R E S E A R C H
Open Access
The existence and multiplicity of positive
solutions for a class of nonlocal elliptic
problems
Baoqiang Yan
*and Tianfu Ma
*Correspondence: [email protected]
School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, P.R. China
Abstract
In this paper, we prove the existence of a solution between a well-ordered
subsolution and supersolution of a class of nonlocal elliptic problems and give some degree information. Using the method and bifurcation theory, we present the existence and multiplicity of positive solutions for the nonlocal problems with the changes of the parameter.
MSC: 35J60; 35J75; 47H10
Keywords: sub-supersolution method; bifurcation theory; nonlocal elliptic equations; degree; existence; multiplicity
1 Introduction
In this paper, we consider the following problem:
⎧ ⎪ ⎨ ⎪ ⎩
–a(|u|γdx)u=f
λ(x,u), xin,
u> , xin, u= , xon∂,
(.)
where⊆RNis a smooth bounded domain,γ ∈(, +∞), anda: [, +∞)→(, +∞) is a continuous function with
inf
t∈[,+∞)a(t)≥a=a() > .
Chipot and Lovat [] considered the following model problem:
⎧ ⎪ ⎨ ⎪ ⎩
ut–a(u(z,t)dz)u=f in×(,T), u(x,t) = on×(,T),
u(x, ) =u(x) on.
Hereis a bounded open subset in RN,N≥, with smooth boundary, andT is an arbitrary time. The diffusion coefficientais a function fromRinto (, +∞), which depends on the entire population in the domain rather than on the local density, andudescribes
the density of a population subject to spreading. If γ = , then we get the well-known Carrier equation. The fact that (.) appears in some applied mathematics attracts a lot of attention. With the aid of the Krasnoselskii fixed point theorem and the Schaefer fixed point theorem, by the monotonicity offλ, Corrêa [] considered the existence of positive
solutions of (.) forγ ≥. By establishing a comparison principle, Corrêaet al.[] proved the existence of positive solutions to (.) also forγ = . Under the assumption thatA(x,u) (which is generalized from the nonlocal terma(s)) is bounded, there are some results on the existence of positive solutions and the existence ofndistinct solutions; see [, ].
Another nonlocal elliptic equations are the Kirchhoff elliptic problems like
⎧ ⎪ ⎨ ⎪ ⎩
–a(u)u=f
λ(x,u), xin,
u> , xin, u= , xon∂,
(.)
related to nonlinear vibrations of beams, wherea:R→Ris a given function, and · denotes the usual norm inH(). In this case, variational methods are used to consider the
existence of the solutions to (.) because the nonlocal operatoru→a(u)upossesses a variational structure; see [–] and the references therein. Especially, for the Kirchhoff elliptic equations
⎧ ⎪ ⎨ ⎪ ⎩
–a(u)u=λf(x)|u|q–u+g(x)|u|p–u, xin,
u> , xin, u= , xon∂,
(.)
Chenet al.[] examined in detail the number of solutions admitted subject to the vari-ations of parameters embedded in nonlinear terms. For the casea(t)≡, the existence and multiplicity of positive solutions for the elliptic equations has been extensively inves-tigated; see [–]. Especially, Ambrosettiet al.[] studied the equation
⎧ ⎪ ⎨ ⎪ ⎩
–u=λuq+up, xin, u> , xin,
u= , xon∂,
(.)
and established multiple results for differentλ, whereis a bounded domain inRN with <q< <p≤∗(∗= N
N– ifN≥ and ∗= +∞ifN= , ) andλ> .
Naturally, we hope that there are some interesting results for (.) that are similar to those in (.) and (.) in [, ] and references therein. Notice that the methods used in [–, –, , –] are the sub-suppersolution method, theory of topological degree, and the variational method. Unfortunately, the operatoru→a(|u(x)|γdx)uhas no
variational structure. Up to now, the tools to study (.) are a fixed point result with Leray-Schauder condition and the Schaefer fixed point theorem. Very recently, Alves and Covei [] established the sub-supersolution method, which can be used to study the existence of weak solutions for a large class of nonlocal problems.
be-tween well-ordered subsolution and supersolution to guarantee the existence of classical solution to (.) and give a formula to calculate the degree. Section presents the exis-tence and multiplicity of positive solutions to (.)λwhenp> >q> or >p>q> ,
which improves the results in [], where thea(t) is bounded, oris an annular region. In Section , when nonlinearity is linear atu= , by bifurcation theory we discuss the un-bounded connected component for (.)λand present sufficient and necessary conditions
for the existence of positive solutions to (.)λ. In Section , in the case where the
non-linear term is singular atu= , we consider the existence of positive solutions to (.)λ.
In Section , sufficient and necessary conditions of positive solutions to (.) are given to guarantee that positive solutions to (.) are inC[, ] orC[, ] whenN= .
Notation In this paper we use the following notation.
Letu:→Ris continuous, and|u|∞=maxx∈|u(x)|;
C() ={u:→R|u(x)is continuous on}with normu=|u|∞;
C() ={u∈C()|∇u(x)is continuous on}with normu=max{|u|
∞,|∇u|∞}.
2 Sub-supersolution method
Now we consider the general problem
–a(|u|γdx)u=F(x,u), xin,
u= , xon∂, (.)
where⊆RN is a smooth bounded domain,γ∈(, +∞), anda: [, +∞)→(, +∞) is a continuous function with
a=a() > .
