Positive solutions of singular Dirichlet problems via variational methods

http://dx.doi.org/10.4134/JKMS.2013.50.4.797

POSITIVE SOLUTIONS OF SINGULAR DIRICHLET PROBLEMS VIA VARIATIONAL METHODS

Juntao Sun and Jifeng Chu

Abstract. In this paper, we establish the existence results for second order singular Dirichlet problems via variational methods. Some recent results are extended and improved. Examples are also given to illustrate the new results.

1. Introduction

We are devoted to establish the existence results for the following second- order non-autonomous Dirichlet problems

(1.1) ( − u

^′′

= λf (t, u), 0 < t < 1, u(0) = u(1) = 0,

where λ ≥ 1 is a parameter and f ∈ C((0, 1) × (0, ∞), [0, ∞)) satisfies the following condition

(A) there exist some positive constants ǫ, C (C could be dependent of ǫ) and α ∈ (0, 1) such that

2ǫ ≤ f (t, u) ≤ Cu

^−α

, (t, u) ∈ (0, 1) × (0, ǫ).

Therefore, f may be singular at u = 0. A typical example is

(1.2) f (t, u) = u

^−α

+ g(t, u)

with g ∈ C((0, 1) × [0, ∞), [0, ∞)).

Problem (1.1) arises in the transport of coal slurries down conveyor belts [19], boundary layer theory [10] and chemical reactor theory [6, 7]. Due to the

Received June 9, 2011; Revised February 14, 2013.

2010 Mathematics Subject Classification. 34B15.

Key words and phrases. positive solutions, singular Dirichlet problems, variational meth- ods, critical points.

Juntao Sun was supported by the National Natural Science Foundation of China (Grant No. 11201270, No.11271372), Shandong Natural Science Foundation (Grant No.

ZR2012AQ010), and Young Teacher Support Program of Shandong University of Technology.

Jifeng Chu was supported by the National Natural Science Foundation of China (Grant No.

11171090), the Program for New Century Excellent Talents in University (Grant No. NCET- 10-0325), China Postdoctoral Science Foundation funded project (Grant 2012T50431).

c

2013 The Korean Mathematical Society 797

important applications, problem (1.1) has received considerable attention dur- ing the last few decades. We refer the reader to [2, 3, 13, 16, 20]. Some classical tools have been used to study (1.1) in literatures, including the method of up- per and lower solutions [4, 14], some fixed point theorems [11, 17], a nonlinear Leray-Schauder alternative principle [3, 11] and the variational approach [5].

Among those results, an important existence result was established by Agarwal and O’Regan in [3], in which it was proved that the scalar problem

(1.3) ( u

^′′

+ q(t)f (t, u) = 0, 0 < t < 1, u(0) = u(1) = 0

has at least two positive solutions under some natural conditions, based on the Leray-Schauder alternative principle and a well-known fixed point theorem on cones. In particular, they obtained that the following problem







u

^′′

+ 1

α + 1 (u

^−α

+ u

^β

+ 1) = 0, 0 < t < 1, u(0) = u(1) = 0, α > 0, β > 1

admits two positive solutions u

1

, u

2

∈ C[0, 1] ∩ C

²

(0, 1) with ku

1

k < 1 <

ku

2

k. Recently, Chu and O’Regan [11] extended the above results to the non- autonomous Dirichlet system

(1.4) ( u

^′′

+ q(t)f (t, u) + e(t) = 0, 0 < t < 1, u(0) = u(1) = 0,

in which e is allowed to take negative values.

The present work is mainly motivated by [5], in which Agarwal, Perera and

O’Regan established the existence results for problem (1.1) with λ = 1 by

variational methods. In particular, they proved that problem (1.1) possesses

two positive solutions when the nonlinear term satisfies the asymptotically

linear and superlinear conditions at infinity, respectively. Inspired by [5], in

this paper we prove that problem (1.1) has at least one positive solution and

two positive solutions under the different conditions, respectively. Moreover,

we also give the existence result of infinitely many positive solutions for singular

problem (1.1), in which the symmetric condition on the nonlinear term is not

assumed. The proofs are based on variational methods. It is worth noting

that there are some works concerning on the results of infinitely many positive

solutions for singular problems up to now. For example, Cid et al. [12] obtained

the existence of infinitely many solutions for a second-order singular problem

with initial value condition. He and Zou [15] studied a singular elliptic equation

involving critical exponents in the whole space R

^N

, and obtained the existence

of infinitely many solutions. However, their results cannot cover our case and

our results are new compared with those in the literature.

