Nonlinear Analysis 67 (2007) 1316–1328
www.elsevier.com/locate/na
On the evolution and qualitative behaviors of bifurcation curves for a
boundary value problemI
Shin-Hwa Wang ∗
Department of Mathematics, National Tsing Hua University, 300 Hsinchu, Taiwan, ROC
Received 23 February 2006; accepted 21 July 2006
Abstract
We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for
00
−u (x) = λ u(1 − sin u) + u p , −1 < x < 1,
u(−1) = u(1) = 0,
where λ > 0 is a bifurcation parameter and p ≥ 1 is an evolution parameter. On the (λ, kuk∞ )-plane, we prove that the bifurcation
curve has exactly one turning point where the curve turns to the left for p > 2, it is a monotone curve for p = 2, it has at least two
cos 1−sin 1
turning points for 1 < p ≤ p̃ ≡ 1+2
1−2 cos 1+sin 1 ≈ 1.629, and it has infinitely many turning points for p = 1. Hence we are able
to determine the (exact) number of positive solutions. In particular we give complete descriptions of the structure of bifurcation
curves when p ≥ 2. We also give some numerical simulations of bifurcation curves for p ≥ 1.
c 2006 Elsevier Ltd. All rights reserved.
MSC: 34B15; 74G35
Keywords: Bifurcation curve; Exact multiplicity; Positive solution; Turning point; Time map
1. Introduction
In this paper we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for the two
point boundary value problem
00
−u (x) = λ f p (u) = λ u(1 − sin u) + u p , −1 < x < 1,
(1.1)
u(−1) = u(1) = 0,
where λ > 0 is a bifurcation parameter and p ≥ 1 is an evolution parameter.
The nonlinearity in (1.1) f p (u) = u(1 − sin u) + u p can be considered as adding an “oscillation” term −u sin u
into the nonlinearity fˆp (u) = u + u p . For the problem
I Work partially supported by the National Science Council, Republic of China.
∗ Tel.: +886 3 5731042; fax: +886 3 5723888.
E-mail address: shwang@math.nthu.edu.tw.
c 2006 Elsevier Ltd. All rights reserved.
0362-546X/$ - see front matter doi:10.1016/j.na.2006.07.019
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
−u 00 (x) = λ fˆp (u) = λ u + u p ,
u(−1) = u(1) = 0,
p > 1, −1 < x < 1,
(1.2)
it is easy to see that the bifurcation curve is a monotone curve on the (λ, kuk∞ )-plane since fˆp (u) − u fˆp0 (u) =
(1 − p)u p < 0 on (0, ∞). More precisely, problem (1.2) has exactly one positive solution for 0 < λ <
π2
4 ,
and no
π2
4 . Note that many authors have extensively studied the existence and uniqueness of positive
positive solution for λ ≥
solutions for the N -dimensional Dirichlet problem
−∆u = λ fˆp (u) = λ u + u p , p > 1 in B,
u = 0 on ∂ B,
where B is a finite ball in R N (N ≥ 2). See e.g. Kwong and Li [11], Erbe and Tang [7] and Brezis and Nirenberg [1],
and references therein.
This research is motivated by a well-known survey paper of Lions [13]. In 1982, for a variety of nonlinearities f ,
Lions gave many “bifurcation curves” to describe the set of (classical) positive solutions of
−∆u = λ f (u) in Ω ,
u = 0 on ∂Ω ,
where Ω is a bounded regular domain in R N (N ≥ 1) and λ > 0 is a bifurcation parameter. Those “bifurcation curves”
were “minimal” in some sense and were formally discussed. In particular, in [13, Remark 1.7], for
f = f p (u) = u(1 − sin u) + u p ,
1 < p < N ∗,
where N ∗ = ∞ if N = 1, 2, and N ∗ = (N + 2)/(N − 2) if N ≥ 3, it was claimed that the “bifurcation curve” has
exactly one turning point where the curve turns to the left on the (λ, kuk∞ )-plane. In this paper, for when N = 1 and
for (1.1), on the (λ, kuk∞ )-plane, we find and prove that the bifurcation curve has exactly one turning point where
the curve turns to the left if and only if p > 2, and the number of turning points of the bifurcation curve varies with
respect to the values of p ≥ 1. More precisely, for p ≥ 1, we find and prove rigorously that, on the (λ, kuk∞ )-plane,
(i) For p > 2, the bifurcation curve has exactly one turning point where the curve turns to the left at some
(λ∗ , ku λ∗ k∞ ) which satisfies parts (A)–(D) in Corollary 2.2(i) stated below.