Definition . The pair functionsα,β∈C()∩C() are subsolution and supersolution
of (.) if
–α(x)≤a(
|χ(x,u(x))|γdx)F(x,α(x)), xin,∀u∈C
()∩C(),
α|∂≤,
and
–β(x)≥a(
|χ(x,u(x))|γdx)F(x,β(x)), xin,∀u∈C
()∩C(),
β|∂≥,
whereχ(x,u) =α(x) + (u–α(x))+– (u–β(x))+.
Definition . Letu,v∈C(). We say thatu≺vifu(x) <v(x) onandu(x)≤v(x) for
allx∈∂, and ifu(x) =v(x) for somex∈⊆∂, then we write∂u
∂n|x∈∂>
∂v
∂n|x∈.
Remark . S={u∈C() :α≺u≺β}is an open set ifα≺β.
In this case, we define
deg(I–A,S,θ) =degI–A,S∩B(,R),θ ,
whereRis such that every fixed pointuofAinSsatisfiesu<R. By excision property this degree does not depend onR.
To be able to associate a degree with a pair of subsolution and supersolution, we have to reinforce the definition.
Definition . A subsolutionαof (.) is said to be strict if every solutionuof (.) such thatα≤usatisfiesα≺u.
In the same way, a strict supersolutionβof (.) is a supersolution such that every solu-tionuof (.) such thatu≤βsatisfiesu≺β.
Definition . The functionF:×Ris anLp-Carathéodory function if
. F(·,u)is measurable for allu∈; . F(x,·)is continuous for a.e.x∈;
. for all bounded setB⊆RN, there existshB∈Lp()such that for a.e.x∈and all u∈B,
F(x,u)≤hB(x).
Remark . The idea of the above definitions comes from [].
IfFis anLp-Carathéodory function withp>N, then the operator
N:C()→Lp() :u→ F(x,u(x)) a(|u(x)|γdx)
is well defined, continuous, and maps bounded sets to bounded sets. Then the operator A:C()→C() defined as
Au= (–+λ)–(Nu+λu), λ> ,
is completely continuous, and problem (.) is
u=Au, λ> .
Theorem . Let⊆RN(N≥)be a smooth bounded domain,andγ ∈(, +∞).Suppose that F:×R→R is a continuous function.Assume thatαandβare the subsolution and supersolution of(.),respectively.If there exists h∈Lp() (p>N)such that
F(x,u)≤h(x), x∈,α(x)≤u≤β(x). (.)
Then problem(.)has at least one solution u such that,for all x∈,
If,moreover,α(x)andβ(x)are strict and satisfyα≺β,then
S=u∈C()|α≺β
is admissible for the degree,and
deg(I–A,S,θ) = .
Proof ByLp-theory there existsR> greater thanmax{α,β}such that, for everyF satisfying (.) and every solution of (.) withα≤u≤β, we have
u<R.
Let
F(x,u) = ⎧ ⎪ ⎨ ⎪ ⎩
F(x,α(x)) ifu<α(x), F(x,u) ifα(x)≤u≤β(x), F(x,β(x)) ifu>β(x).
We will study the modified problem (λ> )
–u+λu=a( F(x,u)
|χ(x,u(x))|γdx)+λχ(x,u), x∈, u|∂= ,
(.)
whereχ(x,u) =α(x) + (u–α(x))+– (u–β(x))+.
Step . Every solutionuof (.) is such thatα(x)≤u(x)≤β(x),x∈.
We prove thatα(x)≤u(x) on. By contradiction assume thatmaxx∈(α(x) –u(x)) = M> . Note thatα(x) –u(x)≡Mon(α(x) –u(x)≤,x∈∂). Ifx∈is such that
α(x) –u(x) =M, then
≤–α(x) –u(x)
≤
a(|χ(x,u(x))|γdx)
Fx,α(x)
–
a(|χ(x,u(x))|γdx)
Fx,u(x) –λχ
x,u(x) +λu(x)
= –λα(x) –u(x)
< .
This is a contradiction.
Now we prove thatβ(x)≥u(x) on. By contradiction assume thatmaxx∈(β–u(x)) = –m< . Note thatβ(x) –u(x)≡–mon(β(x) –u(x)≥,x∈∂). Ifx∈is such that:
β(x) –u(x) = –m, then
≥–β(x) –u(x)
≥
a(|χ(x,u(x))|γdx)
–
a(|χ(x,u(x))|γdx)
Fx,u(x) –λχ
x,u(x) +λu(x)
=λβ(x) –u(x)
> .
This is a contradiction. Consequently,
α(x)≤u(x)≤β(x), x∈.
Step . Every solution of (.) is a solution of (.). Every solution of (.) is such that α(x)≤u(x)≤β(x). SinceFsatisfies (.), we also have thatu<R. Hence,
Fx,u(x) =Fx,u(x) ,
a(|χ(x,u(x))|γdx)=
a(|u(x)|γdx),
anduis a solution of (.).
Step . Problem (.) has at least one solution. Define the operator
N:C()→Lp() :u→ F(x,u(x)) a(|u(x)|γdx).
It is easy to see thatN is well defined, continuous, and maps bounded sets to bounded sets. Then the operatorA:C()→C() defined as
Au= (–+λ)–(Nu+λu)
is completely continuous.
By the hypothesis onFand the construction ofFthere existsh∈Lp() such that, for everyu∈C(),
a( F(x,u(x))
|χ(x,u(x))|γdx)
+λχx,u(x)<h(x), (.)
which guarantees that there existsK> large enough such that, for allv∈A(C()),
v ≤K.
Then there existsK>max{α,β}large enough such that
ABC(,K) ⊆BC(,K)
and, by a classical result of degree theory [],
Therefore, there existsu∈BC(,K) such that
u=Au.
Steps and yield that
α(x)≤u(x)≤β(x), x∈.
