Volume 2010, Article ID 325415,16pages doi:10.1155/2010/325415
Research Article
Variational Approach to Impulsive Differential Equations with Dirichlet Boundary Conditions
Huiwen Chen and Jianli Li
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Correspondence should be addressed to Jianli Li,[email protected] Received 18 September 2010; Accepted 9 November 2010 Academic Editor: Zhitao Zhang
Copyrightq2010 H. Chen and J. Li. 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.
We study the existence of n distinct pairs of nontrivial solutions for impulsive differential equations with Dirichlet boundary conditions by using variational methods and critical point theory.
1. Introduction
Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Such processes are naturally seen in control theory1,2, population dynamics3, and medicine4,5. Due to its significance, a great deal of work has been done in the theory of impulsive differential equations. In recent years, many researchers have used some fixed point theorems6,7, topological degree theory8, and the method of lower and upper solutions with monotone iterative technique9to study the existence of solutions for impulsive differential equations.
On the other hand, in the last few years, some researchers have used variational methods to study the existence of solutions for boundary value problems10–16, especially, in 14–16, the authors have studied the existence of infinitely many solutions by using variational methods.
However, as far as we know, few researchers have studied the existence ofndistinct pairs of nontrivial solutions for impulsive boundary value problems by using variational methods.
Motivated by the above facts, in this paper, our aim is to study the existence of n distinct pairs of nontrivial solutions to the Dirichlet boundary problem for the second-order impulsive differential equations
ut λht, ut 0, t /tj, a.e. t∈0, T,
−Δu tj
Ij
u tj
, j1,2, . . . , p,
u0 uT 0,
1.1
where 0t0 < t1 <· · ·< tp < tp1T,λ >0,h∈C0, T×R, R,Ij ∈CR, R,j 1,2, . . . , p, Δutj utj−ut−j,utjandut−jdenote the right and the left limits, respectively, of utjatttj,j1,2, . . . , p.
2. Preliminaries
Definition 2.1. Suppose thatEis a Banach space andϕ ∈C1E, R. If any sequence{uk} ⊂ E for whichϕukis bounded andϕuk → 0 ask → ∞possesses a convergent subsequence inE, we say thatϕsatisfies the Palais-Smale condition.
LetEbe a real Banach space. Define the setΣ {A|A⊂E\ {θ}as symmetric closed set}.
Theorem 2.2 see 17, Theorem 3.5.3. Let E be a real Banach space, and let ϕ ∈ C1E, R be an even functional which satisfies the Palais-Smale condition, ϕ is bounded from below and ϕ0 0; suppose that there exists a setK ⊂ Σ and an odd homeomorphismh : K → Sn−1n− one-dimensional unit sphereand supx∈Kϕx<0, thenϕhas at least n distinct pairs of nontrivial critical points.
To begin with, we introduce some notation. Denote byX the Sobolev spaceH010, T, and consider the inner product
u, v T
0
utvtdt 2.1
and the norm
u T
0
ut2dt 1/2
. 2.2
Hence,Xis reflexive. We define the norm inC0, Tasx∞maxt∈0,T|xt|.
Foru∈H20, T, we have thatuanduare absolutely continuous andu ∈L20, T. Hence,Δut ut−ut− 0 for everyt ∈ 0, T. If u∈ H010, T, thenuis absolutely continuous andu ∈ L20, T. In this case, the one-sided derivatives ut−,ut may not exist. As a consequence, we need to introduce a different concept of solution. Suppose that u ∈ C0, Tsuch that for everyj 1,2, . . . , p,uj u|tj,tj1satisfiesuj ∈ H2tj, tj1, and it satisfies the equation in problem1.1fort /tj, a.e.t∈0, T, the limitsutj, ut−j, and
j1,2, . . . , pexist, and impulsive conditions and boundary conditions in problem1.1hold, we say it is a classical solution of problem1.1.
Consider the functional
ϕ:X−→R, 2.3
defined by
ϕu 1
2u2−λ T
0
Ht, utdt− p
j1
utj
0
Ijsds, 2.4
whereHt, u 0uht, sds. Clearly,ϕis a Fr´echet differentiable functional, whose Fr´echet derivative at the pointu∈Xis the functionalϕu∈X∗given by
ϕuv
T
0
utvtdt−λ T
0
ht, utvtdt− p j1
Ij
u tj
v tj
, 2.5
for anyv∈X. Obviously,ϕis continuous.
