Open Access

Quasilinear boundary value problem with impulses: variational approach to resonance problem

Boundary Value Problems20142014:64

DOI: 10.1186/1687-2770-2014-64

Received: 5 December 2013

Accepted: 7 March 2014

Published: 24 March 2014

Abstract

This paper deals with the resonance problem for the one-dimensional p-Laplacian with homogeneous Dirichlet boundary conditions and with nonlinear impulses in the derivative of the solution at prescribed points. The sufficient condition of Landesman-Lazer type is presented and the existence of at least one solution is proved. The proof is variational and relies on the linking theorem.

MSC:34A37, 34B37, 34F15, 49K35.

Keywords

quasilinear impulsive differential equations Landesman-Lazer condition variational methods critical point theory linking theorem

1 Introduction

Let p > 1 be a real number. We consider the homogeneous Dirichlet boundary value problem for one-dimensional p-Laplacian
( | u ( x ) | p 2 u ( x ) ) λ | u ( x ) | p 2 u ( x ) = f ( x ) for a.e.  x ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 ,
(1)

where λ R is a spectral parameter and f L p ( 0 , 1 ) , 1 p + 1 p = 1 , is a given right-hand side.

Let 0 = t 0 < t 1 < < t r < t r + 1 = 1 be given points and let I j : R R , j = 1 , 2 , , r , be given continuous functions. We are interested in the solutions of (1) satisfying the impulse conditions in the derivative
Δ p u ( t j ) : = | u ( t j + ) | p 2 u ( t j + ) | u ( t j ) | p 2 u ( t j ) = I j ( u ( t j ) ) , j = 1 , 2 , , r .
(2)
For the sake of brevity, in further text we use the following notation:
φ ( s ) : = | s | p 2 s , s 0 ; φ ( 0 ) : = 0 .

For p = 2 this problem is considered in [1] where the necessary and sufficient condition for the existence of a solution of (1) and (2) is given. In fact, in the so-called resonance case, we introduce necessary and sufficient conditions of Landesman-Lazer type in terms of the impulse functions I j , j = 1 , 2 , , r , and the right-hand side f. They generalize the Fredholm alternative for linear problem (1) with p = 2 .

In this paper we focus on a quasilinear equation with p 2 and look just for sufficient conditions. We point out that there are principal differences between the linear case ( p = 2 ) and the nonlinear case ( p 2 ). In the linear case, we could benefit from the Hilbert structure of an abstract formulation of the problem. It could be treated using the topological degree as a nonlinear compact perturbation of a linear operator. However, in the nonlinear case, completely different approach must be chosen in the resonance case. Our variational proof relies on the linking theorem (see [2]), but we have to work in a Banach space since the Hilbert structure is not suitable for the case p 2 .

It is known that the eigenvalues of
( φ ( u ( x ) ) ) λ φ ( u ( x ) ) = 0 , u ( 0 ) = u ( 1 ) = 0
(3)

are simple and form an unbounded increasing sequence { λ n } whose eigenspaces are spanned by functions { ϕ n ( x ) } W 0 1 , p ( 0 , 1 ) C 1 [ 0 , 1 ] such that ϕ n has n 1 evenly spaced zeros in ( 0 , 1 ) , ϕ n L p ( 0 , 1 ) = 1 , and ϕ n ( 0 ) > 0 . The reader is invited to see [[3], p.388], [[4], p.780] or [[5], pp.272-275] for further details. See also Example 1 below for more explicit form of λ n and ϕ n .

Let λ λ n , n = 1 , 2 ,  , in (1). This is the nonresonance case. Then, for any f L p ( 0 , 1 ) , there exists at least one solution of (1). In the case p = 2 , this solution is unique. In the case p 2 , the uniqueness holds if λ 0 , but it may fail for certain right-hand sides f L p ( 0 , 1 ) if λ > 0 . See, e.g., [6] (for 2 < p < ) and [7] (for 1 < p < 2 ).

The same argument as that used for p = 2 in [[1], Section 3] for the nonresonance case yields the following existence result for the quasilinear impulsive problem (1), (2).

Theorem 1 (Nonresonance case)

Let λ λ n , n = 1 , 2 ,  , I j : R R , j = 1 , 2 , , r , be continuous functions which are ( p 1 ) -subhomogeneous at ±∞, that is,
lim | s | I j ( s ) | s | p 2 s = 0 .

