)

From the weak convergence of (un ) in X, and since |u0 |p(t)−2 u0 ∈ X 0 = W0 p(t) with q(t) = p(t)−1 , we deduce that ZT

|u0 (t)|p(t)−2 u0 (t)(u0n (t) − u0 (t)) dt → 0

as n → ∞.

(0, T )

(2.8)

0

Hence, from (2.8) and since Sn → 0, we get 

ZT a+b 0

 ZT
1 0 p(t) |un (t)| dt |u0n (t)|p(t)−2 u0n (t)−|u0 (t)|p(t)−2 u0 (t) (u0n (t)−u0 (t)) dt → 0 p(t) 0

as n → ∞. Since a, b > 0, then we have ZT


|u0n (t)|p(t)−2 u0n (t) − |u0 (t)|p(t)−2 u0 (t) (u0n (t) − u0 (t)) dt → 0 as n → ∞.

(2.9)

0

To complete the proof of the Palais-Smale condition we follow the argument in the proof of Theorem 3.1 in [3]. We divide I = (0, T ) into two parts I1 = {t ∈ (0, T ) : 1 < p(t) < 2},

I2 = {t ∈ (0, T ) : p(t) ≥ 2}.

We will let | · |p(t),Ii , i = 1, 2, to denote the norm in Lp(t) (Ii ).

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 639

On I1 , from Holder’s inequality (See Proposition 1.2) and using the inequality (1.5) we get Z |u0n (t) − u0 (t)|p(t) dt I1

≤

√

2 p− − 1

Z 

 p(t)
2 |u0n (t)|p(t)−2 u0n (t) − |u0 (t)|p(t)−2 u0 (t) (u0n (t) − u0 (t))

I1

×



|u0n (t)|p(t)

0

p(t)

+ |u (t)|

 2−p(t) 2

dt

  p(t)
2 ≤ c |u0n (t)|p(t)−2 u0n (t) − |u0 (t)|p(t)−2 u0 (t) (u0n (t) − u0 (t))   2−p(t) 2 × |u0n (t)|p(t) + |u0 (t)|p(t) √

where c =

2(2+p+ −p− ) . 2(p− −1)

2 ,I 2−p(t) 1

,

From (2.9) we get

  p(t)
2 |u0n (t)|p(t)−2 u0n (t) − |u0 (t)|p(t)−2 u0 (t) (u0n (t) − u0 (t)) Now using

R I1

2 ,I p(t) 1

2 ,I p(t) 1


|u0n (t)|p(t) + |u0 (t)|p(t) dt is bounded we get Z |u0n (t) − u0 (t)|p(t) dt → 0 as n → ∞.

→0

as n → ∞.

(2.10)

I1

Z

On I2 , from the inequality (1.4) we have Z
+ |u0n (t)−u0 (t)|p(t) dt ≤ 2p |u0n (t)|p(t)−2 u0n (t)−|u0 (t)|p(t)−2 u0 (t) (u0n (t)−u0 (t)) dt,

I2

so

I2

Z

|u0n (t) − u0 (t)|p(t) dt → 0

as n → ∞.

(2.11)

I2

From (2.10) and (2.11) and by Proposition 1.4, we conclude that kun − uk → 0 as n → ∞, so ϕ satisfies the Palais-Smale condition. + 1,p(t) Notice that W01,p (0, T ) ⊂ W0 (0, T ). Consider {e1 , e2 , . . .}, a Schauder basis + 1,p of the space W0 (0, T ) (see [18]), and for each k ∈ N, consider Xk , the subspace of + W01,p (0, T ) generated by the k vectors {e1 , e2 , . . . , ek }. Clearly Xk is a subspace 1,p(t) of W0 (0, T ). For r > 0, consider k n o X Kk (r) = u ∈ Xk : kuk2 = ξi2 = r2 . i=1

640

A. Mokhtari, T. Moussaoui, D. O’Regan

For any r > 0. We consider the odd homeomorphism χ : Kk (r) → S k−1 defined by χ(u) = (ξ1 , ξ2 , . . . , ξk ), where S k−1 is the sphere in Rk . From Theorem 1.7 and Proposition 1.8, we conclude that γ(Kk (r)) = k. Let 0 < r < c10 where c0 is the best constant of the embedding of X in C([0, T ]), so kuk∞ ≤ c0 kuk < 1. It follows from RT hypothesis (h2 ) that 0 H(t, u(t)) dt > 0 for any u ∈ Kk (r). Then ZT µk =

