Variational Method to the Impulsive Equation with Neumann Boundary Conditions

Boundary Value Problems

Volume 2009, Article ID 316812,17pages doi:10.1155/2009/316812

Research Article

Variational Method to the Impulsive Equation with Neumann Boundary Conditions

Juntao Sun and Haibo Chen

Department of Mathematics, Central South University, Changsha, 410075 Hunan, China

Correspondence should be addressed to Juntao Sun,[email protected] Received 28 August 2009; Accepted 28 September 2009

Recommended by Pavel Dr´abek

We study the existence and multiplicity of classical solutions for second-order impulsive Sturm- Liouville equation with Neumann boundary conditions. By using the variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions, and infinitely many solutions under some different conditions, respectively. Some examples are also given in this paper to illustrate the main results.

Copyrightq2009 J. Sun and H. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we consider the boundary value problem of second-order Sturm-Liouville equation with impulsive effects

−

ptut

rtut qtut gt, ut, t /tk, a.e. t∈0,1,

−Δ

ptkutk

I_kutk, k1,2, . . . , p−1, u0 u

1⁻ 0,

1.1

where 0 t₀ < t₁ < t₂ < · · · < t_p−1 < t_p 1, p ∈ C¹0,1, r, q ∈ C0,1 withp and q positive functions,g :0,1×R → Ris a continuous function,I_k:R → R,1≤k≤p−1 are continuous,−Δptkutk −ptkut_k−ut⁻_k,ut_kandut⁻_kdenote the right and the left limits, respectively, ofutattt_k,u0is the right limit ofu0, andu1⁻is the left limit ofu1.

In the recent years, a great deal of work has been done in the study of the existence of solutions for impulsive boundary value problemsIBVPs, by which a number

of chemotherapy, population dynamics, optimal control, ecology, industrial robotics, and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph1. For some general and recent works on the theory of impulsive differential equations, we refer the reader to2–9. Some classical tools or techniques have been used to study such problems in the literature. These classical techniques include the coincidence degree theory of Mawhin10, the method of upper and lower solutions with monotone iterative technique11, and some fixed point theorems in cones12–14.

On the other hand, in the last two years, some researchers have used variational methods to study the existence of solutions for impulsive boundary value problems.

Variational method has become a new powerful tool to study impulsive differential equations, we refer the reader to 15–20. More precisely, in 15, the authors studied the following equation with impulsive effects:

− ρtφp

ut

stφput ft, ut, t /tj, a.e. t∈a, b,

−Δ ρ

t_j φ_p

u t_j

I_j u

t_j

, j1,2, . . . , l, αua−βua A, γub σub B,

1.2

wheref :a, b×0,∞ → 0,∞is continuous,I_j:0,∞ → 0,∞, j 1,2, . . . , l, are continuous, andα, β, γ, σ >0. They essentially proved that IBVP1.2has at least two positive solutions via variational method. Recently, in16, using variational method and critical point theory, Nieto and O’Regan studied the existence of solutions of the following equation:

−ut λut ft, ut, t /t_j, a.e. t∈0, T, Δ

u tj

Ij

u tj

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

u0 uT 0,

1.3

where f : 0, T×R → Ris continuous, and Ij : R → R, j 1,2, . . . , l are continuous.

They obtained that IBVP1.3has at least one solution. Shortly, in17, authors extended the results of IBVP1.3.

In19,Zhou and Li studied the existence of solutions of the following equation:

−ut gtut ft, ut, t /t_j, a.e. t∈0, T, Δ

u t_j

I_j u

t_j

, j 1,2, . . . , p,

u0 uT 0,

1.4

wheref : 0, T×R → Ris continuous, and I_j : R → R, j 1,2, . . . , p, are continuous.

They proved that IBVP1.4has at least one solution and infinitely many solutions by using variational method and critical point theorem.

Motivated by the above facts, in this paper, our aim is to study the variational structure of IBVP1.1in an appropriate space of functions and obtain the existence and multiplicity of solutions for IBVP1.1by using variational method. To the best of our knowledge, there

is no paper concerned impulsive differential equation with Neumann boundary conditions via variational method. In addition, this paper is a generalization of21, in which impulse effects are not involved.

In this paper, we will need the following conditions.

