EXPONENTS FOR TWO-PARAMETER NONLINEAR EIGENVALUE PROBLEMS

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EXPONENTS FOR TWO-PARAMETER NONLINEAR EIGENVALUE PROBLEMS

TETSUTARO SHIBATA Received 17 January 2002

We study the nonlinear two-parameter problem −u(x) +λu(x)^q=µu(x)^p, u(x)>0,x∈(0,1),u(0)=u(1)=0. Here, 1< q < pare constants andλ,µ >0 are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurveµ(λ) asλ→ ∞. We emphasize that the critical case con- cerning the decaying rate of the second term isp=(3q−1)/2 and this kind of criticality is new for two-parameter problems.

1. Introduction

We consider the following nonlinear two-parameter problem:

−u(x) +λu(x)^q=µu(x)^p, x∈I=(0,1), u(x)>0, x∈I,

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

(1.1)

where 1< q < pandλ,µ >0 are parameters.

The purpose of this paper is to establish the asymptotic formulas for the eigencurveµ=µ(λ) with the exact second term asλ→ ∞by using a variational method. We also establish the critical relationship betweenpandqfrom a view- point of the decaying rate of the second term ofµ(λ).

The study of two-parameter eigenvalue problems began with the oscillation theory and has been investigated by many authors. We refer to [1,2,3, 4,5, 6,7,8,9,10,11] and the references therein. One of the main problems in this area is to analyze the structure of the solution set{(λ,µ,u)}of (1.1), and the eﬀective approach to this problem is to study the structure of the setSλ,µ:= {(λ,µ,up+1)} ⊂R³for largeλ. In Shibata [7], by using a standard variational framework (seeSection 2), the variational eigencurve µ=µ(λ) was defined to

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analyzeSλ,µand the following asymptotic formula forµ(λ) asλ→ ∞was estab- lished:

µ(λ)=C1λ⁽^p⁺³⁾^/⁽²^p⁻^q⁺³⁾+oλ⁽^p⁺³⁾^/⁽²^p⁻^q⁺³⁾, (1.2) where

C1=

(p+ 1)(q+ 3) (p+ 3)(q+ 1)

1 γ^p⁺¹

2 p−q

π(q+ 1) 2

p+ 1 q+ 1

(q+3)/2(p−q)

×Γ(q+ 3)/2(p−q) Γ(p+ 3)/2(p−q)

2(p−q)/(2p−q+3)

, Γ(r)=

_∞

0 y^r⁻¹e⁻^ydy (r >0).

(1.3)

By this formula, we understood the first term ofµ(λ) asλ→ ∞. However, the remainder estimate ofµ(λ) has not been obtained. The purpose here is to ob- tainthe exact second termofµ(λ) asλ→ ∞. We emphasize that the second term depends deeply on the relationship between p andq, and the critical case is p=(3q−1)/2. More precisely, if p=(3q−1)/2, then the asymptotic behavior of the second term of µ(λ) is completely diﬀerent from that of the case wherep=(3q−1)/2. As far as we know, this kind of criticality is new for two- parameter problems and great interest by itself. Finally, it should be mentioned that the asymptotic behavior of such eigencurve is also eﬀected by the variational framework (cf. [6,8]).

2. Main results

We explain notations before stating our results. Let H0¹(I) be the usual real Sobolev space. Leturdenote the usualL^r-norm. Foru∈H0¹(I),

Eλ(u) :=1

2u²2+ 1

q+ 1λu^q_q⁺¹+1, M_γ:= u∈H0¹(I) :u_p+1=γ,

(2.1) whereγ >0 is a fixed constant. For a given λ >0, we call µ(λ) the variational eigenvalue when the following conditions are satisfied:

λ,µ(λ),uλ

∈R+×R+×Mγ satisfies (1.1), Eλ

uλ

= inf

u∈MγEλ(u). (2.2)

Thenµ(λ) is obtained as a Lagrange multiplier and is represented explicitly as follows:

