Branches of solutions for an asymptotically linear
elliptic problem onԹܰ
Youyan Wan
Department of Mathematics
Jianghan University
Wuhan, Hubei, China
Abstract—We consider the following nonlinear schrК
െο࢛+ࣅࢂ()࢛=ࢌ(࢞,࢛)with࢛אࡴ)and ࢛ء૙,(כ)
whereࣅ>0and ࢌ(࢞,࢙)is asymptotically linear withrespect to at
origin and infinity. The potential ࢂ(࢞)satisfies ()൒ࢂ>0for
all ࢞אԹand ()=ࢂ
࢒࢏࢓ (λ)א(૙,+λ).We provethat
problem (כ)has two connected sets of positive and negative
solutions inԹ×ࢃ૛,࢖)for some࢖א[૛,+λ)ת(
Keywords-Bifurcation, asymptotically linear, Fredholm opera-
tor of index zero.
In this paper, we consider the following nonlinear
where ɉ>0and the functions Vand ݂satisfy the following
1) ܸ(ݔ)אܥ(ܴܰ,ܴ) and there existsܸ0>0such that
ܸ(ݔ)൒ܸ0>0 for all ݔאԹܰ;
2) ܸ(ݔ)=ܸ (λ )א(0 ,+λ)
݈݅݉ and ݁ܽݏ{ݔאԹܰ(ݔ)<
1)݂ (ݔ,ݏ)אܥ( Թܰ×Թ,Թ)and (ݔ,ή)אܥ1(Թ,Թ);
2)there exist two functions ݄,݃אܮλܰ)such that
݈݅݉ and ݂(ݔ,ݏ)
݈݅݉ uniformly inݔאԹܰ,
where ݄and ݃satisfy
(G) Setting Ȟ=inf༌{׬(|׏ݑ|2+ܸ(ݔ)ݑ2)݀ݔ:ݑא
ܪ1ܰ) ܽ݊݀ ׬ݑ2݀ݔ=1}
Թܰ,there existsߙא(Ȟ,ܸ(λ))such
that ݃(ݔ)
݈݅݉ ;
(H) |݄ |λ<ߙܸ0
3) ݄(ݔ)൑݂(ݔ,ݏ)
ݏ൑݃(ݔ)for all(ݔ,ݏ )אԹܰ×Թ\{0}.
The existence of solutions of problem (1.1) has
beeninvestigated extensively. For problem (1.1)with
potentialwell and various conditions on ݂(ݔ,ݑ)ء݂ (ݑ),
several authorshave obtained the existence of solutions for
large ߣbyvariational methods,for example, [1], [2], [3], [5].
And other authors have got the existence of solutions forߣis
not necessarily large by concentration compactnessargument
and mountain pass geometry, for instance, [7], [8]. Stuart and
Zhou [10] havestudied how the positive and negative solutions
of problem(1.1)depend on ߣby topological methods.
Inspired by the results we mentioned above, the main object
of thisarticle is to investigate the relation between the positive
andnegative solutions of problem (1.1)and the parameterߣ,
where the potential need not be well potential and݂(ݔ,ݑ)is
asymptotically linear with respect to ݑat origin andinfinity.
For this purpose, we use the following global branch
theorem established in [10].
Theorem 1.1:Let ܺand ܻbe real Banach spaces, (ܺ,ܻ)
be thespace of bounded linear operators from ܺinto ܻwith its
usualnorm, ܲ:Թ×ܺ՜Թdenote theprojection ܲ(ߣ,ݑ),
{ܮאࣜ(ܺ,ܻ):ܮ ݅ݏ ܽ ܨݎ݄݁݀݋݈݉ ݋݌݁ݎܽݐ݋ݎ ݋݂ ݅݊݀݁ݔ ݖ݁ݎ݋}.
Let ܮאܥ1(ܬ,ࣜ(ܺ,ܻ))where ܬis an open interval and
ܮ(ߣ)אȰ0(ܺ ,ܻ)for allߣאܬ. Suppose that there exists ߣ0אܬ
suchthat dim݇݁ݎܮ(ߣ 0)is odd and
ܮԢ(ߣ0)݇݁ݎܮ(ߣ0)۩ݎ݁݃ܮ(ߣ0) (1.2)
Letܭאܥ(ܺ,ܻ)be such that ܭ:ܺ՜ܻis compact and
||ݑ||ܺ=0. (1.3)
Let ܼ=ܼ׫{(ߣ0,0)}where ܼ={(ߣ,ݑ)אܬ×ܺ:ݑ്
0 ܽ݊݀ ܮ(ߣ)ݑ+ܭ(ݑ)=0}be considered with the metric
inherited from Թ×ܺ ,and let denote the connected
component ofܼcontaining 0,0). Then possesses at least
one of the followingproperties:
(i)is an unbounded subset of Թ×ܺ;
(ii)ҧת[ܬ×{0}]്{(ߣ0,0)}, whereҧis the closure of in
(iii) eitherݏݑ݌ܲࣝ=ݏݑ݌ܬor ݂݅݊ܲࣝ=݂݅݊ܬ .
REMARK 1.1:For ܭאܥ(ܺ,ܻ),the condition (1.3) is
equivalent to theproperties ܭ(0)=0 and ܭ:ܺ՜ܻ is
FrÉchetdifferentiable at zero with ܭԢ(0)=0.
By 1)and 2)we may define a function ݇having the
following properties
Identify applicable sponsor/s here. (sponsors)
Open Journal of Applied Sciences
Supplement2012 world Congress on Engineering and Technology
ht © 2012 SciRes.187
ݏ (1.4)
with ݇(ݔ,ݏ)(ݔ)݄(ݔ )
݈݅݉ ,݇(ݔ,ݏ)=0
݈݅݉ uniformly
inݔאԹܰand 0൑݇(ݔ,ݏ)൑݃(ݔ)݄(ݔ).From the above
notation (1.4),problem(1.1)is equivalent to
prove the asymptotic bifurcation result, first we study the
following formal asymptotic linearization of (1.5):
ߣ>0. (1.6).