H1There is constantsβ >2, M >0 such that for everyt∈0,1andu∈Rwith|u| ≥M,

0< βGt, u≤ugt, u, 0< β _u

0

I_ksds≤uI_ku, 1.5

whereGt, u _u

0gt, sds.

H2lim_u→0gt, u/u0 uniformly fort∈0,1, and limu→0Iku/u0.

H3There exist numbersh1, h2>0 andp1 >1 such that

gt, u≤h1h2|u|^p1 foru∈R, t∈0,1. 1.6

H4There exist numbersa_k, b_k>0 andγ_k∈0,1such that

I_ku≤a_kb_k|u|^γk foru∈R. 1.7

H5There exist numbersr1, r2>0 andμ∈0,1such that

gt, u≤r1r2|u|^μ foru∈R, t∈0,1. 1.8

H6There exist numbersa_k, b_k >0 andγ_k ∈1,∞such that

Iku≤a_kb_k|u|^γk foru∈R. 1.9

This paper is organized as follows. InSection 2, we present some preliminaries. In Section 3, we discuss the existence and multiplicity of classical solutions to IBVP1.1. Some examples are presented in this section to illustrate our main results in the last section.

2. Preliminaries

Take Lt _t

0rs/psds. Then e^−Lt ∈ C¹0,1. We transform IBVP 1.1 into the following equivalent form:

−

e^−Ltptut

e^−Ltqtut e^−Ltgt, ut, t /t_k, a.e. t∈0,1,

−Δ

e^−Ltkptkutk

e^−LtkI_kutk, k1,2, . . . , p−1, u0 u

1⁻ 0.

2.1

Obviously, the solutions of IBVP2.1are solutions of IBVP 1.1. So it suffices to consider IBVP2.1.

In this section, the following theorem will be needed in our argument. Suppose thatE is a Banach spacein particular a Hilbert spaceandϕ∈C¹E,R. We say thatϕsatisfies the Palais-Smale condition if any sequence{uj} ⊂Efor whichϕujis bounded andϕuj → 0 asj → ∞possesses a convergent subsequence inX. LetB_r be the open ball inXwith the radiusrand centered at 0 and∂Br denote its boundary.

Theorem 2.1 22, Theorem 38.A. For the functional F : M ⊆ X → −∞,∞ with M /∅,min_u∈MFu αhas a solution for which the following hold:

iXis a real reflexive Banach space;

iiMis bounded and weakly sequentially closed;

iiiF is weakly sequentially lower semicontinuous on M; that is, by definition, for each sequence{un}in Msuch thatu_n uasn → ∞, one has Fu ≤ lim inf_{n→ ∞}Fun holds.

Theorem 2.216, Theorem 2.2. LetEbe a real Banach space and letϕ ∈ C¹E,Rsatisfy the Palais-Smale condition. Assume there existu₀, u₁∈Eand a bounded open neighborhoodΩofu₀such thatu1∈E\Ωand

max

ϕu0, ϕu1 < inf

x∈∂Ωϕu. 2.2

Let

Γ {h|h:0,1−→Eis continuous andh0 u0, h1 u1}, cinf

h∈Γmax

s∈0,1ϕhs. 2.3

Thencis a critical value ofϕ; that is, there existsu^∗∈Esuch thatϕu^∗ Θandϕu^∗ c, where c >max{ϕu0, ϕu1}.

Theorem 2.3 23. Let Ebe a real Banach space, and let ϕ ∈ C¹E,Rbe even satisfying the Palais-Smale condition andϕ0 0. IfEV⊕Y, whereV is finite dimensional, andϕsatisfies that

A1there exist constantsρ, α >0 such thatϕ|∂Br∩Y ≥α,

A2for each finite dimensional subspaceW ⊂E, there isR RWsuch thatϕu≤0 for all u∈Wwithu ≥R.

Thenϕpossesses an unbounded sequence of critical values.

Let us recall some basic knowledge. Denote byXthe Sobolev spaceW^1,20,1, and consider the inner product

u, v ₁

0

utvtdt ₁

0

utvtdt 2.4

which induces the usual norm

u ₁

0

ut²dt ₁

0

|ut|²dt _1/2

. 2.5

We also consider the inner product

u, v_{X 1}

0

e^−Ltptutvtdt ₁

0

e^−Ltqtutvtdt, 2.6

and the norm

u_{X 1}

0

e^−Ltptut²dt ₁

0

e^−Ltqt|ut|²dt _1/2

, 2.7

then the norm · Xis equivalent to the usual norm · inW^1,20,1. Hence,Xis reflexive.

