Generalized elliptic functions and their application to a nonlinear eigenvalue problem with $p$-Laplacian

0.21 MB·English

by  Shingo Takeuchi

GENERALIZED ELLIPTIC FUNCTIONS AND THEIR
APPLICATION TO A NONLINEAR EIGENVALUE
PROBLEM WITH p-LAPLACIAN
0
1
0 SHINGO TAKEUCHI
2
Dedicated to Professor Yoshio Yamada on occasion of his 60th birthday
n
a
J
Abstract. The Jacobian elliptic functions are generalized and
9
applied to a nonlinear eigenvalue problem with p-Laplacian. The
1
eigenvalue and the corresponding eigenfunction are represented

] in terms of common parameters, and a complete description of
P
the spectra and a closed form representation of the corresponding
A
eigenfunctionsareobtained. Asaby-productoftherepresentation,
h. it turns out that a kind of solution is also a solution of another
t eigenvalue problem with p/2-Laplacian.
a
m
[

2 1. Introduction
v
7 In this paper we generalize the Jacobian elliptic functions and apply
7
them to a nonlinear eigenvalue problem
3
0
. (φ (u′))′ +λφ (u)(1−|u|q) = 0, t ∈ (0,T),
1 (PE ) p q
pq
0 u(0) = u(T) = 0,
(
0
1 where T, λ > 0, p, q > 1 and φ (s) = |s|m−2s (s 6= 0), = 0 (s = 0).
: m
v Problem (PE ) appears frequently in various articles as station-
pq
i
X ary problems. In particular, the equation for p = q = 2 is called,
r e.g., the Allen-Cahn equation, the Chafee-Infante equation [3], and a
a
bistable reaction-diffusion equation with logistic effect. The equation
for p = 2 < q is said to be a bistable reaction-diffusion equation with
Allee effect. In case p = n and q = 2 with an n-dimensional domain,
an equation of this type is known as the Euler-Lagrange equation of
functional related to models introduced by Ginzburg and Landau for
the study of phase transitions (cf. Problem 17 in [2]).
As to (PE ) for general p > 1, we have to mention the work [8] by
pq
Guedda and V´eron [8]. They showed that if p = q > 1 then there exists
a positive increasing sequence {λ } such that a pair of solutions ±u of
n n
1991 Mathematics Subject Classification. Primary 34L40, Secondary 33E05.
This work was supported by KAKENHI (No. 20740094).
1
2 SHINGOTAKEUCHI
(PE )with(n−1)-zeros z = jT/n(j = 1,2,...,n−1) bifurcates from
pq j
the trivial solution at λ = λ and |u | → 1 uniformly on any compact
n n
set of (0,T)\{z ,z ,...,z } as λ → ∞. Moreover, they proved that
1 2 n−1
if p = q > 2 then for each n ∈ N there exists Λ > λ such that λ > Λ
n n n
implies |u | = 1 on flat cores [z + T (Λn)1/p,z − T (Λn)1/p] (j =
n j−1 2n λ j 2n λ
1,2,...,n) of u , where z = 0 and z = T. This is a great contrast
n 0 n
to case 1 < p = q ≤ 2, where |u | < 1 in [0,T]. Since the equation in
n
(PE ) is autonomous, if u (n ≥ 2) has flat cores, then there exists
pq n
uncountable solution with (n − 1)-zeros near u , which is produced
n
by expanding and contracting the flat cores with preserving its total
length T(1−(Λn)1/p). In this sense, the n-th branch (λ,u ) bifurcating
λ n
from (λ ,0) causes the second bifurcation at (Λ ,u ) for each n ≥ 2.