A number ߣ>0is said to be an eigenvalue of (1.6)if there
exists ݑאܪ1(Թܰ)\{0}such that
׬[׏ݑ ׏ݒെ݃ (ݔ)ݑݒ+
Թܰߣܸݑݒ ]݀ݔ=0 ݂݋ݎ ݈݈ܽ ݒאܪ1(Թܰ).
For the discussion of equation (1.5), we take advantage ofthe
additional regularity of solutions that follows from
ourassumptions(see Proposition 2.1 in [10]).
Proposition 1.1: (1) Assume that the
conditions 1) 2) 1) 2)hold and ݑאܪ1(Թܰ)satisfies
(1.5),then ݑאܹ2,݌(Թܰ)for all ݌א[2,+λ) andhence
ݑאܥ1(Թܰ)with ݑ(ݔ)=0
݈݅݉ and ׏ݑ (ݔ)=0
݈݅݉ .
(2) If ܸsatisfies 1) 2)and ݒאܪ1(Թܰ)is an
eigenfunction of (1.6), then ݒאܹ2,݌(Թܰ)for all ݌א[2,+λ).
Our first result concerning the linearized equation (1.6) is the
following :
Theorem 1.2: Assume that ܸand ݃satisfies 1) 2)and
(G), then
(i) there exists an unique eigenvalue ߣ=Ȧ(ߙ)of (1.6)
having a positive eigenfunction. FurthermoreȦ(ߙ)>1, and it
is simple in the sense that ker༌(ܣȦ(ߙ))=ݏ݌ܽ݊ {ݑ Ȧ(ߙ)}whereܣߣ
denotes the SchrÖdinger operator ܣߣݑ=െȟݑെ݃(ݔ)ݑ+
ߣܸݑ and ݑȦ(ߙ)>0onԹܰ. All other eigenvalues of (1.6) are
less thanȦ(ߙ )and their eigenfunctions change sign.
(ii)Ȧ(ߙ)is the unique value of ߣin theinterval [ߙ
for which 0is theinfimum of the spectrum of the
SchrÖdinger operatorܣߣ.
Now we can state our main result concerning the
nonlinear problem (1.1).
Theorem 1.3: Let the conditions 1) 2) 3)(ܸ1) 2)
hold and fix
2,+λ). Thenthere exist two
connected subsets σ+and σof Թ×ܹ2,݌(Թܰ),
whoseelements (ߣ,ݑ) are, respectively, positive and
negativesolutions ofproblem (1.1), such that ݂݅݊{ߣ:(ߣ,ݑ)א
ܸ(λ)andݑ݌{ߣ:(ߣ,ݑ)אσ±}=Ȧ(ߙ), where Ȧ(ߙ)is given
by Theorem 1.2. Furthermore, σ±is bounded away from
the line of trivial solutions of Թ×{0}and if {(ߣ݊݊)}ؿ
σ±with ߣ݊݊
՜ߣ> ߙ
ܸ(λ),then ݊(ݔ)|݊
݉ܽݔ if and only
if ߣ=Ȧ(ߙ).
In this section, we prove Theorem 1.2. It follows from
Proposition 1.1that anyeigenfunctionݑof equation (1.6)
belongs toܥ(Թܰ)תܪ2(Թܰ), and this leads us to introduce a
SchrÖdinger operator having ݑas an eigenfunction.Define
Then ܣߣis a self-adjoint operator in ܮ2(Թܰ)with spectrum
ߪ(ܣߣ)and essential spectrumߪ݁(ܣߣ)=[ߣܸ (λ)
ߙ,+λ).Furthermore, settingσ(ߣ)=݂݅݊ߪ (ܣߣ),we have
σ(ߣ)=inf༌ߣ(ݑ):ݑאܪ1(Թܰ)ܽ݊݀ ׬ݑ2݀ݔ=1}
Lemma 2.1: Suppose that ܸsatisfies1) (ܸ2)andȞ<ߙ,
thenσ(1)<0. Moreover, there exists ߣ1>1such thatσ(ߣ)<
0for all ߣא(െλ,ߣ1] .
proof :SinceȞ<ߙ, there exists ݑאܪ1(Թܰ)\{0}such
that׬(|׏ݑ| 2+ܸ(ݔ)ݑ2)݀ݔ<ߙ׬ݑ2݀ݔ
Թܰ.This means that ܽ1(ݑ)<0and σ(1)<0.
Hence there existsݑ1אܪ1(Թܰ) ݓ݅ݐ݄ ׬ݑ12݀ݔ=1
Թܰsuch that
ܽ1(ݑ1)<0. By thedefinition ofܽߣ, we have
Թܰfor all ߣאԹ. (2.1)
By 1) (ܸ2), we have there exists ܥ>0such that ܸ(ݔ)൑
ܥfor all ݔאԹܰ. From (2.1), we haveܽߣ(ݑ1)൑ܽ1(ݑ1)+
ܥ(ߣെ1).Therefore choosingߣ1=1+െܽ1(ݑ1)
2<0and σ(ߣ1)<0. Since σ(ߣ)is
increasing with respectto ߣאԹ, we have σ(ߣ)<0for all
Lemma 2.2:Let ܸsatisfies 1) (ܸ2).ForȞ<ߙ<ܸ(λ), if
we set ܵ׷={ߣ൒ ߙ
ܸ(λ)(ߣ)<0}andȦ(ߙ)=sup༌{ߣ :ߣאܵ} ,
Proof :From Lemma 2.1, we haveȦ(ߙ)>1. It is clear that ܵ
is an interval since σ(ߣ)is increasing in ߣ. Therefore,
ifȦ(ߙ)=+λ, we have ܵ=[ߙ
ܸ(λ),+λ)and for any integer
݊൒ ߙ
ܸ(λ), there existsݑ݊אܪ1(Թܰ)with׬ݑ݊
Թܰ<0. (2.2)
By condition 1)we see that (2.2) is impossible when݊൒ߙ
Lemma 2.3:Assume that 1) (ܸ2)hold andȞ< ߙ<ܸ(λ).