2. Preliminaries Define f

ǫ

∈ C((0, 1) × R, [0, ∞)) by

(2.1) f

ǫ

(t, u) = f (t, (u − ϕ

ǫ

(t))

⁺

+ ϕ

ǫ

(t)), where u

^±

= max{±u, 0} and ϕ

ǫ

(t) = ǫt(1 − t) is the solution of (2.2) ( − u

^′′

= 2ǫ, 0 < t < 1,

u(0) = u(1) = 0.

Consider

(2.3) ( − u

^′′

= λf

ǫ

(t, u), 0 < t < 1, u(0) = u(1) = 0.

By condition (A) and (2.1), one has

(2.4) 2ǫ ≤ f

ǫ

(t, u) ≤ Cϕ

^−α_ǫ

, (t, u) ∈ (0, 1) × (−∞, ǫ), and

(2.5) f

ǫ

(t, u) = f (t, u), (t, u) ∈ (0, 1) × [ǫ, ∞).

An important observation is that if u is a solution of problem (2.3), then u ≥ ϕ

ǫ

, and hence u is also a solution of problem (1.1). In fact, if not, then suppose that

(2.6) u(t

0

) < ϕ

ǫ

(t

0

) for some t

0

∈ (0, 1).

According to [1, Lemma 2.8.1], we obtain

(2.7) t(1 − t)kuk

_∞

≤ u(t) t ∈ [0, 1],

where kuk

_∞

= max

_t∈[0,1]

|u(t)|. So (2.6) implies kuk

_∞

< ǫ. But by (2.4),

−u

^′′

≥ 2λǫ ≥ 2ǫ = −ϕ

^′′_ǫ

, so u ≥ ϕ

ǫ

, contradicting (2.6).

Since ϕ

^−α_ǫ

∈ L

¹

(0, 1), from (2.4) we see that the solutions of problem (2.3) are the critical points of the C

¹

functional

(2.8) J (u) = Z

1

0

 1

2 |u

^′

(t)|

²

− λF

ǫ

(t, u)



dt, u ∈ H = H

₀¹

(0, 1), and

(2.9) hJ

^′

(u), vi = Z

1

0

u

^′

(t)v

^′

(t)dt − λ Z

1

0

f

ǫ

(t, u(t))v(t)dt for all v ∈ H, where

F

ǫ

(t, u) = Z

u

ǫ

f

ǫ

(t, y)dy

and H

₀¹

(0, 1) is the usual Sobolev space normed by

(2.10) kuk =

Z

1 0

|u

^′

(t)|

²

dt



1 2

. Define two functionals Φ and Ψ as follows

(2.11) Φ(u) = 1

2 kuk

²

and Ψ(u) = Z

1

0

F

ǫ

(t, u)dt for each u ∈ H. Then by (2.8),

(2.12) J (u) = Φ(u) − λΨ(u).

Lemma 2.1. Let u ∈ H. Then kuk

_∞

≤ kuk.

Proof. For any u ∈ H,

|u(t)| = | Z

t

0

u

^′

(s)ds| ≤ Z

1

0

|u

^′

(s)|ds ≤

Z

1 0

|u

^′

(s)|

²

ds



1 2

,

which implies the conclusion. 

In the next section, we will use the following three theorems in nonlinear functional analysis and critical point theory.

