Czechoslovak Mathematical Journal, 62 (137) (2012), 951–967

IMPULSIVE BOUNDARY VALUE PROBLEMS FOR p(t)-LAPLACIAN’S VIA CRITICAL POINT THEORY Marek Galewski, Lód´z, Donal O’Regan, Galway (Received March 23, 2011)

Abstract. In this paper we investigate the existence of solutions to impulsive problems with a p(t)-Laplacian and Dirichlet boundary value conditions. We introduce two types of solutions, namely a weak and a classical one which coincide because of the fundamental lemma of the calculus of variations. Firstly we investigate the existence of solution to the linear problem, i.e. a problem with a fixed rigth hand side. Then we use a direct variational method and next a mountain pass approach in order to get the existence of at least one weak solution to the nonlinear problem. Keywords: p(t)-Laplacian, impulsive condition, critical point, variational method, Dirichlet problem MSC 2010 : 34B37, 47J30

1. Introduction The study of impulsive boundary value problems is important due to its various applications in which abrupt changes at certain times in the evolution process appear. The dynamics of evolving processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of “impulses”. Such problems arise in physics, population dynamics, biotechnology, pharmacokinetics, industrial robotics, see [1]. Another vital area of research within boundary value problems is the investigation of the so called p(x)-Laplacian problems which began in [3] and was later developed by many authors, see [7] for an up to date study of such boundary value problems. Such problems model various phenomena arising in the study of elastic mechanics (see [16]), electrorheological fluids (see [11]) or image restoration (see [2]). 951

Therefore we believe it is important to investigate a Dirichlet problem with a p(t)Laplacian, i.e. a one—dimensional counterpart of the p(x)-Laplacian, subject to some impulsive changes. In our research we mainly follow the approach applied in [10] with one significant difference. The existence results for problems with a fixed right hand side in [10] were proved via the Lax-Milgram Lemma and in our paper we apply a direct method of the calculus of variations together with the Fundamental Lemma of the calculus of variations which we prove in the case of functions from relevant Orlicz-Sobolev spaces. It is the variational approach for boundary value problems with a p(x)-Laplacian that prevails in the literature, see again [7], while for impulsive problems, the variational approach has only recently begun and most results have been obtained by other methods, see [5], [8]. The variational investigation of impulsive problems inspired by [10] has received a lot of attention recently. In [6] another variational framework for the SturmLiuville boundary value problem is developed in the case of second order impulsive ordinary differential equation of p-Laplacian type. Boundary value problems with dependence on a first order derivative are developed in [12], while [15] considers problems similar to those of [10] but for a slightly more general problem. Periodic solutions with impulses are considered via critical point theory in [14] within the framework sketched in [10]. The paper is organized as follows. Firstly, we consider the Fundamental Lemma of calculus of variations for the Orlicz-Sobolev spaces. Next, we investigate the problem with a fixed right hand side. Later we investigate nonlinear problems by the direct method of calculus of variations and by Mountain Pass Geometry. 2. Mathematical preliminaries Let p, q ∈ C([0, π], R+ ), 1/p(t) + 1/q(t) = 1 for t ∈ [0, π]. In this paper we assume p = inf p(t) > 1, p+ = sup p(t) 6 2. By Lp(t) (0, π) we mean the space −

t∈[0,π]

t∈[0,π]

 p(t) L ([0, π]) = u ; u : [0, π] → R

is measurable,

Z

π p(t)

|u(t)|

0

 dt < +∞

equipped with the norm kukLp(t)

 Z = inf λ > 0 ;

π 0

 u(t) p(t) dt 6 1 λ

and this is a Banach space called the generalized Lebesgue space, see [4]. C0∞ (0, π) denotes the space of infinitely many times differentiable functions with compact support on [0, π]. W 1,p(t) (0, π) is the generalized Orlicz-Sobolev space, 952

namely W 1,p(t) (0, π) =

 Z u ; u ∈ Lp(t) (0, π),

0

π

 d p(t) dt < +∞ , u(t) dt

where the derivative d/dt stands for the weak one, i.e. (d/dt)u is an element of Lp(t) (0, π) which satisfies Z π Z π d d (1) u(t)v(t) dt = − u(t) v(t) dt dt 0 dt 0 for all v ∈ C0∞ (0, π). It is apparent that any function belonging to W 1,p(t) (0, π) is in fact absolutely continuous and so the weak derivative can be considered as an a.e. derivative which is what we understand for the remainder of this paper. Consider W 1,p(t) (0, π) with the following norm (2)

kukW 1,p(t) 1,p(t)

