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First, we deal with the more general domains instead of the exterior domains,and second, we prove thatλ ∗is finite, and third, we also obtain the behavior of the twosolutions on 0,λ ∗ an

Volume 2007, Article ID 14731, 25 pages

doi:10.1155/2007/14731

Research Article

Eigenvalue Problems and Bifurcation of Nonhomogeneous

Semilinear Elliptic Equations in Exterior Strip Domains

Tsing-San Hsu

Received 19 July 2006; Revised 10 October 2006; Accepted 20 October 2006

Recommended by Patrick J Rabier

We consider the following eigenvalue problems:− Δu + u = λ( f (u) + h(x)) in Ω, u > 0

inΩ, u ∈ H1(Ω), where λ > 0, N = m + n ≥2,n ≥1, 0∈ ω ⊆ R mis a smooth boundeddomain, S = ω × R n,D is a smooth bounded domain in RN such that D ⊂⊂ S, Ω= S\––D Under some suitable conditions on f and h, we show that there exists a positive

constantλ ∗ such that the above-mentioned problems have at least two solutions ifλ ∈

(0,λ ∗), a unique positive solution ifλ = λ ∗, and no solution if λ > λ ∗ We also obtainsome bifurcation results of the solutions atλ = λ ∗

Copyright © 2007 Tsing-San Hsu This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited

1 Introduction

Throughout this article, letN = m + n ≥2,n ≥1, 2∗ =2N/(N −2) forN ≥3, 2∗ = ∞for

N =2,x =(y, z) be the generic point ofRNwithy ∈ R m,z ∈ R n

In this article, we are concerned with the following eigenvalue problems:

− Δu + u = λ

f (u) + h(x)

inΩ, u in H1

0(Ω), u > 0 in Ω, N≥2, (1.1)λ

where λ > 0, 0 ∈ ω ⊆ R m is a smooth bounded domain,S = ω × R n, D is a smooth

bounded domain inRNsuch thatD ⊂⊂ S,Ω= S \ D is an exterior strip domain inRN,

h(x) ∈ L2(Ω)∩ L q0(Ω) for some q0> N/2 if N ≥4,q0=2 ifN =2, 3,h(x) ≥0,h(x) ≡0and f satisfies the following conditions:

(f3) limt →0t −1f (t) =0;

(f4) there is a numberθ ∈(0, 1) such that

θt f (t) ≥ f (t) > 0 fort > 0; (1.2)(f5) f ∈ C2(0, +∞) and f (t) ≥0 fort > 0;

C p S(p+1)/2(p −1) andh ≥0,h ≡0 inRN, whereS is the best Sobolev constant and C p =

solu-method cannot know whetherλ ∗is bounded or infinite

In the present paper, motivated by [19], we extend and improve the paper by Zhu andZhou [19] First, we deal with the more general domains instead of the exterior domains,and second, we prove thatλ ∗is finite, and third, we also obtain the behavior of the twosolutions on (0,λ ∗) and some bifurcation results of the solutions atλ = λ ∗ Now, we stateour main results

Theorem 1.1 LetΩ= S \ D orΩ= R N \ D Suppose h(x) ≥ 0, h(x) ≡ 0, h(x) ∈ L2(Ω)∩

L q0(Ω) for some q0> N/2 if N ≥ 4, q0= 2 if N = 2, 3, and f (t) satisfies (f1)–(f5) Then there exists λ ∗ > 0, 0 < λ ∗ < ∞ such that

(i) equation (1.1) λ has at least two positive solutions u λ , U λ , and u λ < U λ if λ ∈(0,λ ∗ ), where u λ is the minimal solution of (1.1) λ and U λ is the second solution of (1.1) λ

constructed in Section 5 ;

(ii) equation (1.1) λ has at least one minimal positive solution u λ ∗ ;

(iii) equation (1.1) λ has no positive solutions if λ > λ ∗

Moreover, assume that condition (f5) ∗ holds, then (1.1) λ ∗ has a unique positive solution u λ ∗ Theorem 1.2 Suppose the assumptions of Theorem 1.1 and condition (f5) ∗ hold, then