Then (1), (2) has a solution for arbitrary f L p ( 0 , 1 ) .

Variational approach to impulsive differential equations of the type (1), (2) with p = 2 was used, e.g., in paper [8]. The authors apply the mountain pass theorem to prove the existence of a solution for λ < λ 1 . Our Theorem 1 thus generalizes [[8], Theorem 5.2] in two directions. Firstly, it allows also λ > λ 1 ( λ λ n , n = 2 , 3 , ) and, secondly, it deals with quasilinear equations ( p 2 ), too.

Let λ = λ n for some n N . This is the resonance case. Contrary to the linear case ( p = 2 ), there is no Fredholm alternative for (1) in the nonlinear case ( p 2 ). If λ = λ 1 , then
f ϕ 1 : = { h L ( 0 , 1 ) : 0 1 h ( x ) ϕ 1 ( x ) d x = 0 }

is the sufficient condition for solvability of (1), but it is not necessary if p 2 . Moreover, if f ϕ 1 but f is ‘close enough’ to ϕ 1 , problem (1) has at least two distinct solutions. The reader is referred to [3] or [9] for more details. It appears that the situation is even more complicated for λ = λ n , n 2 (see, e.g., [10]).

In the presence of nonlinear impulses which have certain asymptotic properties (to be made precise below), we show that the fact f ϕ n might still be the sufficient condition for the existence of a solution to (1) (with λ = λ n ) and (2). For this purpose we need some notation. Let 0 < x 1 < x 2 < < x n 1 < 1 denote evenly spaced zeros of ϕ n , let I + = ( 0 , x 1 ) ( x 2 , x 3 ) and I = ( x 1 , x 2 ) ( x 3 , x 4 ) denote the union of intervals where ϕ n > 0 or ϕ n < 0 , respectively. We arrange t j , j = 1 , 2 , , r , into three sequences: 0 < τ 1 < τ 2 < < τ r + < 1 , τ i I + , i = 1 , 2 , , r + ; 0 < σ 1 < σ 2 < < σ r < 1 , σ j I , j = 1 , 2 , , r ; ξ k { x 1 , x 2 , , x n 1 } , k = 1 , 2 , , r 0 . Obviously, we have r + + r + r 0 = r and r 0 n 1 . Assume that r + + r > 0 , i.e., r 0 < n 1 . The impulse condition (2) can be written in an equivalent form
Δ p u ( τ i ) = I i τ ( u ( τ i ) ) , i = 1 , 2 , , r + , Δ p u ( σ j ) = I j σ ( u ( σ j ) ) , j = 1 , 2 , , r , Δ p u ( ξ k ) = I k ξ ( u ( ξ k ) ) , k = 1 , 2 , , r 0 .
(4)
We assume that I i τ , I j σ , I k ξ : R R , i = 1 , 2 , , r + ; j = 1 , 2 , , r ; k = 1 , 2 , , r 0 , are continuous, bounded functions and there exist limits lim s ± I i τ ( s ) = I i τ ( ± ) , lim s ± I j σ ( s ) = I j σ ( ± ) . We consider the following Landesman-Lazer type conditions: either
i = 1 r + I i τ ( ) ϕ n ( τ i ) + j = 1 r I j σ ( + ) ϕ n ( σ j ) < 0 1 f ( x ) ϕ n ( x ) d x < i = 1 r + I i τ ( + ) ϕ n ( τ i ) + j = 1 r I j σ ( ) ϕ n ( σ j )
(5)
or
i = 1 r + I i τ ( + ) ϕ n ( τ i ) + j = 1 r I j σ ( ) ϕ n ( σ j ) < 0 1 f ( x ) ϕ n ( x ) d x < i = 1 r + I i τ ( ) ϕ n ( τ i ) + j = 1 r I j σ ( + ) ϕ n ( σ j ) .
(6)

Our main result is the following.

Theorem 2 (Resonance case)

Let λ = λ n for some n N in (1). Let the nonlinear bounded impulse functions I j : R R , j = 1 , 2 , , r , and the right-hand side f L p ( 0 , 1 ) satisfy either (5) or (6). Then (1), (2) has a solution.

The result from Theorem 2 is illustrated in the following special example.