We define the norm inC0,1, L²0,1asu_∞ max_t∈0,1|ut|andu_{2 1}

0|u|²dt^1/2, respectively.

Foru∈W^2,20,1, we have thatu, uare absolutely continuous, andu ∈L²0,1, hence −Δe^−Ltkptkutk −e^−Ltkptkut_k−ut⁻_k 0,for any tk ∈ 0,1. Ifu ∈ X, thenuis absolutely continuous and u ∈ L²0,1. In this case, the one-side derivatives u0, u1⁻, ut_k, ut⁻_k, k 1,2, . . . , p−1 may not exist. As a consequence, we need to introduce a different concept of solution. We say thatu ∈ C0,1is a classical solution of IBVP2.1if it satisfies the equation in IBVP 2.1a.e. on0,1, the limitsut_k, ut⁻_k, k 1,2, . . . , p−1 exist and impulsive conditions in IBVP2.1hold,u0, u1⁻exist andu0 u1⁻ 0. Moreover, for everyk0,1, . . . , p−1, u_ku|tk,tk1satisfyu_k∈W^2,2tk, t_k1.

For eachu∈X, consider the functionalϕdefined onXby

ϕu 1

2u²_X−^p−1

k1

e^−Ltk _utk

0

Iksds− ₁

0

e^−LtGt, udt. 2.8

It is clear thatϕis differentiable at anyu∈Xand

ϕuv

₁

0

e^−Ltptutvt e^−Ltqtutvt dt

−^p−1

k1

e^−LtkI_kutkvtk− ₁

0

e^−Ltgt, utvtdt

2.9

for anyv∈X. Obviously,ϕis continuous.

Lemma 2.4. Ifu ∈ X is a critical point of the functionalϕ, thenuis a classical solution of IBVP 2.1.

Proof. Letu∈Xbe a critical point of the functionalϕ. It shows that ₁

0

e^−Ltptutvte^−Ltqtutvt dt

−^p−1

k1

e^−LtkIkutkvtk− ₁

0

e^−Ltgt, utvtdt0

2.10

holds for any v ∈ X. Choose any j ∈ {0,1,2, . . . , p−1} and v ∈ X such that vt 0 if t∈tk, t_k1fork /j. Equation2.10implies

_tj1

tj

e^−Ltptutvt e^−Ltqtutvt−e^−Ltgt, utvt

dt0. 2.11

This means, for anyw∈W₀^1,2tj, t_j1, _tj1

tj

e^−Ltptu_jtwt e^−Ltqtujtwt−e^−Ltg t, u_jt

wt

dt0, 2.12

whereu_ju|tj,tj1. Thusu_jis a weak solution of the following equation:

−

e^−Ltptut

e^−Ltqtut e^−Ltgt, ut t∈ t_j, t_j1

, 2.13

and therefore uj ∈ W₀^1,2tj, t_j1 ⊂ Ctj, t_j1.Let ht : e^−Ltgt, u−qu, then 2.13 becomes the following form:

−

e^−Ltptut

hton tj, t_j1

, j0,1,2, . . . , p−1. 2.14

Then the solution of2.14can be written as

ujt C1C2

_t

tj

e^Ls−lnpsds− _t

tj

e^Ls−lnps _s

tj

hr

pre^lnprdr

ds t∈ tj, t_j1

, 2.15

whereC₁ andC₂ are two constants. Thenu_j ∈ Ctj, t_j1andu_j ∈ Ctj, t_j1. Therefore,u_j is a classical solution of2.13andusatisfies the equation in IBVP2.1a.e. on0,1. By the

previous equation, we can easily get that the limitsut_j, ut⁻_j, j 1,2, . . . , p−1, ut₀and ut⁻_pexist. By integrating2.10, one has

₁

0

e^−Ltptutvt e^−Ltqtutvt dt

−^p−1

k1

e^−LtkIkutkvtk− ₁

0

e^−Ltgt, utvtdt

−^p−1

k1

Δ

e^−Ltkptkutk

vtk e^−L1p1u 1⁻

v1

−e^−L0p0u0v0−^p−1

k1

e^−LtkIkutkvtk

₁

0

−

e^−Ltptut

e^−Ltqtut−e^−Ltgt, ut

vtdt

−^p−1

k1

Δ

e^−Ltkptkutk

e^−LtkI_kutk vtk

e^−L1p1u 1⁻

v1−e^−L0p0u0v0

₁

0

−

e^−Ltptut

e^−Ltqtut−e^−Ltgt, ut

vtdt0,

2.16

and combining with2.13we get

−^p−1

k1

Δ

e^−Ltkptkutk

e^−LtkI_kutk vtk

e^−L1p1u 1⁻

v1−e^−L0p0u0v00.