(i)u λ is strictly increasing with respect to λ, u λ is uniformly bounded in L ∞(Ω)∩ H1(Ω)

for all λ ∈(0,λ ∗ ], and

u λ −→0 in L ∞(Ω)∩ H1(Ω) as λ −→0+, (1.5)(ii)U λ is unbounded in L ∞(Ω)∩ H1(Ω) for λ∈(0,λ ∗ ), that is,

In this paper, we denote byC and C i(i =1, 2, ) the universal constants, unless otherwise

specified Now, we will establish some analytic tools and auxiliary results which will beused later We set

First, we give some properties of f (t) The proof can be found in Zhu and Zhou [19]

Lemma 2.1 Under conditions (f1), (f4), and (f5),

(i) let ν =1 +θ −1> 2, one has that t f (t) ≥ νF(t) for t > 0;

(ii)t −1/θ f (t) is monotone nondecreasing for t > 0 and t −1f (t) is strictly monotone creasing if t > 0;

in-(iii) for any t1,t2∈(0, +∞ ), one has

In order to get the existence of positive solutions of (1.1) λ , consider the energy functional

I : H1(Ω)→ R defined by

I(u) =1

2

 Ω

Now, establish the following decomposition lemma for later use.

Proposition 2.2 Let conditions (f1), (f2), and (f4) be satisfied and suppose that { u k } is a

(PS)α -sequence of I in H1(Ω), that is, I(u k)= α + o(1) and I (u k)= o(1) strong in H −1(Ω) Then there exist an integer l ≥ 0, sequence { x i k } ⊆ R N of the form (0,z i k)∈ S , a solution u of

(1.1) λ , and solutions u i of (2.4) λ , 1 ≤ i ≤ l, such that for some subsequence { u k } , one has

3 Asymptotic behavior of solutions

In this section, we establish the decay estimate for solutions of(1.1)λand(2.4)λ In order

to get the asymptotic behavior of solutions of(1.1)λ, we need the following lemmas First,

we quote regularity Lemma 1 (see Hsu [12] for the proof) Now, letXbe aC1,1domain

inRN

Lemma 3.1 (regularity Lemma 1) Let g : X × R → R be a Carath´eodory function such that for almost every x ∈ X , there holds

Now, we quote Regularity Lemmas 2–4, (see Gilbarg and Trudinger [9, Theorems 8.8,

9.11, and 9.16] for the proof).

Lemma 3.2 (regularity Lemma 2) Let X ⊂ R N be a domain, g ∈ L2(X), and u ∈ H1(X)

a weak solution of the equation − Δu + u = g inX Then for any subdomainX ⊂⊂ X with

Lemma 3.4 (regularity Lemma 4) Let g ∈ L2(X)∩ L q(X) for some q ∈[2,∞ ) and let u ∈

H1(X) be a weak solution of the equation − Δu + u = g inX Then u ∈ W2,q(X) satisfies

By Lemmas3.1and3.4, we obtain the first asymptotic behavior of solution of(1.1)λ

Lemma 3.5 (asymptotic Lemma 1) Let condition (f2) hold and let u be a weak solution of

(1.1) λ , then u(y, z) → 0 as | z | → ∞ uniformly for y ∈ ω Moreover, if h(x) is bounded, then

u ∈ C1,α(Ω) for any 0 < α < 1.

Proof Suppose that u is a solution of(1.1)λ, then− Δu + u = λ( f (u) + h(x)) inΩ Since

f satisfies condition (f2) and h ∈ L2(Ω)∩ L q0(Ω) for some q0> N/2 if N ≥4,q0=2 if

N =2, 3, this implies thath ∈ L2(Ω)∩ L N/2(Ω) for N≥4 andh ∈ L2(Ω) for N=2, 3 ByLemma 3.1, we conclude that

Hence,λ( f (u) + h(x)) ∈ L2(Ω)∩ L q0(Ω) and byLemma 3.4, we have

u ∈ W2,2(Ω)∩ W2,q0(Ω), q0> N

2 ifN ≥4,q0=2 ifN =2, 3. (3.6)