µ(λ)=u_λ²₂+λu_λ^q_q⁺¹₊₁

γ^p⁺¹ . (2.3)

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Indeed, multiply the equation in (1.1) byuλ. Then integration by parts yields u_λ²2+λuλ^q⁺¹

q+1=µ(λ)uλ^p⁺¹

p+1=µ(λ)γ^p⁺¹. (2.4) This implies (2.3). The existence ofµ(λ) for a givenλ >0 is ensured in [7, Theo- rem 2.1] andµ(λ) is continuous forλ >0 (cf. [7, Theorem 2.2]). Finally, let

K1:= √

2 q+ 1

p+ 1

(q−1)/(2(p−q))Γ1/(q+ 1)Γ(q−1)/2(q+ 1) π(q+ 1)

×C⁽1^q⁻¹⁾^/⁽²⁽^p⁻^q⁾⁾

2(q+1)/(q−1)

, K2:=1

2 1

0

s⁽²^p⁻³^q⁻¹⁾^/²(1−s^p⁺¹) 1−s^p⁻^q³^/² ds, K3:=2²⁽^p⁺²⁾^/⁽^q⁺¹⁾

q+ 1 1

0

y⁽²^p⁻²^q⁺²⁾^/⁽^q⁺¹⁾

(1 +y)²⁽^p⁺²⁾^/⁽^q⁺¹⁾(1−y)⁽²^p⁻²^q⁺²⁾^/⁽^q⁺¹⁾dy, J0=

√π p−q

q+ 3 p+ 3

Γ(q+ 3)/2(p−q) Γ(p+ 3)/2(p−q).

(2.5)

Now, we state our results.

Theorem2.1. (1)Assumep >(3q−1)/2. Then the following asymptotic formula holds asλ→ ∞:

µ(λ)=C1λ⁽^p⁺³⁾^/⁽²^p⁻^q⁺³⁾ 1 +C2

1 +o(1)λ⁻²⁽^p⁺¹⁾⁽^q⁺¹⁾^/⁽⁽²^p⁻^q⁺³⁾⁽^q⁻¹⁾⁾, (2.6) where

C2=K1

1− 2(p−q)K2

(2p−q+ 3)J0

. (2.7)

(2)Assumep <(3q−1)/2. Then asλ→ ∞, µ(λ)=C1λ⁽^p⁺³⁾^/⁽²^p⁻^q⁺³⁾ 1−C3

1 +o(1)λ⁻⁽^p⁺¹⁾^/⁽^q⁻¹⁾, (2.8) where

C3= 2(p−q)

(2p−q+ 3)J0K3K1⁽²^p⁻^q⁺³⁾^/⁽²⁽^q⁺¹⁾⁾. (2.9) (3)Assumep=(3q−1)/2. Then asλ→ ∞,

µ(λ)=C1λ⁽^p⁺³⁾^/⁽²^p⁻^q⁺³⁾ 1−C4

1 +o(1)λ⁻²⁽^p⁺¹⁾⁽^q⁺¹⁾^/⁽⁽²^p⁻^q⁺³⁾⁽^q⁻¹⁾⁾logλ, (2.10)

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where

C4= 2(p−q)(p+ 1)

(q−1)(2p−q+ 3)²J0K1. (2.11) We briefly explain the idea of the proof. Put

ν(λ)=λ⁽^p⁻¹⁾^/²⁽^p⁻^q⁾µ(λ)⁽¹⁻^q⁾^/²⁽^p⁻^q⁾, w_λ(t)=

µ(λ) λ

1/(p−q)

u_λ(x), t=ν(λ)

x−1 2

. (2.12)

Then it follows from (1.1) thatwλsatisfies

−w_λ(t)=wλ(t)^p−wλ(t)^q, t∈Iν(λ):=

−1 2ν(λ),1

2ν(λ), w_λ(t)>0, t∈Iν(λ),

w_λ

±1

2ν(λ)=0.