2.17

Next we will show thatusatisfies the impulsive conditions in IBVP2.1. If not, without loss of generality, we assume that there existsi∈ {1,2, . . . , p−1}such that

e^−LtiI_iuti Δ

e^−Ltiptiuti

/0. 2.18

Let

vt p k0, k /i

t−t_k. 2.19

Obviously,v∈X. Substituting them into2.17, we get

Δe^−Ltiptiuti e^−LtiI_iuti

vti 0 2.20

which contradicts2.18. Sousatisfies the impulsive conditions in IBVP2.1. Thus,2.17 becomes the following form:

e^−L1p1u 1⁻

v1−e^−L0p0u0v0 0, 2.21

for all v ∈ X. Since v0, v1 are arbitrary, 2.21 shows that e^−L1p1u1⁻ e^−L0p0u0 0,and it impliesu1⁻ u0 0. Therefore, uis a classical solution of IBVP2.1.

Lemma 2.5. Letu∈X. Thenu_∞≤M₁u_X, where

M12^1/2max

1

min_t∈0,1e^−Ltpt_1/2, 1

min_t∈0,1e^−Ltqt_1/2

. 2.22

Proof. By using the same methods of15, Lemma 2.6, we easily obtain the above result, and we omit it here.

3. Main Results

In this section, we will show our main results and prove them.

Theorem 3.1. Assume that (H1) and (H2) hold. Moreover,gt, uand the impulsive functionsI_ku are odd aboutu, then IBVP1.1has infinitely many classical solutions.

Proof. Obviously,ϕis an even functional andϕ0 0. We divide our proof into three parts in order to showTheorem 3.1.

Firstly, We will show that ϕ satisfies the Palais-Smale condition. Let {ϕun} be a bounded sequence such that lim_n→∞ϕun 0. Then there exists constants C₃ > 0 such that

ϕun≤C3, ϕun

X≤C3. 3.1

By2.8,2.9,3.1, andH1, we have β

2 −1

un²_X β

2un²_X− un²_X βϕun−ϕununβ

p−1 k1

e^−Ltk _untk

0

Iksdsβ ₁

0

e^−LtGt, undt

−^p−1

k1

e^−LtkI_kuntkuntk− ₁

0

e^−Ltgt, unundt

^p−1

k1

e^−Ltk

β _untk

0

Iksds−Ikuntkuntk

₁

0

e^−Lt

βGt, un−gt, unun

dtβϕun−ϕunun

≤βC₃M²₁C₃un_X

₁

0

e^−Ltdt max

t∈0,1, unt∈−M,MβGt, un−gt, unun ^p−1

k1

e^−Ltk max

untk∈−M,M

β

_untk

0

Iksds−Ikuntkuntk .

3.2

It follows that{un} is bounded in X. From the reflexivity of X, we may extract a weakly convergent subsequence that, for simplicity, we call{un}, un uinX. In the following we will verify that{un}strongly converges touinX. By2.9we have

ϕun−ϕu

un−u un−u²_X

−^p−1

k1

e^−LtkIkuntk−I_kutnuntk−utk

− ₁

0

e^−Lt

gt, unt−gt, ut

unt−utdt.

3.3

Byu_n uinX, we see that{un}uniformly converges touinC0,1. So ₁

0

e^−Lt

gt, unt−gt, ut

unt−utdt−→0, p−1

k1

e^−LtkIkuntk−I_kutkuntk−utk−→0, ϕun−ϕu

un−u−→0 asn−→∞.

3.4

By3.3,3.4, we obtainun−u_X → 0 asn → ∞. That is,{un}strongly converges touin X, which means the that P. S. condition holds forϕ.