Now, by the Sobolev embedding theorem, we obtain thatu ∈ C b(Ω) It is well known thatthe Sobolev embedding constants are independent of domains (see [1]) Thus there exists

a constantC such that, for R > 0,

 u  L ∞( Ω\ B R)≤ C  u  W2,q0(Ω\ B R) forN ≥2, (3.7)whereB R = { x =(y, z) ∈Ω| | z | ≤ R } From this, we conclude thatu(y, z) →0 as| z | → ∞

uniformly fory ∈ ω ByLemma 3.4and condition (f2), we also have that

Moreover, ifh(x) is bounded, then we have u ∈ W2,q(Ω) for q∈[2,∞) Hence, by theSobolev embedding theorem, we obtain thatu ∈ C1,α(Ω) for α∈(0, 1) 

We useLemma 3.5, and modify the proof in Hsu [11] We obtain the following preciseasymptotic behavior of solutions of(1.1)λand(2.4)λat infinity

Lemma 3.6 (asymptotic Lemma 2) Let w be a positive solution of (2.4) λ , let u be a positive solution of (1.1) λ , and let ϕ be the first positive eigenfunction of the Dirichlet problem − Δϕ =

λ1ϕ in ω, then for any ε > 0 with 0 < ε < 1 + λ1, there exist constants C, C ε > 0 such that

w(y, z) ≤ C ε ϕ(y) exp

Proof (i) First, we claim that for any ε > 0 with 0 < ε < 1 + λ1, there existsC ε > 0 such that

w(y, z) ≤ C ε ϕ(y) exp

−1 +λ1− ε | z | as| z | −→ ∞, y ∈ (3.10)Without loss of generality, we may assumeε < 1 Now given ε > 0, by condition (f3) and

Lemma 3.5, we may chooseR0large enough such that

λ f

w(y, z)

≤ εw(y, z) for| z | ≥ R0. (3.11)Letq =(q y,q z),q y ∈ ∂ω, | q z | = R0, andB a small ball in Ω such that q ∈ ∂B Since ϕ(y) >

0 for x =(y, z) ∈ B, ϕ(q y)=0,w(x) > 0 for x ∈ B, w(q) =0, by the strong maximumprinciple (∂ϕ/∂y)(q y)< 0, (∂w/∂x)(q) < 0 Thus

lim

x → q

| z |= R0

w(x) ϕ(y) = (∂w/∂x)(q)

(∂ϕ/∂y)

q y> 0. (3.12)Note thatw(x)ϕ −1(y) > 0 for x =(y, z), y ∈ ω, | z | = R0 Thusw(x)ϕ −1(y) > 0 for x =

(y, z), y ∈ | z | = R0 Sinceϕ(y) exp( −1 +λ1− ε | z |) andw(x) are C1(ω × ∂B R(0)), if

The strong maximum principle implies thatw(x) −Φ1(x) ≤0 forx =(y, z), y ∈

| z | ≥ R0, and therefore we get this claim

4 Existence of minimal solution

In this section, by the barrier method, we prove that there exists someλ ∗ > 0 such that

forλ ∈(0,λ ∗),(1.1)λhas a minimal positive solutionu λ(i.e., for any positive solutionu

of(1.1)λ, thenu ≥ u λ)

Lemma 4.1 If conditions (f1) and (f2) hold, then for any given ρ > 0, there exists λ0> 0 such that for λ ∈(0,λ0), one has I(u) > 0 for all u ∈ S ρ = { u ∈ H1(Ω)|  u  = ρ }

For the proof, see Zhu and Zhou [19]

Remark 4.2 For any ε > 0, there exists δ > 0 (δ ≤ ρ) such that I(u) ≥ − ε for all u ∈ { u ∈

H1(Ω)| ρ − δ ≤  u  ≤ ρ }and forλ ∈(0,λ0) if λ0 is small enough (see Zhu and Zhou[19])

For the numberρ > 0 given inLemma 4.1, we denote

B ρ =u ∈ H1(Ω)|  u  < ρ

Thus we have the following local minimum result

Lemma 4.3 Under conditions (f1), (f2), and (f4), if λ0 is chosen as in Remark 4.2 and

λ ∈(0,λ0), then there is a u0∈ B ρ such that I(u0)=min{ I(u) | u ∈ B ρ } < 0 and u0 is a positive solution of (1.1) λ

Proof Since h ≡0 andh ≥0, we can choose a functionϕ ∈ H1(Ω) such thatΩhϕ > 0.

