Electronic Journal of Differential Equations, Vol. 2007(2007), No. 74, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
INVERSE SPECTRAL PROBLEMS FOR NONLINEAR
STURM-LIOUVILLE PROBLEMS
TETSUTARO SHIBATA
Abstract. This paper concerns the nonlinear Sturm-Liouville problem
−u00 (t) + f (u(t)) = λu(t),
u(t) > 0,
t ∈ I := (0, 1),
u(0) = u(1) = 0,
where λ is a positive parameter. We try to determine the nonlinear term
f (u) by means of the global behavior of the bifurcation branch of the positive
solutions in R+ × L2 (I).
1. Introduction
We consider the nonlinear Sturm-Liouville problem
−u00 (t) + f (u(t)) = λu(t),
t ∈ I := (0, 1),
(1.1)
t ∈ I,
(1.2)
u(0) = u(1) = 0,
(1.3)
u(t) > 0,
where λ is a positive parameter. We assume that f (u) satisfies the following conditions:
(A1) f (u) is a function of C 1 for u ≥ 0 satisfying f (0) = f 0 (0) = 0,
(A2) g(u) := f (u)/u is strictly increasing for u ≥ 0 (g(0) := 0),
(A3) g(u) → ∞ as u → ∞.
Then for each given α > 0, there exists a unique solution (λ, u) = (λ(α), uα ) ∈
¯ with kuα k2 = α. The set {(λ(α), uα ); α > 0} gives all solutions of
R+ × C 2 (I)
(1.1)–(1.3) and is an unbounded curve of class C 1 in R+ × L2 (I) emanating from
(π 2 , 0) (cf. [1, 7]).
Typical examples of f (u) are as follows:
f (u) = up
(u ≥ 0),
1
f (u) = up 1 −
(u ≥ 0),
q
1+u
where p > 1 and q > 0 are constants.
The equation (1.1)–(1.3) is motivated by the logistic equation of population
dynamics and vibration of string with self-interaction, and has been extensively
investigated by many authors. We refer to [1, 7, 11, 12, 13] and the references therein
2000 Mathematics Subject Classification. 34B15.
Key words and phrases. Inverse spectral problem; L2 -bifurcation diagram; logistic equations.
c
2007
Texas State University - San Marcos.
Submitted August 1, 2006. Published May 15, 2007.
1
2
T. SHIBATA
EJDE-2007/74
for the works in L∞ -framework from a viewpoint of local and global bifurcation
theory. On the other hand, since (1.1)–(1.3) is regarded as an eigenvalue problem,
it seems important to study (1.1)–(1.3) in L2 -framework. We refer to [2, 3, 4, 5, 6,
8, 9, 10, 14] for other works in this direction. In particular, the asymptotic formulas
for λ(α) as α → 0 have been established in [4, 5]. Therefore, the problem we have
to consider here is a global behavior of uα as α → ∞, and it is known from [1] that
uα (t)
−1
g (λ(α))
→1
(1.4)
locally uniformly on I as α → ∞. By this, it is easy to see that for α 1
α = kuα k2 = (1 + o(1))g −1 (λ(α)).
It follows from this that, in many cases, as α → ∞
λ(α) = g(α) + o(g(α)).
(1.5)
Note that if f (u) = up (p > 1), then g(α) = αp−1 . Motivated by (1.5), the following
asymptotic formula for λ(α) as α → ∞ has been given in [14].
Theorem 1.1 ([14]). Let f (u) = up (p > 1). Let n be an arbitrary, fixed, number
in N0 = {0, 1, 2, . . . }. Then the following asymptotic formula holds as α → ∞:
λ(α) = αp−1 + C0 α(p−1)/2 +
n
X
k=0
ak (p)
C k+2 αk(1−p)/2 + o(αn(1−p)/2 ), (1.6)
(p − 1)k+1 0
where
1
r
p−1
2 p+1
− ξ2 +
ξ
dξ
p+1
p+1
0
and ak (p) is the polynomial (deg ak (p) ≤ k + 1) which is determined by a0 =
1, a1 , . . . , ak−1 .
Z
C0 = (p + 3)
Consider now the implication of Theorem 1.1 and (1.6) from the standpoint of
inverse spectral problems.
Problem A. Assume that (1.6) holds for any n ∈ N0 . Then does f (u) = up
(p > 1) hold?