(2.13)

Then by [7, Lemma 5.1],

ν(λ)−→ ∞ (2.14)

asλ→ ∞. Putz_λ=w_λ/w_λ∞. Then it is easy to see from (2.3) that

µ(λ)=λ^(p+3)/(2(p⁻^q))µ(λ)⁻^(q+3)/(2(p⁻^q))w_λ²₂+w_λ^q_q⁺¹₊₁ γ^p⁺¹

=λ⁽^p⁺³⁾^/⁽²⁽^p⁻^q⁾⁾µ(λ)⁻⁽^q⁺³⁾^/⁽²⁽^p⁻^q⁾⁾wλ^p⁺¹

p+1

γ^p+1

=λ⁽^p⁺³⁾^/⁽²⁽^p⁻^q⁾⁾µ(λ)⁻⁽^q⁺³⁾^/⁽²⁽^p⁻^q⁾⁾wλ^p⁺¹

∞ zλ^p⁺¹

p+1

γ^p⁺¹ .

(2.15)

Therefore, it is crucial to study the asymptotic behavior ofwλ∞ andzλp+1

asλ→ ∞.

3. Asymptotic behavior ofwλ∞

In this section, we study the asymptotic behavior ofwλ∞asλ→ ∞. We put wλ

∞= p+ 1

q+ 1

1 +(λ)

1/(p−q)

. (3.1)

Then by [7, (5.10), Lemma 5.2], we know that(λ)>0 and(λ)→0 asλ→ ∞.

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Lemma3.1. The following equality holds forλ >0:

ν(λ)= 2(q+ 1)

p+ 1 q+ 1

1 +(λ)

−(q−1)/(2(p−q))

L(λ), (3.2) where

L()= ₁

0

1 m(,s)ds, m(,s)=

s^q⁺¹−s^p⁺¹+

1−s^p⁺¹ (>0).

(3.3)

Proof. Multiply the equation in (2.13) byw_λ. Then fort∈Iν(λ),

w_λ(t)w_λ(t) +w_λ(t)^pw_λ(t)−w_λ(t)^qw_λ(t)=0, (3.4) which implies that

d dt

1 2

w_λ(t)²+ 1

p+ 1wλ(t)^p⁺¹− 1

q+ 1wλ(t)^q⁺¹

=0. (3.5)

We know thatw_λ(0)=w_λ∞andw_λ(0)=0 sinceu_λ(1/2)=u_λ∞andu_λ(1/2)=0.

Then putt=0 to obtain 1

2w_λ(t)²+ 1

p+ 1wλ(t)^p⁺¹− 1

q+ 1wλ(t)^q⁺¹≡ 1

p+ 1wλ^p+1

∞ − 1

q+ 1wλ^q+1

∞ . (3.6) Note that w_λ(t)<0 fort∈(0,ν(λ)/2) sinceu_λ(x)<0 forx∈(1/2,1). Then it follows from this and (3.1) that fort∈(0,ν(λ)/2),

−z_λ(t)=w_λ⁽_∞^q⁻¹⁾^/² 2

q+ 1

z_λ(t)^q⁺¹−z_λ(t)^p⁺¹+(λ)1−z_λ(t)^p⁺¹

=wλ⁽^q⁻¹⁾^/²

∞

2

q+ 1m(λ),zλ(t).

(3.7)

Puts=zλ. Then (3.1) and (3.7) yield ν(λ)

2 ⁼

_ν(λ)/2 0

−z_λ(t)

2/(q+ 1)w_λ⁽_∞^q⁻¹⁾^/²m(λ),z_λ(t)dt

= q+ 1

2

p+ 1 q+ 1

1 +(λ)

−(q−1)/(2(p−q))1 0

1 m(λ),sds.

(3.8)

This implies (3.2).

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In order to study the asymptotic behavior of(λ) asλ→ ∞, we investigate the asymptotic behavior ofL() as→0.