n n Λn
The phenomena of flat corein [8] above was generalized to case p > 2
and q > 1 by the author and Yamada [11]. They also studied change
in bifurcation depending on the relation between p and q (as far as
the first bifurcation is concerned, their proof can be applied to case
1 < p ≤ 2), and showed that for each n ∈ N, if p > q then there exists
a pair of solutions ±u of (PE ) with (n−1)-zeros for λ > 0; if p = q
n pq
then there exists λ > 0 such that (PE ) has no solution with (n−1)-
n pq
zeros for λ ≤ λ and (PE ) has a pair of solutions ±u for λ > λ
n pq n n
(the same result as [8]); if p < q then there exists λ∗ > 0 such that
n
(PE ) has no solution with (n−1)-zeros for λ < λ∗ and (PE ) has a
pq n pq
pair of solutions ±u for λ = λ∗ and (PE ) has two pairs of solutions
n n pq
±u , ±v satisfying |u (t)| > |v (t)| with t 6= z (j = 0,1,...,n)
n n n n j
for λ > λ∗n. In this sense, the point (λ∗n,uλ∗n) causes the spontaneous
bifurcation. In any case, each solution u has flat cores for sufficiently
n
large λ.
The purpose of this paper is to obtain a complete description of the
spectra and a closed form representation of the corresponding eigen-
functions of (PE ), while the studies [8] and [11] above are done in
pq
the way of phase-plane analysis and no exact solution is given there.
For the description and representation, we first recall that the Jaco-
bian elliptic function sn(t,k) with modulus k ∈ [0,1) satisfies
(1.1) u′′ +u(1+k2 −2k2u2) = 0.
(e.g., Example 4 of p.516 in the book [13] of Whittaker and Watson).
Eq.(1.1) reminds that the solution of nonlinear eigenvalue problem
(PE ) with p = q = 2 can be represented explicitly by using sn(t,k).
pq
Indeed, for any given k ∈ (0,1), the set of eigenvalues of (PE ) is given
22
GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 3
by
2
2nK(k)
(1.2) λ (k) = (1+k2)
n
T
(cid:18) (cid:19)
for each n ∈ N, with corresponding eigenfunctions ±u , where
n,k
2k2 2nK(k)
(1.3) u (t) = sn t,k
n,k
1+k2 T
r (cid:18) (cid:19)
and K(k) is the complete elliptic integral of the first kind
1 ds
K(k) =
(1−s2)(1−k2s2)
Z0
(cf. Section 2 in [1]). Converseply, all nontrivial solutions are given by
Eqs.(1.2) and (1.3), and in particular, it follows from Eq.(1.3) that all
solutions satisfy |u| < 1.
In our study on (PE ), after the fashion of Jacobi’s sn(t,k), we
pq
introduce a new transcendental function sn (t,k) with modulus k ∈
pq
[0,1). This satisfies
q
(1.4) (φ (u′))′ + φ (u)(1+kq −2kq|u|q) = 0,
p p∗ q
where p∗ := p/(p−1). Using sn (t,k), we can obtain a complete de-
pq
scription of the set of eigenvalues and the corresponding eigenfunctions
of (PE ) as Eqs.(1.2) and (1.3) with
pq
1 ds
K (k) = .
pq
p (1−sq)(1−kqsq)
Z0
It is important that Kpq(k) conpverges to Kp,q(0) as k → 1 −0 if and
2
only if p > 2. Indeed,
1 ds
k→lim1−0Kpq(k) = Z0 (1−sq)p2 = Kp2,q(0).
Similarly, snpq(t,k) converges to snp,q(t,0) as k → 1−0. These conver-
2
gent properties yield the existence of special solutions, not necessarily
|u| < 1, and we can really construct the solutions of (PE ) with flat
pq
cores. Moreover, snp,q(t,0) satisfies Eq.(1.4) with k = 0 and p replaced
2
by p/2 as well as Eq.(1.4) with k = 1. Thus, we obtain the following
(curious) property: a kind of solution of (PE ) is also a solution of the
pq
nonlinear eigenvalue problem with p/2-Laplacian
(φp(u′))′ +λφq(u) = 0, t ∈ (0,T),
2
u(0) = u(T) = 0.