For the first step to solve this problem, we simplified the problem as follows in
[15]:
Problem B. Assume that the following asymptotic formula is valid as α → ∞.
λ(α) = αp−1 + C0 α(p−1)/2 + o(α(p−1)/2 ).
Then does f (u) = up hold?
The answer to Problem B was given in [15].
Theorem 1.2 ([15]). Let f (u) = up (1 − 1/(1 + u2 )). Furthermore, assume that
1 < p < 5. Then as α → ∞
λ(α) = αp−1 + C0 α(p−1)/2 + o(α(p−1)/2 ).
(1.7)
Therefore, we regret to say that the answer to the Problem B is negative. However, it seems worth considering the following problem on which it is imposed
stronger conditions than those in Problem B.
EJDE-2007/74
INVERSE SPECTRAL PROBLEMS
3
Problem C. Assume that the following asymptotic formula is valid as α → ∞.
1
λ(α) = αp−1 + C0 α(p−1)/2 +
C 2 + o(1).
(1.8)
p−1 0
Then does f (u) = up hold?
The purpose of this paper is to answer this question.
Theorem 1.3. Let f (u) = up (1 − 1/(1 + uq )), where p > 1.
(i) Assume that (p − 1)/2 < q < p + 1. Then (1.7) holds as α → ∞.
(ii) Assume that p − 1 < q < p + 1. Then (1.8) holds as α → ∞.
Therefore, unfortunately, the answer to the Problem C is not valid, either. It
seems that the assumption in Problem C is still weak to solve the inverse spectral
problem. However, it seems that Theorem 1.3 certainly is the meaningful step to
give the answer to Problem A.
Our arguments here to prove Theorem 1.3 are quite straightforward and are
different from those of Theorem 1.2, which are variant of the proof of Theorem
1.1. Therefore, in [15], it seems that the restriction 1 < p < 5 for the case q = 2
is technical. However, it follows from Theorem 1.3 (i) that this restriction is not
technical but optimal.
2. Proof of Theorem 1.3
In this section, C denotes various positive constants independent of λ 1. We
begin with the fundamental tools which play important roles in what follows. We
denote by (λ, uλ ) the solution pair of (1.1)–(1.3) for λ > π 2 . We know from [1] that
¯ of (1.1)–(1.3) for a given λ > π 2 . We use
there exists a unique solution uλ ∈ C 2 (I)
this notation in what follows. Therefore, α = kuλ k2 . It is well known that
uλ (t) = uλ (1 − t),
t ∈ I,
1
u0λ (t) > 0, 0 ≤ t < ,
2
1
kuλ k∞ = uλ ( ).
2
(2.1)
(2.2)
(2.3)
We know by [1] that
kuλ kq∞
+ λ1 ,
(2.4)
1 + kuλ kq∞
where λ1 > 0 is the remainder term of λ with respect to kuλ k∞ and depends on λ.
For λ 1, we have
√
λ1 ≤ Cλe− κλ/2 .
(2.5)
Here κ > 0 is a constant. For completeness, we give the proof of (2.5) in Appendix.
Now multiply (1.1) by u0λ (t). Then
upλ (t) 0
u00λ (t) + λuλ (t) − uλ (t)p +
u (t) = 0.
1 + uqλ (t) λ
p−1
λ = kuλ k∞
This along with (2.3) implies that
Z uλ (t)
1 0 2 1
1
ξp
uλ (t) + λuλ (t)2 −
uλ (t)p+1 +
dξ ≡ constant
2
2
p+1
1 + ξq
0
Z kuλ k∞
ξp
1
1
2
p+1
= λkuλ k∞ −
kuλ k∞ +
dξ (put t = 1/2).
2
p+1
1 + ξq
0
(2.6)
4
T. SHIBATA
EJDE-2007/74
Let
Z kuλ k∞
2
ξp
p+1
dξ. (2.7)
(kuλ kp+1
−
θ
)
+
2
∞
p+1
1 + ξq
θ
p
This along with (2.2) and (2.6) implies that u0λ (t) = Lλ (uλ (t)) for 0 ≤ t ≤ 1/2.