(ii) For p = 2, the bifurcation curve is a monotone curve.
1+2 cos 1−sin 1
(iii) For 1 < p ≤ p̃ ≡ 1−2
cos 1+sin 1 ≈ 1.629, the bifurcation curve has at least two turning points.
(iv) For p = 1, the bifurcation curve has infinitely many turning points. Moreover,
any positive integer n, there
for
2
π2
exists δn > 0 such that (1.1) has at least n distinct positive solutions for λ ∈ 8 − δn , π8 + δn , and infinitely
many positive solutions for λ =
π2
8 .
See Theorem 2.1 and Corollary 2.2 stated below for detailed results. Hence we are able to determine the (exact)
number of positive solutions. See Fig. 1 for the partially analyzed evolution of bifurcation curves with respect to
p ≥ 1; also see Fig. 2 (resp. Fig. 3) for the numerical evolution of bifurcation curves with respect to p ≥ 1 (resp.
p ≥ 2) for 0 < α = kuk∞ ≤ 40 (resp. 0 < α = kuk∞ ≤ 1). In particular, Fig. 2 suggests some interesting evolution
results and a phenomenon of a common intersection point of bifurcation curves of (1.1) for p ≥ 1; we state them
in Remark 2 below. For problem (1.1) with this simple nonlinearity f p (u) = u(1 − sin u) + u p , the structure of
bifurcation curves exhibits rich and complicated phenomena which are mainly due to the “oscillation” term −u sin u
in the nonlinearity f p (u). For 1 < p < 2, further investigations on the structure and oscillation of bifurcation curves
are needed. Our results lead to the expectation that similar results might also hold in higher-dimensional spaces.
When p = 1, as stated above, the existence of infinitely many positive solutions of (1.1) with
f p (u) = u(1 − sin u) + u p = u(2 − sin u)
for λ =
π2
8
can be obtained. Problems similar to (1.1) with the existence of infinitely many positive solutions have
√
2
been studied by Cheng [4, Theorem 6] for nonlinearity f (u) = u + sin u and λ = π4 , and by Korman and Li [9,
Theorem 1] for nonlinearity f (u) = u + sin u and λ = π4 . For similar results, also see Schaaf and Schmitt [14] for
ODE’s and Costa et al. [6] and Galstian et al. [8] for PDE’s.
2
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
1+2 cos 1−sin 1 ≈ 1.629, λ = T (α)2 .
Fig. 1. Partially analyzed evolution of bifurcation curves of (1) with respect to p ≥ 1. p̃ = 1−2
p
cos 1+sin 1
Fig. 2. Numerical simulations of T p (α) (=
intersection point (α̂, T p (α̂)) ≈ (1.16, 1.45).
√
λ) for 0 < α = kuk∞ ≤ 40. p = 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4, 2.6, 2.8, 3. The common
Note that numerical simulation results for the N -dimensional Dirichlet problem of (1.1) have been studied by
Weber [16] for when Ω = (0, 1)2 , N = 2 and p = 3, and very recently by Chang et al. [2] for when Ω = (0, 1) N ,
N = 1, 2 and 3, and p ≥ 1. See also Chien and Jeng [5] for similar results for when Ω = (0, 1)2 , N = 2 and
p = 1.3, 1.6, 2, 3. See [16, Sect. 8.3], [2, Sect. 5] and [5, Sect. 5] for details and also see Remark 2 stated below.
2. Main results
The main results in this paper are the next Theorem 2.1 and Corollary 2.2. Our proofs are based on the timemapping method (quadrature method). Consider the two point boundary value problem
00
−u (x) = λ f (u), −1 < x < 1,
(2.1)
u(−1) = u(1) = 0,
where Lipschitz continuous function f satisfies f (u) > 0 for u > 0. The time map for (2.1) takes a form as follows:
Z α
√
−1/2
λ=2
(F(α) − F(u))−1/2 du ≡ T (α) for 0 < α < ∞,
(2.2)
0
√
where F(u) ≡ 0 f (x)dx. We note that positive solutions u of (2.1) correspond to kuk∞ = α and T (α) = λ;
see Laetsch [12]. Thus studying the number of positive solutions of (2.1) is equivalent to studying the shape of the
Ru
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
Fig. 3. Numerical simulations of T p (α) (=
p = ∞.)