Then ߣא[ߙ
ܸ(λ),+λ)and σ(ߣ)=0if and only if ߣ=Ȧ(ߙ)ˈ
whe r eȦ(ߙ)is given by Lemma 2.2.
188 Cop
ht © 2012 SciRes.
Proof :SinceȞ<ߙ<ܸ(λ)and σ(ߣ)is increasing in ߣ, it
follows from Lemma2.1 that σቀߙ
ܸ(λ)ቁ൑σ(1)<0. If
ܸ(λ),+λቁand σ(ߣ)=0,then ߣ> ߙ
ܸ(λ). Nowwe have
σ(ߣ)=݂݅݊ߪ(ܣߣ)=0and ݂݅݊ߪ݁(ܣߣ)=ߣܸ(λ)െߙ>
ܸ(λ)ܸ(λ)െߙ=0. Hence 0 is aneigenvalue of ܣߣand there
exists ݑߣאܥ(Թܰ)תܪ2(Թܰ)such that ݇݁ݎܣߣ=ݏ݌ܽ݊ {ݑ ߣ}
andݑߣ>0onԹܰ(see [9],Theorem 3.20] for example). We
mayassume that ׬ݑߣ2݀ݔ=1
Թܰsuch thatܽߣ(ݑߣ)=0. Then
from the definition of ܽߣwe have for any ߝ>0
and thismeans that ߣെߝאܵfor any ߝ>0. Thereforeߣ
=supS=Ȧ(ߙ ).
Conversely, ifߣ=Ȧ(ߙ), by Lemma 2.2 wehaveȦ(ߙ)>
1> ߙ
ܸ(λ). Hence it issufficient to proveȦ(ߙ)בܵ׫ܶ,whereܶ
׷={ߣ൒ ߙ
ܸ(λ): σ(ߣ)>0}. Indeed, if Ȧ(ߙ)אܵ, then
σ൫ Ȧ(ߙ)൯<0. By the proof of Lemma 2.1 we see that there
existsߣ2(ߙ)such that σ(ߣ)<0for all ߣא(െλ,ߣ2]. This
contradicts the definition of Ȧ(ߙ). On the other hand, ifȦ(ߙ)א
ܶ, then σ൫ Ȧ(ߙ)൯>0. By the definition of ܽߣand ܸ(ݔ)൑ܥ
for allݔאԹܰ, we see that for anyߝ>0and ݑאܪ1(Թܰ)
wit h ׬ݑ2݀ݔ=1
Թܰ൒σ൫ Ȧ(ߙ)൯െߝܥ
Therefore we can chooseߝ=σ൫ Ȧ(ߙ)
that ܽȦ(ߙ)െߝ(ݑߣ)σ൫ Ȧ(ߙ)
2>0for allݑאܪ1(Թܰ)with
Թܰ. This means that σ( Ȧ(ߙ)െߝ)>0and also
contradicts thedefinition ofȦ(ߙ).
Proof of Theorem 1.2(i) From Lemma2.2 and 2.3 we know
thatȦ(ߙ)>1and σ൫ Ȧ(ߙ)൯=݂݅݊ ߪ ൫ܣ Ȧ(ߙ)൯=0. Since
ߙ<ܸ(λ), we have ݂݅݊ ߪ݁൫ܣȦ(ߙ)൯=Ȧ(ߙ)ܸ(λ)െߙ>0.
Hence0 is aneigenvalue of ܣ Ȧ(ߙ)and there existsݑȦ(ߙ)א
ܥ(Թܰ)תܪ2(Թܰ)such that ݇݁ݎܣȦ(ߙ)=ݏ݌ܽ݊{ݑȦ(ߙ)}and
ݑȦ(ߙ)>0on Թܰ. Suppose now thatȦ1്Ȧ(ߙ)is another
eigenvalue of(1.6) with eigenfunctionݑ1אܪ1(Թܰ). Then 0is
an eigenvalue of ܣȦ1and σ(Ȧ1)=݂݅݊ߪ൫ܣȦ1൯൑0. It follows
thatȦ1൑ Ȧ(ߙ). Otherwise, if Ȧ1> Ȧ(ߙ)andσ(Ȧ1)൑0,we
divide two cases to deduce thecontradiction. One hand, if
Ȧ1> Ȧ(ߙ)and σ(Ȧ1)=0, it contradicts Lemma 2.3. On
theother hand, if Ȧ1> Ȧ(ߙ)and σ(Ȧ1)<0, by the proof of
Lemma 2.1 we see thatthere exists ߣ31such that σ(ߣ)<
0for all ߣא(െλ,ߣ3]. This contradicts thedefinition ofȦ(ߙ).
ThereforeȦ(ߙ)isthe largest eigenvalue of (1.6). Furthermore,
integratingby parts, we have
For Ȧ1(ߙ)and ܸ(ݔ)ݑȦ(ߙ)>0on Թܰ, it follows that ݑ1
changes sign.
(ii) This follows from Lemma 2.3.
Let Pא[2,+λ)ת(N
2,+λ)be fixed and we set
ܺ=ܹ2,݌(Թܰ)with ||ή||=||ή|| ܹ2,݌൫Թܰ, and
For ݇defined in (1.4), it can be shown that (see [10],
Lemma B.1] )|݇(ݔ,ݑ)ݑ|݌
uniformly in ݔאԹܰ. Hence in order to make use ofTheorem
1.1 we need to introduce the following truncatedproblem
߰݊(ݔ)=൜1, ݂݅ |ݔ|൑݊,
0, ݂݅ |ݔ|>݊.