By this and (2.1), we obtain
Z 1/2
(kuλ k2∞ − u2λ (t))u0λ (t)
p
kuλ k2∞ − α2 = 2
dt
Lλ (uλ (t))
0
Z kuλ k∞
(kuλ k2∞ − θ2 )
p
=2
dθ
Lλ (θ)
0
Z 1
2kuλ k2
1 − s2
(2.8)
p
= √ ∞
ds
λ
Bλ (s)
0
Z
Z 1
2kuλ k2∞ n 1 1 − s2
1 − s2
1 − s2 o
p
p
= √
ds +
−p
ds
λ
A(s)
Bλ (s)
A(s)
0
0
Lλ (θ) = λ(kuλ k2∞ − θ2 ) −
=
2kuλ k2∞
√
(C1 + Mλ ),
λ
where
2
(1 − sp+1 ),
(2.9)
p+1
Z
kuλ k∞
ξp
2 kuλ kp−1
2
∞
Bλ (s) := 1 − s2 −
(1 − sp+1 ) +
dξ, (2.10)
p+1
λ
λkuλ k2∞ kuλ k∞ s 1 + ξ q
Z 1
1 − s2
p
C1 :=
ds,
(2.11)
A(s)
0
Z 1
1 − s2
1 − s2 p
Mλ :=
−p
ds.
(2.12)
Bλ (s)
A(s)
0
A(s) := 1 − s2 −
By (2.8), we prove Theorem 1.3. Therefore, it is important to obtain the asymptotic
formula for Mλ as λ → ∞. To this end, we first prove the following lemma.
Lemma 2.1. Let 0 < 1 be fixed. Then there exists a constant 0 < δ 1 such
that for 1 − ≤ s ≤ 1 and λ 1,
A(s) = K0 (s)(1 − s)2 ,
(2.13)
Bλ (s) = K1 (λ)(1 − s) + K2 (λ, s)(1 − s)2 ,
(2.14)
(p − 1)(1 − δ) ≤ K0 (s) ≤ p − 1 ,
(2.15)
where
2λ1
,
λ
(p − 1)(1 − δ) ≤ K2 (λ, s) ≤ p − 1 .
K1 (λ) =
(2.16)
(2.17)
Proof. We have A(1) = 0. Furthermore,
A0 (s) = −2s + 2sp ,
A00 (s) = −2 + 2psp−1 .
(2.18)
EJDE-2007/74
INVERSE SPECTRAL PROBLEMS
5
By this and Taylor expansion, for 1 − ≤ s ≤ 1 and λ 1, we obtain
1
A(s) = A(1) + A0 (1)(s − 1) + A00 (s1 )(s − 1)2
2
1 00
2
= A (s1 )(s − 1) ,
2
where 1 − ≤ s < s1 < 1. This along with (2.18) implies (2.13). Next, we have
B(1) = 0. Furthermore,
Bλ0 (s) = −2s + 2sp
p
p−1
2
kuλ kp+1
kuλ k∞
∞ s
−
q q.
2
λ
λkuλ k∞ 1 + kuλ k∞ s
Then by (2.4),
kuλ kp+1
kuλ kp−1
2
∞
∞
−
λ
λkuλ k2∞ 1 + kuλ kq∞
kuλ kp+q−1
∞
= −2 1 −
λ(1 + kuλ kq∞ )
2λ1
.
=−
λ
Bλ0 (1) = −2 + 2
Furthermore,
Bλ00 (s) =
p−1
p+q−1
− λ) 2(p − q)sp+q−1 kuλ k∞
2(pkuλ kp−1
+ psp−1 kuλ kp−1
∞ s
∞
−
.
q q 2
λ
λ(1 + kuλ k∞ s )
Therefore, by (2.4), 2(p − 1)(1 − δ) ≤ Bλ00 (s) ≤ 2(p − 1) for 1 − ≤ s ≤ 1 and λ 1.
By this and Taylor expansion, we obtain
1
Bλ (s) = Bλ (1) + Bλ0 (s)(s − 1) + Bλ00 (s2 )(s − 1)2
2
= K1 (λ)(1 − s) + K2 (λ, s)(1 − s)2 ,
where 1 − < s < s2 < 1. Thus the proof is complete.
Lemma 2.2. For λ 1,
Mλ = C2 (q)kuλ k−q
∞ (1 + o(1)).
(2.19)
Here
Z
C2 (q) =
0
1
(1 − s2 ){(1 − sp+1 )/(p + 1) − (1 − sp−q+1 )/(p − q + 1)}
ds.
