R E S E A R C H
Open Access
Existence and multiplicity of positive
solutions for
p
-Kirchhoff type problem with
singularity
Dechen Wang and Baoqiang Yan
**Correspondence:
Yanbqcn@aliyun.com
School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, P.R. China
Abstract
In this paper, we consider a class ofp-Kirchhoff type problems with a singularity in a bounded domain inRN. By using the variational method, the existence and
multiplicity of positive solutions are obtained.
Keywords: p-Kirchhoff type problem; singularity; Nehari manifold
1 Introduction and main results
The purpose of this paper is to investigate the existence of multiple positive solutions to the following problem:
⎧ ⎨ ⎩
–M(|∇u|pdx)
pu=λf(x)u–r+g(x)uq– in,
u= on∂,
(.)
wherepu=div(|∇u|p–∇u),is a smooth bounded domain inRN, <r< <p<q<p∗
(p∗=NNp–p ifN>pandp∗=∞ifN≤p),M(s) =asp–+banda,b,λ> .f,g∈C() are nontrivial nonnegative functions.
Problem (.) is related to the stationary problem introduced by Kirchhoff in []. More precisely, it is the model
ρ∂
u
∂t –
ρ
h + E
L
L
∂∂uxdx ∂ u
∂x = ,
where ρ,ρ,E,L are constants, which was proposed as an extension of the classical
D’Alembert’s wave equation for free vibrations of elastic strings to describe transversal oscillations of a stretched string. For more details and backgrounds, we refer to [, ].
The existence and multiplicity of solutions for the following problem: ⎧
⎨ ⎩
–M(|∇u|pdx)
pu=h(x,u) in,
u= on∂, (.)
on a smooth bounded domain⊂RN has been studied in many papers. Liu and Zhao
[] proved (.) has at least two nontrivial weak solutions by Morse theory under some
restriction onM(s) andh(x,u). In [], the authors considered the following problem: ⎧
⎨ ⎩
–M(|∇u|pdx)pu=λf(x)|u|q–u+g(x)|u|r–u in,
u= on∂, (.)
whereM(s) =as+b, <q<p<r≤p∗, they proved the existence of multiplicity nontrivial solutions by using the Nehari manifold when the weight functionsf(x) andg(x) change their signs. For more results, we refer to [–] and the references therein.
Whenp= andN= , problem (.) reduces to the following singular Kirchhoff type problem:
⎧ ⎨ ⎩
–(a+b|∇u|dx)u=λf(x)u–r+μg(x)uq in,
u= on∂, (.)
where <q≤, the existence of solutions for problem (.) has been widely studied (see [–]). When <q< and λ= , Liu and Sun [] proved that problem (.) has at least two positive solutions forμ> small enough. Liaoet al.showed the multiplicity of positive solutions by the Nehari mainfold in the case ofq= in []. Whenqis a critical exponents, at least two positive solutions are obtained by variational and perturbation methods in [].
However, the singularp-Kirchhoff type problems have few been considered, especially
p= ,N= . Here we focus on extending the results in [] and []. In fact, the extension
is nontrivial and requires a more careful analysis. Our method is based on the Nehari manifold; see [, , , ].
Before starting our main theorems, we make use of the following notations:
•LetW,p() be the Sobolev space with norm u = (|∇u|pdx)p, the norm inLp() is denoted by p;
•LetSzbe the best Sobolev constant for the embedding ofW,p() inLz() with <
z<p∗. Then, for allu∈W,p()\{},
u z≤S
–p
z u .
In general, we say that a functionu∈W,p() is a weak solution of problem (.) if
M u p
|∇u|p–∇u∇ϕdx–λ
f|u|–rϕdx–
g|u|q–ϕdx=
for allϕ∈W,p(). Thus, the functional corresponding to problem (.) is defined by
J(u) =
pM
u p– λ –r
f|u|–rdx–
q
g|u|qdx, ∀u∈W,p(),
whereM(s) =sM(t)dt.
To obtain the existence results, we introduce the Nehari manifold:
Nλ=
u∈W,p()\ {}:M u p u p–λ
f|u|–rdx–
g|u|qdx=
and we define
K(u) = (p– )M u p u p+pM u p u p+λr
f|u|–rdx– (q– )
g|u|qdx.