√
λ) for 0 < α = kuk∞ ≤ 1. p = 2, 3, 5, 10, 20, 50, 100, 1000, ∞. (Note that T p (α) ≡ T∞ (α) when
Fig. 4. Graphs of min
p−2
p ,
p−2 1/4 −2/ p
p
p
<
p−2
p
for p > 2.
time map T (α) on (0, ∞). Note that, for T (α) in (2.2), to make clearer for (1.1) the dependence on the nonlinearity
f = f p (u) = u(1 − sin u) + u p with finite p ≥ 1, we write T p (α) instead of T (α). We also let
Z α
−1/2
(F∞ (α) − F∞ (u))−1/2 du for 0 < α ≤ 1,
T∞ (α) ≡ 2
0
where f ∞ (u) ≡ u(1−sin u) and F∞ (u) ≡
which is used in Theorem 2.1(i).
Ru
0
f ∞ (x)dx =
u2
2
+u cos u −sin u. Note that the constant T∞ (1) ≈ 3.203,
Theorem 2.1 (See Figs. 1–5). Consider (1.1) where f p (u) = u(1 − sin u) + u p , p ≥ 1.
(i) For p > 2,
π
,
lim T p (α) = 0,
(2.3)
α→∞
2
and T p (α) has exactly one critical point, a maximum, on (0, ∞). Moreover, let α ∗ = α ∗ ( p) be the critical point
for T p (α) and M( p) = T p (α ∗ ) = maxα∈(0,∞) T p (α) > π2 , then
lim T p (α) =
α→0+
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
(A) 0 < min
p−2
p ,
∗
α ( p)
p−2
p
1/4
p −2/ p
< α ∗ ( p) <
p−2
p
1/4
< 1 for p > 2.
α ∗ ( p)
= 0 and lim p→∞
= 1.
(B) lim p→2+
(C) M( p) is a strictly increasing and continuous function on (2, ∞).
(D) (1.571 ≈) π2 < M( p) < T∞ (1) ≈ 3.203 for p > 2.
(E) lim p→2+ M( p) = π2 and lim p→∞ M( p) = T∞ (1).
(ii) For p = 2, in addition to (2.3), T2 (α) is a strictly decreasing function on (0, ∞).
(iii) For 1 < p < 2, in addition to (2.3), T p (α) is strictly decreasing for small α > 0 and T p (α) <
In particular, for
π
2
for all α > 0.
1 + 2 cos 1 − sin 1
≈ 1.629,
1 − 2 cos 1 + sin 1
T p (α) has at least two critical points, a local minimum and a local maximum, on (0, ∞). Let α1 be the smallest
positive critical point, a local minimum, on (0, ∞), then α1 = α1 ( p) ∈ (0, 1) and lim p→1+ α1 ( p) = 0.
(iv) For p = 1,
π
π
(F) limα→0+ T1 (α) = limα→∞ T1 (α) = √
(≈1.111) and (0.907≈) √
< T1 (α) < π2 (≈1.571) for all α > 0.
2 3
2 2
(G) T1 (α) is strictly increasing for small α > 0.
(H) T1 (α) has infinitely many critical points on (0, ∞).
(I)

π

> √ , n ∈ N is odd,
2 2
T1 (nπ )
π

< √ , n ∈ N is even.
2 2
1 < p ≤ p̃ ≡
Remark 1. In Theorem 2.1(i)(A),
(
min
p−2
,
p
p−2
p
1/4
)
p −2/ p

p−2


for 2 < p ≤ p̄ ≈ 3.225,

p
=
p − 2 1/4 −2/ p



p
for p > p̄.
p
The next corollary on the existence and (exact) multiplicity of positive solutions of (1.1) follows by Theorem 2.1
immediately.
Corollary 2.2 (See Figs. 1–5). Consider (1.1) where f p (u) = u(1 − sin u) + u p , p ≥ 1.