Example 1 It follows from the first integral associated with the equation in (3) that the eigenvalues and the eigenfunctions of (3) have the form
λ n = ( p 1 ) ( n π p ) p , ϕ n ( x ) = sin p ( n π p x ) sin p ( n π p x ) L p ( 0 , 1 ) ,
where π p = 2 π p sin π p and x = 0 sin p x d s ( 1 s p ) 1 p , x [ 0 , π p 2 ] , sin p x = sin p ( π p x ) , x [ π p 2 , π p ] , sin p x = sin p ( 2 π p x ) , x [ π p , 2 π p ] , see [[3], p.388]. Let us consider λ = λ 2 in (1) and t 1 = π p 4 , t 2 = 3 π p 4 , I j ( s ) = arctan s , s R , j = 1 , 2 , in (2). Since sin π p 2 = 1 p 1 , sin 3 π p 2 = 1 p 1 , condition (6) reads as follows:
π p 1 < 0 1 f ( x ) sin p 2 π p x d x < π p 1 .

2 Functional framework

We say that u is the classical solution of (1), (2) if the following conditions are fulfilled:

  • u C [ 0 , 1 ] , u C 1 ( t j , t j + 1 ) , φ ( u ( ) ) is absolutely continuous in ( t j , t j + 1 ) , j = 0 , 1 , , r ;

  • the equation in (1) holds a.e. in ( 0 , 1 ) and u ( 0 ) = u ( 1 ) = 0 ;

  • one-sided limits u ( t j + ) , u ( t j ) exist finite and (2) holds.

We say that u W 0 1 , p ( 0 , 1 ) is a weak solution of (1), (2) if the integral identity
0 1 φ ( u ( x ) ) v ( x ) d x λ 0 1 φ ( u ( x ) ) v ( x ) d x + j = 1 r I j ( u ( t j ) ) v ( t j ) = 0 1 f ( x ) v ( x ) d x
(7)

holds for any function v W 0 1 , p ( 0 , 1 ) .

Integration by parts and the fundamental lemma in calculus of variations (see [[11], Lemma 7.1.9]) yields that every weak solution of (1), (2) is also a classical solution and vice versa. Indeed, let u be a weak solution of (1), (2), v D ( t j , t j + 1 ) (the space of smooth functions with a compact support in ( t j , t j + 1 ) , j = 0 , 1 , , r ), v 0 elsewhere in ( 0 , 1 ) , then
t j t j + 1 ( φ ( u ( x ) ) + 0 x [ λ φ ( u ( τ ) ) + f ( τ ) ] d τ ) v ( x ) d x = 0 .
Since v is arbitrary, we have φ ( u ( x ) ) + 0 x [ λ φ ( u ( τ ) ) + f ( τ ) ] d τ = 0 for a.e. x ( t j , t j + 1 ) . Then φ ( u ( ) ) is absolutely continuous in ( t j , t j + 1 ) and
( φ ( u ( x ) ) ) λ φ ( u ( x ) ) = f ( x )
(8)
for a.e. x ( t j , t j + 1 ) , j = 0 , 1 , , r . Taking now v W 0 1 , p ( 0 , 1 ) arbitrary, integrating by parts in the first integral in (7) and using (8), we get
j = 1 r [ φ ( u ( t j + ) ) φ ( u ( t j ) ) ] v ( t j ) = j = 1 r I j ( u ( t j ) ) v ( t j ) ,

and hence also (2) follows. Similarly, we show that every classical solution is a weak solution at the same time.

Let X : = W 0 1 , p ( 0 , 1 ) with the norm u = ( 0 1 | u ( x ) | p d x ) 1 p , X be the dual of X and , be the duality pairing between X and X. For u X , we set
A ( u ) : = 1 p 0 1 | u ( x ) | p d x , B ( u ) : = 1 p 0 1 | u ( x ) | p d x , F ( u ) = 0 1 f ( x ) u ( x ) d x , J ( u ) : = j = 1 r 0 u ( t j ) I j ( s ) d s .
Then, for u , v X , we have
A ( u ) , v = 0 1 φ ( u ( x ) ) v ( x ) d x , B ( u ) , v = 0 1 φ ( u ( x ) ) v ( x ) d x , F , v = 0 1 f ( x ) v ( x ) d x , J ( u ) , v = j = 1 r I j ( u ( t j ) ) v ( t j ) .
Lemma 1 The operators A , B , J : X X have the following properties:
  1. (A)

    A is ( p 1 ) -homogeneous, odd, continuously invertible, and A ( u ) = u p 1 for any u X .