Now we splitNλinto three disjoint parts as follows:
Nλ+=
u∈Nλ:K(u) >
;
Nλ=
u∈Nλ:K(u) =
;
Nλ–=
u∈Nλ:K(u) <
.
Letλ∗=max{(–r)λ(a)
p
p+
p
,(–r)λ
p }, whereλ(a) andλare given by
λ(a) = pS
–r p
–r p
abp–(q–p)(q–p
p–)p–
(q+r– ) f ∞
pS
q p
q p
abp–(p+r– )
(q+r– ) g ∞
p+r–
q–p+
and
λ= bS
–r p
–r(q–p)
(q+r– ) f ∞
bS
q p
q(p+r– )
(q+r– ) g ∞ p+r–
q–p ,
then we state the main theorems.
Theorem . Assume that p<q<p∗and N< p.Then,for each a> and <λ<λ∗,the problem(.)has at least two positive solutions u+λ∈Nλ+and u–λ∈Nλ–.
Define
=inf
u p:u∈W,p(),
g|u|pdx=
, (.)
then> is obtained by someφ∈W,p() with
g|φ|
p
dx= . In particular,
g|u|pdx≤ u p (.)
and ⎧ ⎨ ⎩
– u p
pu=μg|u|p
–
in,
u= on∂, (.)
whereμis an eigenvalue of (.),u∈W,p() is nonzero and an eigenvector corresponding toμsuch that
u p
|∇u|p–∇u∇(uϕ)dx=μ
we write
I(u) = u p, foru∈E=
u∈W,p() :
g|u|pdx=
,
and all distinct eigenvalues of (.) denoted by <μ<μ<· · ·, we have
μ=inf
u∈EI(u) > ,
whereμis simple, isolated and can be obtained at someψ∈Eandψ> in(see []).
Theorem . Assume that p=q<p∗and N< p.Then
(i) for eacha≥ andλ> ,the problem(.)has at least one positive solution; (ii) for eacha< and <λ<–prλ,ˆ where
ˆ λ= bS
–r p
–r(p–p)
f ∞(p+r– )
b(p+r– ) ( –a)(p+r– )
p+r–
p–p ,
the problem(.)has at least two positive solutionsu+λ∈Nλ+,u–λ∈Nλ–and
lim
a→
–uinf∈N–
λ
J(u) =∞.
This paper is organized as follows: In Section , we present some lemmas which will be used to prove our main results. In Section and Section , we will prove Theorems . and ., respectively.
2 Preliminaries
Lemma . (i)If q≥p,then the energy functional J(u)is coercive and bounded below in Nλ;
(ii)if q<p,then the energy functional J(u)is coercive and bounded below in W,p
().
Proof (i) Foru∈Nλ, we have
M u p u p–λ
f|u|–rdx–
g|u|qdx= .
By the Sobolev inequality,
J(u) =
pM
u p– λ –r
f|u|–rdx–
q
g|u|qdx
=
pM
u p–
qM
u p u p–λq+r–
q( –r)
f|u|–rdx
≥ u p
pq
a(q–p)
p u
p–p+b(q–p) –λq+r–
q( –r) f ∞S r–
p
–r u –r
≥b(q–p)
pq u
p–λq+r–
q( –r) f ∞S r–
p
–r u –r.
(ii) Foru∈W,p(), we have
J(u) =
pM
u p– λ –r
f|u|–rdx–
q
g|u|qdx
≥ a
p u
p+b
p u
p–λ f ∞S
r–
p
–r
–r u
–r– g ∞S
–qp
q
q u
q
=
a p u
p–q
– g ∞S
–qp
q
q u
q+
b p u
p+r––λ f ∞S
r–
p
–r
–r u
–r.
Thus,J(u) is coercive and bounded below inW,p().
Lemma . If q>pand <λ<max{λ(a),λ},then,for all a> ,
(i) the submanifoldN
λ=∅; (ii) the submanifoldNλ±=∅.