Step . Ifα(x) andβ(x) are the strict subsolution and supersolution, then we have
deg(I–A,S,θ) = .
Sinceα(x) andβ(x) are the strict subsolution and supersolution,Ahas no fixed point on∂S, and sodeg(I–A,S,θ) is well defined. Step guarantees thatAhas no fixed point inB(,K) –S. Then
deg(I–A,S,θ) =degI–A,BC(,K),θ = .
The proof is complete.
Remark . If we do not define the topological degree, we may useC() and obtain sim-ilar results.
Remark . The difference between our Theorem . and Theorem in [] is thatF(x,u) can change sign and we get the existence of classical solutions to (.).
Remark . In the particular caseN= , we can also allowp= , and it is classical that, in this case,Aalso is completely continuous.
Remark . The difference between Definition . and the definitions in [] is that we define a special functionχand the classical supersolutions and that in [] the sub-supersolutions are in the sense of distribution.
In the following sections, we suppose thata(t) : [, +∞) is continuous and increasing on [, +∞) for convenience.
3 The existence of positive solutions with concave and convex nonlinearities
In this section, we consider the problem ⎧
⎪ ⎨ ⎪ ⎩
–u=
a(|u|γdx)(λuq+up), x∈, u(x) > , x∈,
u|∂= ,
(.)λ
whereγ > , >q> ,p> ,={x∈RN||x|< }.
In order to consider the existence of positive solutions for (.)λ, we list some previous
results. Letϕbe the eigenfunction corresponding to the principle eigenvalue of
It is found thatλ> andϕ(x) > forx∈; see []. Moreover, there existu,u∗∈C()
that satisfy
⎧ ⎪ ⎨ ⎪ ⎩
–u= , x∈, u> , x∈, u|∂= ,
and ⎧ ⎪ ⎨ ⎪ ⎩
–u=uq, x∈, u> , x∈, u|∂= ,
respectively. By [] the following results are true:
ϕ
e ∈C(), ϕ
u∗∈C(). (.)
Suppose thatuλis a positive solution to (.)λ. Let
c=a
uγλdx –p–
and v=cuλ. (.)
Thenvsatisfies
–v=λ
(a(|u(x)|γdx))(p–q)/(p–)vq+vp, x∈, v|∂= ,
(.)
and the transform (.) will be used later. Let
K=u∈C()|u(x)≥,∀x∈.
Obviously,Kis cone inC().
Using Theorem ., we have following theorems.
Theorem . Assume that NN– >p> andlimt→+∞t
(p–)/γ
a(t) = +∞.Then there exist≥
> such that
() (.)λhas at least two positive solutions ifλ∈(,);
() (.)λhas at least one positive solution ifλ=andλ=;
() (.)λhas no positive solutions ifλ>.
Moreover,
|uλ|∞≤C, ∀positive solutions uλto(.)λ,λ∈[,].
Theorem . Assume that <q<p< .Then(.)λhas at least one positive solution for
Now we consider ⎧
⎪ ⎨ ⎪ ⎩
–u=a (λuq+up), u(x) > , x∈, u|∂= ,
(.)λ
wherea=inft∈[,+∞)a(t).
Lemma .(see []) Assume that <q< ,p> .Then there existaand Ca> such that
() (.)λhas at least two positive solutions ifλ∈(,a);
() (.)λhas at least one positive solution ifλ=a;
() (.)λhas no positive solutions ifλ>a. Moreover,
|uλ|∞≤Ca, ∀positive solutions uλto(.)λ,λ∈[,a].
Lemma .(see []) Suppose that f :×R+→R is a continuous function such that
s–f(x,s)is strictly decreasing for s> at each x∈.Let w,v∈C()∩C()satisfy:
(a) w+f(x,w)≤≤v+f(x,v)in; (b) w,v> in,andw≥von∂;
(c) v∈L().
Then w≥v in.
Proof of Theorem. () We show that forλ∈(,a), (.)λhas at least one positive
solution.
Foru∈P, we define the operator
(Aλu)(x) =
a(|u(x)|γdx)
G(x,y)λu(y)q+u(y)pdy, x∈,
whereG(x,y) is the Green function for –u=h.
Forλ∈(,a), by [] there exists auλ∈C() such that
⎧ ⎪ ⎨ ⎪ ⎩
–uλ=a(λuqλ+u
p
λ), x∈,
uλ(x) > , x∈,
u|∂= ,
with ∂uλ
∂n < ,x∈∂. Letβ(x) =uλandb=supt∈[,|β|γ∞||]a(t). Since < <qandλ> ,
we can chooseε> small enough such that
ελϕ(x) <
b
λεϕ(x)
q
, x∈,
εϕ(x) <β(x), ∀x∈,
and
∂εϕ(x)
∂n > ∂β(x)
Letα(x) =εϕ(x). Then
⎧ ⎪ ⎨ ⎪ ⎩
–(εϕ(x)) =ελϕ(x)
<b(λ(εϕ(x))q+ (εϕ(x))p), x∈,
εϕ(x)|∂= .