Theorem 2.2 ([21, Theorem 38.A]). Let X be a Banach space. For the func- tional ϕ : M ⊆ X → [−∞, +∞] with M 6= ∅, min

_u∈M

ϕ(u) = α has a solution in case the following hold:

(i) X is a real reflexive Banach space;

(ii) M is bounded and weak sequentially closed;

(iii) ϕ is weak sequentially lower semi-continuous on M , i.e., by definition, for each sequence {u

n

} in M such that u

n

⇀ u as n → ∞, we have ϕ(u) ≤ lim inf

_n→∞

ϕ(u

n

) holds.

Theorem 2.3 ([18, Theorem 4.10]). Let X be a Banach space and let ϕ ∈ C

¹

(X, R). Assume that there exist x

0

, x

1

∈ X and a bounded open neighbour- hood Ω of x

0

such that x

1

∈ X \ Ω and

max{ϕ(x

0

), ϕ(x

1

)} < inf

x∈∂Ω

ϕ(x).

Let

Γ = {h ∈ C([0, 1], X) : h(0) = x

0

, h(1) = x

1

} and

c = inf

h∈Γ

max

s∈[0,1]

ϕ(h(s)).

If ϕ satisfies the (PS)-condition, i.e., a sequence {u

n

} in X satisfying ϕ(u

n

) is

bounded and ϕ

^′

(u

n

) → 0 as n → ∞ has a convergent subsequence, then c is a

critical value of ϕ and c > max{ϕ(x

0

), ϕ(x

1

)}.

Theorem 2.4 ([8, 9]). Let X be a reflexive real Banach space; let Φ, Ψ : X → R be two Gˆateaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf

X

Φ, let us put

ϕ(r) := inf

u∈Φ⁻¹((−∞,r))

sup

_v∈Φ⁻¹_((−∞,r))

Ψ(v) − Ψ(u) r − Φ(u)

and

γ := lim inf

r→+∞

ϕ(r), δ := lim inf

r→(inf^XΦ)⁺

ϕ(r).

Then one has

(a) for every r > inf

X

Φ and every λ ∈ (0,

_ϕ(r)¹

), the restriction of the functional I

λ

= Φ − λΨ to Φ

⁻¹

((−∞, r)) admits a global minimum, which is a critical point (local minimum) of I

λ

in X.

(b) If γ < +∞, then for each λ ∈ (0,

_γ¹

), the following alternative holds:

either

(b

1

) I

λ

has a global minimum; or

(b

2

) there exists a sequence {u

n

} of critical points (local minima) of I

λ

such that lim

_n→∞

Φ(u

n

) = +∞.

(c) If δ < +∞, then for each λ ∈ (0,

¹_δ

), the following alternative holds:

either

(c

1

) there exists a global minimum of Φ which is a local minimum of I

λ

; or (c

2

) there exists a sequence of pairwise distinct critical points (local minima) of I

λ

, which weakly converges to a global minimum of Φ.

3. Main results Theorem 3.1. Assume that (A) holds and

(S

1

) there exist constants c, d > 0 and θ ∈ (0, 1) such that (3.1) f (t, u) ≤ cu

^θ

+ d, (t, u) ∈ (0, 1) × [ǫ, ∞).

Then problem (1.1) admits a positive solution in H for λ ≥ 1.

Proof. Clearly, Φ is continuous and convex, and hence it is weakly sequentially lower semicontinuous. The functional Ψ is well-defined, continuously Gˆateaux differentiable and has compact derivative, and hence it is sequentially weakly continuous. Therefore, the functional J = Φ − λΨ is weakly sequentially lower semicontinuous.

Next we verify that the functional J is coercive in H for λ ≥ 1.

By (2.4), (2.5) and (3.1), one has

(3.2) F

ǫ

(t, u) ≤

( 0, u < ǫ,

c

2 u

^θ+1

+ du, u ≥ ǫ.

Therefore,

J (u) = 1

2 kuk

²

− λ Z

1

0

F

ǫ

(t, u)dt

= 1

2 kuk

²

− λ Z

u<ǫ

F

ǫ

(t, u)dt − λ Z

u≥ǫ

F

ǫ

(t, u)dt

≥ 1

2 kuk

²

− λ Z

u≥ǫ

F

ǫ

(t, u)dt

≥ 1

2 kuk

²

− λc

2 kuk

^θ+1

− λdkuk → ∞ as kuk → ∞

since 0 < θ < 1. By using [18, Theorem 1.1], we obtain problem (1.1) admits

a positive solution in H for λ ≥ 1. 