H(t, u(t)) dt

inf u∈Kk (r) 0

is strictly positive (note the compactness of Kk (r)). If we set

νk =

inf u∈Kk (r)

u(t Z j) l X Ij (s) ds, j=1 0

we see that νk ≤ 0. Let λk =

1 µk



 a p− b 2p− r r + − ν k , p− 2(p− )2

and note λk > 0. Then when λ > λk , we take 0 < r ≤ 1, and then for any u ∈ Kk (r) we have kuk ≤ 1 and a ϕ(u) ≤ − p

ZT

0

|u (t)|

p(t)

b  dt + 2(p− )2

0

ZT

2 |u0 (t)|p(t) dt − νk − λµk

0

− − a b ≤ − kukp + kuk2p − νk − λµk p 2(p− )2 − b a − r2p − νk − λk µk = 0. < − rp + p 2(p− )2

Theorem 1.10 guarantees that the functional ϕ has at least k pairs of different critical points. Hence, Problem (1.1) has at least k distinct pairs of nontrivial solutions. Corollary 2.2. Assume that (h2 ) and (h3 ) hold, and (h4 ) h(t, u) is bounded. Then for any k ∈ N, there exists a λk such that for any λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions. Theorem 2.3. Assume that (h1 ) holds, and assume the following are satisfied. (h5 ) There exist αj , βj > 0 and γj with 0 < γj < 2p− − 1 (j = 1, 2, . . . , l) such that |Ij (u)| ≤ αj + βj |u|γj

for any u ∈ R (j = 1, . . . , l).

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 641

(h6 ) h(t, u) and Ij (u) (j = 1, 2, . . . , l) are odd with respect to u and H(t, u) > 0 for every (t, u) ∈ [0, T ] × R \ {0}. Then for any k ∈ N, there exists a λk such that for any λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions. Proof. From assumptions (h1 ) and (h6 ), we see that ϕ ∈ C 1 (X, R) is an even functional and ϕ(0) = 0. We now show that ϕ is bounded from below. Let α0 = max{α1 , α2 , . . . , αl }, β0 = max{β1 , β2 , . . . , βl }. We have for any u ∈ X with kuk > 1 that ZT ϕ(u) = a

ZT 2 1 1 b 0 p(t) |u (t)| |u0 (t)|p(t) dt dt + p(t) 2 p(t)

0

0

ZT −λ

H(t, u(t)) dt −

l X

u(t Z j)

Ij (s) ds

j=1 0

0

− − b a kuk2p − ≥ + kukp + p 2(p+ )2

ZT

(α(t)|u(t)| + β(t)|u(t)|)γ(t)+1

(2.12)

0

−

l X

(αj |u(tj )| + βj |u(tj )|γj +1 )

j=1 + − − b a kuk2p − |α|L1 c0 kuk − |β|L1 ckukγ +1 ≥ + kukp + + 2 p 2(p )

− α0 lc0 kuk − β0 c0

l X

kukγj +1 .

j=1

Now, we show that ϕ satisfies the Palais-Smale condition. For any u ∈ X with kuk ≤ 1, we have − + + a b ϕ(u) ≥ + kukp + kuk2p − |α|L1 c0 kuk − |β|L1 ckukγ +1 + 2 p 2(p ) (2.13) l X − α0 lckuk − β0 c0 kukγj +1 . j=1

Let (un ) ⊂ X be a sequence such that (ϕ(un )) is a bounded sequence and ϕ0 (un ) → 0 in X 0 . From (2.12), (2.13), and since γ + , γj < 2p− − 1, in all cases we deduce that (un ) is bounded in X. The rest of the proof of the Palais-Smale condition is similar to that in Theorem 2.1. Consider Kk (r) as in Theorem 2.1. For any r > 0, there exits an odd homeomorphism χ : Kk (r) → S k−1 . From assumption (h6 ) we have ZT µk =

H(t, u(t)) dt > 0.