r

d = u p(t) + kukLp(t) . dt L

Now W0 (0, π) is the closure of C0∞ (0, π) in W 1,p(t) (0, π), see [4]. The norm in 1,p(t) W0 (0, π) is

d

kukW 1,p(t) = u 0 dt Lp(t) which is equivalent to (2). Moreover, from [4] we see that there exist constants C1 , C2 > 0 such that (the Poincaré inequality)

and

d

kukLp(t) 6 C1 u p(t) dt L

1,p(t)

for all u ∈ W0

(0, π)

d

1,p(t) max |u(t)| 6 C2 u for all u ∈ W0 (0, π). dt Lp(t) t∈[0,π] Rπ 1,p(t) The functional u → 0 |(d/dt)u(t)|p(t) dt is called the modular for W0 (0, π). We have the following relation between the modular and the norm p(t)

d p+ o Z π d n d p−



min u , u 6 u(t) dt dt Lp(t) dt Lp(t) dt 0

d p+ o n d p−



6 max u p(t) , u p(t) . dt L dt L 1,p(t)

Let us consider an operator L : W0 (3)

hL(u), vi =

Z

0

π

1,p(t)

(0, π) → (W0

(0, π))∗ given by

d p(t)−2 d d u(t) v(t) dt u(t) dt dt dt

953

1,p(t)

for u, v ∈ W0 (0, π). Then L is a homeomorphism ([3], Theorem 3.1) and the Rπ Gˆateaux derivative of u → 0 |(d/dt)u(t)|p(t) dt is given by (3). 3. Remarks on the fundamental lemma of the calculus of variations Now we provide a classical lemma of the calculus of variations which we formulate for generalized Orlicz-Sobolev spaces in order to get some regularity results for p(t)Laplacian problems different from known results in the literature. Note that the proofs of subsequent lemmas are modifications of well known proofs in the area of the calculus of variations (see [13]). We note that in this paper we have assumed p+ 6 2 1,q(t) 1,p(t) and p− > 1 so Lq(t) (0, π) ⊂ Lp(t) (0, π). Thus we see that W0 (0, π) ⊂ W0 (0, π). Lemma 1. If h ∈ Lq(t) (0, π) and Z π d (4) h(t) v(t) dt = 0 dt 0 1,p(t)

for all v ∈ W0 [0, π].

(0, π), then there exists a constant c ∈ R such that h(t) = c a.e. on

Rπ Rt P r o o f. Put c = π−1 0 h(t) dt. For this c the function v(t) = 0 (h(τ ) − c) dτ 1,p(t) 1,q(t) 1,q(t) is in W0 (0, π) ∩ W0 (0, π) = W0 (0, π). Indeed, (d/dt)v(t) = h(t) − c a.e. on [0, π], v(0) = v(π) = 0 and the derivative is understood here as a classical a.e. derivative. From (4) we obtain that Z π Z π d 06 (h(t) − c)2 dt = (h(t) − c) v(t) dt dt 0 Z0 π Z π d d = h(t) v(t) dt + c v(t) dt = 0. dt 0 0 dt Hence it follows that for almost all t ∈ [0, π] we have h(t) − c = 0.



Lemma 2. If g ∈ L1 (0, π), h ∈ Lq(t) (0, π) and Z π  d g(t)v(t) + h(t) v(t) dt = 0 dt 0 1,p(t)

(0, π), then (d/dt)h = g a.e. on [0, π] and (d/dt)h ∈ L1 (0, π). Rt P r o o f. We define G(t) = 0 g(τ ) dτ . Then G is absolutely continuous and (d/dt)G(t) = g(t) for a.e. t ∈ [0, π]. Integrating by parts we obtain that Z π Z π  d d g(t)v(t) + h(t) v(t) dt = (h(t) − G(t)) v(t) dt + G(t)v(t)|π0 dt dt 0 Z0 π d = (h(t) − G(t)) v(t) dt = 0. dt 0

for all v ∈ W0

954

Thus by Lemma 1 we obtain that h(t)− G(t) = c and the assertion of the proposition follows. 