     
  2. (B)

    B is ( p 1 ) -homogeneous, odd and compact.

     
  3. (J)

    J is bounded and compact.

     

By the linearity of F : X R , F X is a fixed element.

Proof See [[12], Lemma 10.3, p.120]. □

With this notation in hands we can look for (classical) solutions of (1), (2) either as for solutions u X of the operator equation
A ( u ) λ B ( u ) + J ( u ) = F
(9)
or, alternatively, as for critical points of the functional F : X R ,
F ( u ) : = A ( u ) λ B ( u ) + J ( u ) F ( u ) .
(10)

As mentioned already above, in the nonresonance case ( λ λ n , n N ), we can use the Leray-Schauder degree argument and prove the existence of a solution of the equation (9) exactly as in [[1], proof of Thm. 1]. Note that the ( p 1 ) -subhomogeneous condition on I j is used here instead of the sublinear condition imposed on I j in [1] and the proof of Theorem 1 follows the same lines. For this reason we skip it and concentrate on the resonance case ( λ = λ n for some n N ) in the next section.

3 Resonance problem, variational approach

We use the following definition of linked sets and the linking theorem (cf. [13]).

Definition 1 Let be a closed subset of X and let Q be a submanifold of X with relative boundary ∂Q. We say that and ∂Q link if
  1. (i)

    E Q = and

     
  2. (ii)

    for any continuous map h : X X such that h | Q = id , there holds h ( Q ) E .

     

(See [[14], Def. 8.1, p.116].)

Theorem 3 (Linking theorem)

Suppose that F C 1 ( X ) satisfies the Palais-Smale condition. Consider a closed subset E X and a submanifold Q X with relative boundary ∂Q, and let Γ : = { h C 0 ( X , X ) : h | Q = id } . Suppose that and ∂Q link in the sense of Definition  1, and
inf u E F ( u ) > sup u Q F ( u ) .

Then β = inf h Γ sup u Q F ( h ( u ) ) is a critical value of .

(See [[14], Thm. 8.4, p.118].)

The purpose of the following series of lemmas is to show that the hypotheses of Theorem 3 are satisfied provided that either (5) or (6) holds. From now on we assume that λ = λ n (for some n N ) in (1).

Lemma 2 If either (5) or (6) is satisfied, then satisfies the Palais-Smale condition.

Proof Suppose that { u k } X such that | F ( u k ) | c and F ( u k ) 0 in X . We must show that { u k } has a subsequence that converges in X. We prove first that { u k } is a bounded sequence. We proceed via contradiction and suppose that u k and consider v k : = u k u k . Without loss of generality, we can assume that there is v 0 X such that v k v 0 (weakly) in X (X is a reflexive Banach space). Since
0 F ( u k ) = A ( u k ) λ n B ( u k ) + J ( u k ) F ,
dividing through by u k p 1 , we have
A ( v k ) λ n B ( v k ) + J ( u k ) u k p 1 F u k p 1 0 .

By the boundedness of J we know that J ( u k ) u k p 1 0 . We also have F u k p 1 0 . By the compactness of B we get B ( v k ) B ( v 0 ) in X . Thus v k v 0 = ( A ) 1 ( λ n B ( v 0 ) ) in X by Lemma 1(A). It follows that v 0 = ± 1 λ n 1 p ϕ n .