Lemma 2.3. Ifu∈Xis a critical point of the functionalϕ, then u is a classical solution of problem 1.1.
Proof. The proof is similar to the proof of16, Lemma 2.4, and we omit it here.
Lemma 2.4. Letu∈X, thenu∞≤√ Tu.
Proof. Foru∈X, thenu0 uT 0. Hence, fort∈0, T, by H ¨older’s inequality, we have
|ut|
t
0
usds ≤
T
0
usds≤√ T
T
0
us2ds 1/2
√
Tu, 2.6
which completes the proof.
3. Main Results
Theorem 3.1. Suppose that the following conditions hold.
iThere exista, b >0 andγ∈0,1such that
|ht, u| ≤ab|u|γ for anyt, u∈0, T×R. 3.1
iiht, uis odd about u andHt, u>0 for everyt, u∈0, T×R\ {0}.
iiiIju j1,2, . . . , pare odd and 0uIjsds≤0 for anyu∈Rj 1,2, . . . , p.
Then for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. By2.4,ii, andiii,ϕ∈C1X, Ris an even functional andϕ0 0.
Next, we will verify thatϕis bounded from below. In view ofi,iii, andLemma 2.4, we have
ϕu 1
2u2−λ T
0
Ht, utdt− p
j1
utj
0
Ijsds
≥ 1
2u2−λ T
0
a|ut|b|ut|γ1 dt
≥ 1
2u2−λaT3/2u −λbTγ3/2uγ1
>−∞,
3.2
for anyu∈X. That is,ϕis bounded from below.
In the following we will show thatϕsatisfies the Palais-Smale condition. Let{uk} ⊂X, such that{ϕuk}is a bounded sequence and limk→ ∞ϕuk 0. Then, there existsM > 0 such that
ϕuk≤M. 3.3
In view of3.2, we have M≥ 1
2uk2−λaT3/2uk −λbTγ3/2ukγ1. 3.4 So {uk} is bounded inX. From the reflexivity of X, we may extract a weakly convergent subsequence that, for simplicity, we call {uk},uk uinX. Next, we will verify that {uk} strongly converges touinX. By2.5, we have
ϕuk−ϕu
uk−u uk−u2−λ T
0
ht, ukt−ht, utukt−utdt
p j1
Ij
uk
tj
−Ij
u tj
uk
tj
−u tj
.
3.5
Byuk uinX, we see that{uk}uniformly converges touinC0, T. So,
λ T
0
ht, ukt−ht, utukt−utdt−→0, p
j1
Ij
uk
tj
−Ij
u tj
uk
tj
−u tj
−→0 ask−→ ∞.
3.6
By limk→ ∞ϕuk 0 anduk u, we have ϕuk−ϕu
uk−u−→0 ask−→ ∞. 3.7
In view of3.5,3.6, and3.7, we obtainuk−u → 0 as k → ∞. Then,ϕsatisfies the Palais-Smale condition.
Letvmt √
2T/mπsinmπ/Tt,m1,2, . . . , n, then
vm2 1 m2π2 T2
T
0
|vmt|2dt, m1,2, . . . , n. 3.8
Define
Knr n
m1
cmvm|n
m1
c2mr2
, r >0. 3.9
Then, for anyr >0, there exists an odd homeomorphismf:Knr → Sn−1. Let 0< r <1/√ T, thenu∞≤√
Tu√
Tr <1 for anyu∈Knr. Byii, we have
Ht, ut ut
0
ht, sds >0 asut/0, 3.10
then 0THt, utdt >0 for anyu∈Knr.
Letαn infu∈Knr 0THt, utdt,βninfu∈Knrp j1 utj
0 Ijsds, thenαn >0,βn ≤0.
Letλn 1/2r2−βnα−1n >0, then whenλ > λn, for anyu∈Knr, we have ϕu≤ 1
2r2−λαn−βn
< 1
2r2−λnαn−βn
0.
3.11
By Theorem 2.2, ϕ possesses at least n distinct pairs of nontrivial critical points. That is, problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Corollary 3.2. Let the following conditions hold:
iht, uis bounded,
iiht, uis odd about u andHt, u>0 for everyt, u∈0, T×R\ {0}, iiiIju j1,2, . . . , pare odd and 0uIjsds≤0 for anyu∈Rj 1,2, . . . , p.