(
4 SHINGOTAKEUCHI
Thispaperisorganizedasfollows. InSection2,weintroduceagener-
alized trigonometric function sin (t) given by Dr´abek and Mana´sevich
pq
[7] and define a new transcendental function sn (t,k), which is a gen-
pq
eralization of the Jacobian elliptic function sn(t,k) and an extension
of sin (t) as sn (t,0) = sin (t). In Section 3, we apply them to non-
pq pq pq
linear eigenvalue problems, particularly to the problem considered in
[8] and [11], and obtain complete descriptions of the set of eigenvalues
and the corresponding eigenfunctions.
2. Transcendental Functions
2.1. Generalized trigonometric functions. Generalizedtrigonomet-
ric functions were introduced by Dr´abek and Mana´sevich [7] (see also
[6]). For σ ∈ [0,1], we define (in a slightly different way from [7])
σ ds
(2.1) arcsin (σ) := ,
pq 1
Z0 (1−sq)p
where p > 1, q > 0. Letting s = z1/q, we have
1 σq 1 1 1
arcsin (σ) = z1q−1(1−z)−1p dz = B˜ , ,σq ,
pq q q q p∗
Z0 (cid:18) (cid:19)
where B˜(s,t,u) denotes the incomplete beta function
u
B˜(s,t,u) = zs−1(1−z)t−1dz.
Z0
We define the constant π as
pq
2 1 1
π := 2arcsin (1) = B , ,
pq pq
q q p∗
(cid:18) (cid:19)
where B(s,t) denotes the beta function
1
B(s,t) = B˜(s,t,1) = zs−1(1−z)t−1dz.
Z0
We have that arcsin : [0,1] → [0,π /2], and is strictly increasing.
pq pq
Let us denote its inverse by sin . Then, sin : [0,π /2] → [0,1] and
pq pq pq
is strictly increasing. We extend sin to all R (and still denote this
pq
extension by sin ) in the following form: for t ∈ [π /2,π ], we set
pq pq pq
sin (t) := sin (π −t), then for t ∈ [−π ,0], we define sin (t) :=
pq pq pq pq pq
−sin (−t), and finally we extend sin to all R as a 2π periodic
pq pq pq
function.
When 0 < p ≤ 1, we also define arcsin as Eq.(2.1) for σ ∈ [0,1).
pq
We have that arcsin : [0,1) → [0,∞), and is strictly increasing. Let
pq
us denote its inverse by sin . Then, sin : [0,∞) → [0,1) and is
pq pq
GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 5
strictly increasing. We extend sin to all R as sin (t) := −sin (−t)
pq pq pq
for t ∈ (−∞,0] and still denote this extension by sin .
pq
Remark 2.1. We immediately find that sin (t) = sin(t) and π = π
22 22
from the properties of the beta function. Moreover, sin (t) = sin (t)
pp p
and π = π = 2π , where sin and π are the generalized sine
pp p psinπ p p
p
function and its half-period, respectively, appearing in [4], [5] and [6].
We define for t ∈ [0,π /2] (in case 0 < p ≤ 1, for t ∈ [0,∞))
pq
cospq(t) := (1−sinqpq(t))p1,
then we obtain
cosp (t)+sinq (t) = 1,
pq pq
d
sin (t) = cos (t).
pq pq
dt
Proposition 2.1. For p, q > 1, sin satisfies for all R
pq
q
(2.2) (φ (u′))′ + φ (u) = 0.
p p∗ q
Proof. For t ∈ (0,π /2) we have
pq
(φ (u′))′ = (φ (cos (t)))′
p p pq
= ((1−sinq (t))p1∗)′
pq
1
= p∗(1−sinqpq(t))−p1 ·(−qsinpq−q1(t))·cospq(t)
q
= − φ (u).
p∗ q
By symmetry of sin , Eq.(2.2) holds true for t 6= t := nπ /2, n ∈ Z.