inf u∈Kk (r) 0

642

A. Mokhtari, T. Moussaoui, D. O’Regan

Let u(t  Z j) l  X b 1  a p− 2p− r + r − νk . νk = inf Ij (s) ds and λk = max 0, µk p− 2(p− )2 u∈Kk (r) j=1 0

Then when λ > λk , we take 0 < r ≤ 1, and then for any u ∈ Kk (r) we have kuk ≤ 1 and − b a p− r + r2p − νk − λµk − − 2 p 2(p ) − a p− b < −r + r2p − νk − λk µk ≤ 0. − 2 p 2(p )

ϕ(u) ≤

From Theorem 1.10, ϕ has at least k pairs of different critical points. Consequently, Problem (1.1) has at least k distinct pairs of nontrivial solutions. Corollary 2.4. Assume that the assumptions (h4 ), (h6 ) hold, and (h7 ) Ij (u) (j = 1, 2, . . . , l) are bounded. Then for any k ∈ N, there exists a λk such that for any λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions. Theorem 2.5. Assume that (h3 ) holds, and (h8 ) There exists a constant σ > 0 such that h(t, σ) = 0, h(t, u) > 0 for every u ∈ (0, σ), (h9 ) h(t, u) is odd with respect to u. Then for any k ∈ N, there exists a λk such that when λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions. Proof. Define the bounded function ( 0, f (t, u) = h(t, u), Consider   RT    − a + b 0

1 0 p(t) p(t) |u (t)|

−∆u0 (tj ) = Ij (u(tj )),     u(0) = u(T ) = 0,

if |u| > σ, if |u| ≤ σ.

 0 dt |u0 (t)|p(t)−2 · u0 (t) = λf (t, u(t)), j = 1, 2, . . . , l,

t 6= tj , t ∈ [0, T ], (2.14)

and we now show that solutions of Problem (2.14) are also solutions of Problem (1.1). Let u0 be a solution of Problem (2.14). We now prove that −σ ≤ u0 (t) ≤ σ. Suppose

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 643

that max0≤t≤T u0 (t) > σ, then there exists an interval [d1 , d2 ] ⊂ [0, T ] such that u0 (d1 ) = u0 (d2 ) = σ and for any t ∈ (d1 , d2 ) we have u0 (t) > σ, and so ZT   0 1 − a+b |u0 (t)|p(t) dt |u0 (t)|p(t)−2 · u0 (t) = λf (t, u(t)) = 0, p(t)

t ∈ (d1 , d2 ).

0

We deduce that there is a constant c such that |u00 (t)|p(t)−2 ·u00 (t) = c for any t ∈ [d1 , d2 ], and since u00 (d1 ) ≥ 0 and u00 (d2 ) ≤ 0, then we have |u00 (d1 )|p(d1 )−2 · u00 (d1 ) = c ≥ 0, |u00 (d2 )|p(d2 )−2 · u00 (d2 ) = c ≤ 0, so, u00 (t) = 0 for any t ∈ [d1 , d2 ], i.e. u0 (t) = σ for any t ∈ [d1 , d2 ], which is a contradiction. From a similar argument we see that min0≤t≤T u0 (t) ≥ −σ. The functional ϕ1 : X → R defined by ZT ϕ1 (u) = a

ZT 2 b 1 1 0 p(t) |u (t)| dt + |u0 (t)|p(t) dt p(t) 2 p(t) 0

0

−

l X

u(t Z j)

(2.15)

ZT

Ij (s) ds − λ

j=1 0

F (t, u(t)) dt 0

is continuously Fréchet differentiable at any u ∈ X, where F (t, u) = We have ϕ01 (u)

ZT



·v = a+b

−

0

f (t, s) ds.