A(s)3/2
Proof. It is easy to see that
Z
Mλ =
0
1
(1 − s2 )(A(s) − Bλ (s))
p
p
p
ds.
A(s) Bλ (s)( A(s) + Bλ (s))
p
6
T. SHIBATA
EJDE-2007/74
Then by (2.4), (2.9) and (2.10), for 0 < s ≤ 1 and λ 1,
A(s) − Bλ (s)
=
2
kuλ kp−1
2
∞
− 1 (1 − sp+1 ) −
p+1
λ
λkuλ k2∞
Z
kuλ k∞
kuλ k∞ s
ξp
dξ
1 + ξq
p−q−1
2
2
kuλ k∞
p+1
(1 + o(1))kuλ k−q
)−
(1 + o(1))
(1 − sp−q+1 )
∞ (1 − s
p+1
p−q+1
λ
1
1
(1 − sp+1 ) −
(1 − sp−q+1 ) .
= 2kuλ k−q
∞ (1 + o(1))
p+1
p−q+1
(2.20)
Furthermore, since q < p + 1, as λ → ∞,
=
p−1
2
kuλ k∞
2
−1 −
|A(0) − Bλ (0)| ≤ p+1
λ
λkuλ k2∞
kuλ k∞
ξp
dξ ≤ Ckuλ k−q
∞.
q
1+ξ
0
(2.21)
By this and (2.20), for 0 ≤ s ≤ 1, as λ → ∞, Bλ (s) → A(s). We apply Lebesgue’s
convergence theorem to our situation. Let an arbitrary 0 < 1 be fixed. Then
1−
Z
Mλ =
0
Z
(1 − s2 )(A(s) − Bλ (s))
p
p
p
p
ds
A(s) Bλ (s)( A(s) + Bλ (s))
1
(1 − s2 )(A(s) − Bλ (s))
p
p
p
p
ds
A(s) Bλ (s)( A(s) + Bλ (s))
1−
:= M1,λ + M2,λ .
Z
+
(2.22)
We know that A(s) and Bλ (s) is strictly decreasing for 0 ≤ s ≤ 1 and Bλ (1) = 0
(cf. Appendix). So we see from (2.14) that for 0 ≤ s ≤ 1 − and λ 1,
A(s) ≥ A(1 − ) ≥ (p − 1)(1 − δ)2 ,
Bλ (s) ≥ Bλ (1 − ) ≥ (p − 1)(1 − δ)2 > 0.
By this, there exists a constant C > 0 such that for 0 ≤ s ≤ 1 − and λ 1,
(1 − s2 )(A(s) − Bλ (s))
p
p
p
p
≤ C .
A(s) Bλ (s) A(s) + Bλ (s)
Therefore, by (2.4), (2.20), (2.21), (2.22) and Lebesgue’s convergence theorem, we
obtain
1−
Z
M1,λ =
0
Z
→
0
1−
(1 − s2 )(A(s) − Bλ (s))
p
p
p
p
ds
A(s) Bλ (s)( A(s) + Bλ (s))
(1 − s2 ){(1 − sp+1 )/(p + 1) − (1 − sp−q+1 )/(p − q + 1)}
ds.
A(s)3/2
(2.23)
EJDE-2007/74
INVERSE SPECTRAL PROBLEMS
7
By Lemma 2.1, for 1 − ≤ s ≤ 1 and λ 1,
(1 − s2 )(A(s) − Bλ (s))
p
p
p
p
A(s) Bλ (s)( A(s) + Bλ (s))
≤
(1 − s2 ){K1 (λ)(1 − s) + |K0 (s) − K2 (λ, s)|(1 − s)2 }
p
(K1 (λ)(1 − s) + K2 (λ, s)(1 − s)2 ) K0 (s)(1 − s)2
(K1 (λ)(1 − s) + |K0 (s) − K2 (λ, s)|(1 − s)2 )
p
{K1 (λ)(1 − s) + K2 (λ, s)(1 − s)2 } K0 (s)
K1 (λ) + |K0 (s) − K2 (λ, s)|(1 − s)
p
≤ C.
≤2
{K1 (λ) + K2 (λ, s)(1 − s)} K0 (s)
≤2
By this, we apply Lebesgue’s convergence theorem to M2,λ to obtain
Z 1
(1 − s2 )(A(s) − Bλ (s))
p
p
p
p
M2,λ =
ds
A(s) Bλ (s)( A(s) + Bλ (s))
1−
Z 1
(1 − s2 ){(1 − sp+1 )/(p + 1) − (1 − sp−q+1 )/(p − q + 1)}
ds.