Lemma3.2. For0<1,

L()=Γ1/(q+ 1)Γ(q−1)/2(q+ 1)

(q+ 1)^√π

−(q−1)/(2(q+1))+o⁻⁽^q⁻¹⁾^/⁽²⁽^q⁺¹⁾⁾ . (3.9) Proof. Put

L1() :=L()− ₁

0

√ 1

s^q+1+ds. (3.10)

Puts=¹^/⁽^q⁺¹⁾tan²^/⁽^q⁺¹⁾θ. Then 1

0

√ 1

s^q⁺¹+ds

= 2 q+ 1

−(q−1)/(2(q+1))

tan⁻¹(1/^√)

0 sin⁻⁽^q⁻¹⁾^/⁽^q⁺¹⁾θcos⁻²^/⁽^q⁺¹⁾θ dθ

= 2 q+ 1

1 +o(1)⁻⁽^q⁻¹⁾^/⁽²⁽^q⁺¹⁾⁾ _π/2

0 sin⁻⁽^q⁻¹⁾^/⁽^q⁺¹⁾θcos⁻²^/⁽^q⁺¹⁾θ dθ

= 1 q+ 1

1 +o(1)⁻⁽^q⁻¹⁾^/⁽²⁽^q⁺¹⁾⁾B 1

q+ 1, q−1 2(q+ 1)

= 1 q+ 1

1 +o(1)⁻⁽^q⁻¹⁾^/⁽²⁽^q⁺¹⁾⁾Γ1/(q+ 1)Γ(q−1)/2(q+ 1) Γ(1/2)

= 1 q+ 1

1 +o(1)⁻⁽^q⁻¹⁾^/⁽²⁽^q⁺¹⁾⁾Γ1/(q+ 1)Γ(q−1)/2(q+ 1)

√π .

(3.11) We use here the formula

2 _π/2

0 sin²^m⁻¹θcos²ⁿ⁻¹θdθ=B(m,n)=Γ(m)Γ(n)

Γ(m+n) (m,n >0), (3.12) whereB(m,n) is the beta function. Next, we calculateL1(). Note that for 0≤ s≤1,

m(,s)=

s^q⁺¹1−s^p⁻^q+

1−s^p⁺¹≥

s^q⁺¹+

1−s^p⁻^q. (3.13)

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By this, we obtain L1()

= ₁

0

(1 +)s^p+1 m(,s)^√s^q⁺¹+

m(,s) +^√s^q⁺¹+ds

≤ 1

0

(1 +)s^p⁺¹ s^q+1+

1−s^p⁻^q^√s^q+1+

s^q+1+

1−s^p⁻^q+^√s^q+1+ds

≤(1 +) 1

0

s^p⁺¹ s^q⁺¹+3/2√

1−s^p⁻^q1 +^√1−s^p⁻^qds

≤2 1

0

s^p⁺¹ s^q+1+_3/2√

1−s^p⁻^qds

=2 _δ

0

s^p+1 s^q⁺¹+3/2√

1−s^p⁻^qds+ 2 ₁

δ

s^p+1 s^q⁺¹+3/2√

1−s^p⁻^qds :=I+II,

(3.14) where 0< δ1 is a fixed constant. LetCj,δ>0 (j=1,2,...) be constants de- pending only onδ. Puts=sin²^/⁽^p⁻^q⁾θ. Then

II≤ 2 δ³⁽^q⁺¹⁾^/²

1 δ

√ 1

1−s^p⁻^qds

= 2 δ³⁽^q⁺¹⁾^/²

2 p−q

1

sin⁻¹δ^(p⁻^q)/2sin⁽²⁺^q⁻^p⁾^/⁽^p⁻^q⁾θ dθ

≤C1,δ.

(3.15)

Moreover, puts=¹^/⁽^q⁺¹⁾t. Then for 0<1, I≤_√ 2

1−δ^p⁻^q _δ

0

⁽^p⁺¹⁾^/⁽^q⁺¹⁾t^p⁺¹ ³^/²(t^q⁺¹+ 1)³^/²

1/(q+1)dt

≤2_√ δ^p⁺¹ 1−δ^p⁻^q

(2p−3q+1)/(2(q+1))=o⁻⁽^q⁻¹⁾^/⁽²⁽^q⁺¹⁾⁾ .