Proof (i) SupposeN
λ=∅. Then, foru∈Nλ, we have
(q+r– ) g ∞S
–qp
q u q≥(q+r– )
g|u|qdx
=ap+r– u p+b(p+r– ) u p
≥ ⎧ ⎨ ⎩
ppabp–(p+r– ) u p–,
b(p+r– ) u p, (.)
and
λ(q+r– ) f ∞S
––pr
–r u –r≥λ(q+r– )
f|u|–rdx
=aq–p u p+b(q–p) u p
≥ ⎧ ⎨ ⎩
ppabp–(q–p)(q–p
p–)p– u p–,
b(q–p) u p. (.)
By (.) and (.), for allu∈Nλ, we have
pS
q p
q p
abp–(p+r– )
(q+r– ) g ∞
q–p+
≤ u ≤
λ(q+r– ) f ∞
pS
–r p
–r p
abp–(q–p)(q–p
p–)p–
p+r–
and
bS
q p
q(p+r– )
(q+r– ) g ∞
q–p
≤ u ≤
λ(q+r– ) f ∞
bS
–r p
–r(q–p)
p+r–
.
(ii) Fixu∈W,p(). Let
ha(t) =atp
–(–r)
u p+btp–(–r) u p–tq–(–r)
g|u|qdx fora,t≥.
We see thatha() = andha(t)→–∞ast→ ∞. Sinceq>pand
ha(t) =tp+r–
ap+r– tp–p u p+b(p+r– ) u p
– (q+r– )tq–p
g|u|qdx ,
there is a uniqueta,max> such thatha(t) reaches its maximum atta,max, increasing for t∈[,ta,max) and decreasing fort∈(ta,max,∞) withlimt→∞ha(t) = –∞. Clearly, iftu∈Nλ, thentu∈Nλ+(orNλ–) if and only ifha(t) > (or < ). Moreover,
t,max=
b(p+r– ) u p
(q+r– )g|u|qdx
q–p
and
h(t,max)
=b
q+r–
q–p
p+r–
q+r– p+r–
q–p –
p+r–
q+r– q+r–
q–p u p(q+r–)
q–p
(g|u|qdx)p+q–r–p ≥ b(q–p)
q+r–
bS
q p
q(p+r– )
(q+r– ) g ∞ p+r–
q–p
u –r.
On the other hand, since
ha() = <λ
f|u|–rdx≤λ f ∞S
––pr –r u –r
<b(q–p)
q+r–
bS
q p
q(p+r– )
(q+r– ) g ∞ p+r–
q–p
u –r
≤h(t,max) <ha(ta,max), (.)
there exist uniquet+andt–such that <t+<t
a,max<t–, ha
t+=λ
f|u|–rdx=ha
t–
and
hat+> >hat–.
Lemma . (i)If q=pand a≥
,then,for allλ> ,N
+
λ=Nλ=∅; (ii)if q=p,a<
and <λ<λ,ˆ then Nλ=N
+
λ∪Nλ–and Nλ±=∅.
Proof First, we show thatNλ+=Nλ. Indeed, for allu∈Nλ, we have
ap+r– u p+b(p+r– ) u p–p+r–
g|u|pdx
≥(a– )(p+r– ) u
p+b(p+r– ) u p> .
Therefore,u∈N+
λ. Next, we declareNλ+=∅. Fixu∈W,p(). Let
¯
h(t) =tp+r–
a u p–
g|u|pdx +btp+r– u p fora,t≥.
Obviously,h¯() = andlimt→∞h¯(t) =∞. Since
¯
h(t) =p+r– tp+r–
a u p–
g|u|pdx
+b(p+r– )tp+r– u p,
we can deduce thath¯(t) is increasing fort∈[,∞). Thus, there is a uniquet+> such that
¯
h(t+) =λ
f|u|
–rdxandh¯(t+) > . That is,t+u∈N+
λ.
(ii) The proof is similar to Lemma ., we omit it here.
We writeNλ=Nλ+∪Nλ–and define
α+= inf
u∈Nλ+J(u); α
–= inf
u∈Nλ–J(u),
then we have the following lemma.
Lemma . Suppose that q>pand <λ<λ∗,then we have
(i) α+< ; (ii) α–>C
,for someC> . In particularα+=infu∈NλJ(u).