 ZT 1 0 p(t) |u (t)| dt |u0 |p(t)−2 u0 (t)v 0 (t) dt p(t) 0

0 l X

Ru

(2.16)

ZT Ij (u(tj ))v(tj ) − λ

j=1

f (t, u(t))v(t) dt 0

for all v ∈ X. It is clear that ϕ1 is an even functional, ϕ1 (0) = 0 and bounded from below. To see this note for u ∈ X with kuk ≥ 1 we have a ϕ1 (u) ≥ + p ≥

a p+

ZT

b  |u0 (t)|p(t) dt + 2(p+ )2

ZT

0

0

ZT

 ZT

0

|u0 (t)|p(t) dt +

b 2(p+ )2

0

u(t) ZT Z 2 f (t, s) dsdt |u0 (t)|p(t) dt − λ 0

2 |u0 (t)|p(t) dt − λ

f (t, s) dsdt 0

| ≥

− − a b kukp + kuk2p − λτ, + + 2 p 2(p )

0

ZT Zσ 0

{z

=τ >0

}

644

A. Mokhtari, T. Moussaoui, D. O’Regan

so it follows that ϕ1 is bounded from below. Let (un ) be a Palais-Smale sequence. It is easy to see that (un ) is bounded in X and the rest of the proof of the Palais-Smale condition is similar to that in the proof in Theorem 2.1. Consider Kk (r) = {u ∈ Xk : kuk = r}. For any r > 0 the odd homeomorphism χ : Kk (r) → S k−1 gives γ(Kk (r)) = k. Let 0 < r < min{1, cσ0 }, where c0 is the best constant of the embedding of X in C([0, T ]), so kuk∞ ≤ c0 kuk < σ for any u ∈ Kk (r). Using assumptions (h8 ) and (h9 ), we have F (t, u(t)) > 0 as u(t) 6= 0. Then RT F (t, u(t)) dt > 0 for any u ∈ Kk (r). If we set 0 ZT µk =

F (t, u(t)) dt

inf u∈Kk (r)

and νk =

0

inf u∈Kk (r)

u(t Z j) l X Ij (s) ds, j=1 0

then µk > 0 and νk ≤ 0. Let λk =

1 µk



 a p− b 2p− r + r − ν k , p− 2(p− )2

so λk > 0, and for any λ > λk and any u ∈ Kk (r) with kuk < 1 we have − a p− b r + r2p − νk − λµk − − 2 p 2(p ) − a p− b < −r + r2p − νk − λk µk = 0. − 2 p 2(p )

ϕ(u) ≤

From Theorem 1.10, ϕ1 has at least k pairs of different critical points. Then, Problem (2.14) has at least k distinct pairs of nontrivial solutions. Consequently, Problem (1.1) has at least k distinct pairs of nontrivial solutions. A similar argument to that in Theorem 2.5, yields the following result. Theorem 2.6. Let conditions (h5 ), (h8 ) hold and (h10 ) h(t, u) and Ij (u) (j = 1, 2, . . . , l) are odd with respect to u. Then for any k ∈ N, there exists a λk such that for any λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions. Theorem 2.7. Suppose that (h5 ) holds, and (h11 ) There exist a constant σ1 > 0 such that h(t, σ1 ) ≤ 0, (h12 ) h(t, u) and Ij (u) (j = 1, 2, . . . , l) are odd with respect to u and limu→0 uniformly for t ∈ [0, T ].

h(t,u) u

=1

Then for any k ∈ N, there exists a λk such that for any λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions.

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 645

Proof. Define the bounded function   h(t, σ1 ), g(t, u) = h(t, u),   h(t, −σ1 ),

if u > σ1 , if |u| ≤ σ1 , if u < −σ1 .

We will verify that the solutions of the problem   RT   − a + b 


0 dt |u0 (t)|p(t)−2 · u0 (t) + λg(t, u(t)) = 0, t 6= tj , t ∈ [0, T ], 0 (2.17) 0 −∆u (t ) = Ij (u(tj )), j = 1, 2, . . . , l,  j    u(0) = u(T ) = 0, 1 0 p(t) p(t) |u (t)|

are solution of Problem (1.1). Let u0 be a solution of Problem (2.17). We prove that −σ1 ≤ u0 (t) ≤ σ1 for any t ∈ [0, T ]. Suppose that max0≤t≤T u0 (t) > σ1 , then there exists an interval [d1 , d2 ] ⊂ [0, T ] such that u0 (d1 ) = u0 (d2 ) = σ1 and for any t ∈ (d1 , d2 ) we have u0 (t) > σ1 , and then when t ∈ (d1 , d2 ) we obtain 

ZT a+b

 0 1 |u00 (t)|p(t) dt |u00 (t)|p(t)−2 · u00 (t) = −λg(t, u0 (t)) = −λh(t, σ1 ) ≥ 0. p(t)