(i) For p > 2, problem (1.1) has exactly two positive solutions for
π2
π2
4
< λ < λ∗ ≡ (M( p))2 , exactly one positive
solution for 0 < λ ≤ 4 and λ = λ∗ , and no positive solution for λ > λ∗ . In addition, λ∗ = λ∗ ( p) is a strictly
increasing and continuous function on (2, ∞), and
(A) (2.467≈) π4 < λ∗ < (T∞ (1))2 ≈ 10.259.
2
(B) lim p→2+ λ∗ ( p) = π4 , lim p→∞ λ∗ ( p) = (T∞ (1))2 .
Moreover, let u λ∗ be the unique positive
solution for λ = λ∗ , then
1/4
1/4
p−2
−2/ p < ku ∗ k < p−2
(C) 0 < min p−2
,
p
< 1.
λ
∞
p
p
p
2
(D) lim p→2+ ku λ∗ k∞ = 0, lim p→∞ ku λ∗ k∞ = 1.
(ii) For p = 2, problem (1.1) has exactly one positive solution for 0 < λ <
π2
4 ,
and no positive solution for λ ≥
(iii) For 1 < p < 2, problem (1.1) has at least one positive solution for 0 < λ <
λ≥
π2
4 .
In particular, for
1 < p ≤ p̃ =
1 + 2 cos 1 − sin 1
≈ 1.629,
1 − 2 cos 1 + sin 1
π2
4 ,
π2
4 .
and no positive solution for
S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
1321
there exist positive numbers λ1 < λ2 with
2
2
π2
λ1 ≡ T p (α1 ) < λ2 ≡
max T p (α) <
α∈(0,∞)
4
such that (1.1) has at least three positive solutions for λ1 < λ < λ2 , at least two positive solutions for λ = λ1
2
2
and λ = λ2 , at least one positive solution for λ2 < λ < π4 and 0 < λ < λ1 , and no positive solution for λ ≥ π4 .
(iv) For p = 1, there exists numbers λ3 < λ4 with
2
2
π2
π2
< λ3 ≡
min T1 (α) < λ4 ≡
max T1 (α) <
α∈(0,∞)
α∈(0,∞)
12
4
such that
(E) Problem (1.1) has at least one positive solution for λ3 ≤ λ ≤ λ4 , and no positive solution for λ < λ3 and
λ > λ4 .
(F) For any
n distinct positive solutions for
2 positive 2integer n, there exists δn > 0 such that (1.1) has at least
2
π
π
π2
λ ∈ 8 − δn , 8 + δn , and infinitely many positive solutions for λ = 8 . More precisely, for λ = π8 , for
any positive integer n ≥ 1, there exists a positive solution u n (x) satisfying ku n k∞ ∈ (nπ, (n + 1) π).
Conjecture 2.3 (See Figs. 3 and 5). For Theorem 2.1(i)(A)–(B), we conjecture that α ∗ ( p) is a strictly increasing
2
function on (2, ∞). Also, for Corollary 2.2(iv)(F), we conjecture that, for p = 1 and λ = π8 , problem (1.1) has a
unique positive solution u n (x) satisfying ku n k∞ ∈ (nπ, (n + 1) π). But we are not able to give proofs.
We finally give the next remark on problem (1.1).
Remark 2 (See Figs. 2 and 3 and 5 and 6). Consider (1.1) where f p (u) = u(1 − sin u) + u p , p ≥ 1.
(i) (Existence of a common intersection point) For any numbers 1 < p1 < p2 ,
f p1 (u) = u(1 − sin u) + u p1 > u(1 − sin u) + u p2 = f p2 (u)
for 0 < u < 1.
Thus
T p1 (α) < T p2 (α)
for 0 < α < 1
(2.4)
by a comparison theorem of [12, Theorem 2.3]. In addition, by [12, Theorem 2.7], we obtain that
T p1 (α) > T p2 (α)
for any α > 1, large enough.