Remark 3.2. Theorem 3.1 gives the existence result of positive solutions for problem (1.1) when the nonlinear term f satisfies sublinear condition at infinity, which is not contained in [5]. Therefore, we extend the results in [5].

Example 1. Consider the following problem

(S) ( − u

^′′

= λ(a(t)u

^−α

+ u

^θ

), 0 < t < 1, u(0) = u(1) = 0,

where λ ≥ 1, a ∈ L

¹

(0, 1), 0 < α < 1, 0 < θ < 1.

Take ǫ = 1. Then (A) holds. Moreover, it is easy to verify that the condition (S

1

) holds. Therefore, by Theorem 3.1, problem (S) has at least a positive solution for λ ≥ 1.

Theorem 3.3. Assume (A) holds. Moreover, assume that (S

2

) There exist two constants µ > 2 and M

0

> ǫ such that

µF

ǫ

(t, u) ≤ uf (t, u), (t, u) ∈ (0, 1) × [M

0

, ∞);

(S

3

) There exist constants κ > 2, δ > 0 such that

F

ǫ

(t, u) ≤ δu

^κ

, (t, u) ∈ (0, 1) × [ǫ, ∞).

Then problem (1.1) with λ = 1 has at least two positive solutions in H.

Proof. Step 1. We verify that the functional ϕ satisfies (PS)-condition.

Let a sequence {u

n

} in X satisfying J(u

n

) be bounded and J

^′

(u

n

) → 0. By a standard argument, it suffices to show that {u

n

} is bounded when verifying (PS)-condition.

By (2.4), (2.9) and Lemma 2.1, it is easy to verify that ku

⁻_n

k

²

= −

Z

1 0

f

ǫ

(t, u

n

(t))u

⁻_n

(t)dt + hJ

^′

(u

n

), u

⁻_n

i



≤ ku

⁻_n

k

_∞

Z

un<0

f

ǫ

(t, u

n

(t))dt + kJ

^′

(u

n

)kku

⁻_n

k

≤ ku

⁻_n

k Z

un<0

f

ǫ

(t, u

n

(t))dt + kJ

^′

(u

n

)kku

⁻_n

k

≤ Cku

⁻_n

k Z

1

0

ϕ

ǫ

(t)

^−α

dt + kJ

^′

(u

n

)kku

⁻_n

k, and hence

(3.3) ku

⁻_n

k

_∞

≤ ku

⁻_n

k ≤ C Z

1

0

ϕ

ǫ

(t)

^−α

dt + o(1).

Therefore, ku

⁻_n

k

_∞

is bounded.

On the other hand, by (2.9) and (S

2

), one has

(3.4)

 µ 2 − 1 

ku

n

k

²

= Z

1

0

(µF

ǫ

(t, u

n

) − u

n

f (t, u

n

))dt + µJ(u

n

)

− hJ

^′

(u

n

), u

⁻_n

i

≤ C

Z

un<ǫ

ϕ

ǫ

(t)

^−α

|u

n

(t)|dt + 1



≤ C

 ku

⁻_n

k

_∞

Z

1 0

ϕ

ǫ

(t)

^−α

dt + 1

 . Combining this with (3.3), we obtain {u

n

} is bounded in H.

Step 2. We apply Theorem 2.2 to show that there exists M > 0 such that the functional J has a local minimum u

0

∈ B

M

= {u ∈ H : kuk ≤ M }.