pq n pq
Since lim (φ (u′))′ exists, φ (u′) is differentiable also at t = t and
t→tn p p n
satisfies Eq.(2.2) for all R in the classical sense. (cid:3)
2.2. Generalized Jacobian elliptic functions. We shall introduce
new transcendental functions, which generalize the Jacobian elliptic
functions. For σ ∈ [0,1] and k ∈ [0,1), we define
σ ds
(2.3) arcsn (σ) = arcsn (σ,k) := ,
pq pq
p (1−sq)(1−kqsq)
Z0
where p > 1, q > 0. We define the constantpKpq(k) as
1 ds
K = K (k) := arcsn (1,k) =
pq pq pq
p (1−sq)(1−kqsq)
Z0
p
6 SHINGOTAKEUCHI
We have that arcsn : [0,1] → [0,K ], and is strictly increasing.
pq pq
Let us denote its inverse by sn (·) = sn (·,k). Then, sn : [0,K ] →
pq pq pq pq
[0,1] and is strictly increasing. We extend sn to all R (and still
pq
denotethis extension by sn ) inthefollowing form: for t ∈ [K ,2K ],
pq pq pq
we set sn (t) := sn (2K −t), then for t ∈ [−2K ,0], we define
pq pq pq pq
sn (t) := −sn (−t), and finally we extend sn to all R as a 4K
pq pq pq pq
periodic function.
When 0 < p ≤ 1, we also define arcsn as Eq.(2.3) for σ ∈ [0,1).
pq
We have that arcsn : [0,1) → [0,∞), and is strictly increasing. Let
pq
us denotes its inverse by sn (·) = sn (·,k). Then, sn : [0,∞) →
pq pq pq
[0,1) and is strictly increasing. We extend sn to all R as sn (t) :=
pq pq
−sn (−t) for t ∈ (−∞,0] and still denote this extension by sn .
pq pq
The following proposition is crucial to our study.
Proposition 2.2. For p, q > 0, K is continuous and strictly increas-
pq
ing in [0,1), 2K (0) = π and sn (t,0) = sin (t). Moreover,
pq pq pq pq
πp,q if p > 2,
lim 2K (k) = 2
pq
k→1−0 (∞ if 0 < p ≤ 2,
lim snpq(t,k) = sinp,q(t).
k→1−0 2
Proof. The first half is trivial from the definitions of K and sn . If
pq pq
p > 2, then the monotone convergence theorem of Beppo Levi gives
1 ds
k→lim1−02Kpq(k) = 2Z0 (1−sq)p2 = 2arcsinp2,q(1) = πp2,q.
If 0 < p ≤ 2, then 2K (k) diverges to ∞ as k → 1 − 0 by Fatou’s
pq
lemma.
The last property is proved as follows. By the symmetry of sn (·,k),
pq
we may assume t > 0. Suppose p > 2 and that there exist t , ε > 0
0
and {k } such that k → 1 as j → ∞ and
j j
(2.4) |σkj −sinp2,q(t0)| ≥ ε,
where σ = sn (t ,k ). Let n ∈ Z be the number satisfying t ∈
kj pq 0 j 0
In := [nπp,q/2,(n+1)πp,q/2) and j ∈ N a large number satisfying t0 ∈
2 2
(n)
I (k ) := [nK (k ),(n+1)K (k )). We write sn (·,k ) as sn (·,k )
n j pq j pq j pq j pq j
on I (k ) and sin(n)(·) as sin (·) on I . Now, since σ is bounded, we
n j pq pq n kj
can choose a subsequence {kj′} of {kj} such that σkj′ → σ for some
σ ∈ [−1,1] as j′ → ∞. Thus, as j′ → ∞
nπp,q
t0 = nKpq(kj′)+arcsnpq(σkj′) → 22 +arcsinp2,q(σ),
GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 7
(n)
and hence σ = sin (t ), which contradicts (2.4). The proof to case
p,q 0
0 < p ≤ 2 is similar2and we omit it. (cid:3)
Remark 2.2. In case p > 2, 2K (k) and sn (·,k) converge to the finite
pq pq
valueπp,q andto thefinite-periodicfunction sinp,q ask → 1−0, respec-
2 2
tively. This is quite different from case p = 2, where 2K (k) diverges
2q
to ∞ and sn (t,k) converges to the monotone increasing function
22
sin (t) = tanh(t) as k → 1−0.