4. Existence with fixed right hand side Let the numbers 0 < t1 < t2 < . . . < tm < π be fixed throughout the paper. Let 1,p(t) us consider the following problem in W0 (0, π) −

(5)

p(t)−2 d  d  d x(t) = f (t) x(t) dt dt dt x(0) = x(π) = 0

with impulsive conditions (6)

d p(t)−2 d d p(t)−2 d x(t+ x(t− x(t− ) = dj x(t+ j ) j )− j ) dt dt dt dt j

for j = 1, 2, . . . , m

and where f ∈ L1 (0, π). In the condition (6) it is assumed that both limits lim+ |(d/dt)x(t)|p(t)−2 (d/dt)x(t), lim− |(d/dt)x(t)|p(t)−2 (d/dt)x(t) exist and the ret→tj

t→tj

quired equality holds. We introduce two types of solution for the problem (5)–(6). 1,p(t) Weak solution: We call a function x ∈ W0 (0, π) a weak solution to (5)–(6) if it satisfies (7)

Z

0

π

Z π m d p(t)−2 d X d f (t)v(t) dt = 0 x(t) x(t) v(t) dt + d v(t ) dt − j j dt dt dt 0 j=1 1,p(t)

for all v ∈ W0 (0, π). 1,p(t) Classical solution: A function x ∈ W0 (0, π) is called a classical solution to (5)–(6) if it is a weak solution such that the function |(d/dt)x(·)|p(·)−2 (d/dt)x(·) is absolutely continuous on [0, π], the limits in (6) are defined and the relation (6) holds together with the boundary condition x(0) = x(π) = 0 and −

p(t)−2 d  d  d x(t) = f (t) x(t) dt dt dt

is satisfied for a.e. t ∈ [0, π] and (d/dt)(|(d/dt)x(t)|p(t)−2 (d/dt)x(t)) ∈ L1 (0, π).

955

Lemma 3. If u ¯ is a weak solution to (5)–(6), then it is also a classical one. P r o o f. Let the function u ¯ satisfy (7), i.e. (8)

Z

0

π

Z π m d p(t)−2 d X d ¯(t) u ¯(t) v(t) dt + dj v(tj ) − f (t)v(t) dt = 0 u dt dt dt 0 j=1 1,p(t)

for all v ∈ W0 (0, π). Now, we shall show that u ¯ is a classical solution. Let us take any interval (tj , tj+1 ) 1,p(t) 1,p(t) and a function v ∈ W0 (tj , tj+1 ). Extend function v to W0 (0, π) by taking 0 outside (tj , tj+1 ). Then we have from (8) Z

tj+1

tj

Z tj+1 d p(t)−2 d d ¯(t) u ¯(t) v(t) dt − f (t)v(t) dt = 0. u dt dt dt tj

An application of Lemma 2 shows that the function |(d/dt)¯ u(·)|p(·)−2 (d/dt)¯ u(·) is absolutely continuous, its derivative exists for a.e. t ∈ (tj , tj+1 ) and p(·)−2 d  d  d u ¯(·) ∈ L1 (tj , tj+1 ). u¯(·) dt dt dt (tj ,tj+1 )

Therefore both limits

d p(t)−2 d lim+ u ¯(t) u ¯(t), dt t→tj dt

d p(t)−2 d lim u ¯ (t) u ¯(t) dt dt t→t− j+1

exist. Taking intervals (tj , tj+1 ) for j = 0, 1, 2, . . . , m we see that (5) is satisfied a.e. on [0, π] and that (d/dt)(|(d/dt)¯ u(·)|p(·)−2 (d/dt)¯ u(·)) ∈ L1 (0, π). Hence, we may 1,p(t) multiply (5) with x = u ¯ by any v ∈ W0 (0, π) and obtain Z

0

π

−

Z π p(t)−2 d d d ¯(t) u ¯(t)v(t) dt − f (t)v(t) dt = 0. u dt dt dt 0

By integrating by parts it follows for a fixed interval [tj , tj+1 ] that (9)

Z

tj+1

tj

956

p(t)−2 d  d  d ¯(t) u ¯(t) v(t) dt u dt dt dt Z tj+1 d d p(t)−2 d = ¯(t) u ¯(t) v(t) u dt dt dt tj d p(t)−2 d d p(t)−2 d + + − + u¯(t+ ) u ¯ (t )v(t ) − u ¯ (t ) u ¯(t− )v(t− j j+1 ). dt j dt j dt j+1 dt j+1 −

Note that m p(t)−2 d X d − u ¯(t− u¯(t− j+1 ) j+1 )v(tj+1 ) dt dt j=0

m p(t)−2 d d X + + ) u ¯ (t )v(t ) = dj v(tj ). − u¯(t+ j dt j dt j j=1

Summing (9) for j = 0, 1, 2, . . . , m we get recalling that v is continuous (10)

Z π d p(t)−2 d d ¯(t) u ¯(t) v(t) dt − f (t)v(t) dt u dt dt dt 0 0 m  p(t)−2 d p(t)−2 d d  X d + − − + ¯(t+ ) u ¯ (t ) − u ¯ (t ) u ¯ (t ) v(tj ) = 0. u dt j dt j dt j+1 dt j+1 j=1