Therefore, by the strict monotonicity ofawe have ⎧
⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
–β(x) =a (λβ(x)q+β(x)p)
≥
a(χ(x,u(x))γdx)(λβ(x)
q+β(x)p), x∈,u∈C()∩C(),
β(x) > , x∈, β(x)|∂= ,
and ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
–(α(x)) =ελϕ(x)
<b(λ(εϕ(x))q+ (εϕ(x))p)
≤
a(χ(x,u(x))γdx)(λ(εϕ(x))q+ (εϕ(x))p), x∈,u∈C()∩C(),
α|∂= ,
with
∂α ∂n >
∂β(x)
∂n , ∀x∈∂,
which implies thatαandβare the subsolution and supersolution of (.) withα≺β. Now Theorem . implies that (.)λhas at least one positive solutionuλwithα(x)≤uλ(x)≤
β(x),x∈. Moreover, ifW={u∈K⊆C()|α≺u≺β}andu∈[α,β] is a solution to (.), then we have
–(u(x) –α(x)) > , x∈, (u–α)|∂= ,
which, together with the maximum principle, means that
∂u(x) ∂n <
∂α(x)
∂n , x∈∂,
that is,
α≺u.
A similar argument shows that
u≺β.
Now Theorem . guarantees that
Let
=sup
λ> : (.)λhas at least one positive solution
.
Obviously,> .
() We show that< +∞.
Assume thatuλis a solution to (.)λ. Letc=a(
|uλ|
γdx)–p– andv=cu
λ. Then we get
(.). By Lemma . there existC> and> such that equation
–v=λvq+vp, v|∂=
has at least one positive solution for all ≤λ≤and
sup λ∈[,]
|vλ|∞≤C, (.)
which, together with (.), implies that
λ
(a(|uλ(x)|γdx))(p–q)/(p–)
≤, (.)
|v|∞=a
uλ(x) γ
dx –p–
u
∞≤
C,
and
|u|∞≤a
uλ(x) γ
dx
p–
C. (.)
Now we show that{|uλ(x)|γdx:λ∈(,)}is bounded.
In fact, if{|uλ(x)|γdx:λ∈(,)}is unbounded, then there exists a sequence{uλn} such that
lim
n→+∞
uλn(x)
γ
dx= +∞.
Now (.) means that
≤a
uλn(x)
γ
dx –p–
uλn(x)≤C, x∈.
Then
≤a
uλn(x)
γ
dx –pγ–
uγλn(x)≤C
γ
, x∈.
Integration onyields that
≤a
uλn(x)
γ
dx –pγ–
uλn(x)
γ
Letsn=|uλn(x)|
γdx. Then
s(np–)/γ
a(sn) γ
p–
≤ ||Cγ,
which contradicts to
lim
t→+∞
t(p–)/γ
a(t) = +∞.
Sincea(t) > is continuous on [, +∞) withinft≥a(t) =a> , the boundedness of
{|uλ(x)|
γdx:λ∈(,
)}means that
a
uλ(x) γ
dx
is bounded,
which, together with (.), means that
λ≤sup λ∈
a
uλ(x) γ
dx
(p–q)/(p–)
.
Hence,
< +∞. (.)
From (.) we have
|uλ|∞≤a
uλ(x) γ
dx
p–
Cdef=C< +∞. (.)
() We show that there existsusatisfying (.). By the definition of> there exists a sequenceλn→anduλnis a positive solution of (.)λn. From (.), there existsC>
such that
uλn(x)≤ a
λuλn(x)q+uλn(x)p
≤C, ∀λn∈[,],
which guarantees that{uλn}is relatively compact inC(). Then there existsu∈C() such that
lim
ni→+∞
uλni(x) =u(x) uniformly on.
A standard bootstrap argument shows thatu∈C
()∩C() is a nonnegative solution
for (.).
() We show that forλ∈(,a), (.)λhas at least two positive solutions.
By (.) and the Green formula there existsC> such that
∇uλ(x)=
a(|uλ(x)|γdx)
Gx(x,y)λuλ(y)q+uλ(y)p
dy
LetR>max{C,C}andλ>. LetH(τ,u) =u– (–)–((τ λ+ ( –τ)λ)uq+up) andBR=
{u|u<R}. If there existτ∈[, ] andu∈K∩∂BRsuch that
H(τ,u) =u– (–)–
τλ+ ( –τ)λ uq+u
p
= ,
then we have ⎧ ⎪ ⎨ ⎪ ⎩
–u=a(
|u(x)|γdx)((τλ+ ( –τ)λ)u q+up
), x∈,
u(x) > , x∈,
u|∂= ,
which, together with (.) and (.), means thatu=max{|u|∞,|∇u|∞}<R. This
contradicts tou∈(∂BR)∩K. The homotopy ofH,
degI–H(,·),BR∩K,θ =degI–H(,·),BR∩K,θ .
Next, we claim that
degI–H(,·),BR∩K,θ = . (.)
In fact, suppose that there exist ≥μ≥ andu∈∂BR∩Ksuch thatH(,u) =μu.
Obviously,μ> andusatisfy
⎧ ⎪ ⎨ ⎪ ⎩
–u=μa(
|u(x)|γdx)(λu q
+u
p
), x∈,
u(x) > , x∈,
u|∂= .
Letv=μ– p–
u. Thenvsatisfies that
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–v=a( |u(x)|γdx)(
μ(p–q)/(p–)λv
q+vp), x∈,
v(x) > , x∈, v|∂= .
Since
μ(p–q)/(p–)λ>λ,
this contradicts the definition of, which means that
H(,u)=μu, ∀μ∈[, ],u∈∂BR∩K.
Therefore, (.) is true, and so,
Now forλ∈(,a), we consider ⎧
⎪ ⎨ ⎪ ⎩
–u=λuq, x∈, u(x) > , x∈, u|∂= .
(.)
From [], (.) has one positive solutionv. Letr=v. Let <r<min{a(Cγ||)
q–v
,
e}. Forτ∈[, ], define
H(τ,u) =u– (–)–λuq+up+τ , u∈C()∩K.
We claim that
H(τ,u)=θ, τ∈[, ],u∈K∩∂B(,r).