Let M > 0, which will be determined later. First we claim that B

M

is bounded and weak sequentially closed. In fact, let {u

n

} ⊆ ¯ B

M

and u

n

⇀ u as n → ∞. By Mazur Theorem ([18]), there exists a sequence of convex combinations

v

n

=

n

X

k=1

β

nk

u

k

,

n

X

k=1

β

nk

= 1, β

nk

≥ 0, k ∈ N

such that v

n

→ u in H. Since B

M

is a closed convex set, we have {v

n

} ⊆ ¯ B

M

and u ∈ ¯ B

M

. Now we claim that ϕ has a minimum u

0

∈ ¯ B

M

. It is easy to verify that ϕ is weak sequentially lower semi-continuous on B

M

. Furthermore, H is a reflexive Banach space, so our claim follows from Theorem 2.2.

Without loss of generality, we assume that J(u

0

) = min

_{u∈ ¯BM}

J (u). Now we show that

(3.5) J (u

0

) < inf

u∈∂BM

J (u).

If this is true, then the result of Step 2 holds. In fact, take M = ǫ

1

> 0 satisfying

(3.6) 1

2 ǫ

²₁

− δǫ

^κ₁

≥ 0.

For any u ∈ ∂B

M

, kuk = M = ǫ

1

. By (2.8), (3.6), Lemma 2.1 and (S

3

), one has

J (u) = 1 2

Z

1 0

|u

^′

|

²

dt − Z

1

0

F

ǫ

(t, u)dt

≥ 1 2 ǫ

²₁

−

Z

u≥ǫ

F

ǫ

(t, u)dt

≥ 1 2 ǫ

²₁

− δ

Z

1 0

|u|

^κ

dt

≥ 1

2 ǫ

²₁

− δkuk

^κ_∞

≥ 1

2 ǫ

²₁

− δkuk

^κ

= 1

2 ǫ

²₁

− δǫ

^κ₁

≥ 0

for any u ∈ ∂B

M

. On the other hand, J(u

0

) ≤ J(0) ≤ −2ǫ

²

. So J(u

0

) ≤ J (0) ≤ −2ǫ

²

<

¹₂

ǫ

²₁

− δǫ

^κ₁

≤ J(u) for any u ∈ ∂B

M

. So (3.5) holds and u

0

∈ B

M

.

Step 3. We show that there exists u

1

with ku

1

k > M such that J(u

1

) <

inf

_u∈∂BM

J (u).

Let ϕ

1

> 0 be the normalized eigenfunction associated with ρ

1

, where ρ

1

is the first eigenvalue of

( − u

^′′

= ρu, 0 < t < 1, u(0) = u(1) = 0.

By (2.4) and (S

2

), one has

(3.7) F

ǫ

(t, u) ≥



 

 

− Cϕ

ǫ

(t)

^−α

, 0 ≤ u < ǫ, F

ǫ

(t, M

0

)

 u M

0



µ

, u ≥ M

0

for some M

0

> ǫ, and it follows that J(Rϕ

1

) → −∞ as R → ∞. Therefore, from Step 2 and Step 3, we have max{J(u

0

), J(u

1

)} < inf

_u∈∂M

J (u). Theorem 2.3 gives the critical point u

^∗

. Therefore, u

0

, u

^∗

are two critical points of J, and hence they are two positive solutions of problem (1.1) with λ = 1.  Remark 3.4. In [5], Agarwal et al. obtained the existence of two positive solu- tions for problem (1.1) with λ = 1 under the condition (S

2

) and the following abstract condition:

(F ) There is a constant M

1

> 0, independent of λ, such that kuk 6= M

1

for every solution u > 0 to

( − u

^′′

= λf (t, u), 0 < t < 1, u(0) = u(1) = 0

for each λ ∈ (0, 1].

However, in Theorem 3.3, we use the condition (S

3

) instead of (F ) and

obtain the same result. In general, (F ) holds if there exists an a priori bound

of the solutions of the problem, but in most cases, it is not easy to obtain such a priori bound of the solutions. Our condition (S

3

) is independent of the bound of the possible solutions and it is clear to be more easily verified.

Theorem 3.5. Suppose (A) holds. Assume that

(S

4

)

lim inf

ξ→+∞

2 R

1

0

max

_ǫ≤u≤ξ

F

ǫ

(t, u)dt + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt ξ

²

< 1

4 · lim sup

ξ→+∞

R

3/4

1/4

F

ǫ

(t, ξ)dt ξ

²

≤ 1.