Fort ∈(0, +∞), then

I(tϕ) = t2

2

 Ω

B ρ of(1.1)λsuch thatu k  u0weakly inH1(Ω) and α= I(u0) +l

i =1I ∞(u i) Note that

I ∞(u i)≥ S ∞ > 0 for i =1, 2, , m Since u0∈ B ρ, we have I(u0)≥ α We conclude that

By the standard barrier method, we prove the following lemma

Lemma 4.4 Let conditions (f1), (f2), and (f4) be satisfied, then there exists λ ∗ > 0 such that (i) for any λ ∈(0,λ ∗ ), (1.1) λ has a minimal positive solution u λ and u λ is strictly increas- ing in λ;

(ii) if λ > λ ∗ , (1.1) λ has no positive solution.

Proof Set Q λ = {0< λ < + ∞ |(1.1)λis solvable}, byLemma 4.3, we haveQ λis nonempty.Denotingλ ∗ =supQ λ > 0, we claim that(1.1)λhas at least one solution for allλ ∈(0,λ ∗)

In fact, for anyλ ∈(0,λ ∗), by the definition ofλ ∗, we know that there existsλ > 0 and

0< λ < λ < λ ∗such that (1.3)λ has a solutionu λ > 0, that is,

(1.1)λsuch that 0≤ u λ ≤ u λ Since 0 is not a solution of(1.1)λandλ > λ, the maximum

principle implies that 0< u λ < u λ Using the result of Graham-Eagle [10], we can choose

Letu λbe the minimal positive solution of(1.1)λforλ ∈(0,λ ∗), we study the followingeigenvalue problem

then we have the following lemma

Lemma 4.5 Under conditions (f1)–(f5), the first eigenvalue μ λ of ( 4.4 ) is defined by

(ii)μ λ > λ and is strictly decreasing in λ, λ ∈(0,λ ∗ );

(iii)λ ∗ < + ∞ and (1.1) λ ∗ has a minimal positive solution u λ ∗

Proof (i) Indeed, by the definition of μ λ, we know that 0< μ λ < + ∞ Let{ v k } ⊂ H1(Ω)

be a minimizing sequence ofμ λ, that is,



∇ v k 2

+v2

k dx −→ μ λ ask −→ ∞ (4.6)

This implies that{ v k }is bounded inH1(Ω), then there is a subsequence, still denoted by

{ v k }and somev0∈ H1(Ω) such that

v k v0 weakly inH01(Ω),

Thus,

 Ω



∇ v0  2

+v2 dx ≤lim inf

 Ω

It follows from the Sobolev embedding theorem that there existsk1, such that fork ≥ k1,

(ii) We now proveμ λ > λ Setting λ > λ > 0 and λ ∈(0,λ ∗), byLemma 4.4,(1.1)λ has

a positive solutionu λ Sinceu λis the minimal positive solution of(1.1)λ, thenu λ > u λas

λ > λ By virtue of(1.1)λ and(1.1)λ, we see that

Applying the Taylor expansion and noting thatλ > λ, h(x) ≥0 and f (t) ≥0, f (t) > 0

for allt > 0, we get

and there ist, with 0 < t < 1 such that

showing thatμ λis strictly decreasing inλ, for λ ∈(0,λ ∗)

(iii) We show next thatλ ∗ < + ∞ Letλ0∈(0,λ ∗) be fixed For anyλ ≥ λ0, we have

μ λ > λ and by (4.21), then

μ λ0≥ μ λ > λ (4.22)for allλ ∈[λ0,λ ∗) Thusλ ∗ < + ∞

By (4.4) andμ λ > λ, we have

 Ω

for allλ ∈(0,λ ∗) Sinceλ ∗ < + ∞, by (4.26) we can obtain that u λ  ≤ C < + ∞for all

λ ∈(0,λ ∗) Thus, there existsu λ ∗ ∈ H1(Ω) such that



∇ u λ ∗ · ∇ ϕ + u λ ∗ ϕ

dx λ

From I λ(u λ),ϕ  =0 and letλ → λ ∗, we deduceI λ ∗(u λ ∗)=0 inH −1(Ω) Hence, uλ ∗ is apositive solution of(1.1)λ ∗.