0

Therefore, we deduce that 

0 |u00 (t)|p(t)−2 · u00 (t) ≥ 0,

t ∈ (d1 , d2 ),

thus t 7→ |u00 (t)|p(t)−2 · u00 (t) is nondecreasing in (d1 , d2 ), so then 0 ≤ |u00 (d1 )|p(d1 )−2 u00 (d1 ) ≤ |u00 (t)|p(t)−2 u00 (t) ≤ |u00 (d2 )|p(d2 )−2 u00 (d2 ) ≤ 0, for every t ∈ [d1 , d2 ]. Hence u00 = 0 on [d1 , d2 ], so, since u0 (d1 ) = u0 (d2 ) = σ1 , then u(t) = σ1 for every t ∈ [d1 , d2 ], which is a contradiction. From a similar argument we see that min0≤t≤T u0 (t) ≥ −σ1 , i.e. u0 is a solution of Problem (1.1). We consider the functional ϕ2 : X → R defined by ZT ϕ2 (u) = a

ZT 2 1 b 1 0 p(t) |u (t)| dt + |u0 (t)|p(t) dt p(t) 2 p(t)

0

−

l X

0 u(t Z j)

j=1 0

ZT

Ij (s) ds − λ

G(t, u(t)) dt, 0

(2.18)

646

A. Mokhtari, T. Moussaoui, D. O’Regan

where G(t, u) =

Ru

g(t, s) ds. Obviously, ϕ2 is continuously Fréchet differentiable at

0

any u ∈ X and ZT   ZT 1 ϕ02 (u) · v = a + b |u0 (t)|p(t) dt |u0 (t)|p(t)−2 u0 (t)v 0 (t) dt p(t) 0

0

−

l X

(2.19)

ZT Ij (u(tj ))v(tj ) − λ

j=1

g(t, u(t))v(t) dt, 0

for all v ∈ X. It is clear that critical points of ϕ2 are solutions of Problem (2.17). Now ϕ2 ∈ C 1 (X, R) is an even functional and ϕ2 (0) = 0. Let α0 = max{α1 , α2 , . . . , αl }, β0 = max{β1 , β2 , . . . , βl }, and we see that ZT

u(t) ZT Z ZT Zσ1 G(t, u(t)) dt = g(t, s) dsdt ≤ g(t, s) dsdt = η.

0

0

0

0

(2.20)

0

Using assumption (h5 ) and (2.20), we have for any u ∈ X with kuk > 1 that ZT ϕ2 (u) = a

ZT 2 b 1 1 |u0 (t)|p(t) dt + |u0 (t)|p(t) dt p(t) 2 p(t) 0

0

−

l X

u(t Z j)

ZT

Ij (s) ds − λ

j=1 0

≥

≥

G(t, u(t)) dt (2.21)

0 l X

− − a b kukp + kuk2p − (αj |u(tj )| + βj |u(tj )|γj +1 ) − λη p+ 2(p+ )2 j=1

l X b a p− 2p− 0 kuk + kuk − α lc kuk − β c kukγj +1 − λη, 0 0 0 p+ 2(p+ )2 j=1

so it follows that ϕ2 is bounded from below. Now we show that ϕ2 satisfies the Palais-Smale condition. For any u ∈ X with kuk ≤ 1, we have ϕ(u) ≥

+ + a b kukp + kuk2p − α0 lc0 kuk + + 2 p 2(p )

− β0 c0

l X

(2.22) kukγj +1 − λη.

j=1

Let (un ) ⊂ X be a sequence such that (ϕ(un )) is a bounded sequence and ϕ0 (un ) → 0 as n → +∞. From (2.21), (2.22), and since γ, γj < 2p− − 1, in all cases we deduce

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 647

that (un ) is bounded in X. The proof of the Palais-Smale condition is now similar to that in Theorem 2.1. Consider Kk (r) as in Theorem 2.1. From assumption (h12 ), for any ε > 0, there exists δ > 0, when |u| ≤ δ, we have h(t, u) ≥ u − ε|u|. Take 0 < r ≤ 1 sufficiently small such that kuk∞ < min{σ1 , δ} for any u ∈ Kk (r). Then, taking 0 < ε < 1 we have ZT

u(t) ZT Z ZT 1 G(t, u(t)) dt = g(t, s) dsdt ≥ (1 − ε)|u(t)|2 dt > 0, 2

0

0

0

0

for any u ∈ Kk (r). Set ZT µk =

G(t, u(t)) dt

inf u∈Kk (r)

and νk =

inf u∈Kk (r)