Z

π

Since u ¯ is a weak solutions we get equating (8) and (10) that (11)

m  m p(t)−2 d p(t)−2 d d  X X d + − − ¯(t+ ) u ¯ (t )− u ¯ (t ) u ¯ (t ) v(t ) = dj v(tj ). u j dt j dt j dt j+1 dt j+1 j=1 j=1

Since (11) holds for arbitrary v it follows that d p(t)−2 d p(t)−2 d d ¯(t+ u¯(t+ u ¯(t− u ¯(t− ) = dj u j ) j )− j+1 ) dt dt dt dt j+1

for j = 1, 2, . . . , m and therefore the impulsive conditions (6) are also satisfied. 1,p(t)

The action functional J : W0 J(u) =

Z

0

π



(0, π) → R corresponding to (5)–(6) is as follows

Z π m p(t) X 1 d dt + dj u(tj ) − f (t)u(t) dt. u(t) p(t) dt 0 j=1

Lemma 4. J is a Gˆ ateaux differentiable, weakly l.s.c. and coercive functional and its critical points correspond to the classical solutions of (5)–(6). 1,p(t)

P r o o f. Note that J is differentiable. Now we take an arbitrary u ∈ W0 1,p(t) and fix an arbitrary v ∈ W0 (0, π). Then the Gˆateaux derivative is J ′ (u; v) =

Z

0

π

(0, π)

Z π m d p(t)−2 d X d u(t) v(t) dt + dj v(tj ) − f (t)v(t) dt. u(t) dt dt dt 0 j=1

Therefore each critical point of J is a weak solution of (5)–(6). 957

1,p(t)

Let us take any weakly convergent sequence {uk }∞ (0, π). This means k=1 ⊂ W0 ∞ p(t) that {(d/dt)uk }k=1 is weakly convergent in L (0, π). Then {uk }∞ k=1 has a subsep(t) quence {ukn }∞ which is strongly convergent in L (0, π) and convergent in C[0, π]. n=1 1,p(t) ∞ Denote by u ¯ ∈ W0 (0, π) the weak limit of {uk }k=1 . Hence, Z π p(t) 1 d lim inf J(ukn ) > lim inf dt ukn (t) n→∞ n→∞ 0 p(t) dt X  Z π m dj ukn (tj ) dt − f (t)ukn (t) dt > J(¯ u). + lim n→∞

0

j=1

1,p(t)

Hence J is weakly l.s.c. on W0 (0, π). Moreover, we see that for any u ∈ 1,p(t) W0 (0, π) Z π Z π m p(t) X 1 d dt + dj u(tj ) − f (t)u(t) dt u(t) 0 p(t) dt 0 j=1

d p+ o n d p− 1



, min u

u p(t)

p+ dt Lp(t) dt L m X −C2 dj kukW 1,p(t) C1 kukW 1,p(t) kf kLq(t) . >

j=1

0

0

Since p− > 1 we see that J is coercive. 1,p(t) Thus J is Gˆ ateaux differentiable, weakly l.s.c. and coercive on W0 (0, π). There1,p(t) fore there exists u ¯ ∈ W0 (0, π) such that J(¯ u) = inf J(u) and thus u ¯ 1,p(t)

u∈W0

(0,π)

satisfies (8). An application of Lemma 3 finishes the proof.



5. Existence of solutions for a nonlinear problem by a direct variational method In this section we will apply a direct variational argument to a so called nonlinear problem. Let Ij : R → R be continuous functions, j = 1, 2, . . . , m, and let f : [0, π] × R be Rv a Caratheodory function. Let F (t, v) = 0 f (t, τ ) dτ . Assume that (H1) there exist constants Mj for j = 1, 2, . . . , m such that |Ij (v)| 6 Mj

for all v ∈ R;

(H2) there exist a constant α > 0 and functions l ∈ L1 (0, π), s ∈ C[0, π], with 1 < s− 6 s+ < p− , such that for all v ∈ R and a.e. t ∈ [0, π] F (t, v) > −α|v|s(t) + l(t); 958

(H3) for each r > 0 there exist functions gr ∈ L1 (0, π) and hr ∈ L1 (0, π) such that 1,p(t) for all v ∈ W0 (0, π) satisfying kvkW 1,p(t) 6 r and a.e. t∈ [0, π] 0

|F (t, v(t))| 6 gr (t) and |f (t, v(t))| 6 hr (t). 1,p(t)

Now we consider in W0 (12)