In fact, suppose (τ,u)∈[, ]×K∩∂B(,r). Then
–u=a(
|u|γdx)(λu q
+u
p
+τ), x∈,
u|∂= .
Letc=a(uγdx)–q– andv=cu
. Then
–v=λvq+cq–pvp+cqτ
,
v|∂= .
(.)
By Lemma . we have
v=cuλ≥v, ∀λ≥,x∈,
and so
u≥a
Cγ||
q–v
,
which contradicts tou=r<a(Cγ||)
q–v
. From the homotopy ofHit follows that
deg(I–A,Br∩K,θ) =degH(,·),Br∩K,θ =degH(,·),Br∩K,θ = ,
which, together with (.), implies that
degI–A,BR– (W∪Br) ∩K,θ = –.
Consequently,Ahas another fixed pointuλ,∈(BR– (W∪Br))∩K, that is, (.)λhas
an-other positive solutionuλ,for allλ∈(,a). Consequently,
+∞>≥≥a.
Proof of Theorem. For givenλ> , since <q<p< , there existsK> such that
>λK q– |e|
q
∞+Kp–|e|p∞
a
,
that is,
K>
λKq|e|q∞+Kp|e|p∞
a
.
Letβ(x) =Ke(x). Then
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
–(β(x)) = –(Ke(x))
=K
>λK q |e|
q ∞+Kp|e|∞
a , x∈, β(x) = , x∈∂.
(.)
Letb=a(
β
γ(x)dx). Chooseε> small enough such that
ελϕ(x) <
b
λεϕ(x)
q
, x∈, (.)
and
εϕ≤Ke(x), ∀x∈. (.)
Letα(x) =εϕ(x). Now (.), (.), and (.) guarantee that
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–(β(x)) >λK q |e|
q ∞+Kp|e|∞
a ≥ λ(β(x))q+(β(x))q
a(χ(x,u(x))γdx), x∈,u∈C()∩C(), β(x) = , x∈∂,
and ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
–(α(x)) =ελϕ(x)
<
b(λ(εϕ(x)) q+ (εϕ
(x))p)
≤
a(χ(x,u(x))γdx)(λ(α(x))q+ (α(x))p), x∈,u∈C()∩C(), α(x)|∂= ,
which guarantees thatαandβare the subsolution and supersolution to (.)λ. Now
The-orem . implies that (.)λhas at least one positive solution for allλ≥. The proof is
complete.
4 The existence of positive solutions when the nonlinearity is linear atu= 0
In this section, we consider the problem
⎧ ⎪ ⎨ ⎪ ⎩
–u=a(
|u|γdx)(λu+f(x,u)), x∈, u(x) > , x∈,
u|∂= ,
whereγ > , and⊆RNis a bounded smooth domain. Now we list following conditions for convenience:
(H) f(x,u)is continuous on×(–∞, +∞), and
lim |u|→+
f(x,u)
u = uniformly onx∈. (.)
Foru∈C(), we define the operator
Fλ,u(x) = a(|u(x)|γdx)
G(x,y)λu(y) +fy,u(y) dy
= λ a
G(x,y)u(y)dy+
G(x,y)
λu(y) +f(y,u(y)) a(|u(x)|γdx) –
λ a
u(y)
dy
def
= λ(Lu)(x) +Hλ,u(x) , x∈,
whereG(x,y) is the Green function for –u=h.
Of course, under these new notation, (λ,u) solves (.)λif and only if
u=F(λ;u) :=λLu+H(λ,u). (.)
Suppose that (H) holds. It is easy to see thatL:C()→C() is a compact and
con-tinuous linear operator andH(λ,·) :C()→C() is a compact and continuous nonlinear operator. Moreover, (.) guarantees that
Hλ,u(x) =
G(x,y)
λu(y) +f(y,u(y)) a(|u(x)|γdx) –
λ a
u(y)
dy
=
G(x,y)
λu(y) +f(y,u(y)) a+o()
– λ a
u(y)
dy
=
G(x,y)
λu(y) +f(y,u(y)) a
+o()–
λ a
u(y)
dy
=
G(x,y)
λu(y) +f(y,u(y)) a
+o()– λ a
u(y)
dy
=
G(x,y)f(y,u(y)) a
+o()dy
=ou , asu →. (.)
Now, we state the following result.
Lemma .(see []) Let E be a Banach space.Suppose that L is a compact linear operator and thatλ–∈σ(L)with odd multiplicity.If H satisfies condition(.),then the set
=(λ;u)∈R×E:u=λLu+H(λ,u);u=
has a closed connected component C=Cλsuch that(λ, )∈C and (i) Cis unbounded inR×E,or
Suppose thatλis the principle eigenvalue to the problem
–u=λau, x∈, u|∂= .
It is well known that the first eigenfunctionφassociated toλ can be chosen positive.
Moreover,λis an eigenvalue with odd multiplicity.
By the global bifurcation theorem, (H) guarantees that there exists a closed connected
componentC=Cλof solutions for (.)λthat satisfies (i) or (ii).
Lemma . There existsδ> such that if(λ,u)∈C with|λ–λ|+|u|<δand u= ,then
u has a defined sign,that is,
u(x) > , x∈ or u(x) < , x∈.
Proof Take{un}inC() andλn→λsuch that
un= , un →, un=λnLuλn+H(λn,un). (.)
Consideringwn=un/un, we get ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–wn=λn awn+ [
λnwn+f(xun,un)
a(|un|γdx)– λn awn] =λn
awn+ [
λnwn+o()
a+o() – λn awn] =λn
awn+ [ o()
a+o()], x∈,
wn(x) = , x∈∂.