Then, for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. Letγ0 inTheorem 3.1, thenCorollary 3.2holds.
Theorem 3.3. Suppose that the following conditions hold.
iThere existsa, b >0 andγ∈0,1such that
|ht, u| ≤ab|u|γ for anyt, u∈0, T×R. 3.12
iiThere existsaj, bj>0 andγj ∈0,1 j1,2, . . . , psuch that Iju≤ajbj|u|γj for anyu∈R
j1,2, . . . , p
. 3.13
iiiht, u and Iju j 1,2, . . . , p are odd about u and Ht, u > 0 for everyt, u ∈ 0, T×R\ {0}.
Then, for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. By2.4andiii,ϕ∈C1X, Ris an even functional andϕ0 0.
Next, we will verify thatϕis bounded from below. LetM1max{a1, a2, . . . , ap},M2 max{b1, b2, . . . , bp}. In view ofi,ii, andLemma 2.4, we have
ϕu 1
2u2−λ T
0
Ht, utdt
p j1
utj
0
Ijsds
≥ 1
2u2−λ T
0
a|ut|b|ut|γ1 dt
− p j1
aju
tjbju
tjγj1
≥ 1
2u2−λaT3/2u −λbTγ3/2uγ1−pM1
√Tu
−M2
p j1
Tγj1/2uγj1
>−∞,
3.14
for anyu∈X. That is,ϕis bounded from below.
In the following, we will show thatϕ satisfies the Palais-Smale condition. As in the proof ofTheorem 3.1, by3.3and3.14, we have
M≥ 1
2uk2−λaT3/2uk −λbTγ3/2ukγ1−pM1
√Tuk −M2
p j1
Tγj1/2ukγj1. 3.15
It follows that{uk}is bounded inX. In the following, the proof of the Palais-Smale condition is the same as that inTheorem 3.1, and we omit it here.
Take the same Knr as in Theorem 3.1, then for any r > 0, there exists an odd homeomorphismf : Knr → Sn−1. Let 0 < r < 1/√
T, then u∞ ≤ √
Tu √
Tr < 1 for anyu∈Knr. Byiii, we have
Ht, ut ut
0
ht, sds >0 asut/0. 3.16
Then, 0THt, utdt >0 for anyu∈Knr.
Let αn infu∈Knr 0THt, utdt,βn infu∈Knrp j1
utj
0 Ijsds, thenαn > 0. Let λnmax{0,1/2r2−βnα−1n }, then whenλ > λn, for anyu∈Knr, we have
ϕu≤ 1
2r2−λαn−βn< 1
2r2−λnαn−βn≤0. 3.17 By Theorem 2.2, ϕ possesses at least n distinct pairs of nontrivial critical points. That is, problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Corollary 3.4. Let the following conditions hold:
iht, uis bounded,
iiIju j1,2, . . . , pare bounded,
iiiht, u and Iju j 1,2, . . . , p are odd about u and Ht, u > 0 for everyt, u ∈ 0, T×R\ {0}.
Then, for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. Letγ0 andγj0 j1,2, . . . , pinTheorem 3.3, thenCorollary 3.4holds.
Theorem 3.5. Suppose that the following conditions hold.
iThere exist constantsσ >0 such thatht, σ 0, ht, u>0 for everyu∈0, σ.
iiht, uis odd aboutu.
iiiIju j1,2, . . . , pare odd and 0uIjsds≤0 for anyu∈Rj 1,2, . . . , p.
Then, for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. Let
h1t, u
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
ht, σ, u > σ, ht, u, |u| ≤σ, ht,−σ, u <−σ,
3.18
thenh1t, uis continuous, bounded, and odd. Consider boundary value problem ut λh1t, ut 0, t /tj, a.e. t∈0, T,
−Δu tj
Ij
u tj
, j1,2, . . . , p,
u0 uT 0.
3.19
Next, we will verify that the solutions of problem3.19are solutions of problem1.1. In fact, letu0tbe the solution of problem3.19. If max0≤t≤Tu0t> σ, then there exists an interval a, b⊂0, Tsuch that
u0a u0b σ, u0t> σ for anyt∈a, b. 3.20
Whent∈a, b, byi, we have
u0t −λh1t, u −λht, σ 0. 3.21 Thus, there exist constantscsuch thatu0t cfor anyt∈a, b. We consider the following two possible cases.