Proof (i) Letu∈Nλ+, it follows that
M u p u p–λ
f|u|–rdx–
g|u|qdx=
and
λ(q+r– )
Substituting this intoJ(u), we have
J(u) =
pM
u p– λ –r
f|u|–rdx–
q
g|u|qdx
=
pM
u p–
qM
u p u p–λq+r–
q( –r)
f|u|–rdx
<a(q–p
)( –r–p)
pq( –r) u
p+b(q–p)( –r–p)
pq( –r) u
p< ,
and thenα+< . (ii) Letu∈N–
λ. We divide the proof into two cases.
Case(A):λ∗=(–r)λ
p . Sinceu∈Nλ–, and by the Sobolev inequality,
b(p+r– ) u p≤ap+r– u p+b(p+r– ) u p
< (q+r– )S–
q p g
∞ u q, which implies
u >
bS
q p
q(p+r– )
(q+r– ) g ∞
q–p
for allu∈Nλ–.
Hence,
J(u)≥a(q–p
) u p
pq +
b(q–p) u p pq –λ
q+r–
q( –r) f ∞S
––pr –r u –r
≥ u –r
b(q–p)
pq u
p+r––λq+r– q( –r) f ∞S
––pr –r
>
bS
q p
q(p+r– )
(q+r– ) g ∞
–r
q–pb(q–p)
pq
bS
q p
q(p+r– )
(q+r– ) g ∞ p+r–
q–p
–λq+r–
q( –r) f ∞S
––pr –r
=C.
Thus, if <λ<(–r)λ
p , thenα–>C> .
Case(B):λ∗=(–r)λ(a)
p
p+
p
. By (.), one has
pp
abp–p+r– u p–≤(q+r– ) g
∞S
–qp
q u q,
which implies
u >
pS
q p
q p
abp–(p+r– )
(q+r– ) g ∞
q–p+
for allu∈Nλ–.
Repeating the argument of case (A), we conclude ifλ< (–r)λ(a)
p
p+
p
, thenα–>C
for some
Lemma . Suppose that q=p,a<
and <λ<
–r
p λ,ˆ then we have
(i) αˆ+< ;
(ii) αˆ–>C,for someC> . In particularαˆ+=inf
u∈NλJ(u).
Proof (i) Repeating the same argument of Lemma .(i), we conclude thatαˆ+< .
(ii) Letu∈N–
λ. By (.), one has
b(p+r– ) u p<p+r–
g|u|pdx–a u p
≤( –a)(p+r– ) u
p,
which implies that
u >
b(p+r– ) ( –a)(p+r– )
p–p
for allu∈Nλ–. (.)
Then we have
J(u) =
pM
u p– λ –r
f|u|–rdx–
q
g|u|qdx
≥ u –r
(p– )b
p u
p+r––λp+r– p( –r) f ∞S
––pr
–r
>
b(p+r– ) ( –a)(p+r– )
–r
p–p(p– )b
p
b(p+r– ) ( –a)(p+r– )
p+r–
p–p
–λp
+r–
p( –r) f ∞S ––pr
–r
. (.)
Thus, ifλ<–prλ, thenˆ αˆ–>C
for someC> .
Lemma . For each u∈Nλ+(resp.u∈Nλ–),there existε> and a continuous function
f :B(;ε)⊂W,p()→R+such that
f() = ,f(ω) > ,f(ω)(u+ω)∈Nλ+resp. u∈Nλ–, for allω∈B(;ε),
where B(;ε) ={ω∈W,p() : ω <ε}.
Proof Foru∈Nλ+, defineF:W
,p
()×R→Ras follows:
F(ω,t) =atp+r–
∇(u+ω)pdx
p
+btp+r–
∇(u+ω)pdx
–tq+r–
g|u|qdx–λ
f|u|–rdx.
Sinceu∈N+
λ, it is easily seen thatF(, ) = andFt(, ) > . Then by the implicit function
f :B(;ε)⊂W,p()→R+such that
f() = ,f(ω) > ,f(ω)(u+ω)∈Nλ+, for allω∈B(;ε).
In the same way, we can prove the caseu∈N–
λ.
Remark . The proof of Lemma . is inspired by [].