(3.16)

By (3.14), (3.15), and (3.16), we have

L1()=o⁻^(q⁻1)/(2(q+1))

. (3.17)

By this, (3.10), and (3.11), we obtain (3.9).

Now, we study the asymptotic behavior of(λ) asλ→ ∞. Lemma3.3. Asλ→ ∞,

(λ)=K1

1 +o(1)λ⁻²⁽^p⁺¹⁾⁽^q⁺¹⁾^/⁽⁽^q⁻¹⁾⁽²^p⁻^q⁺³⁾⁾. (3.18)

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Proof. By (1.2) and (2.12), we have ν(λ)=λ⁽^p⁻¹⁾^/⁽²⁽^p⁻^q⁾⁾µ(λ)⁽¹⁻^q⁾^/⁽²⁽^p⁻^q⁾⁾

=λ⁽^p⁻¹⁾^/⁽²⁽^p⁻^q⁾⁾C1λ⁽^p⁺³⁾^/⁽²^p⁻^q⁺³⁾⁽¹⁻^q⁾^/⁽²⁽^p⁻^q⁾⁾1 +o(1)

=C⁽¹1⁻^q⁾^/⁽²⁽^p⁻^q⁾⁾

1 +o(1)λ⁽^p⁺¹⁾^/⁽²^p⁻^q⁺³⁾.

(3.19)

On the other hand, by Lemmas3.1and3.2and Taylor expansion, we have ν(λ)=

2(q+ 1) p+ 1

q+ 1

₋(q−1)/(2(p−q))

1 +(λ)⁻⁽^q⁻¹⁾^/⁽²⁽^p⁻^q⁾⁾L(λ)

= 2(q+ 1)

p+ 1 q+ 1

−(q−1)/(2(p−q))

1− q−1

2(p−q)(λ) +o(λ)

×

Γ1/(q+ 1)Γ(q−1)/2(q+ 1)

(q+ 1)^√π (λ)⁻^(q⁻1)/(2(q+1))

+o(λ)⁻⁽^q⁻¹⁾^/⁽²⁽^q⁺¹⁾⁾

=√ 2

p+ 1 q+ 1

−(q−1)/(2(p−q))Γ1/(q+ 1)Γ(q−1)/2(q+ 1) π(q+ 1)

×(λ)⁻^(q⁻1)/(2(q+1))

1 +o(1).

(3.20)

By this and (3.19), we obtain (3.18).

4. Asymptotic behavior ofzλp+1

In this section, we calculatezλp+1. Note thatzλ(t)=zλ(−t) fort∈Iν(λ). Then by (3.7) and puttings=z_λ(t), we have

z_λ^p_p⁺¹₊₁=2 _ν(λ)/2

0 z_λ(t)^p⁺¹dt

=2 _ν(λ)/2

0 z_λ(t)^p⁺¹ ⁻z_λ(t)

wλ⁽^q⁻¹⁾^/²

∞

2/(q+ 1)m(λ),zλ(t)dt

=

2(q+ 1) wλ^(q⁻^1)/2

∞

J(λ),

(4.1)

where

J() := ₁

0

s^p⁺¹

m(,s)ds (>0). (4.2)

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Therefore, we study the precise asymptotics ofJ() as→0. Puts=sin²^/⁽^p⁻^q⁾θ.

Then as→0,

J()−→J(0)= 1

0

s⁽²^p⁻^q⁺¹⁾^/²

√1−s^p⁻^qds

= 2 p−q

_π/2

0 sin⁽^p⁺³⁾^/⁽^p⁻^q⁾θ dθ

=

√π p−q

q+ 3 p+ 3

Γ(q+ 3)/2(p−q) Γ(p+ 3)/2(p−q)⁼J0.