Case 1. c≥0, thenu0is nondecreasing ina, b. Byu0a≥0 andu0b≤0, we have
0≤u0a≤u0t≤u0b≤0 for everyt∈a, b. 3.22 That is,u0t ≡ 0 for anyt ∈a, b. So, there exists a constantdsuch thatu0t ≡ d, which contradicts3.20. Then, max0≤t≤Tu0t≤σ. Similarly, we can prove that min0≤t≤Tu0t≥ −σ.
Case 2. c <0, the arguments are analogous, thenu0tis solution of problem1.1.
For everyu∈X, we consider the functional
ϕ1:X−→R, 3.23
defined by
ϕ1u 1
2u2−λ T
0
H1t, utdt− p j1
utj
0
Ijsds, 3.24
whereH1t, u 0uh1t, sds.
It is clear thatϕ1is Fr´echet differentiable at anyu∈Xand
ϕ1uv T
0
utvtdt−λ T
0
h1t, utvtdt− p j1
Ij
u tj
v tj
, 3.25
for anyv∈X. Obviously,ϕ1is continuous. By Lemma2.3, we have the critical points ofϕ1as solutions of problem3.19. By3.24,ii, andiii,ϕ1 ∈C1X, Ris an even functional and ϕ10 0.
In the following, we will show thatϕ1 is bounded from below. sinceh1t, u 0 for
|u| ≥σ, thus
T
0
H1t, utdt T
0
ut
0
h1t, sds dt≤ T
0
σ
0
h1t, sds dte >0. 3.26
Byiii, we have
ϕ1u 1
2u2−λ T
0
H1t, utdt− p j1
utj
0
Ijsds
≥ 1
2u2−λe≥ −λe,
3.27
for anyu∈X. That is,ϕ1is bounded from below.
In the following we will show thatϕ1satisfies the Palais-Smale condition. Let{uk} ⊂X such that{ϕ1uk}is a bounded sequence and limk→ ∞ϕ1uk 0. Then, there existsM3 >0 such that
ϕ1uk≤M3. 3.28
By3.27, we have
1
2uk2≤M3λe. 3.29
It follows that{uk}is bounded inX. In the following, the proof of the Palais-Smale condition is the same as that inTheorem 3.1, and we omit it here.
Take the same Knr as in Theorem 3.1, then, for any r > 0, there exists an odd homeomorphismf : Knr → Sn−1. Let 0 < r < σ/√
T, thenu∞ ≤ √
Tu √
Tr < σ for anyu∈Knr. Byiandii, we have
H1t, ut ut
0
h1t, sds ut
0
ht, sdt >0 asut/0. 3.30
Then, 0TH1t, utdt >0 for anyu∈Knr.
Letαninfu∈Knr 0TH1t, utdt,βninfu∈Knrp j1 utj
0 Ijsds, thenαn>0,βn≤0.
Letλn 1/2r2−βnα−1n >0, then whenλ > λn, for anyu∈Knr, we have ϕ1u≤ 1
2r2−λαn−βn
< 1
2r2−λnαn−βn
0.
3.31
By Theorem 2.2, ϕ1 possesses at least n distinct pairs of nontrivial critical points. Then, problem3.19has at leastndistinct pairs of nontrivial classical solutions, that is, problem 1.1has at leastndistinct pairs of nontrivial classical solutions
Theorem 3.6. Let the following conditions hold.
iThere exist constantsσ >0 such thatht, σ 0, ht, u>0 for everyu∈0, σ.
iiThere existaj, bj>0, andγj∈0,1 j 1,2, . . . , psuch that Iju≤ajbj|u|γj for anyu∈R
j1,2, . . . , p
. 3.32
iiiht, uandIju j1,2, . . . , pare odd aboutu.
Then, for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. The proof is similar to the proof ofTheorem 3.5, and we omit it here.
Theorem 3.7. Let the following conditions hold.
iThere exist constantsσ1>0 such thatht, σ1≤0.
iiThere existaj, bj>0, andγj∈0,1 j 1,2, . . . , psuch that Iju≤ajbj|u|γj for anyu∈R
j1,2, . . . , p
. 3.33
iiiht, uandIju j1,2, . . . , pare odd aboutuand limu→0ht, u/u1 uniformly for t∈0, T.