−

(0, π) the following problem

p(t)−2 d  d  d x(t) + f (t, x(t)) = 0 x(t) dt dt dt x(0) = x(π) = 0

with impulsive conditions p(t)−2 d d p(t)−2 d d + − ) x(t ) − x(t ) x(t− ) (13) x(t+ dt j dt j dt j dt j = Ij (x(tj )) for j = 1, 2, . . . , m. As before in the condition (13) it is assumed that both limits d p(t)−2 d lim x(t) x(t), dt dt t→t+ j

d p(t)−2 d lim x(t) x(t) dt dt t→t− j

exist and the given equality holds. 1,p(t) We call a function x ∈ W0 (0, π) a weak solution to (12)–(13) if it satisfies (14)

Z

0

π

m d p(t)−2 d X d x(t) v(t) dt + Ij (x(tj ))v(tj ) dt x(t) dt dt dt j=1 Z π + f (t, x(t))v(t) dt = 0 0

1,p(t)

for all v ∈ W0 (0, π). 1,p(t) A function x ∈ W0 (0, π) is called a classical solution to (12)–(13) if it is a weak solution such that the function |(d/dt)x(·)|p(·)−2 (d/dt)x(·) is absolutely continuous on [0, π], the limits in (13) are defined and the relation (13) holds together with the boundary condition x(0) = x(π) = 0 and −

p(t)−2 d  d  d x(t) + f (t, x(t)) = 0 x(t) dt dt dt

is satisfied for a.e. t ∈ [0, π] and (d/dt)(|(d/dt)x(t)|p(t)−2 (d/dt)x(t)) ∈ L1 (0, π). 1,p(t) The action functional J : W0 (0, π) → R corresponding to (12)–(13) is (15)

J(u) =

Z

0

π

Z π m Z u(tj ) p(t) X 1 d dt + Ij (t) dt + F (t, u(t)) dt. u(t) p(t) dt 0 j=1 0 959

Lemma 5. If u ¯ is a weak solution to (12)–(13), then it is also a classical one. P r o o f. Let a function u ¯ satisfy (14). As in the proof of Lemma 3 we take any 1,p(t) 1,p(t) interval (tj , tj+1 ) and a function v ∈ W0 (tj , tj+1 ) extended to W0 (0, π) by taking 0 outside (tj , tj+1 ). Then we have in (14) Z

tj+1 tj

Z tj+1 d p(t)−2 d d ¯(t) u¯(t) v(t) dt + f (t, u ¯(t))v(t) dt = 0. u dt dt dt tj

It follows from (H3) that f (·, u ¯(·))|(tj ,tj+1 ) ∈ L1 (tj , tj+1 ). Then, by Lemma 2 p(·)−2 |(d/dt)¯ u(·)| (d/dt)¯ u(·) is absolutely continuous on [tj , tj+1 ], its derivative exists for a.e. t ∈ (tj , tj+1 ) and belongs to L1 (tj , tj+1 ). Next we obtain Z π Z π p(t)−2 d d d ¯(t) u ¯(t) v(t) dt + f (t, u ¯(t))v(t) dt u dt dt dt 0 0 m  p(t)−2 d d p(t)−2 d  X d + − − ¯(t+ ) u ¯ (t ) − u ¯ (t ) u ¯ (t ) v(tj ) = 0. + u dt j dt j dt j+1 dt j+1 j=1 Since u ¯ is a weak solutions we get equating the above relation and (14) that m  p(t)−2 d p(t)−2 d d  X d + − − ) u ¯ (t ) − u ¯ (t ) u ¯ (t ) v(tj ) u¯(t+ dt j dt j dt j+1 dt j+1 j=1

=

m X

Ij (¯ u)v(tj ).

j=1

Hence for all j = 1, 2, . . . , m we have d p(t)−2 d d p(t)−2 d u ¯(t+ u ¯(t− u ¯(t− ) = Ij (¯ u). u¯(t+ j ) j )− j ) dt dt dt dt j



Lemma 6. J is a Gˆ ateaux differentiable, weakly l.s.c. and coercive functional and its critical points correspond to the classical solutions of (12)–(13). 1,p(t)

P r o o f. By the assumption (H3) we see that J is well defined on W0 (0, π). Again by (H3) we see that J is Gˆ ateaux differentiable. Indeed, it suffices to show Rπ that u → 0 F (t, u(t)) dt is differentiable in the sense of Gˆateaux. Let us fix u ∈ 1,p(t)

1,p(t)