We assume v 0 = 1 λ n 1 p ϕ n and remark that a similar argument follows if v 0 = 1 λ n 1 p ϕ n . Next we estimate
p F ( u k ) F ( u k ) , u k = p J ( u k ) J ( u k ) , u k + ( 1 p ) 0 1 f ( x ) u k ( x ) d x .
(11)
Our assumption | F ( u k ) | c yields
c p p F ( u k ) c p
(12)
and the Cauchy-Schwarz inequality implies
u k F ( u k ) F ( u k ) , u k u k F ( u k ) ,
(13)
where denotes the norm in X . It follows from (11)-(13) that
c p u k F ( u k ) p J ( u k ) J ( u k ) , u k + ( 1 p ) 0 1 f ( x ) u k ( x ) d x c p + u k F ( u k ) .
Dividing through by u k and writing 0 u k ( t j ) I j ( s ) d s u k = I ˆ j ( u k ( t j ) ) v k ( t j ) , where
I ˆ j ( σ ) : = { 0 σ I j ( s ) d s σ for  σ 0 , 0 for  σ = 0 ,
j = 0 , 1 , , r , we get
| p j = 1 r I ˆ j ( u k ( t j ) ) v k ( t j ) j = 1 r I j ( u k ( t j ) ) v k ( t j ) + ( 1 p ) 0 1 f ( x ) v k ( x ) d x | c p u k + F ( u k ) 0 .
(14)
Since 0 1 f ( x ) v k ( x ) d x 1 λ n 1 p 0 1 f ( x ) ϕ n ( x ) d x as k , we obtain from (14):
lim k j = 1 r ( p I ˆ j ( u k ( t j ) ) I j ( u k ( t j ) ) ) v k ( t j ) = p 1 λ n 1 p 0 1 f ( x ) ϕ n ( x ) d x .
(15)
Recall that X embeds compactly in C [ 0 , 1 ] , so, without loss of generality, we assume that v k ( t j ) 1 λ n 1 p ϕ n ( t j ) , j = 0 , 1 , , r , as k . Hence, u k ( t j ) ± for t j I ± , which implies I j ( u k ( t j ) ) I j ( ± ) as well as I ˆ j ( u k ( t j ) ) I j ( ± ) as k by an application of the l’Hospital rule to 0 σ I j ( s ) d s σ . Notice that by the boundedness of I j we have
( p I ˆ j ( u k ( t j ) ) I j ( u k ( t j ) ) ) v k ( t j ) 0 as  k
if t j is a zero point of ϕ n for some j { 1 , 2 , , r } . Thus, passing to the limit in (15) as k , we get
i = 1 r + I i τ ( + ) ϕ n ( τ i ) + j = 1 r I j σ ( ) ϕ n ( σ j ) = 0 1 f ( x ) ϕ n ( x ) d x ,

which contradicts (5) or (6). Hence { u k } is bounded.

By compactness there is a subsequence such that B ( u k ) and J ( u k ) converge in X (see Lemma 1(B), (J)). Since F ( u k ) 0 by our assumption, we also have that A ( u k ) converges in X . Finally, u k = ( A ) 1 ( A ( u k ) ) converges in X by Lemma 1(A). The proof is finished. □

With the Palais-Smale condition in hands, we can turn our attention to the geometry of the functional . To this end we have to find suitable sets which link in the sense of Definition 1. Actually, we use the sets constructed in [13] and explain that they fit with the hypotheses of Theorem 3 if either (5) or (6) is satisfied.

Consider the even functional
E ( u ) : = A ( u ) B ( u ) for  u X { 0 }
and the manifold
S : = { u W 0 1 , p ( 0 , 1 ) : B ( u ) = 1 } .
For any n N , let F n : = { A S :  continuous odd surjection  h : S n 1 A } , where S n 1 represents the unit sphere in R n . Next we define
λ n : = inf A F n sup u A E ( u ) , n N .
(16)

It is proved in [[15], Section 3] that { λ n } is a sequence of eigenvalues of homogeneous problem (3). It then follows from the results in [16] that this sequence exhausts the set of all eigenvalues of (3) with the properties described in Section 1.

Now consider the functions ϕ n , i = χ [ i 1 n , i n ] ϕ n for i = 1 , 2 , , n , where χ [ i 1 n , i n ] is a characteristic function of the interval [ i 1 n , i n ] , and let
Λ n : = { α 1 ϕ n , 1 + + α n ϕ n , n : α i R  and  | α 1 | p B ( ϕ n , 1 ) + + | α n | p B ( ϕ n , n ) = 1 } .
Observe that Λ n is symmetric and is homeomorphic to the unit sphere in R n . Moreover, for u Λ n , we have
B ( u ) = B ( α 1 ϕ n , 1 + + α n ϕ n , n ) = B ( α 1 ϕ n , 1 ) + + B ( α n ϕ n , n ) = | α 1 | p B ( ϕ n , 1 ) + + | α n | p B ( ϕ n , n ) = 1 .
Notice that the second equality holds thanks to the fact
{ x : ϕ n , i ( x ) 0 } { x : ϕ n , j ( x ) 0 } =
for i j , i , j = 1 , 2 , , n , while the third one follows from the p-homogeneity of B. Thus Λ n S and so Λ n F n . A similar computation then shows that E ( u ) = A ( u ) = λ n for all u Λ n . For a given T > 0 , we let
Q n , T : = { s u : 0 s T , u Λ n } .
Then Q n , T is homeomorphic to the closed unit ball in R n . For a given c R , we denote by
E c : = { u X : A ( u ) c B ( u ) } = { u X { 0 } : E ( u ) c } { 0 }
a super-level set, and
K c : = { u X { 0 } : E ( u ) = c , E ( u ) = 0 } .