Define ܮ(ߣ):ܺ՜ܻby
ܮ(ߣ)ݑ=െοݑെ݃ (ݔ)ݑ+ߣܸ(ݔ)ݑ (3.2)
Using the inversion, ݑհݒ= ݑ
|ݑ|2, problem (3.1) is equivalent
to ܮ(ߣ)ݒ+ܭ݊(ݒ)=0 (3.3)
|ݒ|2൱ݒ, ݂݋ݎ ݒאܺ\{0}
0, ݂݋ݎ ݒؠ0
In the sequel we show that Theorem 1.1 is applicable tothe
inverted truncated problem (3.3). First, it follows
fromTheorem 4.3 of [6]thatܮ(ߣ)אȰ0(ܺ,ܻ)for all>ߙ
where Ȱ0(ܺ,ܻ)is defined in Theorem 1.1 and
Y= ݇݁ݎܮ(ߣ)۩ݎ݃݁ܮ (ߣ)for allߣ> ߙ
ܸ(λ) (3.4)
From (3.4) and Theorem1.2 we can prove (1.2) holds with
ߣ0=Ȧ(ߙ)(see the proof of Theorem 4.2in [10]). By Remark
1.1 the following lemma(Lemma 3.2in [10]) can verify
condition (1.3) for ܭ݊defined in(3.3) .
Lemma 3.1: For all݊אܰ,ܭ݊אܥ(ܺ,ܻ)תܥ1(ܺ\{0},ܻ),
ܭ݊:ܺ՜ܻis compact and it is FrÉchetdifferentiable at 0
Now we can apply Theorem 1.1for ܮ(ߣ)defined by (3.2) on
the interval ܬ=(ߙ
ܸ(λ),+λ)and at the point ߣ0=Ȧ(ߙ). Set
ܸ(λ),+λ)×ܺ:ݒis a nontrivial solution of
ht © 2012 SciRes.189
(3.3)}. Let ݊denote the connected component of݊׫
{( Ȧ(ߙ),0)}containing ( Ȧ(ߙ),0). In order to get some initial
informationabout ݊from Theorem 1.1, we establish
someestimates about the solutions of (3.2) that will help us
toobtain more precise information about ݊.
Lemma 3.2: Let the conditions 1) 2)hold and ߙ=݈+
ߤSuppose thatȞ<ߙ<ܸ(λ)and 0൑ߤ< ݈ܸ0
ܸ(λ)െܸ0. Then, there
exists ܶ>0, such that for anyߚ> ߙ
ܸ(λ)there is ܰߚא
Գsatisfying ||V||൑ܶfor all (ߣ,ݒ)אࣴ݊with ߣ൒ߚand
Proof :Firstwe define the bound ܶ. Since0൑ߤ< ݈ܸ0
ߙ=݈+ߤ, we candeduce that ߤ<ߙܸ0
ܸ(λ)and choose ߟ=
ܸ(λ)െߤቁ>0.By (1.4) and 1)we know ݇(ݔ,0)and
there exists ߜ1such that ݇(ݔ,ݏ)൒݈െߟ=1
2݈>0for all |ݏ |൑ߜ1. By Sobolevembeding there is ܥ1
suchthat |ݑ |λ൑ܥ1|ݑ|.Setܶ=ܥ1
ߜ1.Suppose that ݒאܺand
|ݒ|ห൒ܶ. Then, for all ݔאԹܰ, we have(ݔ)|
T1. Consequently, ݇൬ݔ,(ݔ)|
|ݒ|2൰൒݈െߟ. For anyߚ> ߙ
we can choose ߜߚ(λ)ߙ
ߚ>0. By 2), there existsܰߚא
Գ,such that ܸ(ݔ)(λ)െߜߚfor all |ݔ |൒ܰߚ. Hence
ߣV(x) ൒ߚܸ(λ)െߜߚ (3.5)
for all ߣ൒ߚand |ݔ|൒ܰߚ. On the otherhand, for ߣ൒ߚ,
݊൒ܰߚ,|ݔ|൑ܰߚ, and |ݒ|ห൒ܶ, we have
|ݒ|2൰=݇൬ݔ,|ݒ |
andെߙ+ߣܸ(ݔ)݊(ݒ)൒െߙ+ ߙ
Combining the above inequality and (3.5), we get that
for(ߣ,ݒ)אࣴ݊with ߣ൒ߚand ݊൒ܰߚ,if|ݒ|ห൒ܶ, then
ݒ(ݔ)οݒ(ݔ)2(ݔ) [െߙ+ߣܸ(ݔ)݊݇൭ݔ,|ݒ|
for all ݔאԹܰ. The maximum principle now leads toa
Lemma 3.3: Fix ߚ> ߙ
ܸ(λ). Then, for any ߝא(0,ܸ(λ)ߙ
there exists ܥߝ>0such that(ݔ)|൑|ݑ|λ݁ߦ(|ݔ|െܥߝ)for all
ݔאԹܰ,where ߦ=ߚ(ܸ(λ)െߝ)െߙ>0for all
(ߣ,ݑ )א
[ߚ ,+λ)with ݑܮ (ߣ)ݑ൑0onԹܰ.
Proof :Since ܸ(ݔ)=ܸ(λ)
݈݅݉ , for any ߝא(0,ܸ(λ)
ߚ), there exists ܥߝ>0suchthat ܸ(ݔ)൒ܸ(λ)െߝfor all
|ݔ |൒ܥߝ. Set ݍ(ݔ)=|ݑ|λ݁ߦ(|ݔ|െܥߝ)െݑ(ݔ) and ȳߝ=
{ݔאԹܰ: |ݔ|ߝ ܽ݊݀ ݍ(ݔ)<0}.For all ݔאȳߝ, we have
since ൒ߚ. By direct calculations, we have forݔאȳߝ
|ݑ|λ݁ߦܥߝߦ݁ߦ|ݔ|െߦݑ =ߦݍ(ݔ)<0.
Since ݍ(ݔ)՜0as |ݔ|՜+λand ݍ(ݔ)൒0for |ݔ|ߝ, we
have ݍ(ݔ)൒0for ݔאȳߝ. If ȳߝ്׎, the weakmaximum
principle (Theorem 8.1 in[4]) implies that ݍ(ݔ)൒0in ȳߝ, a
contradiction. Thus we see that(ݔ)|൑|ݑ|λ݁ߦ(|ݔ|െܥߝ)for
all|ݔ|൒ܥߝ.Replacing ݑby െݑ , we get the above inequality for
(ݔ)|.Hence, we complete the proof.