3 Proof of Theorem 1.1
By Lemma . and the Ekeland variational principle [], there exists a minimizing se-quence{un} ⊂Nλ+such that
(i) J(un) <α++
n;
(ii) J(u) >J(un) –
n u–un , ∀u∈N +
λ.
Note thatJ(|un|) =J(un). We may assume thatun≥ in. Using Lemma . again, we can
see that there is a constantC> such that, for alln∈N+, un ≤C. Thus, there exist a
subsequence (still denoted by{un}) andu+λinW
,p
() such that
unu+λweakly inW
,p
(),
un→u+λstrongly inL–r(),
un→u+λstrongly inLq(),
un→u+λa.e. in.
Now we conclude thatu+λ∈Nλ+is a positive solution of (.). The proof is inspired by Liu and Sun []. In order to prove the claim, we divide the arguments into six steps.
Step:u+
λis not identically zero.
Indeed, it is an immediate conclusion of the following inequalities:
Ju+λ
≤ lim
n→∞J(un) =α
+< .
Step: There existsCsuch that up to a subsequence we have
ap+r– un p
+b(p+r– ) un p– (q+r– )
gu+λqdx>C. (.)
In order to prove (.), it suffices to verify
ap+r– lim
n→∞ un
p+b(p+r– ) lim n→∞ un
p> (q+r– )
gu+λ
q
dx. (.)
Sinceun∈Nλ+,
ap+r– un p
+b(p+r– ) un p≥(q+r– )
gu+λ
q
It follows that
ap+r– lim
n→∞ un
p+b(p+r– ) lim n→∞ un
p≥(q+r– )
gu+λ
q
dx.
Suppose by contradiction that
ap+r– lim
n→∞ un
p
+b(p+r– ) lim
n→∞ un
p= (q+r– )
gu+λqdx. (.)
Then, from (.) and (.), one has
ap+r– lim
n→∞ un
p+b(p+r– ) lim n→∞ un
p= (q+r– )
gu+λ
q
dx.
Thus un pconverges to a positive numberAthat satisfies
ap+r– Ap+b(p+r– )A= (q+r– )
gu+λqdx
and
aq–pAp+b(q–p)A=λ(q+r– )
fu+λ
–r
dx.
On the other hand, by (.), we have
≤λ∗–λ
f|un|–rdx
<bq+q–r–p
p+r–
q+r– p+r–
q–p q–p
q+r–
un
p(q+r–)
q–p
(g|un|qdx)
p+r–
q–p –λ
f|un|–rdx
→b
q+r–
q–p
p+r–
q+r– p+r–
q–p q–p
q+r–
Aqq+–r–p
(a(p+r–)qA+pr+–b(p+r–)A)pq+–r–p –a(q–p
)Ap+b(q–p)A
q+r–
<bq+q–r–p
p+r–
q+r– p+r–
q–p q–p
q+r–
A
q+r–
q–p
(b(qp++rr–)–A)pq+–r–p –a(q–p
)Ap+b(q–p)A
q+r–
= –a(q–p
)
q+r– A
p< ,
which is impossible. Hence, (.) and (.) must hold.
Step: For nonnegative ϕ∈W,p() andt> small, we can findfn(t) :=fn(tϕ) such
thatfn() = andfn(t)(un+tϕ)∈Nλ+for eachun∈Nλ+by Lemma ..fn+()∈[–∞,∞] is
fn+() > –Cfor alln∈N+. Sinceun,fn(t)(un+tϕ)∈Nλ, we deduce that
=a un p
+b un p–λ
f|un|–rdx–
g|un|qdx
and
=afnp(t) un+tϕ p
+bfnp(t) un+tϕ p–λfn–r(t)
f|un+tϕ|–rdx
–fnq(t)
g|un+tϕ|qdx.
Thus
=afnp(t) – un+tϕ p
+a un+tϕ p
– un p
+bfnp(t) – un+tϕ p+b
un+tϕ p– un p
–λfn–r(t) –
f|un+tϕ|–rdx–λ
f|un+tϕ|–r–|un|–r
dx
–fnq(t) –
g|un+tϕ|qdx–
g|un+tϕ|q–|un|q
dx
≤afnp(t) – un+tϕ p
+a un+tϕ p
– un p
+bfnp(t) – un+tϕ p+b
un+tϕ p– un p
–λfn–r(t) –
f|un+tϕ|–rdx–
fnq(t) –
g|un+tϕ|qdx.