Secondly, we verify the conditionA1 in Theorem 2.3. Let V R, Y {u ∈ X | ₁

0utdt 0}, then X V ⊕ Y, where dimV 1 < ∞. In view of H2, take ε min{1/8M²₁₁

0e^−Ltdt,1/8M²_1p−1

k1e^−Ltk} > 0, there exists anδ > 0 such that for every u with|u|< δ,

Gt, u≤ε|u|², _u

0

I_ksds≤ε|u|². 3.5

Hence, for anyu∈Y withuX ≤δ/M₁, by2.8and3.5, we have

ϕu 1

2u²_X−^p−1

k1

e^−Ltk _utk

0

I_ksds− ₁

0

e^−LtGt, udt

≥ 1

2u²_X−^p−1

k1

e^−Ltkε|uktk|²− ₁

0

e^−Ltε|ut|²dt

≥ 1

2u²_X−εM²₁ p−1 k1

e^−Ltku²_X−εM²_{1 1}

0

e^−Ltdtu²_X

≥ 1

2u²_X−1

8u²_X−1 8u²_X 1

4u²_X.

3.6

Takeαδ²/4M₁², ρδ/M1, thenϕu≥α,∀u∈Y ∩∂Bρ.

Finally, we verify conditionA2inTheorem 2.3. According toH1, for anyu≥M >0 andt∈0,1we have that

Gt, u u^β

u

u^βgt, u−βu^β−1Gt, u

u^2β ugt, u−βGt, u

u^β1 ≥0. 3.7

Hence

Gt, u

u^β ≥ Gt, M

M^β ≥M^−βmin

t∈0,1Gt, M C>0 3.8

for allt∈0,1andu≥M >0. This implies thatGt, u≥Cu^βfor allt∈0,1andu≥M >0.

Similarly, we can prove that there is a constantC>0 such thatGt, u≥C|u|^βfor allt∈0,1 andu≤ −M. SinceGt, u−C₄|u|^βis continuous on0,1×−M, M, there existsC₅>0 such

thatGt, u−C4|u|^β>−C5on0, T×−M, M. Thus, we have

Gt, u≥C4|u|^β−C5 ∀t, u∈0,1×R, 3.9

whereC4min{C, C}.

Similarly, there exist constantsC₆, C₇>0 such that _u

0

I_ksds≥C₆|u|^β−C₇ ∀t, u∈0,1×R. 3.10

For everyξ ∈ R\ {0}andu ∈ W\ {0}, by2.8,3.9, and3.10, we have that the following inequality:

ϕξu≤ 1

2ξu²_X−^p−1

k1

e^−Ltk

C₆|ξutk|^β−C₇

− ₁

0

e^−Lt

C₄|ξu|^β−C₅ dt

≤ ξ²

2u²_X−C6|ξ|^β p−1 k1

e^−Ltk|utk|^βC7

p−1 k1

e^−Ltk−C4|ξ|^β ₁

0

e^−Lt|ut|^βdtC5

₁

0

e^−Ltdt 3.11

holds. Takew∈Wsuch thatw_X1, sinceβ >2,3.11implies that there existsξ>0 such thatξw_X > ρandϕξw<0 forξ ≥ξ >0. SinceWis a finite dimensional subspace, there existsRW>0 such thatϕu≤0 onW\B_RW. ByTheorem 2.3,ϕpossesses infinite many critical points; that is, IBVP1.1has infinite many classical solutions.

Theorem 3.2. Assume that (H1) and the first equality in (H2) hold. Moreover,gt, uis odd about uand the impulsive functionsI_kuare odd and nonincreasing. Then IBVP1.1has infinitely many classical solutions.

Proof. We only verifyA1inTheorem 2.3. SinceI_kuare odd and nonincreasing continuous functions, then for anyu ∈R,_u

0Iksds < 0. So we havep−1

k1e^−Ltk_untk

0 Iksds < 0. Take ε1/8M²₁₁

0e^−Ltdt >0, α3δ²/8M²₁, ρδ/M₁, like in3.6we can obtain the result.

Theorem 3.3. Suppose that the first inequalities in (H1), (H3), and (H4) hold. Furthermore, one assumes thatgt, uand the impulsive functionsIkuare odd aboutuand we have the following.

H7There existsA₀>0 such that

A0

2 > M₁ p−1 k1

e^−Ltk

b_kM^γ₁^kA^γ₀^k a_k M₁

h₂M^p₁¹A^p₀¹h₁¹

0

e^−Ltdt. 3.12

Then IBVP1.1has infinitely many classical solutions.