W0 (0, π). Now we fix an arbitrary v ∈ W0 (0, π) and take any ε ∈ (−1, 1). Then there exists a constant r > 0 and functions gr ∈ L1 (0, π) and hr ∈ L1 (0, π) such that ku + εvkW 1,p(t) 6 r and for a.e. t ∈ [0, π] 0

|F (t, v(t))| 6 gr (t) and |f (t, v(t))| 6 hr (t). 960

Rπ Then we can differentiate the auxiliary function g(ε) = 0 F (t, u(t) + εv(t)) dt at 0. This proves the Gˆ ateaux differentiability. 1,p(t) 1,p(t) Let us take an arbitrary u ∈ W0 (0, π) and fix v ∈ W0 (0, π). Then the Gˆateaux derivative is Z π Z π m p(t)−2 d X d d J ′ (u; v) = u(t) v(t) dt + Ij (u)v(tj ) + f (t, u(t))v(t) dt. u(t) dt dt dt 0 0 j=1 Therefore each critical point of J is a weak solution of (12)–(13). 1,p(t) Let us take any weakly convergent sequence {uk }∞ (0, π). This means k=1 ⊂ W0 ∞ p(t) that {(d/dt)uk }k=1 is weakly convergent in L (0, π). Then {uk }∞ k=1 has a subsep(t) which is strongly convergent in L (0, π) and convergent in C[0, π]. quence {ukn }∞ n=1 1,p(t) ∞ Denote by u ¯ ∈ W0 (0, π) the weak limit of {uk }k=1 . Then by continuity we have that m Z u m Z ukn (tj ) ¯(tj ) X X Ij (t) dt → Ij (t) dt. 0

j=1

j=1

0

1,p(t)

Since {ukn }∞ (0, π) there exist a constant r > 0 n=1 is weakly convergent in W0 such that

d



ukn p(t) 6 r dt L for all n ∈ N. Now from (H3) there exists a function gr ∈ L1 (0, π) such that |F (t, ukn (t))| 6 gr (t) for a.e. t ∈ [0, π]. Then by the Lebesgue Dominated Convergence Theorem we get Z π Z π F (t, ukn (t)) dt → F (t, u ¯(t)) dt. 0

0

1,p(t)

Therefore, J is weakly l.s.c. on W0 (0, π). 1,p(t) Moreover, we see that for any u ∈ W0 (0, π) (16)

Z π

d p+ o n d p− 1



J(u) > + min u p(t) , u p(t) − α |u(t)|s(t) dt p dt L dt L 0 Z π m

d X

+ l(t) dt − C2 Mj u p(t) . dt L 0 j=1

Since s+ < p− it follows from (16) that J is coercive. 1,p(t) Thus J is Gˆ ateaux differentiable, weakly l.s.c. and coercive on W0 (0, π). There1,p(t) fore there exists u ¯ ∈ W0 (0, π) such that J(¯ u) = inf J(u) and thus u ¯ 1,p(t)

u∈W0

satisfies (14). An application of Lemma 3 finishes the proof.

(0,π)



961

6. Existence of mountain pass solutions In this section we consider the problem (17)

−

p(t)−2 d  d  d x(t) = f (t, x(t)) x(t) dt dt dt x(0) = x(π) = 0

with impulsive conditions for j = 1, 2, . . . , m (18)

d p(t)−2 d d p(t)−2 d x(t+ x(t− x(t− ) + Ij (x(tj )) = 0, x(t+ j ) j )− j ) dt dt dt dt j

with assumptions which do not yield the coercivity of the action functional J : 1,p(t) W0 (0, π) → R (19)

J(u) =

Z

0

π

Z π m Z u(tj ) p(t) X 1 d u(t) dt − I (t) dt − F (t, u(t)) dt. j p(t) dt 0 j=1 0

Now (H1) reads: Let Ij : R → R be continuous functions, j = 1, 2, . . . , m, and let f : [0, π]×R be a Caratheodory function. We assume that (H1) holds and additionally (H4) there exists a constant θ > p+ such that for v ∈ R, v 6= 0 and a.e. t ∈ [0, π] and all j = 1, 2, . . . , m 0 < θF (t, v) 6 vf (t, v), Z v 0<θ Ij (s) ds 6 vIj (v); 0

(H5) there exist constants β1 , β2 , β1j , β2j , α, αj > 0 with α > p+ , αj > p+ for j = 1, 2, . . . , m and such that for all v ∈ R and a.e. t ∈ [0, π] |f (t, v)| 6 β1 |v|α−1 + β2 , |Ij (v)| 6 β1j |v|αj −1 + β2i ; +

+

(H6) lim |f (t, v)|/|v|p −1 = 0 uniformly for t ∈ [0, π] and lim |Ij (v)|/|v|p −1 = 0 for v→0 v→0 j = 1, 2, . . . , m. Note that with (H1), (H5) the functional J is well defined. The assumption which we employ were already used in [3] in the context of boundary value problems for partial differential equations connected with the so called p(x)-Laplacian. Let us recall some preliminaries. Let E be a Banach space. For any sequence {un } ⊂ E, if {J(un )} is bounded and J ′ (un ) → 0 as n → ∞ possesses a convergent subsequence, then we say J satisfies the Palais-Smale condition—(PS) condition for short.