Then, dividing byt> and lettingt→, we have
≤ap un p
fn+() +ap un p
–p
|∇un|p–∇un∇ϕdx
+bp un pfn+() +bp
|∇un|p–∇un∇ϕdx
–λ( –r)fn+()
f|un|–rdx–qfn+()
g|un|qdx
=fn+()
ap un p
+bp un p–λ( –r)
f|un|–rdx–q
g|un|qdx
+ap un p
–p
|∇un|p–∇un∇ϕdx+bp
|∇un|p–∇un∇ϕdx
=fn+()
ap+r– un p
+b(p+r– ) un p– (q+r– )
g|un|qdx
+ap un p
–p
|∇un|p–∇un∇ϕdx+bp
|∇un|p–∇un∇ϕdx.
One deduces from (.)
fn+()≥–ap
u
n p
–p
|∇un|p–∇un∇ϕdx+bp
|∇un|p–∇un∇ϕdx
a(p+r– ) u
n p
+b(p+r– ) un p– (q+r– )
Therefore, by the boundedness of{un}, we conclude that{fn+()}is bounded from below. Step: Choosen∗large enough such that (–r)C
n < C
for alln>n∗. Then we claim that
there existsCsuch thatfn+() <Cfor eachn>n∗. Without loss of generality, we may
supposefn+()≥. Then from condition (ii), we have fn(t) –
un
n +tfn(t)
ϕ
n
≥
nfn(t)(un+tϕ) –un
≥J(un) –J
fn(t)(un+tϕ)
=a(p
+r– )
p( –r)
fnp(t) – un+tϕ p
+a(p
+r– )
p( –r)
un+tϕ p
– un p
+b(p+r– )
p( –r)
fnp(t) – un+tϕ p+
b(p+r– )
p( –r)
un+tϕ p– un p
–q+r–
q( –r)
fnq(t) –
g|un|qdx–
q+r–
q( –r)f
q n(t)
g|un+tϕ|q–|un|q
dx.
Then, dividing byt> and lettingt→, we deduce
fn+() un
n +
ϕ
n
≥a(p+r– ) –r f
n+() un p
+a(p
+r– )
–r un
p–p
|∇un|p–∇un∇ϕdx
+b(p+r– ) –r f
n+() un p+
b(p+r– ) –r
|∇un|p–∇un∇ϕdx
–q+r– –r f
n+()
g|un|qdx–
q+r– –r
g|un|q–ϕdx. (.)
From (.) and the choice ofn∗, we have
ϕ
n ≥ C
( –r)f
n+() +
a(p+r– )
–r un
p–p
|∇un|p–∇un∇ϕdx
+b(p+r– ) –r
|∇un|p–∇un∇ϕdx–
q+r– –r
g|un|q–ϕdx.
Namely,
C
( –r)f
n+()≤
ϕ
n –
a(p+r– )
–r un
p–p
|∇un|p–∇un∇ϕdx
–b(p+r– ) –r
|∇un|p–∇un∇ϕdx+
q+r– –r
g|un|q–ϕdx.
Therefore, by the boundedness of{un}, we conclude{fn+()}n>n∗is bounded from above.
Step:u+
λ> a.e. inand for nonnegativeϕ∈W
,p
(), we have
au+λp–p+b
∇u+λp–∇u+λ∇ϕdx–λ
fu+λ–rϕdx–
gu+λq–ϕdx
Similar to the argument in Step , one can obtain
fn+() un
n +
ϕ
n
≥–fn+()
a un p
+b un p–
g|un|qdx–λ
f|un|–rϕdx
–a un p
–p
|∇un|p–∇un∇ϕdx–b
|∇un|p–∇un∇ϕdx
+
g|un|q–ϕdx+ lim t→+
λ –r
f(|un+tϕ|–r–|un|–r)
t dx
= –a un p
–p
|∇un|p–∇un∇ϕdx–b
|∇un|p–∇un∇ϕdx
+
g|un|q–ϕdx+ lim t→+
λ –r
f(|un+tϕ|–r–|un|–r)
t dx. (.)