(.)
It is easy to check that
wnC()≤ a
λn+unC()+K
,
whereKis a positive constant.
Since that{un}is bounded inC(),{wn}is also bounded inC(). By the Arzelà-Ascoli
theorem,{wn}converges to somew∈C(), uniformly in, under a convenient
subse-quence. Of course,wC()= , and thusw= in.
Now, by (.) we know that{un}is a Cauchy sequence inC() and
wn=λnLwn+H(λn,un) un .
Lettingn→+∞, we have
w(x) = λ a
G(x,y)w(y)dy,
that is,
–w=λ
Sincew= , by spectral theory we must have
w(x) > , x∈ or w(x) < , x∈.
Without loss of generality, we can suppose thatw(x) > for allx∈. Sincewis the C()-limit of{wn}, we must havewn(x) > for allx∈andnlarge enough. Therefore,
the sign ofuncoincides with that ofwnfornlarge enough. The proof is complete.
Now we decomposeCintoC=C+∪C–, where
C+=(λ,u)∈C|u(x)≥,∀x∈
and
C–=(λ,u)∈C|u(x)≤,∀x∈.
A simple computation gives thatC+={(λ,u)∈C|(λ, –u)∈C–}andC+is unbounded if
and only ifCis also unbounded.
Theorem . If(H)holds,then there exists an unbounded closed connected component
C=Cλof solutions for(.)λ.
Proof In fact, suppose thatCis bounded, which implies thatC+is bounded andCcontains (λˆ, ), whereλˆ=λ,λˆ–∈σ(L).
In this way, we can take{un}inC() andλn→ ˆλsuch that
λn= , un → and un=F(λn,un), (λn,un)∈C+.
Consideringwn=un/unC(), we know that it satisfies problem (.). Moreover, as in
the proof of Lemma ., under an adequate subsequence,{wn}converges towinC(), which is a nonzero solution of the eigenvalue problem
–w=λˆaw, x∈, w= on∂,
that is,wis an eigenfunction related toλˆ. Sinceλˆ =λ,wmust change sign. Then, forn
large, eachwnmust change sign, and the same should hold forun=wnunC(), which
contradicts to (λn,un)∈C+. The proof is complete.
Now we consider the following special problem:
⎧ ⎪ ⎨ ⎪ ⎩
–u=c+c(
|u|γdx)α(λu–|u|
p–u), x∈,
u(x) > , x∈, u|∂= ,
(.)λ
whereγ > ,is a bounded smooth domain, andp> . By Theorem . the connected componentC+ of (.)
λis unbounded. Now we have
Theorem . Suppose that p>max{αγ + ,γ – }.Then at least one positive solution of (.)λexists if and only ifλ>λ.
Proof First, we will show that for any> , there existsr> such that
uH()≤r, ∀(λ,u)∈C
+andλ≤. (.)
From now on, we denote by · the usual norm inH(), that is,
u=uH ()≤r.
Indeed, suppose that (.) is false. Then, there are{un} ∈H
() such that
un →+∞ and un=F(λn,un), λn≤.
Consideringwn=un/un, it follows from (.) that
∇wn· ∇v dx+ un
a(un(x)γdx)
upnv dx
= λn
a(un(x)γdx)
wnv dx, ∀v∈H(). (.)
Since that{wn}is bounded inH
(), without loss of generality, we can suppose that there
isw∈H
() satisfying
wn→w inH(), wn→w inL(),
and
wn(x)→w(x), a.e. in.
Takingv=un/unp–αγ as a test function, (.) is
unp–αγ–+
w
p+
n dx cun–αγ +c(
wn(x)γdx)α
=
wndx unp––αγ[c
+c(
un(x) γdx)α].
Sincep>αγ+ , lettingn→+∞, we derive
lim
n→+∞
w
p+
n dx cun–αγ +c(
wn(x)
γdx)α = .
Since
w
p+
n dx cun–αγ +c(
wn(x)
γdx)α≥
w
p+
n dx cun–αγ+c(
wn(x)p+dx)
αγ p+||
α(p+–γ) p+
we have
lim
n→+∞
w
p+
n dx cun–αγ +c(
wn(x)p+dx)
αγ p+||
α(p+–γ) p+
= ,
which implies that
lim
n→+∞
wpn+dx= .
By the Fatou lemma
w(x)p+dx≤ lim
n→+∞
wn(x)p+dx= .
Therefore, we should havew= . Thereby,{wn}converges to inL(). Takingv=wnas a test function, we see that
|∇wn|+
un[c+c(
un(x)γdx)α]
un(x)pwn(x)dx
=λn
c+c(
un(x) γdx)α
wn(x)dx.
Since{λn} is bounded from above by and c+c(
un(x)γdx)α
un(x)pwn(x)dx≥, we
have
|∇wn|≤ c
wn(x)dx.
Taking the limit, we have thatwn →, which contradicts town= for alln. Then (.) is true, which, together with the boundedness of, implies that
uC()≤r, ∀(λ,u)∈C+andλ≤.
Next, we will show the nonexistence of solution forλ≤λ, proving thatC+does not
intersect [,λ]×H(). Indeed, suppose that
(λ,u)∈[,λ]×H(), (λ,u)∈C+.
Usingv=φas a test function in (.), we get
λ
c+c(
un(x)γdx)α
uφdx>
λuφdx–
u
pφ
dx
c+c(
un(x)γdx)α
=
∇u∇dx=
λ
c
uφdx.
This is a contradiction.