→
A(s)3/2
1−
By this and (2.23), we obtain (2.19). Thus the proof is complete.
Proof of Theorem 1.3. By (2.8) and Lemma 2.2, we obtain
kuλ k2∞ − α2 =
2kuλ k2∞
√
(C1 + C2 kuλ k−q
∞ (1 + o(1))).
λ
By this, (2.4) and the Taylor expansion, for λ 1,
kuλ k2∞ − α2
−1/2
p−q−1
= 2kuλ k2∞ kuλ kp−1
−
ku
k
(1
+
o(1))
C1 + C2 kuλ k−q
λ
∞
∞
∞ (1 + o(1))
= 2kuλ k(5−p)/2
(C1 + (C1 /2 + C2 )kuλ k−q
∞
∞ (1 + o(1))).
(2.24)
By (2.4) and direct calculation, we obtain
1
kuλ k∞ = λ1/(p−1) 1 +
λ−q/(p−1) + o λ−q/(p−1) .
p−1
(2.25)
By this, (2.24) and Taylor expansion,
α2 = kuλ k2∞ − 2kuλ k(5−p)/2
(C1 + (C1 /2 + C2 )kuλ k−q
∞
∞ (1 + o(1)))
2
λ(2−q)/(p−1) + o λ(2−q)/(p−1)
= λ2/(p−1) +
p−1
5−p
1
− 2λ(5−p)/(2(p−1)) C1 +
C1 + C1 + C2 λ−q/(p−1) (1 + o(1)) .
2(p − 1)
2
(2.26)
(i) Assume that q > (p − 1)/2. Then for λ 1,
λ(5−p)/(2(p−1))−q/(p−1) λ(2−q)/(p−1) λ(5−p)/(2(p−1)) .
By this and (2.26), we obtain
α2 = λ2/(p−1) − 2C1 λ(5−p)/(2(p−1)) +
2
λ(2−q)/(p−1) (1 + o(1)).
p−1
8
T. SHIBATA
EJDE-2007/74
Now we put
λ = αp−1 (1 + C3 α−η1 (1 + o(1))).
Then by direct calculation, we obtain
C3 = (p − 1)C1 ,
η1 =
p−1
.
2
Since (p − 1)C1 = C0 , this implies Theorem 1.3 (i).
(ii) Furthermore, assume that q > p − 1. We put
λ = αp−1 (1 + (p − 1)C1 α−(p−1)/2 + C4 α−η2 (1 + o(1))).
Then by a straightforward calculation, we obtain
C4 = (p − 1)C12 ,
η2 = p − 1.
This implies that for λ 1
λ = αp−1 + (p − 1)C1 α(p−1)/2 + (p − 1)C12 + o(1).
Since (p − 1)C1 = C0 , we obtain Theorem 1.3 (ii). Thus the proof is complete. 3. Appendix
We first prove (2.5). We consider
(1.1)–(1.3) with f (u) = up+q /(1+uq ) for p > 1
Ru
and q > 0. We put F (u) := 0 f (s)ds. Furthermore, let
Qλ (θ) = λ(kuλ k2∞ − θ2 ) − 2(F (kuλ k∞ ) − F (θ)).
For 0 ≤ t ≤ 1, (3.1) is equivalent to (2.7). Then for 0 ≤ t ≤ 1/2, we obtain
p
u0λ (t) = Q(uλ (t)).
By this, we obtain
Z 1/2
Z kuλ k∞
Z 1
1
1
u0λ (t)
1
1
p
p
p
=
dt =
dθ = √
ds,
2
λ 0
Q(uλ (t)
Qλ (θ)
Rλ (s)
0
0
(3.1)
(3.2)
(3.3)
where
Rλ (s) := 1 − s2 −
2
(F (kuλ k∞ ) − F (kuλ k∞ s))
λkuλ k2∞
(3.4)
Let an arbitrary 0 < 1 be fixed. Then for 1 − ≤ s ≤ 1, by Taylor expansion,
1
F (kuλ k∞ s) = F (kuλ k∞ ) + f (kuλ k∞ )kuλ k∞ (s − 1) + f 0 (kuλ k∞ s1 )kuλ k2∞ (s − 1)2 ,
2
where s < s1 < 1 and s1 depends on s. Since f 0 (u) = pup−1 (1 + o(1)) for u 1,
there exists a constant δ > 0 such that for 1 − ≤ s ≤ 1
Rλ (s) = 2 1 −
f (kuλ k∞ ) f 0 (kuλ k∞ s1 )
(1 − s) +
− 1 (1 − s)2
λkuλ k∞
λ
(3.5)
≥ 2ξ(1 − s) + δ(1 − s)2 ,
where
ξ := 1 −
f (kuλ k∞ )
> 0.