The existence of a pseudo-gradient vector field with the following properties is proved in [[13], Lemma 6] (cf. [[14], pp.77-79] and [[2], p.55]).

Lemma 3 For ε < min { λ n + 1 λ n , λ n λ n 1 } , there is ε ˜ ( 0 , ε ) and a one-parameter family of homeomorphisms η : [ 1 , 1 ] × S S such that
  1. (i)

    η ( t , u ) = u if E ( u ) ( , λ n ε ] [ λ n + ε , ) or if u K λ n ;

     
  2. (ii)

    E ( η ( t , u ) ) is strictly decreasing in t if E ( u ) ( λ n ε ˜ n , λ n + ε ˜ n ) and u K λ n ;

     
  3. (iii)

    η ( t , u ) = η ( t , u ) ;

     
  4. (iv)

    η ( 0 , ) = id .

     
An important fact is that the flow η ‘lowers’ Q n , T and ‘raises’ E λ n if we modify them as follows:
E ˜ λ n : = { s u : s R , u η ( 1 , E λ n S ) }
and
Q ˜ n , T : = { s u : 0 s T , u η ( 1 , Λ n ) } .
Then, by Lemma 3 and the definition of E λ n , we have
A ( u ) λ n B ( u ) 0
for u E ˜ λ n with equality if and only if u = c ϕ n for some c R . Similarly,
A ( u ) λ n B ( u ) 0

for u Q ˜ n , T with equality if and only if u = c ϕ n for some c R .

It is proved in [[13], Lemma 7] that the couple E : = E λ n + 1 and Q : = Q ˜ n , T satisfies condition (ii) from Definition 1. It is also proved in [[13], Lemma 8] that the couple E : = E ˜ λ n and Q : = Q n 1 , T satisfies the same condition. To show that also other hypotheses of Theorem 3 are satisfied, we need some technical lemmas.

Lemma 4 If (6) is satisfied, then there exist R > 0 and δ > 0 such that F ( s u ) , u δ for any s R and u η ( 1 , Λ n ) .

Proof We proceed via contradiction and assume that there exist s k and u k η ( 1 , Λ n ) such that
lim sup k F ( s k u k ) , u k 0 .
(17)

Since η ( 1 , Λ n ) is compact, we may assume, without loss of generality, that u k u 0 in η ( 1 , Λ n ) for some u 0 η ( 1 , Λ n ) .

If u 0 ± p 1 p ϕ n , then there exists ε > 0 such that
0 1 | u 0 ( x ) | p d x λ n 0 1 | u 0 ( x ) | p d x ε .
Hence, there exists k ε N such that for any k k ε we have
0 1 | u k ( x ) | p d x λ n 0 1 | u k ( x ) | p d x ε 2 .
This implies
F ( s k u k ) , u k ε 2 s k p 1 + j = 1 r I j ( s k u k ( t j ) ) u k ( t j ) 0 1 f ( x ) u k ( x ) d x

for k k ε . However, this contradicts (17).

If u 0 = p 1 p ϕ n , we still have
0 1 | u k ( x ) | p d x λ n 0 1 | u k ( x ) | p d x 0 ,
and so
F ( s k u k ) , u k j = 1 r I j ( s k u k ( t j ) ) u k ( t j ) 0 1 f ( x ) u k ( x ) d x
for all k N . The boundedness of I j , j = 1 , 2 , , r , and uniform convergence u k p 1 p ϕ n as k (due to continuous embedding X C [ 0 , 1 ] ) then yield
lim k F ( s k u k ) , u k p 1 p ( i = 1 r + I i τ ( + ) ϕ n ( τ i ) + j = 1 r I j σ ( ) ϕ n ( σ j ) 0 1 f ( x ) ϕ n ( x ) d x ) < 0
by the first inequality in (6). This contradicts (17) again. Notice that by the boundedness of I j we have
( p I ˆ j ( u k ( t j ) ) I j ( u k ( t j ) ) ) v k ( t j ) 0 as  k

if t j is a zero point of ϕ n for some j { 1 , 2 , , r } . The case u 0 = p 1 p ϕ n is proved similarly using the second inequality in (6). □