12
We define for t ∈ [0,K ] (in case 0 < p ≤ 1, for t ∈ [0,∞))
pq
cnpq(t) := (1−snqpq(t))p1,
dnpq(t) := (1−kqsnqpq(t))p1,
then we obtain
cnp (t)+snq (t) = 1,
pq pq
d
sn (t) = cn (t)dn (t).
pq pq pq
dt
Proposition 2.3. For p, q > 1, sn satisfies for all R
pq
q
(2.5) (φ (u′))′ + φ (u)(1+kq −2kq|u|q) = 0,
p p∗ q
which includes Eq.(2.2) as case k = 0.
Proof. For t ∈ (0,K (k)) we have
pq
(φ (u′))′ = (φ (cn (t)dn (t)))′
p p pq pq
= (((1−sinq (t))(1−kqsinq (t)))p1∗)′
pq pq
1
= ((1−sinq (t))(1−kqsinq (t)))−p1
p∗ pq pq
×(−qsinq−1(t)·(1+kq −2kqsinq (t)))·cn (t)dn (t)
pq pq pq pq
q
= − φ (u)(1+kq −2kquq).
p∗ q
By symmetry of sn , Eq.(2.5) holds true for t 6= t := nK (k), n ∈ Z.
pq n pq
Since lim (φ (u′))′ exists, φ (u′) is differentiable also at t = t and
t→tn p p n
satisfies Eq.(2.5) for all R in the classical sense. (cid:3)
Remark 2.3. Letting s = sin (t) in Eq.(2.3), we have
pq
arcsinpq(σ) dt
arcsn (σ,k) = .
pq
Z0 p 1−kqsinqpq(t)
q
8 SHINGOTAKEUCHI
We define the amplitude function am (·,k) : [0,K (k)] → [0,π /2]
pq pq pq
by
ampq(t,k) dθ
t = ,
Z0 p 1−kqsinqpq(θ)
q
thus sn is represented by sin as
pq pq
sn (t,k) = sin (am (t,k)).
pq pq pq
3. Applications
3.1. The (p,q)-eigenvalue problem. Let T, λ > 0 and p, q > 1. We
consider the nonlinear eigenvalue problem
(φ (u′))′ +λφ (u) = 0, t ∈ (0,T),
p q
(E )
pq
u(0) = u(T) = 0.
(
Problem (E ) has been studied by many authors. In particular, in pa-
pq
per [10] of Oˆtani, the existence of infinitely many multi-node solutions
was proved by using subdifferential operators method and phase-plane
analysis combined with symmetry properties of the solutions. After
that, Dr´abek and Mana´sevich [7] provided explicit forms of the whole
spectrum and the corresponding eigenfunctions for (E ) (see also [6]).
pq
We follow [7] to understand completely the set of all solutions of (E ).
pq
It will be convenient to find first the solution to the initial value
problem
(φ (u′))′ +λφ (u) = 0,
p q
(3.1)
u(0) = 0, u′(0) = α,
(
where without loss of generality we may assume α > 0.
Let u be a solution to Eq.(3.1) and let t(α) be the first zero point of
u′(t). On interval (0,t(α)), u satisfies u(t) > 0 and u′(t) > 0, and thus
u′(t)p u(t)q Rq αp
+λ = λ = ,
p∗ q q p∗
where R = u(t(α)) > 0. Solving for u′ and integrating, we find
q p1 t u′(s)
ds = t,
(cid:18)λp∗(cid:19) Z0 (Rq −u(s)q)p1
which after a change of variable can be written as
1 u(t) 1
q p 1 R ds q p 1 u(t)
t = = arcsin .
(cid:18)λp∗(cid:19) Rpq−1 Z0 (1−sq)p1 (cid:18)λp∗(cid:19) Rpq−1 pq(cid:18) R (cid:19)
GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 9
Thus we obtain the solution to Eq.(3.1) can be written as
1
(3.2) u(t) = Rsinpq λp∗ p Rpq−1t ,
q
!
(cid:18) (cid:19)
where R = ( q )1qαqp.