Lemma 3.4:For each ݊אԳ, there exists an open
neighborhood ܷof(ߙ),0)אԹ×ܺ, such that ݑ2>0on Թܰ
for all (ߣ,ݒ)אܷתࣴ݊.
Proof :By contradiction, suppose that there exists asequence
{(ߣ݅݅)}ؿࣴ݊such thatߣ݅݅
՜Ȧ(ߙ)and ||ݑ||݅݅
continuous functionݑ݅has at least one zero in Թܰ. We prove
this leadsto a contradiction.
Setting ݖ݅=ݑ݅
||ݑ||݅, by the definition of݊, we
have ܮ(ߣ݅)ݖ݅+ܭ݊݅)
||ݑ||݅=0on Թܰ.Since ܸ(ݔ)אܮλܰ), by
Lemma 3.1, wefind that
ܮ൫Ȧ(ߙ)൯ݖ݅=(Ȧ(ߙ)െߣ݅)ܸ (ݔ)ݖ݅ܭ݊݅)
՜0in ܻ.
On the other hand, by passing to a subsequence, we may
supposethat ݖ݅֊ݖweakly in ܺ. SinceȦ(ߙ)>1>ߙ
ܸ(λ), we
have ܮ൫Ȧ(ߙ)൯אȰ0(ܺ,ܻ). By Lemma 3.5 of [10], we know
that ݖ݅݅
՜ݖ strongly inܺ. This means
that |ݖ|ห=1and ܮ൫Ȧ(ߙ)൯ݖ݅=0. By Lemma3.3, wehave
ݖאܪ1(Թܰ). According to Theorem 1.2(i), we may as well
suppose thatZ>0on Թܰ. Sinceߣ݅݅
՜Ȧ(ߙ)and Ȧ(ߙ)>1,
2>0, there exists ݅0אԳsuch that ߣ݅൒1+
ߝfor all ݅൒݅0. By 2)and ߙ<ܸ(λ), thereexists ܴ൒݊such
that ܸ(ݔ)൒ߙfor all|ݔ |൒ܴ. Hence, for all ݅൒݅0and |ݔ|൒ܴ,
we haveെߙ+ߣܸ݅(ݔ)൒െߙ+(1+ߝ)ߙ>0.Sinceܴ൒݊, we
have for |ݔ|,߰݊(ݔ)=0and
0=ܮ(ߣ݅)ݖ݅=െοݖ݅+൫െߙ+ߣܸ݅(ݔ)൯ݖ݅ (3.6)
190 Cop
ht © 2012 SciRes.
Recalling that ܺؿܥ(Թܰ), we have ߜ= ݖ(ݔ)>0
݂݅݊ .
Since ݖ݅݅
՜ݖ strongly in ܺ, we have that
MAXand there exists ݅1൒݅0such that
2ߜfor all |ݔ|൑ܴand݅൒݅1. Since ݖ݅(ݔ)1
|ݔ|and ݖ݅(ݔ)=0
݈݅݉ , we get ݖ݅(ݔ)൒0on ߲ܤܴܿ.
Now we can apply the strong maximumprinciple (Theorem
8.19 in [4]) to equation (3.6) on theregion ܤܴܿand deduce
thatݖ݅>0in ܤܴܿfor all݅൒݅1. This contradicts that each ݑ݅
has at least onezero in Թܰ.
Now we can give more information about ݊.
Theorem 3.1: Let ݊denote the connected component
of݊׫{(Ȧ(ߙ),0)}containing the point(Ȧ(ߙ),0). Then:
(i) ݑ2>0on Թܰand ߣ൑Ȧ(ߙ)for all (ߣ,ݑ)א
(ii) for any ߚ> ߙ
ܸ(λ), there exist ܶ>0and ܰߚאԳsuch that,
for all ݊൒ܰߚ,
݂݅݊ܲ ࣝ݊׷= ݂݅݊{ ߣ: (ߣ,ݑ )אࣝ݊}<ߚand||ݑ||൑ܶ
for all (ߣ,ݒ )אࣝ݊with ߣ൒ߚ.
Proof : The first step is to show that if (ߣ,ݑ)א
݊\{(Ȧ(ߙ),0)}, we haveߣ൑Ȧ(ߙ). Since (ߣ,ݑ )solves
equation (3.3), we have
|ݑ|2൰ݑ=0 (3.7)
By Lemma 3.3, we know ݑאܪ1(Թܰ). Hence
byTheorem1.2 (ii) and (3.7) ,
0=݂݅݊ቐන[|׏ݒ |2െߙݒ2(ߙ)ܸݒ2]݀ݔ:ݒ
2െߙݑ2(ߙ)ܸݑ2݊(ݔ)݇൭ݔ, |ݑ|
= (ߙ)െߣ)׬ܸ(ݔ)ݑ2݀ݔ
Since ׬ܸ(ݔ)ݑ2݀ݔ൒ܸ0
Թܰ, we see that ߣ൑
The next step is to show that if (ߣ,ݑ)אࣝ݊\{(Ȧ(ߙ),0)},
thenݑ2>0onԹܰ. Set
࣫={(ߣ,ݑ)אࣝ݊: ݑ2>0 onԹܰ}ת{(Ȧ(ߙ),0)}.
We prove that is both anopen and closed subset of ݊, then
by theconnectedness of݊we have࣫=ࣝ݊.