Lemma 5 If (6) is satisfied, then there exists T > 0 such that
inf u E λ n + 1 F ( u ) > sup u Q ˜ n , T F ( u ) .
(18)
Proof There exists α R such that for any u E λ n + 1 we have
F ( u ) 1 p ( λ n + 1 λ n ) u L p ( 0 , 1 ) p + j = 1 r 0 u ( t j ) I j ( ζ ) d ζ 0 1 f ( x ) u ( x ) d x > α .
By Lemma 4 there exists c R such that for all s > R and u η ( 1 , Λ n ) we have
F ( s u ) = F ( R u ) + F ( s u ) F ( R u ) = F ( R u ) + R s F ( ζ u ) , u d ζ c δ ( s R ) .
Thus there exists T > R such that
F ( s u ) c δ ( s R ) < α

for all s T , u η ( 1 , Λ n ) . In particular, F ( u ) < α for all u Q ˜ n , T and (18) is proved. □

Now we can finish the proof of Theorem 2 under assumption (6). Indeed, it follows from (18) that E λ n + 1 Q ˜ n , T = and thus the hypotheses of Theorem 3 hold with E : = E λ n + 1 and Q : = Q ˜ n , T . It then follows that has a critical point and hence (1), (2) has a solution.

Next we show that the sets E : = E ˜ λ n and Q : = Q n 1 , T satisfy the hypotheses of Theorem 3 if (5) is satisfied.

The principal difference consists in the fact that, in contrast with η ( 1 , Λ n ) , the set η ( 1 , E λ n S ) is not compact. That is why one more technical lemma is needed.

Lemma 6 For any ε > 0 , there exists δ > 0 such that
E ( u ) λ n + δ
(19)

for u η ( 1 , E λ n S ) B ε ( ± ϕ n ) . (Here B ε ( ± ϕ n ) is the ball in X centered at ± ϕ n with radius  ε .)

Proof We note that the pseudo-gradient flow η from Lemma 3 is constructed as a solution of the initial value problem d d t η ( t , u ) = v ˜ ( η ( t , u ) ) , η ( 0 , ) = id , where
v ˜ ( u ) = { ψ ( u ) dist ( u , K λ n ) v ( u ) for  u S ˜ : = { w S : E ( w ) 0 } , 0 for  u S S ˜ ,

v ( u ) is a locally Lipschitz continuous symmetric pseudo-gradient vector field associated with E on S ˜ and ψ [ 0 , 1 ] is a smooth function such that ψ ( u ) = 1 for u satisfying λ n ε ˜ E ( u ) λ n + ε ˜ and ψ ( u ) = 0 for u satisfying E ( u ) λ n ε or λ n + ε E ( u ) .

Let ε > 0 and u η ( 1 , E λ n S ) B ε ( ± ϕ n ) . Without loss of generality, we may assume that E ( u ) λ n + ε ˜ . Let u 0 E λ n S be such that u = η ( 1 , u 0 ) . Observe that there is a constant M > 0 such that for t [ 1 , 1 ] we have
d d t η ( t , u 0 ) v ˜ ( η ( t , u 0 ) ) dist ( η ( t , u 0 ) , K λ n ) v ˜ ( η ( t , u 0 ) ) < M .
Hence η ( t , u 0 ) B ε 2 ( ± ϕ n ) for t [ 1 , 1 + ε 2 M ] . Since E satisfies the Palais-Smale condition on S (see [[13], Lemma 2]), there exists ρ > 0 such that E ( u ) ρ for all u { w S : λ n E ( w ) λ n + ε ˜ } B ε 2 ( ± ϕ n ) . Then
d d t E ( η ( t , u 0 ) ) = E ( η ( t , u 0 ) ) , d d t η ( t , u 0 ) = ψ ( η ( t , u 0 ) ) dist ( η ( t , u 0 ) , K λ n ) E ( η ( t , u 0 ) ) , v ( η ( t , u 0 ) ) 1 ε 2 min { E ( η ( t , u 0 ) ) , 1 } E ( η ( t , u 0 ) ) ε 2 ρ 2
for all t [ 1 , 1 + ε 2 M ] . The last but one inequality holds due to the following property of v ( u ) :
E ( u ) , v ( u ) > min { E ( u ) , 1 } E ( u )
(see [14] and [2]). We also used the fact that ψ ( η ( t , u 0 ) ) 1 for t [ 1 , 0 ] . Hence
E ( u ) = E ( η ( 1 , u 0 ) ) = E ( η ( 1 + ε 2 M , u 0 ) ) + 1 + ε 2 M 1 d d t E ( η ( t , u 0 ) ) d t E ( η ( 1 + ε 2 M , u 0 ) ) + ε 2 ρ 2 ε 2 M λ n + δ