(4.3)

We use here the formulas _π/2

0 sin^rθ dθ=

√π 2

Γ(r+ 1)/2

Γ(r/2 + 1) (r >−1),

Γ(r+ 1)=rΓ(r). (4.4)

Therefore, put

J1() :=J()−J0:= −J2(), J2() :=

₁

0

s^p⁺¹1−s^p⁺¹

m(,s)m(0,s)m(,s) +m(0,s)ds. (4.5) Now, we study the asymptotic behavior ofJ2() as→0.

Lemma4.1. (1)Ifp >(3q−1)/2, thenJ2()→K2as→0.

(2)Ifp <(3q−1)/2, then as→0, J2()=K3

1 +o(1)⁽²^p⁻³^q⁺¹⁾^/⁽²⁽^q⁺¹⁾⁾. (4.6)

(3)Ifp=(3q−1)/2, then as→0, J2()= − 1

2(q+ 1)

1 +o(1)log. (4.7)

Proof. (1) Since p >(3q−1)/2, we have (2p−3q−1)/2>−1. Therefore, by Lebesgue’s convergence theorem, as→0,

J2()−→1 2

1 0

s⁽²^p⁻³^q⁻¹⁾^/²1−s^p⁺¹

1−s^p⁻^q^3/2 ds=K2. (4.8)

(2) We have the following two steps.

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Step 1. Assume thatp <(3q−1)/2. We introduceJ3() to approximateJ2():

J3() := 1

0

s⁽²^p⁻^q⁺¹⁾^/²

√s^q⁺¹+

s⁽^q⁺¹⁾^/²+^√s^q⁺¹+ds

=J4(,δ) +J5(,δ) :=

_δ

0

s^(2p⁻^q+1)/2

√s^q+1+

s^(q+1)/2+^√s^q+1+ds +

1 δ

s⁽²^p⁻^q⁺¹⁾^/²

√s^q⁺¹+

s⁽^q⁺¹⁾^/²+^√s^q⁺¹+ds,

(4.9)

where 0< δ1 is a fixed small constant. We study the asymptotic behaviors of J3,J4, andJ5as→0. Note that 0<(2p−2q+ 2)/(q+ 1)<1 sincep <(3q−1)/2.

Then puts=¹^/⁽^q⁺¹⁾tan²^/⁽^q⁺¹⁾θandy=tan(θ/2) to obtain J3()= 2

q+ 1

(2p−3q+1)/(2(q+1))

_tan⁻¹_(1/^√₎

0

tan⁽²^p⁻²^q⁺²⁾^/⁽^q⁺¹⁾θ 1 + sinθ dθ

=2²⁽^p⁺²⁾^/⁽^q⁺¹⁾ q+ 1

(2p−3q+1)/(2(q+1))

×

_tan(1_/_2)(tan⁻¹₍₁_/^√₎₎

0

y⁽²^p⁻²^q⁺²⁾^/⁽^q⁺¹⁾

(1 +y)2(p+2)/(q+1)(1−y)^(2p⁻2q+2)/(q+1)dy

=2²⁽^p⁺²⁾^/⁽^q⁺¹⁾ q+ 1

1 +o(1)⁽²^p⁻³^q⁺¹⁾^/⁽²⁽^q⁺¹⁾⁾

× 1

0

y⁽²^p⁻²^q⁺²⁾^/⁽^q⁺¹⁾

(1 +y)²⁽^p⁺²⁾^/⁽^q⁺¹⁾(1−y)⁽²^p⁻²^q⁺²⁾^/⁽^q⁺¹⁾dy

=K3

1 +o(1)⁽²^p⁻³^q⁺¹⁾^/⁽²⁽^q⁺¹⁾⁾.

(4.10)

Similarly, we obtain

J4(,δ)=K3

1 +o(1)⁽²^p⁻³^q⁺¹⁾^/⁽²⁽^q⁺¹⁾⁾, J5(,δ)≤ 1

δ^q⁺¹. (4.11)

Sincep <(3q−1)/2, this along with (4.10) implies thatJ3()/J4(,δ)→1 as→ 0 for a fixedδ.

Step 2. We show that as→0,

J2()

J3()^−→1. (4.12)