First we prove that is open in ݊. Given(ߣ,ݑ)א࣫, we
show that there exists an openneighborhood ܷof (ߣ,ݑ)in
Թܰ×ܺsuchthat ܷתࣝ݊ؿ࣫. For (ߣ,ݑ)=(Ȧ(ߙ),0)this is
established in Lemma 3.4 . For(ߣ,ݑ)אࣝ݊with ݑ2>0 onԹܰ,
wehave that ݑdoes not change sign sinceܺؿܥ(Թܰ).We
suppose that ݑ>0on Թܰ, the case ݑ<0beingsimilar. By
2)and ߣ> ߙ
ܸ(λ), there exist ݎ>0andܴ൒݊such that for all
ߟwith|ߣെߟ |൑ݎ,െߙ+ߟܸ (ݔ)>0for all |ݔ|൒ܴ.Let
ߜ= ݑ(ݔ)
݂݅݊ . Then, ߜ>0andthere exists an open
neighborhood ܷof (ߣ,ݑ)inԹܰ×ܺsuch that |ߣെߟ|൑ݎ
and ݒ(ݔ)൒ߜ2
݂݅݊ for all (ߟ,ݒ)אܷ. Sinceܴ൒݊, for
we have ߰݊(ݔ)=0andfor (ߟ,ݒ)אܷתࣴ݊,ܮ(ߟ)ݒ=െοݒ+
൫െߙ+ߟܸ (ݔ)൯ݒ=0.As the proof of Lemma 3.4, the
maximum principle impliesthat ݒ(ݔ)>0for |ݔ|and then
ݒ>0onԹܰ. Henceܷתࣝ݊ؿ࣫and is open.
Now we show that is closed in ݊. Supposethat (ߣ ,ݑ)א
݊and there exists a sequence{(ߣ݅݅)}ؿ࣫such thatߣ݅݅
and ||ݑ݅െݑ||݅
՜0. Ifݑ=0, we must have ߣ=Ȧ(ߙ)since
݊ת[Թ×{0}]={(Ȧ(ߙ),0)}and so (ߣ ,ݑ )א࣫. If ݑ്0,
passing to a subsequence, we may as wellsuppose that ݑ݅>0
on Թܰfor all݅אԳ. Itfollows that ݑ൒0on Թܰandെοݑ+
ܿ+ݑ=ܿݑ൒0 on Թܰ,where ܿ(ݔ)=െߙ+ߣܸ(ݔ)+
||ݑ||2. Bythe strong maximum principle we have
ݑ>0on Թܰ. Hence(ߣ,ݑ)א࣫and is closed in݊.
Now we know that࣫=ࣝ݊. We claim that thismeans that
case (ii) in Theorem 1.1 cannot occur. Indeed, if݊has the
property (ii), there exist ߣאܬ\{Ȧ(ߙ)}and a
sequence{(ߣ݅݅)}ؿࣝ݊such thatߣ݅݅
՜ߣand ||ݑ݅||݅
||ݑ݅||and arguing as in the proof of Lemma 3.4, we
may assume thatݖ݅݅
՜ݖ strongly in ܺand ܮ(ߣ)ݖ=0with
|ݖ|ห=1. By Lemma 3.3, we know ݖאܪ1(Թܰ). It follows
fromTheorem1.2 (i) that ߣ<Ȧ(ߙ)and ݖchanges sign on Թܰ.
On the other hand, since ࣫=ࣝ݊, the sequence ݖ݅can be
chosen sothat ݖ݅>0on Թܰ. Then we have ݖ൒0on Թܰand
this contradicts the earlier conclusion.
For any ߚ>ߙ
ܸ(λ), by Lemma 3.2 there exists ܰߚאԳsuch
that|ݒ|ห൑ܶfor all (ߣ ,ݒ )אࣝ݊with ߣ൒ߚand ݊൒ܰߚ. Thus,
if ݊൒ܰߚand݂݅݊ܲࣝ݊൒ߚ, we deduce that ݊is boundedin
Թ×ܺ. Since we have shown that ݊has not the property (ii) in
Theorem 1.1, we must have݊satisfying ݂݅݊ܲࣝ݊=݂݅݊ܬ=
ܸ(λ). This contradicts our earlier assumption݂݅݊ܲࣝ݊൒ߚ>
ܸ(λ)and we complete the proof.
In this section, by using the globalbifurcation result for the
inverted truncated problem (3.3), we first prove the bifurcation
result for the following invertedproblem
ht © 2012 SciRes.191
ܮ(ߣ)ݒ+ܭ(ݒ)=0, (4.1)
|ݒ|2൰ݒ, ݂݋ݎ ݒאܺ\{0}
0, ݂݋ݎ ݒؠ0.
Set ࣴ={(ߣ,ݒ)א(ߙ
ܸ(λ),+λ)×ܺ: ݒis a nontrivial
solution of (4.1)}and={(ߣ,ݒ)אࣴ:ݒ2>0 ݋݊ Թܰ}.
Lemma 4.1:Let {(ߣ݊݊)}ؿࣴ
be a sequencesuch that
՜ߣ> ߙ
՜0. Then ߣ=Ȧ(ߙ).
Proof :First we show that
|ܭቆ |ݒ݊|
՜0 (4.2)
for all ݌א(1,+λ). Since(ߣ݊݊)solves(4.1), we have
ݒ݊ܮ(ߣ݊)ݒ݊൑0on Թܰ. From ߣ݊݊
՜ߣ> ߙ
ܸ(λ), we may choose
ߚ= ߙ
2ቀߣെ ߙ
ܸ(λ)ቁ> ߙ
ܸ(λ)such thatߣ݊൒ߚfor ݊large. It
follows from Lemma 3.3 that there exist ܮ>0and ߛ>0
which areindependent of ݊such that
݊(ݔ)|൑ܮ|ݒ݊|λ݁െߛ|ݔ|for allݔאԹܰ. (4.3)
By the Sobolev embedding, there is ܥ>0which is
independent of ݊such that ݊|λ൑ܥห|ݒ݊|. Thus, we have
|ܭቆ |ݒ݊|
|ݒ݊|൑ܮܥ݁െߛ|ݔ|for allݔאԹܰ. (4 .4)
Conseuqently, for every fixed ݔאԹܰ,
|ܭቆ |ݒ݊|
՜0 (4.5)
Combining (4.3) and (4.5), it follows from the
dominatedconvergence that (4.2) holds. Let ߱݊=ݒ݊
||ݒ݊||. Since
ݒ݊2>0on Թܰ, passing to a subsequence, we mayassume that
߱݊>0 on Թܰfor all ݊אԳand for some ߱אܺwith ߱൒0
on Թܰ,߱݊֊߱weakly in ܺ. By (4.2) wededuce that
՜0 ݅݊ ܻ
Since ߣ>ߙ
ܸ(λ), we have ܮ(ߣ)אȰ0(ܺ,ܻ). By Lemma 3.5 of
[1] we know that ߱݊݊
՜߱strongly in ܺ. So ܮ(ߣ)߱=0and
|߱|ห=1. By Lemma 3.3 we have ߱אܪ1(Թܰ). This means
that ߣis an eigenvalue of equation (1.6) and its
correspondingeigenfunction߱does not change sign. Thus it
follows from Theorem 1.2 (i) thatߣ=Ȧ(ߙ).