0

 − Let λk = max 0, µ1k pa− rp + and every u ∈ Kk (r), we have

b 2p− 2(p− )2 r

− νk

u(t Z j) l X Ij (s) ds. j=1 0


, then for all λ such that λ > λk

− a p− b r + r2p − νk − λµk p− 2(p− )2 − − a b < − kukp + kuk2p − νk − λk µk ≤ 0. − 2 p 2(p )

ϕ2 (u) ≤

Theorem 1.10 guarantees that ϕ2 has at least k pairs of different critical points. That is, Problem (2.17) has at least k distinct pairs of nontrivial solutions. Therefore we have the same result for Problem (1.1). Theorem 2.8. Assume that (h11 ) and (h12 ) hold, and Ru (h13 ) 0 Ij (s) ds ≤ 0 for any u ∈ R (j = 1, . . . , l). Then for any k ∈ N, there exists a λk such that for any λ > λk , Problem (1.1) has at least k distinct pairs of nontrivial solutions. Proof. The argument is similar to that in Theorem 2.7.

REFERENCES [1] A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, Cambridge, 2007. [2] A. Castro, Metodos Variacionales y Analisis Funcional no Lineal, X Coloquio Colombiano de Matematicas, 1980. [3] J. Chabrowski, Y. Fu, Existence of solutions for p(t)-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), 604–618.

648

A. Mokhtari, T. Moussaoui, D. O’Regan

[4] H. Chen, J. Li, Variational approach to impulsive differential equation with Dirichlet boundary conditions, Bound. Value Probl. 2010 (2010), 1–16. [5] D.C. Clarke, A variant of the Lusternik-Schnirelmann theory, Indiana Univ. Math. J. 22 (1972), 65–74. [6] F.J.S.A. Corrêa, G.M. Figueiredo, On an elliptic equation of p-Kirchhoff-type via variational methods, Bull. Aust. Math. Soc. 74 (2006), 263–277. [7] F.J.S.A. Corrêa, G.M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii’s genus, Appl. Math. Lett. 22 (2009), 819–822. [8] G. Dai, J. Wei, Infinitely many non-negative solutions for a p(t)-Kirchhoff-type problem with Dirichlet boundary condition, Nonlinear Anal. 73 (2010), 3420–3430. [9] L. Diening, P. Harjulehto, P. Hastö, M. Ružička, Lebesgue and Sobolev spaces with variable exponents, Lectures Notes in Mathematics, 2011. [10] X.L. Fan, Q.H. Zhang, Existence of solutions for p(t)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1852. [11] X.L. Fan, D. Zhao, On the spaces Lp(t) and W m,p(t) , J. Math. Anal. Appl. 263 (2001), 424–446. [12] M. Galewski, D. O’Regan, Impulsive boundary value problems for p(t)-Laplacian’s via critical point theory, Czechoslovak Math. J. 62 (2012), 951–967. [13] M. Galewski, D. O’Regan, On well posed impulsive boundary value problems for p(t)-Laplacian’s, Math. Model. Anal. 18 (2013), 161–175. [14] O. Kavian, Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, Springer-Verlag, 1993. [15] M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964. [16] I. Peral, Multiplicity of solutions for the p-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations, ICTP, Trieste, 1997. [17] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Conference board of the mathematical sciences, American Mathematical Society, Providence, Rhode Island, 1984. [18] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

A. Mokhtari [email protected] Laboratory of Fixed Point Theory and Applications Department of Mathematics E.N.S. Kouba, Algiers, Algeria

Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type. . . 649

T. Moussaoui [email protected] Laboratory of Fixed Point Theory and Applications Department of Mathematics E.N.S. Kouba, Algiers, Algeria

D. O’Regan [email protected] School of Mathematics, Statistics and Applied Mathematics National University of Ireland Galway, Ireland

Received: January 20, 2016. Revised: March 12, 2016. Accepted: March 30, 2016.