λkuλ k∞
(3.6)
EJDE-2007/74
INVERSE SPECTRAL PROBLEMS
9
We know that ξ → 0 as λ → ∞. By this, (3.3) and (3.5), we obtain
√
Z 1
Z 1−
1
1
λ
p
p
ds +
ds
≤
2
Rλ (s)
2ξ(1 − s) + δ(1 − s)2
1−
0
Z 1
p
≤C+
dv
2ξv + δv 2
0
h
i
p
= C + δ −1 log |2δv + 2ξ + 2 δ(δv 2 + 2ξv)|
0
= δ −1 (log C − log 2ξ).
By this, we obtain
2ξ ≤ Ce−δ
√
λ/2
,
which along with (3.6) implies
λ≤
f (kuλ k∞ ) C −δ√λ/2
.
+ λe
kuλ k∞
2
Thus the proof of (2.5) is complete. We next prove that Bλ (s) is decreasing for
0 ≤ s ≤ 1. Indeed, since Bλ (s) = Rλ (s) in (3.4), by (A2) and (2.4),
2s f (kuλ k∞ s)
λ kuλ k∞ s
2s f (kuλ k∞ )
≤ −2s +
< 0.
λ kuλ k∞
Bλ0 (s) = −2s +
Thus the proof is complete.
References
[1] H. Berestycki, Le nombre de solutions de certains problèmes semi-linéares elliptiques, J.
Funct. Anal. 40 (1981), 1–29.
[2] A. Bongers, H.-P. Heinz and T. Küpper, Existence and bifurcation theorems for nonlinear
elliptic eigenvalue problems on unbounded domains, J. Differential Equations 47 (1983), 327–
357.
[3] J. Chabrowski, On nonlinear eigenvalue problems, Forum Math. 4 (1992), 359–375.
[4] R. Chiappinelli, Remarks on bifurcation for elliptic operators with odd nonlinearity, Israel J.
Math. 65 (1989), 285–292.
[5] R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term, Nonlinear Anal. TMA 13 (1989), 871–878.
[6] R. Chiappinelli, Constrained critical points and eigenvalue approximation for semilinear elliptic operators, Forum Math. 11 (1999), 459–481.
[7] J. M. Fraile, J. López-Gómez and J. C. Sabina de Lis, On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems, J. Differential Equations
123 (1995), 180–212.
[8] H.-P. Heinz, Free Ljusternik–Schnirelman theory and the bifurcation diagrams of certain
singular nonlinear problems, J. Differential Equations 66 (1987) 263–300.
[9] H.-P. Heinz, Nodal properties and bifurcation from the essential spectrum for a class of
nonlinear Sturm-Liouville problems, J. Differential Equations 64 (1986) 79–108.
[10] H.-P. Heinz, Nodal properties and variational characterizations of solutions to nonlinear
Sturm-Liouville problems, J. Differential Equations 62 (1986) 299–333.
[11] M. Holzmann and H. Kielhöfer, Uniqueness of global positive solution branches of nonlinear
elliptic problems, Math. Ann. 300 (1994), 221–241.
[12] P. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations,
J. Differential Equations 9 (1971), 536–548.
[13] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7
(1971), 487–513.
10
T. SHIBATA
EJDE-2007/74
[14] T. Shibata, Precise spectral asymptotics for nonlinear Sturm-Liouville problems, J. Differential Equations 180 (2002), 374–394.
[15] T. Shibata, Global behavior of the branch of positive solutions for nonlinear Sturm-Liouville
problems, Ann. Mat. Pura Appl. 186 (2007), 525–537.
Tetsutaro Shibata
Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan
E-mail address: shibata@amath.hiroshima-u.ac.jp