λp∗
Theorem 3.1. All nontrivial solutions of (E ) are given as follows.
pq
For any given R > 0, the set of eigenvalues of (E ) is given by
pq
q nπ p
(3.3) λ (R) = pq Rp−q
n p∗ T
(cid:16) (cid:17)
for each n ∈ N, with corresponding eigenfunctions ±u , where
n,R
nπ
pq
(3.4) u (t) = Rsin t .
n,R pq
T
(cid:16) (cid:17)
Proof. For given R > 0, by imposing that u in Eq.(3.2) satisfies the
boundaryconditions in (E ), we obtain that λ isaneigenvalue of (E )
pq pq
if and only if
1
λp∗ p Rpq−1T = nπpq, n ∈ N,
q
(cid:18) (cid:19)
and hence Eq.(3.3) follows. Expression (3.4) for the eigenfunctions
(cid:3)
follows directly from Eq.(3.2).
3.2. A perturbed (p,q)-eigenvalue problem. Let T, λ > 0 and
p, q > 1. We consider the nonlinear eigenvalue problem
(φ (u′))′ +λφ (u)(1−|u|q) = 0, t ∈ (0,T),
p q
(PE )
pq
u(0) = u(T) = 0.
(
Problem(PE )hasbeenstudiedbyBergerandFraenkel[1]andChafee
pq
and Infante [3] (p = q = 2), Wang and Kazarinoff [12] and Korman, Li
and Ouyang [9] (p = 2 < q), Guedda and V´eron [8] (p = q > 1), and
Takeuchi and Yamada [11] (p > 2, q > 1). However, there is no study
providing explicit forms of the whole spectrum and the corresponding
eigenfunctions for (PE ).
pq
As we have done for (E ), it will be convenient to find first the
pq
solution to the initial value problem
(φ (u′))′ +λφ (u)(1−|u|q) = 0,
p q
(3.5)
u(0) = 0, u′(0) = α,
(
where without loss of generality we may assume α > 0.
10 SHINGOTAKEUCHI
Let u be a solution to Eq.(3.5) and let t(α) be the first zero point of
u′(t). On interval (0,t(α)), u satisfies u(t) > 0 and u′(t) > 0, and thus
u′(t)p F(u) F(R) αp
+λ = λ = ,
p∗ q q p∗
where F(s) = sq − 1s2q and R = u(t(α)). Since we are interested
2
in functions satisfying the boundary condition of (PE ), it suffices to
pq
assume 0 < R ≤ 1, which means |u| ≤ 1. Moreover, we restrict to
0 < R < 1 and concentrate solutions satisfying |u| < 1 for a while.
Solving for u′ and integrating, we find
q 1p t u′(s)
ds = t,
λp∗ p F(R)−F(u(s))
(cid:18) (cid:19) Z0
which after a change of variapble can be written as
1 u(t)
q p R R
(3.6) t = ds.
λp∗ p F(R)−F(Rs)
(cid:18) (cid:19) Z0
It is easy to verify that p
Rq
F(R)−F(Rs) = F(R)(1−sq) 1− sq ,
2−Rq
(cid:18) (cid:19)
and hence
1 u(t)
q p R R ds Rq
t = kq :=
(cid:18)λp∗(cid:19) F(R)p1 Z0 p (1−sq)(1−kqsq) (cid:18) 2−Rq(cid:19)
1
q p R p u(t)
= arcsn ,k .
λp∗ 1 pq R
(cid:18) (cid:19) F(R)p (cid:18) (cid:19)
Then we obtain that the solution to Eq.(3.5) can be written as
λp∗ p1 F(R)p1
(3.7) u(t) = Rsn t,k ,
pq
q R
!
(cid:18) (cid:19)
where
1
Rq q
(3.8) k = ,
2−Rq
(cid:18) (cid:19)
1 −1
q q p 1 1 2q q
R = αq + 1− αp .
λp∗ 2 2 λp∗
(cid:18) (cid:19) (cid:18) r (cid:19)
We first observe the structure of the set of all nontrivial solutions of
(PE ) satisfying |u| < 1.
pq