Proof. Obviously,ϕis an even functional andϕ0 0. Firstly, we will show thatϕsatisfies the Palais-Smale condition. As in the proof of Theorem 3.1, by 2.8, 2.9, 3.1, the first inequalities inH1andH4, we have

β 2 −1

un²_X β

2un²_X− u_n²_X βϕun−ϕununβ

p−1 k1

e^−Ltk _untk

0

I_ksdsβ ₁

0

e^−LtGt, undt

−^p−1

k1

e^−LtkIkuntkuntk− ₁

0

e^−Ltgt, unundt

βϕun−ϕunun ₁

0

e^−Lt

βGt, un−gt, unun

dt

β

p−1

k1

e^−Ltk _untk

0

Iksds−^p−1

k1

e^−LtkIkuntkuntk

≤βC₃M²₁C₃un_{X 1}

0

e^−Ltdt max

t∈0,1, unt∈−M,MβGt, un−gt, unun

β1^p−1

k1

e^−Ltk

a_kM₁un_Xb_kM₁^γk1un^γ_X^k1 .

3.13

It follows that{un}is bounded inX. In the following, the proof of P. S. condition is the same as that inTheorem 3.1, and we omit it here.

Secondly, as in Theorem 3.1, we can obtain that condition A2 in Theorem 2.1 is satisfied.

Take the same direct sum decompositionXV⊕Yas inTheorem 3.1. For anyu∈Y, by2.8,H3, andH4, we obtain

ϕu 1

2u²_X−^p−1

k1

e^−Ltk _utk

0

I_ksds− ₁

0

e^−LtGt, udt

≥ 1

2u²_X−^p−1

k1

e^−Ltk

akM1u_XbkM^γ₁^k1u^γ_X^k1

− ₁

0

e^−Ltdt

h₁M₁u_Xh₂M^p₁¹¹u^p_X¹¹

1

2u²_X−^p−1

k1

e^−Ltkb_kM^γ₁^k1u^γ_X^k1−h₂M^p₁¹¹ ₁

0

e^−Ltdtu^p_X¹¹

−M₁u_{X p−1}

k1

e^−Ltka_kh_{1 1}

0

e^−Ltdt

.

3.14

In view ofH7, setu_Xρ:A0>0, then we have

ϕu≥α 1

2A²₀−^p−1

k1

e^−Ltkb_kM^γ₁^k1A^γ₀^k1−h₂M₁^p11 ₁

0

e^−LtdtA^p₀¹¹

−M1A0

_p−1

k1

e^−Ltkakh1

₁

0

e^−Ltdt

>0.

3.15

Therefore,ϕu≥α,∀u∈Y ∩∂B_ρ.ByTheorem 2.3,ϕpossesses infinite many critical points, that is, IBVP1.1has infinite many classical solutions.

Theorem 3.4. Assume that the second inequalities in (H1), (H5), and (H6) hold, moreover, one assumes the following.

H8There existsA1>0 such that

A1

2 > M₁

p−1

k1

e^−Ltk

b_kM₁^γkA^γ₁^k a_k M₁

r₂M^μ₁A^μ₁r₁¹

0

e^−Ltdt. 3.16

Then IBVP1.1has at least two classical solutions.

Proof. We will use Theorems2.1and2.2to prove the main results. Firstly, we will show that ϕsatisfies the Palais-Smale condition. Similarly, as in the proof ofTheorem 3.1, by2.8,2.9, 3.1, the second inequalities inH1andH5, we have

β 2−1

un²_X β

2un²_X− u_n²_X βϕun−ϕununβ

p−1 k1

e^−Ltk _untk

0

I_ksdsβ ₁

0

e^−LtGt, undt

−^p−1

k1

e^−LtkIkuntkuntk− ₁

0

e^−Ltgt, unundt

≤βC₃M²₁C₃un_{X 1}

0

e^−Ltdt

r₁r₂M₁^μun^μ_X

β ₁

0

e^−Ltdt

M1r1un_Xr2M^μ1₁ un^μ1_X

^p−1

k1

e^−Ltk max

untk∈−M,M

β

_untk

0

I_ksds−I_kuntkuntk

βC3M²₁C3un_{X 1}

0

e^−Ltdt

r1r2M₁^μun^μ_X

βM1un_X1

^p−1

k1

e^−Ltk max

untk∈−M,M

β

_untk

0

Iksds−Ikuntkuntk .