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Lemma 7 (Mountain Pass Lemma [9]). Let J ∈ C 1 (E, R) satisfy the (PS) condition. Suppose that 1. J(0) = 0; 2. there exist ̺ > 0 and α > 0 such that J(u) > α for all u ∈ E with kuk = ̺; 3. there exist u1 in E with ku1 k > ̺ such that J(u1 ) < α. Then J has a critical value c > α. Moreover, c can be characterized as inf

max

g∈Γ u∈g([1,0])

J(u),

where Γ = {g ∈ C([1, 0], E) : g(0) = 0, g(1) = u1 }. Lemma 8. Suppose (H1), (H4), (H5) hold. Then the functional J given by (19) satisfies the (PS) condition. 1,p(t)

P r o o f. Let us take a sequence {uk }∞ (0, π) such that {J(uk )}∞ k=1 ⊂ W0 k=1 ′ is bounded and kJ (uk )kW 1,q(t) (0,π) → 0 as k → ∞. We shall show that {uk }∞ k=1 has 0 a convergent subsequence. Since kJ ′ (uk )kW 1,q(t) (0,π) → 0, we see that for some ε > 0 there exists k0 with 0 kJ ′ (uk )kW 1,q(t) (0,π) 6 ε for k > k0 . Note that for k > k0 0

d

hJ ′ (uk ), uk i 6 ε uk dt Lp(t)

and ′

hJ (uk ), uk i =

Z

0

π

Z π m d p(t) X − Ij (uk (tj ))uk (tj ) − f (t, uk (t))uk (t) dt. uk (t) dt 0 j=1

Then, we see that −

Z

π

F (t, uk (t)) dt −

0

m Z X j=1

1 >− θ

Z

0

u(tj )

Ij (t) dt

0

π

f (t, uk (t))uk (t) dt +

m X j=1

 Ij (uk (tj ))uk (tj )

Z p(t) 1 1 π d = J ′ (uk ; uk ) − uk (t) θ θ 0 dt

d p+ o n d p− ε 1

d



> − uk − max uk , uk . θ dt θ dt dt Lp(t) Lp(t) Lp(t)

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Since {J(uk )}∞ k=1 is bounded, there exists a constant C such that C > J(uk ). Using the above estimates we obtain C > J(uk ) >

d p+ o  1 n d p− 1



− max u ,

uk p(t) − k p+ θ dt dt Lp(t) L

ε

d

u p(t) . θ dt L

p(t) (0, π). Hence {uk }∞ Hence, it follows that {(d/dt)uk }∞ k=1 is bounded in L k=1 has 1,p(t) a weakly convergent subsequence in W0 (0, π). This means that {(d/dt)uk }∞ k=1 p(t) ∞ is weakly convergent in L (0, π). Then {uk }∞ has a subsequence {u } k n n=1 k=1 which is strongly convergent in Lp(t) (0, π) and convergent in C[0, π]. Denote by 1,p(t) u ¯ ∈ W0 (0, π) the weak limit of {uk }∞ k=1 . We see that as k → ∞

hJ ′ (uk ) − J ′ (¯ u), uk − u ¯i → 0

(20)

and further it follows by a direct calculation that as k → ∞ Z

(21)

π

0

Z π d p(t) p(t) d dt → ¯(t) dt. uk (t) u dt dt 0

Indeed, as k → ∞ we obtain m X

Ij (uk (tj ))uk (tj ) −

Z

π

f (t, uk (t))uk (t) dt

0

j=1

→

m X

Ij (¯ u(tj ))¯ u(tj ) −

Ij (uk (tj ))¯ u(tj ) −

f (t, u ¯(t))¯ u(t) dt,

Z

π

f (t, uk (t))¯ u(t) dt

0

j=1

→

m X

Ij (¯ u(tj ))¯ u(tj ) −

Z

π

f (t, u ¯(t))¯ u(t) dt,

0

j=1

m X

π

0

j=1

m X

Z

Ij (¯ u(tj ))uk (tj ) −

Z

π

f (t, u ¯(t))uk (t) dt

0

j=1

→

m X j=1

Ij (¯ u(tj ))¯ u(tj ) −

Z

π

f (t, u ¯(t))¯ u(t) dt.

0

Hence (20) implies (21) and thus {ukn }∞ n=1 is strongly convergent.