Then for every λ ∈ Λ :=





4 lim sup_ξ→+∞

R3/4 1/4Fǫ(t,ξ)dt

ξ²

, 1

lim inf_ξ→+∞²

R1

0maxǫ≤u≤ξFǫ(t,u)dt+ǫCR1 0ϕ^−αǫ (t)dt ξ²



 , where ǫ, C are defined as in (A) and ϕ

ǫ

(t) = ǫt(1 − t), problem (1.1) has an unbounded sequence of positive solutions in H.

Proof. We use part (b) of Theorem 2.4 to prove our result. Φ is continuous and convex, and hence it is weakly sequentially lower semi-continuous. Moreover, Φ is continuously Gˆ ateaux differentiable and its Gˆateaux derivative admits a continuous inverse. The functional Ψ is well-defined, continuously Gˆateaux differentiable and has compact derivative, and hence it is sequentially weakly continuous. In particular, one has

Φ

^′

(u)(v) = Z

1

0

u

^′

(t)v

^′

(t)dt and

Ψ

^′

(u)(v) = Z

1

0

f

_ǫ

(t, u(t))v(t)dt for all v ∈ H.

Next we wish to prove that

γ < +∞.

Let {ξ

n

} be a sequence of positive constants such that for each n ∈ N, ξ

n

> ǫ, ξ

n

→ +∞ and

n→∞

lim 2 R

1

0

max

_ǫ≤u≤ξn

F

ǫ

(t, u)dt + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt ξ

²_n

= lim inf

ξ→+∞

2 R

1

0

max

_ǫ≤u≤ξ

F

ǫ

(t, u)dt + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt

ξ

²

.

Take r

n

=

¹₂

ξ

²_n

for all n ∈ N. By Lemma 2.1, for all v ∈ H with kvk

²

< 2r

n

, one has

t∈[0,1]

max |v(t)| ≤ ξ

n

.

Hence by (2.4), one has ϕ(r

n

) = inf

u∈Φ⁻¹((−∞,rn))

sup

_v∈Φ−1((−∞,rn))

Ψ(v) − Ψ(u) r

n

− Φ(u)

≤ sup

_v∈Φ−1((−∞,rn))

Ψ(v) − Ψ(0) r

n

− Φ(0)

= sup

_v∈Φ−1((−∞,rn))

Ψ(v) + R

1 0

R

ǫ

0

f

ǫ

(t, u)dudt r

n

≤ 2 R

1

0

max

_{ǫ≤|u|≤ξn}

F

ǫ

(t, u)dt + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt

ξ

_n²

.

So

(3.8) γ ≤ lim inf

ξ→+∞

2 R

1

0

max

_ǫ≤u≤ξ

F

ǫ

(t, u)dt + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt

ξ

²

< +∞.

From assumption (S

4

) and (3.8), one has Λ ⊆ (0, 1

γ ).

Fixing λ ∈ Λ, (3.8) assures that conclusion (b) of Theorem 2.4 can be used and either J has a global minimum or there exists a sequence {u

n

} of solutions of problem (1.1) such that lim

_n→∞

u

n

= +∞.

Finally, we verify that the functional J has no global minimum in H.

By the choice of λ, we can choose a constant b such that, for each n ∈ N,

(3.9) 1

4 · sup

ν≥n

R

3/4

1/4

F

ǫ

(t, ν)dt

ν

²

> b > 1 λ and so there exists ν

n

≥ max{n, ǫ} such that

1 4 ·

R

3/4

1/4

F

ǫ

(t, ν

n

)dt ν

_n²

> b.

Consider a function y

n

∈ H defined by

y

n

(t) =



 

 

4ν

n

t, t ∈ [0,

¹₄

] ν

n

, t ∈ (

¹₄

,

³₄

]

4ν

n

(1 − t), t ∈ (

³₄

, 1].