More precisely, it can be proved that
T p1 (α) > T p2 (α)
for any α >
√
e ≈ 1.649
by showing that
∂
T p (α) < 0
∂p
for α >
√
e, p > 1;
√
we omit the proof. So bifurcation curves T p1 (α) and T p2 (α) intersect for some α = ᾱ ∈ (1, e). It is very
interesting to notice that numerical simulations suggest that all bifurcation curves T p (α) with p ≥ 1 intersect
commonly at a point (α̂, T p (α̂)) ≈ (1.16, 1.45), and any two bifurcation curves T p (α), p ≥ 1 have no other
intersection points on (0, ∞) except (α̂, T p (α̂)). This common intersection point (α̂, T p (α̂)) is independent of
p ≥ 1. The problem on existence of this common intersection point remains open, and further investigation is
needed. In particular we note here that this interesting phenomenon on the existence of a common intersection
point was also observed by Chang et al. [2, Sect. 5] for (1.1) (and also for the N -dimensional Dirichlet problem of
(1.1) when Ω = (0, 1) N , N = 2 and 3). However, it was stated that (α̂, T p (α̂)) ≈ (1.33, 1.45) which is slightly
different from ours; see Figs. 2 and 6 and [2, p. 2695].
(ii) The numerical results in [2, p. 2695] show that T p (α) is a monotone function on (0, ∞) for 2 ≤ p ≤ 2.1, but it
is proved rigorously in our Theorem 2.1(i) that, for p > 2, T p (α) has exactly one critical point, a maximum, on
(0, ∞). Also see Fig. 6 for a numerical simulation of T p (α) for p = 2.1.
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
Fig. 5. Numerical simulation of T1 (α) (=
Fig. 6. Numerical simulations of T p (α) (=
(α̂, T p (α̂)) ≈ (1.16, 1.45).
√
λ) for f 1 (u) = u(2 − sin u) and 0 < α = kuk∞ ≤ 200.
√
λ) for 0 < α = kuk∞ ≤ 2. p = 1, 1.5, 2.1, 2.5, 3. The common intersection point
(iii) Let Ncrit ( p) be the number of critical points of the bifurcation curves T p (α), p > 1, on (0, ∞), then numerical
simulations of T p (α) as given in Figs. 2 and 5, together with Theorem 2.1, suggest that Ncrit ( p) is a nonincreasing
function of p ∈ (1, 2). (Notice that Ncrit ( p) → ∞ as p → 1+ and Ncrit ( p) = 0 for p = 2 by Theorem 2.1(iv)
and (ii). Also Ncrit ( p) = 1 for p > 2 by Theorem 2.1(i).)
3. Proof of Theorem 2.1
To prove Theorem 2.1, we need the next Lemmas 3.1 and 3.2. Lemma 3.1 is well known; see Laetsch [12, Theorems
2.7, 2.8 and 2.10]. Lemma 3.2, our key lemma applied to prove uniqueness of the critical point of T p (α) for (1.1) with
p > 2, is due to Korman and Shi [10, Theorem 3].
Lemma 3.1. Consider (2.1). Assume that Lipschitz continuous function f satisfies f (0) = 0 and f (u) > 0 for all
u > 0. Then the time map T (α) defined in (2.2) satisfies
0 ≤ lim T (α) =
α→0+
π
(m 0 )−1/2 ≤ ∞,
2
0 ≤ lim T (α) =
α→∞
π
(m ∞ )−1/2 ≤ ∞,
2
(3.1)
where 0 ≤ m 0 ≡ limu→0+ ( f (u)/u) ≤ ∞ and 0 ≤ m ∞ ≡ limu→∞ ( f (u)/u) ≤ ∞. In particular, if f (u) = mu with
m > 0, then T (α) = π2 m −1/2 for all α > 0.
S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
1323
Lemma 3.2. Consider (2.1). Assume that f ∈ C 2 [0, ∞) satisfies f (u) > 0 for all u > 0, and assume that for some
γ > β > 0 such that
2F(γ ) − γ f (γ ) ≤ 0,
f (u) − u f (u) < 0
0
f (u) < 0
00
(3.2)
for all u > γ ,
for 0 < u < β,
(3.3)
f (u) > 0
00
for β < u < γ .
(3.4)
Then, in addition to (3.1), T (α) has exactly one critical point, a maximum, on (0, ∞).
Proof of Theorem 2.1. First, T p0 (α) can be easily computed from (2.2) for f = f p (u) = u(1 − sin u) + u p , p ≥ 1;
see e.g. [15, p. 273]. We have
Z α
θ p (α) − θ p (u) du
0
−3/2
T p (α) = 2
(3.5)
3/2 α
0 (F p (α) − F p (u))
Ru
1
u p+1 + u cos u − sin u and
where F p (u) = 0 f p (x)dx = 12 u 2 + p+1
1 − p p+1
u
+ 2u cos u + (u 2 − 2) sin u.
p+1
θ p (u) = 2F p (u) − u f p (u) =
(3.6)
(i) Assume that p > 2. We prove that T p (α) has exactly one critical point, a maximum, on (0, ∞) by applying
Lemma 3.2 in which we take f = f p (u) = u(1 − sin u) + u p ,
γ = 1,
and β to be the unique positive zero of
f p00 (u) = p( p − 1)u p−2 − 2 cos u + u sin u
(3.7)
on (0, 1). (The existence and uniqueness of β on (0, 1) is proved below.)