Letu be any positive solution of(1.1)λ ∗ By adopting the argument as inLemma 4.4,

we haveu ≥ u λinΩ for λ ∈(0,λ ∗) Letλ → λ ∗, we deduce thatu ≥ u λ ∗inΩ This implies

5 Existence of second solution

Whenλ ∈(0,λ ∗), we have known that(1.1)λhas a minimal positive solutionu λbyLemma4.4, then we need only to prove that(1.1)λhas another positive solution in the form of

U λ = u λ+v, where v is a solution of the following equation:

Using the monotonicity of f and the maximum principle, we know that the nontrivial

critical points of energy functionalJ are the positive solutions of (5.1)

First, we give an inequality about f and u λ

Lemma 5.1 Under conditions (f1), (f2), and (f5), then for any ε > 0, there exists C ε > 0 such that

where 1 < p < 2 ∗ − 1 and u λ is the minimal solution of (1.1) λ

For the proof, see Zhu and Zhou [19]

Lemma 5.2 Under conditions (f1), (f2), (f4), and (f5), there exist ρ > 0 and α > 0 such that

J(v) | S ρ ≥ α > 0, (5.4)

where S ρ = { u ∈ H1(Ω)|  u  = ρ }

Proof ByLemma 4.5, it is easy to see that, for allv ∈ H1(Ω),

 Ω

Again, byLemma 5.1and Sobolev embedding, we obtain that

Hence, there existρ > 0 and α > 0 such that J(v) | S ρ ≥ α > 0. 

Similar toProposition 2.2, for the energy functionalJ, we also have the following

re-sult

Proposition 5.3 Under conditions (f1), (f2), and (f4), let { v k } be a (PS) c -sequence of J Then there exists a subsequence (still denoted by { v k } ) for which the following holds: there exist an integer l ≥ 0, a sequence { x i

k } ⊆ R N of the form (0,z i

k)∈ S , a solution v of ( 5.1 ), and solutions u i of (2.4) λ , 1 ≤ i ≤ l, such that for some subsequence { v k } , as k → ∞ , one has

where one agrees that in the case l = 0, the above hold without u i , x i k

Now, letδ be small enough, D δ aδ-tubular neighborhood of D such that D δ ⊂⊂ S.Letη(x) : S →[0, 1] be aC ∞cut-off function such that 0≤ η ≤1 and

wherew is a ground state solution of(2.4)λ.

Lemma 5.4 Let conditions (f1)–(f5) be satisfied Then

(i) there exists t0> 0 such that J(tηw τ)< 0 for t ≥ t0, τ ≥ τ0,

(ii) there exists τ ∗ > 0 such that the following inequality holds for τ ≥ τ ∗ :

tηw τ0

u > 0, where ν =1 +θ −1> 2 Thus, for any given constant C > 0, there is u0≥0 such that

Let r0 be a positive constant such thatB m(0;r0)= { y | | y | ≤ r0} ⊂⊂ω, B n(0; 1)= { z |

| z | ≤1},Ω1= B m(0;r0)× B n(0; 1), andΩ1τ = B m(0;r0)× { z + τe N | | z | ≤1} By the nition ofτ0, we have thatΩ1τ ⊂⊂Ω\ D δfor allτ ≥ τ0 This also implies that there exists

Sinceν > 2, we can choose t0> 0 large enough such that (i) holds.

(ii) By (i),J is continuous on H1(Ω), J(0)=0, andLemma 5.2, we know that thereexistst1with 0< t1< t0such that

Forτ ≥ τ0,t1≤ t ≤ t0, by condition (f2), (2.5), Lemmas2.1and3.6, we have

tηw τ0

tηw τ0

It follows from the Taylor’s expansion that

Now byLemma 3.6, forτ ≥max(τ0,τ1) andt1≤ t ≤ t0, we obtain that



Ω 1τ

tηw τ0

Now, letε =(1 +λ1)/2, then we can find some τ ∗large enough such that

Theorem 5.5 Let conditions (f1)–(f5) be satisfied Then ( 5.1 ) has a positive solution v if

Again, byLemma 5. 1and Sobolev embedding, we obtain that

Hence, there existρ > and α > such that J(v) |... ≤lim inf

 Ω

It follows from the Sobolev embedding theorem... get

and there ist, with < t < such that

showing thatμ λis