Sincef(|un+tϕ|–r–|un|–r)≥,∀t> , by Fatou’s lemma, we obtain
f|un|–rϕdx≤ lim t→+
–r
f(|un+tϕ|–r–|un|–r)
t dx. (.)
It follows from (.) and (.) that
λ
f|un|–rϕdx
≤
n
fn+() un + ϕ
+a un p
–p
|∇un|p–∇un∇ϕdx
+b
|∇un|p–∇un∇ϕdx–
g|un|q–ϕdx
≤C·max{C,C}+ ϕ
n +a un
p–p
|∇un|p–∇un∇ϕdx
+b
|∇un|p–∇un∇ϕdx–
g|un|q–ϕdx,
for alln>n∗.
Passing to the limit asn→ ∞, one has
lim
n→∞λ
f|un|–rϕdx≤a lim n→∞ un
p–p
∇u+λ
p–
∇u+λ∇ϕdx
+b
∇u+λp–∇u+λ∇ϕdx–
gu+λq–ϕdx.
Then using Fatou’s lemma again, we infer that
λ
fu+λ
–r
ϕdx
≤a lim
n→∞ un
p–p
∇u+
λ
p–
∇u+λ∇ϕdx
+b
∇u+λ
p–∇
u+λ∇ϕdx–
gu+λ
q–
Sinceun→u+λa.e. in, we getu+λ≥ a.e. in. Thus, one infers from (.) that
λ
fu+λ
–r
dx≤a lim
n→∞
un p
–p
u+λ
p
+bu+λ
p
–
gu+λ
q
dx. (.)
On the other hand
a lim
n→∞
un p
–p
u+λp+bu+λp≤a lim
n→∞ un
p
+b lim
n→∞ un
p
=λ
fu+λ
–r
dx+
gu+λ
q
dx. (.)
Combining (.) and (.), we have
lim
n→∞
un p= lim n→∞ un
p=u+
λ
p
. (.)
Thus, (.) can be obtained by inserting (.) into (.). Moreover, from (.), one has
∇u+λ
p–
∇u+λ∇ϕdx≥, ∀ϕ∈W
,p
(),ϕ≥.
Therefore, using the strong maximum principle for weak solutions (see []), we obtain
u+
λ> a.e. in.
Step:u+λis a weak solution of (.), andu+λ∈Nλ+. By (.), we haveun→u+λstrongly inW,p(), and souλ+∈Nλ+. Assumeφ∈W
,p
() andε> , define∈W ,p
() by:=
(u+λ+εφ)+. Then from Step it follows
≤
au+λ
p–p
+b∇u+λ
p–∇
u+λ∇–λfu+λ
–r
–gu+λ
q–
dx
=
[u+λ+εφ>]
au+λp–p+b∇uλ+p–∇u+λ∇u+λ+εφ
–λfu+λ
–r
u+λ+εφ
–gu+λ
q– u+λ+εφ
dx = –
[u+λ+εφ≤]
au+λ
p–p
+b∇u+λ
p–∇ u+λ∇
u+λ+εφ
–λfu+λ
–r
u+λ+εφ
–gu+λ
q– u+λ+εφ
dx
=au+λp+bu+λp–λ
fu+λ–rdx–
gu+λqdx
+ε
au+λ
p–p
+b∇u+λ
p–
∇u+λ∇φ–λfu+λ
–r
φ–gu+λ
q–
φdx
–
[u+λ+εφ≤]
au+λp–p+b∇uλ+p–∇u+λ∇u+λ+εφ
–λfu+λ
–r
u+λ+εφ
–gu+λ
q– u+λ+εφ
dx =ε
au+λ
p–p
+b∇u+λ
p–
∇u+λ∇φ–λfu+λ
–r
φ–gu+λ
q–
φdx
–
[u+
λ+εφ≤]
au+λ
p–p
+b∇u+λ
p–
∇u+λ∇
–λfu+λ
–r
u+λ+εφ
–gu+λ
q– u+λ+εφ
dx
≤ε
au+λ
p–p
+b∇u+λ
p–
∇u+λ∇φ–λfu+λ
–r
φ–gu+λ
q–
φdx
–εau+λ
p–p
+b
[u+λ+εφ≤]
∇u+λ
p–
∇u+λ∇φdx.