(I) For p > 2, it is easy to see that f p (u) > 0 for u > 0, and
m 0 = lim
u→0+
f p (u)
= 1,
u
m ∞ = lim
u→∞
f p (u)
= ∞.
u
π
2
So, by (3.1), limα→0+ T p (α) = and limα→∞ T p (α) = 0. Hence (2.3) holds.
For θ p (u) in (3.6), we compute that
i
h
θ p0 (u) = f p (u) − u f p0 (u) = u 2 cos u − ( p − 1)u p−2 .
(3.8)
So, by (3.6) and (3.8),
θ p (0) = 0,
θ p0 (0) = 0,
θ p0 (1) = 1 − p + cos 1 < 0,
θ p (1) =
1− p
+ 2 cos 1 − sin 1 < 0
p+1
for p > 2 > p̃ =
1 + 2 cos 1 − sin 1
≈ 1.629.
1 − 2 cos 1 + sin 1
Thus, by (3.8), it is easy to see that there exist some positive numbers A < B < γ = 1 such that

> 0 for 0 < u < B,
θ p (u) = 0 for u = B,

< 0 for u > B,
(3.9)
and

> 0 for 0 < u < A,
θ p0 (u) = 0 for u = A,

< 0 for u > A.
(3.10)
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
So (3.2) and (3.3) hold. We then prove (3.4). By (3.7), for p > 2, we obtain that
f p00 (1) = p( p − 1) − 2 cos 1 + sin 1 > 0,
f p00 (0) = −2 < 0,
f p000 (u) = p( p − 1)( p − 2)u p−3 + 3 sin u + u cos u > 0
for 0 < u < 1.
So, on (0, 1), f p00 (u) has exactly one zero at some β such that (3.4) holds. We conclude that, for p > 2, f p (u) =
u(1 − sin u) + u p satisfies all the hypotheses of Lemma 3.2. So by Lemma 3.2, for p > 2, T p (α) has exactly one
critical point, a maximum, on (0, ∞). Moreover, let α ∗ = α ∗ ( p) be the critical point for T p (α). By (3.5), (3.9) and
(3.10), we obtain 0 < A < α ∗ < B < γ = 1 and T p (α ∗ ) > π2 . Let M( p) = T p (α ∗ ) = maxα∈(0,∞) T p (α) > π2 .
(II) We prove parts (A)–(B). It can be computed that, for p > 2,
h
i
θ p0 (u) = u 2 cos u − ( p − 1)u p−2
)
(
p−2
p − 2 1/4 −2/ p
< 1.
p
,
> 0 for u = min
p
p
So, by (3.10),
(
0 < min
p−2
,
p
p−2
p
)
1/4
p
−2/ p
< A(< α ∗ < 1).
(3.11)
In addition, it can be computed that, for p > 2,
1 − p p+1
u
+ 2u cos u + (u 2 − 2) sin u
p+1
p − 2 1/4
< 1.
< 0 for u =
p
θ p (u) =
So, by (3.9),
(0 < α <)B <
∗
p−2
p
1/4
< 1.
(3.12)
Inequalities (3.11) and (3.12) imply part (A) immediately. In addition, since lim p→2+
lim p→2+ α ∗ ( p) = 0. Also, since
)
(
p − 2 1/4 −2/ p
p − 2 1/4 −2/ p
p−2
,
p
p
= lim
lim min
=1
p→∞
p→∞
p
p
p
p−2
p
1/4
= 0, we obtain
by Remark 1 and by using the l’Hôpital rule, we obtain lim p→∞ α ∗ ( p) = 1 by (3.11). So part (B) holds.
(III) We prove parts (C)–(E). For any numbers 2 < p1 < p2 , we obtain
T p1 (α) < T p2 (α)
for 0 < α < 1
by (2.4). Thus by part (A), we obtain
M( p1 ) = max T p1 (α) = max T p1 (α) < max T p2 (α) = max T p2 (α) = M( p2 ).