Then, for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. Let
h2t, u
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
ht, σ1, u > σ1, ht, u, |u| ≤σ1, ht,−σ1, u <−σ1,
3.34
thenh2t, uis continuous, bounded, and odd. Consider boundary value problem ut λh2t, ut 0, t /tj, a.e. t∈0, T,
−Δu tj
Ij
u tj
, j1,2, . . . , p,
u0 uT 0.
3.35
Next, we will verify that the solutions of problem3.35are solutions of problem1.1. In fact, letu0tbe the solution of problem3.35. If max0≤t≤Tu0t> σ1, then there exists an interval a, b⊂0, Tsuch that
u0a u0b σ1, u0t> σ1 for anyt∈a, b. 3.36
Whent∈a, b, byi, we have
u0t −λh2t, u −λht, σ1≥0. 3.37
Thus,u0tis nondecreasing ina, b. Byu0a≥0 andu0b≤0, we have
0≤u0a≤u0t≤u0b≤0 for everyt∈a, b. 3.38 That is,u0t ≡ 0 for anyt ∈a, b. So, there exists a constantdsuch thatu0t ≡ d, which contradicts3.36. Then max0≤t≤Tu0t≤σ1. Similarly, we can prove that min0≤t≤Tu0t≥ −σ1. Then,u0tis solution of problem1.1.
For everyu∈X, we consider the functional
ϕ2:X−→R, 3.39
defined by
ϕ2u 1
2u2−λ T
0
H2t, utdt− p j1
utj
0
Ijsds, 3.40
whereH2t, u 0uh2t, sds.
It is clear thatϕ2is Fr´echet differentiable at anyu∈Xand
ϕ2uv T
0
utvtdt−λ T
0
h2t, utvtdt− p j1
Ij
u tj
v tj
, 3.41
for anyv ∈X. Obviously,ϕ2 is continuous. ByLemma 2.3, we have the critical points ofϕ2
as solutions of problem3.35. By3.40andiii,ϕ2 ∈ C1X, Ris an even functional and ϕ20 0.
Next, we will show that ϕ2 is bounded from below. Let M1 max{a1, a2, . . . , ap}, M2max{b1, b2, . . . , bp}. sinceuh2t, u≤0 for|u| ≥σ1, thus
T
0
H2t, utdt T
0
ut
0
h2t, sds dt≤ T
0
σ1
0
h2t, sds dte. 3.42
ByiiandLemma 2.4, we have
ϕ2u 1
2u2−λ T
0
H2t, utdt− p
j1
utj
0
Ijsds
≥ 1
2u2−λe−pM1
√Tu −M2
p j1
Tγj1/2uγj1
>−∞,
3.43
for anyu∈X. That is,ϕ2is bounded from below.
In the following we will show thatϕ2satisfies the Palais-Smale condition. Let{uk} ⊂X such that{ϕ2uk}is a bounded sequence and limk→ ∞ϕ2uk 0. Then, there existsM4 >0 such that
ϕ2uk≤M4. 3.44
By3.43, we have
1
2uk2≤M4λepM1
√TukM2
p j1
Tγj1/2ukγj1. 3.45
It follows that{uk}is bounded inX. In the following, the proof of the Palais-Smale condition is the same as that inTheorem 3.1, and we omit it here.
Take the same Knr as in Theorem 3.1, then for any r > 0, there exists an odd homeomorphismf : Knr → Sn−1. By iii, for any 0 < ε < 1, there existsδ > 0, when
|u| ≤δ, we have
h2t, u≥u−ε|u|. 3.46
Let 0< r <min{σ1/√ T, δ/√
T}, thenu∞≤√
Tu√
Tr <min{σ1, δ}for anyu∈Knr.
Then, 0TH2t, utdt 0T 0uth2t, sdt≥ 0T1/21−ε|ut|2dt >0 for anyu∈Knr.
Letαn infu∈Knr 0TH2t, utdt,βn infu∈Knrp j1 utj
0 Ijsds, thenαn > 0. Let λnmax{1/2r2−βnα−1n ,0}, then whenλ > λn, for anyu∈Knr, we have
ϕ1u≤ 1
2r2−λαn−βn
< 1
2r2−λnαn−βn
≤0.
3.47
By Theorem 2.2, ϕ2 possesses at least n distinct pairs of nontrivial critical points. Then, problem3.35has at leastndistinct pairs of nontrivial classical solutions, that is, problem 1.1has at leastndistinct pairs of nontrivial classical solutions.