Consequently, problem (.)λhas at least one positive solution if and only ifλ>λ. The
5 The positive solutions for singular nonlocal elliptic problems
In this section, we consider the singular elliptic equation ⎧
⎪ ⎨ ⎪ ⎩
–a(|u(x)|γdx)u(x) +K(x)u–μ=λuq, x∈, u(x) > , x∈,
u|∂= ,
(.)λ
where γ > , >q> , is a bounded domain inRN, N≥, withC,β boundary∂,
β∈(, ),K∈C,β(), and <q< ,μ∈(, ).
Now we list some previous results for the following equation: ⎧
⎪ ⎨ ⎪ ⎩
–u(x) +K(x)u–μ=λuq, x∈, u(x) > , x∈,
u|∂= ,
(.)λ
whereK∈C() and <q< ,μ∈(, ). Define
E=u∈C,β()∩C() :u–μ∈L().
Theorem .(see []) Let K(x) < ,x∈.Then
(i) (.)λhas a unique solutionuλ∈Efor anyλ∈R; (ii) uλis increasing with respect toλ;
(iii) cd(x)≤uλ(x)≤cd(x)for anyx∈and somecandc> independent ofx;
(iv) uλ∈C,–μ().
Theorem .(see []) Letminx∈K(x) > .Then
(i) there existsλ∗> such that(.)λhas at least one positive solutionuλ∈Efor any
λ>λ∗;
(ii) cd(x)≤uλ(x)≤cd(x)for anyx∈and somecandc> independent ofx;
(iii) uλ∈C,–μ().
Using Theorems . and ., by Theorem . we have the following results for (.)λ.
Theorem . Let K(x) < for all x∈.Then
(i) (.)has at least one solutionuλ∈Efor anyλ≥;
(ii) cd(x)≤uλ(x)≤cd(x)for anyx∈and somecandc> independent ofx,and
uλ∈C,–μ().
Proof () Forλ≥, we consider the problem ⎧
⎪ ⎨ ⎪ ⎩
–a(|u(x)|γdx)u(x) +K(x)(u(x) +
n)
–μ=λuq, x∈, u(x) > , x∈,
u|∂= ,
(.)n
wheren∈ {, , . . .}.
Sinceμ∈(, ) and <q< , there existsK> such that
>maxx∈|K(x)|K
–α–
+λK
q– |e+ |
q
∞
a
that is,
K>
maxx∈|K(x)|K–μ+λKq|e+ |q∞
a
.
Letβ(x) =K(e(x) + ). Then
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–(β(x)) = –K(e(x) + )
=K
>maxx∈|K(x)|K –μ +λK
q |e+|
q ∞ a
≥–K(x)(β(x))–μ+λ(β(x))q
a , x∈, β(x) = , x∈∂.
(.)
Letb=a(
β(x)
γdx). Since <q< , there exists
n>εn> small enough such that
εnλϕ(x) <
b min x∈
–K(x) εnϕ(x) +
n
–μ
+λεn
ϕ(x)
q
, x∈, (.)
and
εnϕ(x) <K
e(x) + , ∀x∈. (.)
Letαn(x) =εnϕ(x). Then, foru∈C()∩C(), (.), (.), and (.) imply that
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–(β(x)) >–K(x)(β(x))a–μ+λβ(x)q
≥–K(x)((β(x)+n)q)–μ+λβ(x)q
a(χ(x,u(x))γdx) , x∈, β(x) = > , x∈∂,
and ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
–(αn(x)) =ελϕ(x)
<b (minx∈(–K(x))(εnϕ(x) +n)–μ+ (εnϕ(x))q)
≤
a(χ(x,u(x))γdx)(–K(x)(αn(x) +n)–μ+λ(αn(x))q), x∈, αn|∂= ,
which means that αn(x) and β(x) are the subsolution and supersolution to (.)n. Now Theorem . implies that (.)nhas at least one positive solutionun.
Now we consider set{un}. Since|un|∞≤K|e+ |∞, it follows that
λ
un(x) + n
q –K(x)
un(x) + n –μ ≥min x∈ K(x) K
|e+ |∞ +
n –μ , and so ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–(un(x)) = a(u
γ
n(x)dx)(λun(x)
q–K(x)(un(x) +
n)
–μ)
≥
bminx∈|K(x)|(K(|e+ |∞) +n)–
μ, x∈,
Lemma ., together with (.), implies there existsc> such that
un(x)≥ b
min
x∈
K(x)K
|e+ |∞+ –μe(x)≥cϕ(x), x∈,
whereb=a(
β
γ(x)dx). Then
∇un(x)≤ a
Gx(x,y)max
x∈
K(x)cϕ(y) – μ
+λK|e+ |∞ q
dy
and
∇un(x) –∇un(x)
≤ a
Gx(x,y) –Gx(x,y)max
x∈
K(x)cϕ(y) –μ
+λK|e+ |∞
q dy.
The same technique as in [], Theorem ., yields
∇un(x) –∇un(x)≤C|x–x|–μ.
Therefore,un∈C,–μ(). The sequence{un}has a subsequence{un
i}such that
lim
ni→+∞
uni=uλ uniformly on.
Now a straightforward calculation yields
–(uλ(x)) =
a(uγλ(x)dx)(λuλ(x)
q–K(x)u(x)–μ), x∈,
uλ|∂= .
() Suppose thatuλsatisfies (.)λ. Letc=a(
u
γ
λ(x)dx) > andv(x) =a(
u
γ
λ(x)dx)×
uλ(x),x∈. Thenv(x) satisfies
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–v(x) +cμK(x)u–μ=
cqλu
q, x∈,
v(x) > , x∈, v|∂= .