Since the measure of the domain of integration [u+λ+εφ≤] tends to zero asε→, it follows[u+
λ+εφ≤]|∇u
+
λ|p–∇u+λ∇φdx→. Dividing byεand lettingε→, we have
au+λ
p–p
+b
∇u+λ
p–
∇u+λ∇φdx–λ
fu+λ
–r
φdx–
gu+λ
q–
φdx≥.
Notice thatφis arbitrary, the inequality also holds for –φ, so it follows thatu+
λis a weak solution of (.). Moreover, from (.) and (.), we deduce thatu+
λ∈Nλ+. A similar argument shows that there exists another solutionu–λ∈Nλ–.
4 Proof of Theorem 1.2
(i) By Lemma .(i), we writeNλ=Nλ+and define
θ+= inf
u∈Nλ+J(u).
Similar to Lemma .(i), we haveθ+< . Applying Lemma .(i) and the Ekeland varia-tional principle, we see that there exists a minimizing sequence{un}forJ(u) inNλ+such that
(i) J(un) <θ++
n;
(ii) J(u) >J(un) –
n u–un , ∀u∈N +
λ.
Repeating the same argument as Theorem ., we can see thatuλ∈Nλ+is a positive solution of the problem (.).
(ii) Similar to the proof of Theorem ., we know that the problem (.) has at least two positive solutionsu+
λ∈Nλ+andu–λ∈Nλ–. Moreover, combining (.) with (.), we have
lim
a→–
u–λ=∞
and
lim
a→–
inf
u∈Nλ–J(u) =∞.
This completes the proof of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by Young Award of Shandong Province (ZR2013AQ008), NNSF (61603226), the Fund of Science and Technology Plan of Shandong Province (2014GGH201010) and NSFC (11671237).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 11 December 2016 Accepted: 9 March 2017
References
1. Kirchhoff, G: Mechanik. Teubner, Leipzig (1883)
2. Alves, CO, Corrêa, FJSA: On existence of solutions for a class of the problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal.8, 43-56 (2001)
3. Chipot, M, Lovat, B: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal.30, 4619-4627 (1997) 4. Liu, DC, Zhao, PH: Multiple nontrivial solutions to ap-Kirchhoff equation. Nonlinear Anal.75, 5032-5038 (2012) 5. Li, YX, Mei, M, Zhang, KJ: Existence of multiple nontrivial solutions for ap-Kirchhoff type elliptic problem involving
sign-changing weight functions. Discrete Contin. Dyn. Syst., Ser. B21, 883-908 (2016)
6. Chen, CY, Kuo, YC, Wu, TF: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ.250, 1876-1908 (2011)
7. Alves, CO, Corrêa, FJSA, Figueiredo, GM: On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl.
2, 409-417 (2010)
8. Cheng, BT, Wu, X: Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal.71, 4883-4892 (2009)
9. Ma, TF, Munoz Rivera, JE: Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett.16, 243-248 (2003)
10. Perera, K, Zhang, Z: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ.221, 246-255 (2006)
11. Liu, X, Sun, YJ: Multiple positive solutions for Kirchhoff type problems with singularity. Commun. Pure Appl. Anal.12, 721-733 (2013)
12. Liao, JF, Zhang, P, Liu, J, Tang, CL: Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity. J. Math. Anal. Appl.430, 1124-1148 (2015)
13. Lei, CY, Liao, JF, Tang, CL: Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents. J. Math. Anal. Appl.421, 521-538 (2015)
14. Brown, KJ, Wu, TF: A fibering map approach to a semilinear elliptic boundary value problem. Electron. J. Differ. Equ.
200769, (2007)
15. Brown, KJ, Wu, TF: A fibering map approach to a potential operator equation and its applications. Differ. Integral Equ.
22, 1097-1114 (2009)
16. Zhang, Z, Perera, K: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl.317, 456-463 (2006)
17. Ekeland, I: On the variational principle. J. Math. Anal. Appl.47, 324-353 (1974)