with δ = ( ε ρ ) 2 4 M . □

The following lemma is a counterpart of Lemma 4 in the case of condition (5).

Lemma 7 If (5) is satisfied, then there exist R > 0 and δ > 0 such that F ( s u ) , u δ for any s R and u η ( 1 , E λ n S ) .

Proof We proceed via contradiction and assume that there exist s k and u k η ( 1 , E λ n S ) such that
lim sup k F ( s k u k ) , u k 0 .
(20)
If there is ε > 0 such that u k η ( 1 , E λ n S ) B ε ( ± ϕ n ) for all k large enough, then Lemma 6 leads to the estimate
F ( s k u k ) , u k δ s k p 1 + j = 1 r I j ( s k u k ( t j ) ) u k ( t j ) 0 1 f ( x ) u k ( x ) d x
contradicting (20). Thus it must be u k ± p 1 p ϕ n as k . If u k p 1 p ϕ n as k , we still have
0 1 | u k ( x ) | p d x λ n 0 1 | u k ( x ) | p d x 0 ,
and so
F ( s k u k ) , u k j = 1 r I j ( s k u k ( t j ) ) u k ( t j ) 0 1 f ( x ) u k ( x ) d x
for all k N . Similar arguments as in the proof of Lemma 4 lead to
lim k F ( s k u k ) , u k p 1 p ( i = 1 r + I i τ ( + ) ϕ n ( τ i ) + j = 1 r I j σ ( ) ϕ n ( σ j ) 0 1 f ( x ) ϕ n ( x ) d x ) > 0

by the second inequality in (5). This contradicts (20) again. The case u k p 1 p ϕ n as k is proved similarly but using the first inequality in (5). □

Lemma 8 If (5) is satisfied, then there exists T > 0 such that
inf u E ˜ λ n F ( u ) > sup u Q n 1 , T F ( u ) .
(21)
Proof By Lemma 7 there exists d R such that for all s > R and u η ( 1 , E λ n S ) we have
F ( s u ) = F ( R u ) + F ( s u ) F ( R u ) = F ( R u ) + R s F ( ζ u ) , u d ζ d + δ ( s R ) .
Hence, there exists α R such that for any u E ˜ λ n we have
F ( u ) > α .
On the other hand, for any s > 0 and u Λ n 1 , we get
F ( s u ) = 1 p ( λ n 1 λ n ) s u L p ( 0 , 1 ) p + j = 1 r 0 s u ( t j ) I j ( ζ ) d ζ s 0 1 f ( x ) u ( x ) d x = ( λ n 1 λ n ) s p + j = 1 r 0 s u ( t j ) I j ( ζ ) d ζ s 0 1 f ( x ) u ( x ) d x .
Thus, there exists T > 0 such that, for u Q n 1 , T ,
F ( u ) < α

and (21) is proved. □

It follows that the sets E : = E ˜ λ n and Q : = Q n 1 , T satisfy the hypotheses of Theorem 3 if (5) is satisfied. The proof of Theorem 2 is thus completed.

Final remark Reviewers of our manuscript suggested to include some recent references on impulsive problems. Variational approach to impulsive problems can be found, e.g., in [1721]. The last reference deals with the p-Laplacian with the variable exponent p = p ( t ) . Singular impulsive problems are treated in [2224]. Impulsive problems are still ‘hot topic’ attracting the attention of many mathematicians and the bibliography on that topic is vast.

Declarations

Acknowledgements

This research was supported by Grant 13-00863S of the Grant Agency of Czech Republic and by the European Regional Development Fund (ERDF), project ‘NTIS - New Technologies for the Information Society’, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.

Authors’ Affiliations

(1)
Department of Mathematics, University of West Bohemia
(2)
NTIS, University of West Bohemia

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