Now Theorem . implies that there existcandc> independent ofxsuch that
cd(x)≤vλ(x)≤cd(x) for anyx∈,
and so
a(uγλ(x)dx)cd(x)≤uλ(x)≤
a(uγλ(x)dx)cd(x) for anyx∈.
Theorem . Letminx∈K(x) > .Suppose that there existsρ>–qsuch that
lim
t→+∞a(t) = +∞ and t→lim+∞
t/(ργ)
a(t) = +∞. (.)
Then
(i) there existsλ∗> ,such that(.)has at least one positive solutionuλ∈Efor any
λ>λ∗;
(ii) cd(x)≤uλ(x)≤cd(x)for anyx∈and somecandc> independent ofx,and
uλ∈C,–μ().
Proof Forϕ, by the Hopf maximum principle, there existδ> and⊂such that
|∇ϕ| ≥δ, x∈, |ϕ| ≥δ, x∈–.
Then there existsM> such that
M( –μ)|∇ϕ|
( +μ)ϕμ/(+μ)
≥ K∗
Mμϕμ/(+μ), x∈
, (.)
and there existsM> such that
K∗ Mμϕμ/(+μ) ≤
λM
+μϕ +μ
, x∈–, (.)
whereK∗=maxx∈K(x). By (.), chooseM>max{M,M}large enough such that
a
Mγ
ϕ(x)
γ +μdx
> . (.)
Combining (.) and (.), we have
–Mϕ +μ
+K(x)
Mϕ +μ –μ = M +μλϕ
+μ
–
( –μ)M|∇ϕ|
( +μ)ϕμ/(+μ)
+K(x) Mμϕ
μ +μ
≤ M +μλϕ
+μ
, x∈–,
and so
–Mϕ +μ
≤
M +μλϕ
+μ
–K(x)
Mϕ +μ –μ
, x∈–.
Therefore, forn> ,
–Mϕ +μ
≤
M +μλϕ
+μ
–K(x)
Mϕ +μ –μ ≤ M +μλ
ϕ
+μ
–K(x)
M ϕ +μ + n –μ
Letu∗(x) satisfy
–u=uq, x∈, u|∂= .
(.)
Now by (.) it follows that
lim
t→+∞
t a(tγρ
u∗(x)γdx)
= +∞,
which implies that there existsT> such that, for allt>T,
t a(tγρ
u∗(x)γdx)
>M
–q
+μλϕ –q
+μ
.
Let
T>max
T,
sup
x∈
M(ϕ/(+μ)(x) + ) u∗(x)
/ρ
,a(–)/( ρ(–q)–)
. (.)
Forλ>T, let
uλ(x) =Mϕ +μ
(x) and uλ,(x) =λρu∗(x), x∈.
It is easy to see that
uλ(x)≤uλ,(x) =λρu∗(x), x∈. (.)
Foru∈C(), let
χx,u(x) =uλ(x) +
u(x) –uλ(x) +
–u(x) –uλ,(x) +
.
Then
a≤a
χx,u(x) γdx
≤a
λγρ
u∗(x)γdx
.
Let
b=a
λγρ
u∗(x) γdx
.
From (.) we know thatb> . By (.), (.), and (.), foru∈C()∩C(), we have
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–(uλ,(x)) = –(λρu∗(x))
=λρ–ρq(λρu∗(x))q =λ(–q)ρ–λ(u
λ,(x))q
≥
aλ(uλ,(x))q–
a((χ(x,u(x)))γdx)K(x)(uλ,(x) +n)–μ
≥
a((χ(x,u(x)))γdx)[λ(uλ,(x))
q–K(x)(u
λ,(x) +n)–μ], x∈,
and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–(uλ(x)) = –(Mϕ +μ
(x))
≤[M
+μλϕ
+μ
(x) –K(x)(Mϕ
+μ
(x))–μ]
=b[bM
–q
+μ λϕ(x)
–q +μ(u
λ(x))q
–K(x)b(Mϕ
+μ
(x))–μ]
≤
a((χ(x,u(x)))γdx)[λ(uλ(x))q–K(x)(uλ(x) +n)
–μ], x∈,
uλ(x)|∂= .
Hence,uλ(x) anduλ(x) are the subsolution and supersolution of (.)n. Now Theorem .
implies that forn∈ {, , . . .}, (.)nhas at least one solutionuλwith
uλ(x)≤uλ,n(x)≤uλ,(x), x∈.
Now we consider set{uλ,n}. From (.) we have
∇uλ,n(x)≤
a
Gx(x,y)max
x∈
K(x)Mϕ(y)
–μ/(+μ)
+λλρu∗ q
dy
and
∇uλ,n(x) –∇uλ,n(x)
≤ a
Gx(x,y) –Gx(x,y)max
x∈
K(x)Mϕ(y)
–μ/(+μ)
+λλρu∗ qdy.
The same technique as in [], Theorem ., yields
∇uλ,n(x) –∇uλ,n(x)≤C|x–x|–μ/(+μ).
Therefore,un∈C,–μ/(+μ)(). The sequence{un}has a subsequence{un
i}such that
lim
ni→+∞
uni=uλ uniformly on.
Now a straightforward calculation yields
–(uλ(x)) =a(
u γ λ(x)dx)
(λuλ(x)q–K(x)uλ(x)–μ), x∈,
uλ|∂= .
() Suppose thatuλsatisfies (.)λ. Letc=a(
u
γ
λ(x)dx) > andv(x) =a(
u
γ
λ(x)dx)×
uλ(x),x∈. Thenv(x) satisfies
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–v(x) +cμK(x)u–μ=
cqλu
q, x∈,