3.17

It follows that{un}is bounded inX. In the following, the proof of P. S. condition is the same as that inTheorem 3.1, and we omit it here.

Let A > 0, which will be determined later. Set BA : {u ∈ X : uX < A}, then B_A : {u ∈ X : u_X ≤ A} is a closed ball. From the reflexivity ofX, we can easily obtain thatBA is bounded and weakly sequentially closed. We will show thatϕ is weakly lower semicontinuous onB_A. Let

ϕ₁u 1 2

₁

0

e^−Ltptut²dt ₁

0

e^−Ltqt|ut|²dt,

ϕ₂u −^p−1

k1

e^−Ltk _utk

0

I_ksds− ₁

0

e^−LtGt, udt.

3.18

Thenϕu ϕ1u ϕ2u. By un u onX we see that{un} uniformly converges to u inC0,1. Soϕ₂is weakly continuous. Clearly, ϕ₁is continuous, which, together with the convexity ofϕ1, implies thatϕ1is weakly lower semicontinuous. Therefore,ϕis weakly lower semi-continuous onB_A. So byTheorem 2.1, without loss of generality, we assume thatϕu0

inf_u∈B

Aϕu. Now we will show that

ϕu0< inf

u∈∂BA

ϕu. 3.19

For anyu∈∂B_A, byH5andH6, we have

ϕu 1

2u²_X−^p−1

k1

e^−Ltk _utk

0

I_ksds− ₁

0

e^−LtGt, udt

≥ 1

2u²_X−M1

p−1 k1

e^−Ltk

b_kM^γ₁^ku^γ_X^k1a_ku_X

−M_{1 1}

0

e^−Ltdt

r₂M₁^μu^μ1_X r₁u_X .

3.20

Hence

u∈∂BinfA

ϕu≥ 1

2u²_X−M₁

p−1

k1

e^−Ltk

b_kM^γ₁^ku^γ_X^k1a_ku_X

−M1

₁

0

e^−Ltdt

r2M^μ₁u^μ1_X r1u_X .

3.21

In view ofH8, takeA A1 > 0, we have inf_u∈∂BA

1ϕu > 0, for anyu ∈∂BA1. Soϕu0 <

ϕ0 0<inf_u∈∂BA1ϕu.

Next we will verify that there exists a u₁ with u1_X > A₁ such that ϕu1 <

inf_u∈∂BA

1ϕu. Letξ∈R\ {0}, Bt 1. Then by3.10andH5, we have ϕξB ξ²

2 ₁

0

e^−Ltqtdt−^p−1

k1

e^−Ltk _ξ

0

I_ksds− ₁

0

e^−LtGt, ξdt

≤ ξ² 2

₁

0

e^−Ltqtdt−C6|ξ|^βp−1

k1

e^−LtkC7

p−1 k1

e^−Ltk

r₂|ξ|^μ1r₁|ξ|¹

0

e^−Ltdt.

3.22

Sinceβ > 2,0 ≤ μ < 1, we have lim_{|ξ| →∞}ϕξB −∞. Therefore, there exists a sufficiently largeξ₀ > 0 withξ0B_X > A₁such that ϕξ0B < inf_u∈∂BA

1ϕu. Setu₁ ξ₀B, thenϕu1 <

inf_u∈∂BA

1ϕu. So byTheorem 2.2, there existsu₂ ∈X such thatϕu2 0. Therefore,u₀and u2are two critical points ofϕ, and they are classical solutions of IBVP1.1.

Remark 3.5. Obviously, ifgis a bounded function, in view ofTheorem 3.4, we can obtain the same result.

Theorem 3.6. Suppose that (H4) and (H5) hold. Then IBVP1.1has at least one solution.

Proof. The proof is similar to that in19, and we omit it here.

Corollary 3.7. Suppose thatgand impulsive functionsIk, k1,2, . . . , p−1 are bounded, then IBVP 1.1has at least one solution.

4. Some Examples

Example 4.1. Consider the following problem:

−ut ut ut gt, ut, t /tk, a.e. t∈0,1,

−Δ utk

I_kutk, k1,2, u0 u

1⁻ 0,

4.1

wheregt, u 4u³6tu⁵, I_ku u³.