Then

Φ(y

n

) = 4ν

_n²

and, using (2.4) and (2.12), we have

(3.10)

J (y

n

) = Φ(y

n

) − λΨ(y

n

) = 4ν

_n²

− λ Z

1

0

F

ǫ

(t, y

n

(t))dt

≤ 4ν

_n²

− λ Z

3/4

1/4

F

ǫ

(t, ν

n

)dt + λǫ

²

2ν

n

< 4ν

_n²

(1 − λb) + λǫ

²

2ν

n

.

Combining this with (3.9), we obtain the functional J is unbounded from below and thus has no global minimum.

Therefore, Theorem 2.4 assures that there is a sequence {u

n

} ⊆ H of crit- ical points of J such that lim

_n→+∞

ku

n

k = +∞ and, taking into account the considerations made in Section 2, the theorem is completely proved.  Remark 3.6. Observe that assumption (S

4

) in Theorem 3.5 can be replaced by the more general condition:

(S

5

) There exist two sequences {a

n

} and {b

n

} with ǫ ≤ a

n

<

¹

2√

2

b

n

for every n ∈ N and lim

_n→+∞

b

n

= +∞ such that

n→+∞

lim R

1

0

max

_ǫ≤u≤bn

F

ǫ

(t, u)dt + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt − R

3/4

1/4

F

ǫ

(t, a

n

)dt +

_2a^ǫ3_n

b²_n 2

− 4a

²_n

< 1

4 · lim sup

ξ→+∞

R

3/4

1/4

F

ǫ

(t, ξ)dt

ξ

²

≤ 1.

Clearly, by choosing a

n

= ǫ for all n ∈ N, from (S

5

) we obtain (S

4

). Moreover, if we assume (S

5

) instead of (S

4

) and take r

n

=

¹₂

b

²_n

for all n ∈ N , similar to the discussion in Theorem 3.5, we obtain

ϕ(r

n

) = inf

u∈Φ⁻¹((−∞,rn))

sup

_v∈Φ−1((−∞,rn))

Ψ(v) − Ψ(u) r

n

− Φ(u)

≤ sup

_v∈Φ−1((−∞,rn))

Ψ(v) − R

1

0

F

ǫ

(t, w

n

(t))dt r

n

−

¹₂

kw

n

k

²

≤ R

1

0

max

_|u|≤bn

F

ǫ

(t, u)dt − R

3/4

1/4

F

ǫ

(t, a

n

)dt +

_2a^ǫ3

n

1

2

b

²_n

− 4a

²_n

≤ R

1

0

max

_{ǫ≤|u|≤bn}

F

ǫ

(t, u)dt − R

3/4

1/4

F

ǫ

(t, a

n

)dt +

_2a^ǫ3

n

1

2

b

²_n

− 4a

²_n

≤ R

1

0

max

_ǫ≤u≤bn

F

ǫ

(t, u)dt + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt − R

3/4

1/4

F

ǫ

(t, a

n

)dt +

_2a^ǫ3

n

b²_n

2

− 4a

²_n

,

where

w

n

(t) =



 

 

4a

n

t, t ∈ [0,

¹₄

] a

n

, t ∈ (

¹₄

,

³₄

]

4a

n

(1 − t), t ∈ (

³₄

, 1].

The conclusion is similar to that of Theorem 3.5 with Λ replaced by Λ

^′

:=





4 lim sup_ξ→+∞

R3/4 1/4Fǫ(t,ξ)dt

ξ²

, 1

lim_n→+∞

R₁

0max_ǫ≤u≤bnFǫ(t,u)dt+ǫCR₁

0ϕ^−αǫ (t)dt−R3/4

1/4Fǫ(t,an)dt+_2an^ǫ3 b2n

2−4a²n



 . Corollary 3.7. Let f ∈ C((0, ∞), [0, ∞)) satisfy

(A)

^′

2ǫ ≤ f (u) ≤ Cu

^−α

, u ∈ (0, ǫ) for some ǫ, C > 0 and α ∈ (0, 1). Moreover, assume that (S

6

) lim inf

ξ→+∞

2 R

ξ

ǫ

f

ǫ

(u)du + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt

ξ

²

< 1

8 · lim sup

ξ→+∞

R

ξ

ǫ

f

ǫ

(u)du ξ

²

≤ 1.