Theorem 3.8. Let the following conditions hold.
iThere exist constantsσ1>0 such thatht, σ1≤0.
iilimu→0ht, u/u1 uniformly fort∈0, T.
iiiht, uandIju j 1,2, . . . , pare odd aboutuand 0uIjsds≤0 for anyu∈R j 1,2, . . . , p.
Then, for anyn∈N, there existsλnsuch thatλ > λn, and problem1.1has at leastndistinct pairs of nontrivial classical solutions.
Proof. The proof is similar to the proof ofTheorem 3.7, and we omit it here.
4. Some Examples
Example 4.1. Consider boundary value problem
ut λ1t3
ut 0, t /tj, a.e. t∈0, π,
−Δu tj
−u tj
, j1,2,
u0 uπ 0.
4.1
It is easy to see that conditionsi,ii, andiiiofTheorem 3.1hold. Let αn inf
u∈Knr
3 4
π
0
1t|ut|4/3dt > inf
u∈Knr
3 4
π
0
|ut|2dt > 3r2 4n2, βn inf
u∈Knr−2
j1
utj
0
sds inf
u∈Knr−2
j1
u tj2
2 ≥ −πr2,
4.2
thenλn 1/2r2 −βnα−1n < 24π/3n2. ApplyingTheorem 3.1, then for anyn ∈ N, whenλ > 24π/3n2, problem4.1has at leastndistinct pairs of nontrivial classical solutions.
Example 4.2. Consider boundary value problem
ut λ1t3
ut 0, t /tj, a.e. t∈0, π,
−Δu tj
−3 u
tj
, j1,2,
u0 uπ 0.
4.3
It is easy to see that conditionsi,ii, andiiiofTheorem 3.3hold. Letr 1/2√ π,
αn inf
u∈Knr
3 4
π
0
1t|ut|4/3dt > inf
u∈Knr
3 4
π
0
|ut|2dt > 3r2 4n2, βn inf
u∈Knr−2
j1
utj
0
s1/3ds inf
u∈Knr−2
j1
3 4u
tj4/3>−3 2,
4.4
thenλn 1/2r2−βnα−1n < 224π/3n2. Applying Theorem 3.3, then for any n ∈ N, whenλ > 224π/3n2, problem4.3has at least ndistinct pairs of nontrivial classical solutions.
Example 4.3. Consider boundary value problem
ut λ 1t2
ut−ut3
0, t /tj, a.e. t∈0, π,
−Δu tj
−u tj
, j1,2,
u0 uπ 0.
4.5
Letσ1, it is easy to see that conditionsi,ii, andiiiofTheorem 3.5hold. Let αn inf
u∈Knr
π
0
1t21
2|ut|2−1 4|ut|4
dt > inf
u∈Knr
1 4
π
0
|ut|2dt > r2 4n2, βn inf
u∈Knr−2
j1
utj
0
sds inf
u∈Knr−2
j1
u tj2
2 ≥ −πr2,
4.6
thenλn 1/2r2−βnα−1n <24πn2. ApplyingTheorem 3.5, then for anyn∈N, whenλ >24πn2, problem4.5has at leastndistinct pairs of nontrivial classical solutions.
Example 4.4. Consider boundary value problem
ut λ
ut−1tut3
0, t /tj, a.e. t∈0, π,
−Δu tj
−3 u
tj
, j1,2,
u0 uπ 0.
4.7
Letσ1 1, it is easy to see that conditionsi,ii, andiiiofTheorem 3.7hold. Let r1/4√
π,
αn inf
u∈Knr
π
0
1
2|ut|2− 1
41t|ut|4
dt > inf
u∈Knr
1 4
π
0
|ut|2dt > r2 4n2, βn inf
u∈Knr−2
j1
utj
0
s1/3ds inf
u∈Knr−2
j1
3 4u
tj4/3>−3 2,
4.8
thenλn 1/2r2 −βnα−1n < 224πn2. Applying Theorem 3.7, then for anyn ∈ N, whenλ > 224πn2, problem4.7has at least ndistinct pairs of nontrivial classical solutions.
Acknowledgments
This work was supported by the NNSF of Chinano. 10871062and a project supported by Hunan Provincial Natural Science Foundation of Chinano. 10JJ6002.
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