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Lemma 9. Suppose that (H1), (H4)–(H6) hold. Then there exist numbers η, 1,p(t) ξ > 0 such that J(u) > ξ for all u ∈ W0 (0, π) such that kukW 1,p(t) = η. Moreover, 1,p(t)

there exists an element v ∈ W0

0

(0, π) with kukW 1,p(t) > η and such that J(v) < 0. 0

P r o o f. From [4] there exist constants C3 , C4 , C41 , . . . , C4m > 0 with kukLp+ 6 C3 kukW 1,p(t) 0

and kukLα 6 C4 kukW 1,p(t) , 0

kukLαj 6 C4j kukW 1,p(t) 0

1,p(t)

for all u ∈ W0 (0, π). Let us take a small ε > 0. By (H5), (H6) we see for some A(ε), A1 (ε), . . . , Am (ε) > 0 that +

F (t, v) 6 ε|v|p + A(ε)|v|α

for a.e. t ∈ [0, π], v ∈ R

and for j = 1, 2, . . . , m Z

v

+

Ij (s) ds 6 ε|v|p + Aj (ε)|v|αj

for (t, v) ∈ [0, π] × R.

0

Note Z

π

0

F (t, u) dt +

m Z X j=1

u(tj )

+

6 2εC3 kukp

Ij (t) dt

0

1,p(t) W0

+ A(ε)C4 kukα 1,p(t) + W 0

m X j=1

Aj (ε)C4j kuk

αj 1,p(t)

W0

and as a result we see that J(u) >

m  1  X α p+ α − 2εC kuk − A(ε)C kuk − Aj (ε)C4j kuk j 1,p(t) . 1,p(t) 3 4 1,p(t) W0 W0 W0 p+ j=1

Taking ε small enough, we see for some η, ξ > 0, η < 1 that J(u) > ξ for all 1,p(t) u ∈ W0 (0, π) such that kukW 1,p(t) = η. 0 Notice for v ∈ R that m Z v X − Ij (s) ds 6 0. j=1

0

From (H4) if u ∈ R it is easy to see that there exists a constant C5 > 0 such that F (t, u) > C5 |u|θ . 965

1,p(t)

Fix v ∈ W0

(0, π) with v 6= 0. For s ∈ R+ note that

Z π m Z sv(tj ) p(t) X 1 d s v(t) dt − I (t) dt − F (t, sv(t)) dt j 0 0 p(t) dt j=1 0 Z π

+ 1 p+

d p θ 6 − s v p(t) − C5 s |v(t)|θ dt. p dt L 0

J(sv) =

Z

π

Since θ > p+ we see that

lim J(sv) = −∞

s→∞

and the condition 3. of Lemma 7 is satisfied.



Theorem 10. Suppose that (H1), (H4)–(H6) hold. Then the problem (17)–(18) admits at least one nontrivial classical solution. P r o o f. By Lemmas 8 and 9 we see that we can apply Lemma 7 to obtain the existence of at least one nontrivial weak solution. Applying Lemma 5 we see that the solution is in fact a classical one.  Remark 1. From the proof above it is easy to see that one could replace |f (t, v)| 6 β1 |v|α−1 + β2 in (H5) with |f (t, v)| 6 β1 |v|α(t)−1 + β2 , for some function α ∈ C[0, π] with α− > p+ .

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[7] P. Harjulehto, P. Hästö, U. V. Lˆe, M. Nuortio: Overview of differential equations with non-standard growth. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 4551–4574. [8] T. Jankowski: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Appl. Math. Comput. 202 (2008), 550–561. [9] J. Mawhin: Probl`emes de Dirichlet Variationnels non Linéaires. Les Presses de l’Université de Montréal, Montreal, 1987. (In French.) [10] J. J. Nieto, D. O’Regan: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10 (2009), 680–690. [11] M. R˚ užička: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000. [12] K. Teng, Ch. Zhang: Existence of solution to boundary value problem for impulsive differential equations. Nonlinear Anal., Real World Appl. 11 (2010), 4431–4441. [13] J. L. Troutman: Variational Calculus with Elementary Convexity. With the assistence of W. Hrusa. Undergraduate Texts in Mathematics. Springer, New York, 1983. [14] H. Zhang, Z. Li: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl. 11 (2010), 67–78. [15] Z. Zhang, R. Yuan: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 11 (2010), 155–162. [16] V. V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 29 (1987), 33–66; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50 (1986), 675–710. Authors’ addresses: M a r e k G a l e w s k i, Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland, e-mail: [email protected]; D o n a l O ’ R e g a n, School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland, e-mail: [email protected].

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