α∈(0,∞)
α∈(0,1)
α∈(0,1)
α∈(0,∞)
So M( p) is a strictly increasing function on (2, ∞). The proof of the assertion that M( p) is a continuous function
on (2, ∞) is easy but tedious; we omit it. So part (C) holds.
It is easy to see that, for p > 2,
f p (u) = u(1 − sin u) + u p > u(1 − sin u) = f ∞ (u)
for 0 < u ≤ 1.
So by a comparison theorem of [12, Theorem 2.3], we obtain
T p (α) < T∞ (α)
for 0 < α ≤ 1.
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
It is easy to see that limα→0+ T∞ (α) =
function on (0, 1] since
π
2
since m 0 = 1 and by Lemma 3.1. Also, T∞ (α) is a strictly increasing
θ∞ (u) ≡ 2F∞ (u) − u f ∞ (u),
0
0
θ∞
(u) = f ∞ (u) − u f ∞
(u) = u 2 cos u > 0
for 0 < u ≤ 1,
and
0
T∞
(α) = 2−3/2
Z
α
0
θ∞ (α) − θ∞ (u) du
>0
(F∞ (α) − F∞ (u))3/2 α
for 0 < α ≤ 1.
Thus we conclude that
π < M( p) = max T p (α) = max T p (α) < T∞ (1)
2
α∈(0,∞)
α∈(0,1)
for p > 2.
Hence part (D) holds.
The proofs of the assertions in part (E) that lim p→2+ M( p) = π2 and lim p→2+ M( p) = T∞ (1) are easy but tedious;
we omit them. So the proof of Theorem 2.1(i) is complete.
(ii) For f p (u) = u(1 − sin u) + u p with p = 2, it is easy to compute that m 0 = 1, m ∞ = ∞, and
< 0, u 6= 2nπ, u > 0,
0
2
0
θ2 (u) = f 2 (u) − u f 2 (u) = −u (1 − cos u)
= 0, u = 2nπ, n = 1, 2, 3, . . . .
So, by (3.1) and (3.5), in addition to (2.3), T2 (α) is a strictly decreasing function on (0, ∞).
(iii) For f p (u) = u(1 − sin u) + u p with 1 < p < 2, we compute that m 0 = 1 and m ∞ = ∞. So, by (3.1),
limα→0+ T p (α) = π2 and limα→∞ T p (α) = 0. By (3.6) and (3.8), it is easy to see that θ p (0) = 0 and θ p (u) is strictly
decreasing for u near 0+ . So, by (3.5), T p (α) is strictly decreasing for small α > 0. In addition, for 1 < p < 2,
f p (u) = u(1 − sin u) + u p = u + (u p − u sin u) > u
for all u > 0.
So by a comparison theorem of [12, Theorem 2.3] and Lemma 3.1, we obtain T p (α) <
1+2 cos 1−sin 1
for 1 < p ≤ p̃ = 1−2
cos 1+sin 1 ≈ 1.629,
θ p (1) =
π
2
for all α > 0. In particular,
1− p
+ 2 cos 1 − sin 1 ≥ 0.
p+1
Hence there exists a positive number α̃ ≤ 1 such that
θ p (α̃) = 0 > θ p (u)
for 0 < u < α̃ ≤ 1.
Thus T p0 (α̃) > 0 by (3.5). Since limα→∞ T p (α) = 0 by (2.3), we conclude that T p (α) has at least two critical points,
a local minimum and a local maximum, on (0, ∞). Moreover, let α1 be the smallest positive critical point of T p (α), a
local minimum, on (0, ∞), then α1 = α1 ( p) ∈ (0, 1). We then prove lim p→1+ α1 ( p) = 0 as follows.
By (3.6), we compute that
−2 p+1 ( p − 1)2+ p − 4( p 2 − 1) cos(2( p − 1)) + 4( p − 1)2 − 2 ( p + 1) sin(2( p − 1))
θ p (2( p − 1)) =
p+1
2
= ( p − 1)3 + O(( p − 1)4 ) as p → 1+ .
3
Thus θ p (2( p − 1)) > 0 when p is sufficiently close to 1+ . It was proved above that θ p (0) = 0 and θ p (u) is strictly
decreasing for u near 0+ , and since α1 ( p) is the smallest positive critical point of T p (α), a local minimum, on (0, ∞).