Theorem 4.1: Let denote the connected component
of׫{( Ȧ(ߙ),0)}containing{( Ȧ(ߙ),0)}. The following hold.
(i) is bounded with ݂݅݊ܲࣝ= ߙ
ܸ(λ)and ݏݑ݌ܲࣝ= Ȧ(ߙ).
(ii) If
{(ߣ݊݊)}ؿࣝwith ߣ݊݊՜+λ
݈݅݉ (ߙ),then ||ݒ݊||
݈݅݉ =
Proof : (i) First we show that if (ߣ,ݒ)אࣴ, then ߣ<Ȧ(ߙ).
Sinceݒ്0solves equation (4.1), by Lemma3.3, we knowݒא
ܪ1(Թܰ)and then we deduce thatݑ= ݒ
|ݒ|2solves equation (1.5).
By Theorem 1.2 (ii) we have
0=݂݅݊ቐන[|׏߱ |2െߙ߱2+ߣܸ߱2]݀ݔ:߱
Թܰאܪ1(Թܰ) ܽ݊݀ න߱
Թܰ. We claim
that ׬݇(ݔ,ݑ)ݑ2݀ݔ
Թܰ>0. Indeed, since݇(ݔ,ݏ)=݈>0
uniformly in ݔאԹܰ,there exists ߜ>0such that ݇(ݔ,ݏ)݈2
for all |ݏ|൑ߜ and ݔאԹܰ. ByProposition 1.1 we have
݈݅݉ =0and there exists ܴ>0such that (ݔ)|൑
ߜfor|ݔ|൒ܴ. Since (ߣ,ݒ)אࣴ, we haveݒ2>0on Թܰand
ݑ2>0on Թܰ. Thereforewe deduce that
|ݔ|൒ܴ ݈2׬ݑ2݀ݔ
|ݔ|൒ܴ .
It follows thatߣ<Ȧ(ߙ). Hence{ߣ: (ߣ,ݒ)אࣝ\{(Ȧ(ߙ),0)}ؿ
ܸ(λ)(ߙ))and ݏݑ݌ܲࣝ= Ȧ(ߙ).
Secondly we show that |ݒ|ห൑ܶfor all (ߣ,ݒ)אࣴ, where
ܶ>0is defined in Lemma 3.2. Bycontradiction, if (ߣ,ݒ)א
and |ݒ|ห൒ܶ, similar to the proof of Lemma 3.2, we have
for allݔאԹܰ,ܭ൬ݒ(ݔ)
|ݒ|2൰൒݈െߟ, whereߟ=1
The maximum principle leads to a contradiction. Therefore we
seethat is bounded in Թ×ܺand sois .
192 Cop
ht © 2012 SciRes.
The next step is to prove that ݂݅݊ܲࣝ= ߙ
ܸ(λ). By
contradiction, supposethatߩ׷=݂݅݊ܲࣝ> ߙ
ܸ(λ). For anyߩҧא
ܸ(λ),ߩ), we setܣ={(ߣ,ݒ)א࣫: ߣ൒ߩҧ },ܣ1={( Ȧ(ߙ),0)},
ܣ2={(ߣ,ݒ)א࣫: ߣ=ߩҧ}, where ࣫=ࣴ
׫{( Ȧ(ߙ),0)}. We
claim that ܣis a compact subset of Թ×ܺ.
Indeed, let{(ߣ݊݊)}ؿܣbe an infinitely sequence. Passing toa
subsequence, we may suppose that ݒ݊>0on Թܰforall
՜ߣא[ߩҧ(ߙ)],ݒ݊֊ݒweakly in ܺ,||ݒ݊||݊
0.To prove this claim, it is sufficient to prove ݒ݊݊
in ܺwith(ߣ ,ݒ )אܣ .If ߬൒0, then ݒ݊݊
՜ݒstrongly in ܺ. By
Lemma 4.1, we must have (ߣ,ݒ)=( Ȧ(ߙ),0)אܣ. If ്߬0,
similar to the proof ofLemma 4.1, we can deduce that||ݒ݊
՜0and hence (ߣ ,ݒ )אࣴ. The strong maximum implies
that (ߣ,ݒ )אܣ . By a result of Whyburn [11] (see Lemma C.2
in [10]), we can prove that there exists a connected subset ܣ0of
ܣsuchthatܣ0תܣ1്׎ ܽ݊݀ ܣ0תܣ2്׎ .It follows that
݂݅݊ܲ ܣ 0ҧ. But sinceܣ0ؿࣝ, we have ݂݅݊ܲࣝ൑ߩҧ,
a contradiction.
(ii) By contradiction, suppose that there exists a
sequence{(ߣ݊݊)}ؿࣝ\{( Ȧ(ߙ),0)}such that ߣ݊݊
՜ Ȧ(ߙ)and
|ݒ݊|ห൒ߜ>0for all ݊אԳ. Bythe proof of part (i) we
have|ݒ݊|ห൑ܶ.Passing to asubsequence, we may assume that
ݒ݊֊ݒweaklyin ܺ. Similar to the proof of Lemma 4.1, we
candeduce that ||ݒ݊െݒ||݊
՜0and ( Ȧ(ߙ))אࣴ. Then by the
strong maximum principle wehave( Ȧ(ߙ))אࣴ
. But at
thebeginning of the proof of part (i) we proved that ߣ<Ȧ(ߙ)
for all (ߣ,ݒ )אࣴ. This is a contradiction.