Then for every λ ∈





8 lim sup

_ξ→+∞

Rξ ǫ fǫ(u)du

ξ²

, 1

lim inf

_ξ→+∞²

Rξ

ǫ fǫ(u)du+ǫCR1

0ϕ^−αǫ (t)dt ξ²



 , the problem

(3.11) ( − u

^′′

= λf (u), 0 < t < 1, u(0) = u(1) = 0,

has an unbounded sequence of positive solutions in H.

Corollary 3.8. Let f ∈ C((0, ∞), [0, ∞)) satisfy (A)

^′

. Assume that lim inf

ξ→+∞

2 R

ξ

ǫ

f

ǫ

(u)du + ǫC R

1

0

ϕ

^−α_ǫ

(t)dt

ξ

²

= 0

and

lim sup

ξ→+∞

R

ξ

ǫ

f

ǫ

(u)du ξ

²

= 8.

Then for every λ ∈ (1, +∞), the problem (3.11) has an unbounded sequence of positive solutions in H.

Example 2. Let f : (0, ∞) → [0, ∞) be the continuous function defined as follows:

f (u) =



 

 



 

 



2

u^α

, u ∈ (0, 1)

− 2u + 4, u ∈ [1, 2]

4β[(n + 1)

²

− 1] min{u − n!n, (n + 1)! − u}, u ∈ [n!n, (n + 1)!]

0, elsewhere

where 0 < α < 1, 0 < β ≤ 8, n ∈ N and n ≥ 2.

Take ǫ = 1, C = 2. Then (A)

^′

holds. Put a

n

= n!, b

n

= n!n for each n ∈ N.

By (2.5), one has F

1

(a

n

) =

Z

an

1

f

1

(u)du = 2 Z

2

1

(−u + 2)du +

n−1

X

i=1

Z

ai+1

bi

f (u)du

= 1 +

n−1

X

i=1

4β[(i + 1)

²

− 1]



(a

i+1

− b

i

)  a

i+1

+ b

i

2 − (i!)i  1 2



= 1 +

n−1

X

i=1

β[(i + 1)

²

− 1](i!)

²

= 1 + β[(n!)

²

− 1]

for each n ∈ N. Therefore, we obtain

n→+∞

lim F

1

(a

n

)

a

²_n

= β and by simple calculations, one has

lim sup

ξ→+∞

F

1

(ξ)

ξ

²

= lim sup

ξ→+∞

R

ξ

1

f

1

(u)du ξ

²

= β.

On the other hand, it is easy to verify that F

1

(b

n

) =

Z

bn

1

f

1

(u)du = 1 + β[(n!)

²

− 1].

One has

n→+∞

lim

2F

1

(b

n

) + 2 R

1

0

ϕ

^−α₁

(t)dt

b

²_n

= 0,

for which lim inf

ξ→+∞

2F

1

(ξ) + 2 R

1

0

ϕ

^−α₁

(t)dt

ξ

²

= lim inf

ξ→+∞

2 R

ξ

1

f

1

(u)du + 2 R

1

0

ϕ

^−α₁

(t)dt

ξ

²

= 0.

Therefore, 0 = lim inf

ξ→+∞

2 R

ξ

1

f

ǫ

(u)du + 2 R

1

0

ϕ

^−α₁

(t)dt

ξ

²

< β

8 = 1

8 · lim sup

ξ→+∞

R

ξ

1

f

ǫ

(u)du ξ

²

≤ 1.

Hence, according to Corollary 3.6, for each λ >

_β⁸

, the problem ( − u

^′′

= λf (u), 0 < t < 1,

u(0) = u(1) = 0,

has an unbounded sequence of pairwise distinct positive solutions.

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Juntao Sun School of Science

Shandong University of Technology Zibo, 255049 Shandong, P. R. China E-mail address: [email protected]

Jifeng Chu

Department of Mathematics College of Science

Hohai University

Nanjing, 210098 Jiangsu, P. R. China E-mail address: [email protected]