So by (3.5) and (3.9), we obtain that
0 < α1 ( p) < 2( p − 1) when p is sufficiently close to 1+ .
Hence lim p→1+ α1 ( p) = 0.
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S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
Fig. 7. Graph of θ1 (u) = 2u cos u + (u 2 − 2) sin u, u > 0.
(iv) For p = 1, f 1 = u(1 − sin u) + u p = u(2 − sin u), it is easy to compute and obtain that
m 0 = 2,
π
+ 2nπ, n ∈ N,
2
> u otherwise,
(
3π
= 3u if u =
+ 2nπ, n ∈ N,
f 1 (u) = u(2 − sin u)
2
< 3u otherwise,
(
f 1 (u) = u(2 − sin u)
if u =
=u
θ1 (u) = 2u cos u + (u 2 − 2) sin u,
θ10 (u) = u 2 cos u;
see Fig. 7 for θ1 (u). So,
π
,
θ1 (0) = 0 and θ1 (u) is strictly increasing on 0,
2
nπ
, n ∈ N is odd,
θ1 (u) has a critical point at
2
nπ nπ
> θ1 (u) for 0 < u <
, n = 4k + 1, k ∈ N ∪ {0},
θ1
2
2
nπ nπ
θ1
< θ1 (u) for 0 < u <
, n = 4k − 1, k ∈ N.
2
2
(3.13)
(3.14)
Therefore, by (3.1) and (3.5), a comparison theorem of [12, Theorem 2.3] and Lemma 3.1, it can be proved that
π
π
π
(F) limα→0+ T1 (α) = √
and √
< T1 (α) < π2 for all α > 0. The assertion that limα→∞ T1 (α) = √
can be
2 3
2 2
2 2
proved as follows:
Z α
−1/2
lim T1 (α) = lim 2
(F1 (α) − F1 (u))−1/2 du
α→∞
α→∞
0
= lim 2−1/2
α→∞
= 2−1/2
1
Z
α(F1 (α) − F1 (αv))−1/2 dv( set u = αv)
0
Z
1
lim α(F1 (α) − F1 (αv))−1/2 dv
0 α→∞
(by applying Lebesgue’s Dominated Convergence Theorem, see e.g. [17, Theorem 5.19])
Z 1
cos α
sin α
v cos (αv) sin (αv) −1/2
−1/2
2
− 2 −
+
=2
lim 1 − v +
dv
α
α
α
α2
0 α→∞
S.-H. Wang / Nonlinear Analysis 67 (2007) 1316–1328
= 2−1/2
1
Z
1 − v2
−1/2
1327
dv
0
= 2−1/2 sin−1 v |10
π
= √ .
2 2
(G) T1 (α) is strictly increasing for small α > 0.
(H) T1 (α) has infinitely many critical points on (0, ∞) since
nπ T10
> 0, n = 4k + 1, k ∈ N ∪ {0}
2
and
nπ < 0, n = 4k − 1, k ∈ N,
T10
2
which follow by (3.13), (3.14) and (3.5).
Finally, we prove part (I) by modifying the proof of [3, Theorem 3(ii)–(iii)]. In the proof of part (F) above, we
obtain
Z 1
cos α
v cos (αv) sin (αv) −1/2
sin α
−1/2
2
T1 (α) = 2
1−v +
dv
− 2 −
+
α
α
α
α2
0
Z 1
≡ 2−1/2
(h(v, α))−1/2 dv.
0
Choosing α = nπ(n ∈ N is even), we get
nπ (1 − v cos (nπ v)) + sin (nπ v)
n2π 2
nπ
−
v)
−
sin
(1
(nπ (1 − v))
≥ 1 − v2 +
2
2
n π
> 1 − v2
h(v, nπ ) = 1 − v 2 +
since nπ (1 − v) − sin (nπ (1 − v)) > 0 for 0 < v < 1. Therefore
Z 1
−1/2
π
T1 (nπ ) < 2−1/2
1 − v2
dv = 2−1/2 sin−1 v |10 = √ .
2 2
0
π
. So part (I) holds.
Analogously, for odd positive integer n, T1 (nπ ) > √
2 2
The proof of Theorem 2.1 is now complete. References
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