We have established the global properties of a connected
subset of׫{( Ȧ(ߙ),0)}containing {( Ȧ(ߙ),0)}. However, in
order to maintainconnectedness under inversion, we need to
find a connected subset ofhaving similar properties.
Set +={(ߣ,ݒ)אࣴ:ݒ>0 ݋݊ Թܰ}and ={(ߣ,ݒ)א
ࣴ:ݒ<0 ݋݊ Թܰ}.
Corollary 4.1: Let the function ݂be odd. Then there exist
two bounded connectedsubsets 0+and 0of +and ,
respectively, satisfying the followingproperties:
(i) ݂݅݊ܲࣝ0±=ߙ
ܸ(λ)and( Ȧ(ߙ),0)אࣝ0±
(ii) ݏݑ݌ܲ ࣝ0±= Ȧ(ߙ)and0<ห|ݒ|ห൑ܶfor all (ߣ ,ݒ)אࣝ0±,
where ܶ>0isgiven by Lemma 3.2.
Proof : The proof is the same as the one of Corollary 5.3 in
Proof of Theorem1.3
Let ݂ܴand ݂ܮbe the odd functions defined by
݂ܴ(ݏ)=൜݂(ݔ,ݏ), ݂݋ݎ ݏ൒0,
െ݂ (ݔ,െݏ), ݂݋ݎ ݏ<0
and݂ܮ(ݏ)=൜െ݂(ݔ,െݏ), ݂݋ݎ ݏ൒0,
݂(ݔ,ݏ), ݂݋ݎ ݏ<0.
By corollary 4.1there exist two bounded connected
subsets0+and 0of positive or negativesolutions for problem
(1.1) with ݂ܴor݂ܮ(ݏ)respectively. Setting
σ±={൬ߣ, ݒ
|ݒ|2൰: (ߣ,ݒ)אࣝ0±},
it follows that σ±areconnected sets of (ߙ
ܹ2,݌(Թܰ)consisting of, respectively, positive and negative
solutions of (1.1) with ݂݅݊ܲσ±=ߙ
ܸ(λ),ݏݑ݌ܲσ±= Ȧ(ߙ)and
ܶfor all (ߣ,ݒ)אσ±.
Suppose that {(ߣ݊݊)}ؿσ±with ߣ݊݊
՜ߣ> ߙ
݉ܽݔ ݊
՜+λby the Sobolev
embedding. Hence(ߣ݊݊)אࣴ
with ݒ݊=ݑ݊
|ݑ݊|2and |ݒ݊|݊
ByLemma 4.1 we have ߣ=Ȧ(ߙ). On the otherhand ,
if{(ߣ݊݊)}ؿσ±with ߣ݊݊
՜ Ȧ(ߙ), by setting ݒ݊=ݑ݊
know that(ߣ݊݊)אࣝ. Then by Theorem 4.1(ii) we have
՜0. This means that|ݑ݊|݊
՜+λ. We claim that
݉ܽݔ ݊
՜+λ. Otherwise, passing to a subsequence, there
is ܥ>0such that݊|
݉ܽݔ ൑ܥfor all݊אԳ. Since(ߣ݊݊)is
a solution of problem (1.1) we have
ܮ(ߣ݊)ݑ݊(ݔ,ݑ݊)ݑ݊=0 (4. 6)
By Lemma 3.3, we know that ݑ݊isbounded inܻ. Therefore,
{(െο+1)ݑ݊}is bounded in ܻby (4.6). Sinceെο+1:ܺ՜ܻ
isan isomorphism, this implies that ݊}is bounded in ܺ, a
[1]T.Bartsch,A.Pankov and Z.Q.Wang, Nonlinear Sch¸dinger equations
with steep potential well, Commun. Contemp. Math., 3(2001),549-569.
[2]Y.Ding and K.Tanaka, Multiplicity of positive solutions of a nonlinear
Sch¸dinger equation, Manuscripta Math., 112(2003),109-135.
[3]D.G.DE Figueiredo and Y.Ding, Solutions of a non-linear Schr¸dinger
equation, Discrete Contin. Dynam. Systems, 8(2002),563-584.
[4]D.Gilbarg and N.S.Trudinger, it Elliptic Partial Diffential Equations of
Second Order, Second edition, Springer-Verlag,Berlin, 1983.
[5]F.A.Van Heerden and Z.Q.Wang, Schr¸dinger tyoe equations with
asymptotically linear nonlinearities, it Differential Integral Equations,
[6]L.Jeanjean, M.Lucia and C.A.Stuart, Branches of solutions to semilinear
ellptic equations on Թܰ, Math. Z., 230 (1999), 79-105.
[7]L.Jeanjean and K.Tanaka, A positive solution for an asymptotically
linear elliptic problem onԹܰautonomous at infinity, WSAIM Control
Optim. Calc. Var., 7(2002), 597-614.
[8]Z. Liu and Z.Q. Wang, Existence of a positive solution of an elliptic
equation onԹܰ, Proc. Roy. Soc. Edinburgh, 134 A (2004), 191-200.
ht © 2012 SciRes.193
[9]C.A.Stuart, An introduction to elliptic equations on Թܰ, in Nonlinear
Functional Analysis and Applications toDifferential Equations, editors
A.Ambrosetti, K.C. Chang, I.Ekeland, World Scientific, Singapore,1998.
[10]C.A.Stuart and Huansong Zhou, Global branch of solutios for nonlinear
Schr¸dinger equations with deepening potential well, Proc.London
Math.Soc., 92 (2006) 655-681.
[11]G.T. Whyburn, Topological Analysis, Princeton University Press,
Preceton 1958.
194 Cop
ht © 2012 SciRes.