Bai Boundary Value Problems (2016) 2016:212 DOI 10.1186/s13661-016-0715-3

RESEARCH

Open Access

Multiplicity results for a fractional Kirchhoff equation involving sign-changing weight function Chuanzhi Bai* *

Correspondence: [email protected] Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, P.R. China

Abstract In this paper, we prove the existence and multiplicity of solutions for a fractional Kirchhoff equation involving a sign-changing weight function which generalizes the corresponding result of Tsung-fang Wu (Rocky Mt. J. Math. 39:995-1011, 2009). Our main results are based on the method of a Nehari manifold. MSC: 35J50; 35J60; 47G20 Keywords: fractional p-Laplacian; Kirchhoff type problem; sign-changing weight; Nehari manifold

1 Introduction In this paper, we consider the following fractional elliptic equation with sign-changing weight functions: ⎧  ⎨M(

RN

⎩u = ,

|u(x)–u(y)|p |x–y|N+sp

dx dy)(–)sp u = λf (x)uq + g(x)ur , x ∈ , x ∈ RN \ ,

(.)

where  is a smooth bounded domain in RN , N > s,  < s < ,  ≤ q <  < r < p∗s –  (p∗s =

pN ); N–ps

λ > , M(t) = a + bt p– , (–)sp is the fractional p-Laplacian operator defined as 

(–)sp u(x) =  lim

ε B (x)c ε

|u(x) – u(y)|p– (u(x) – u(y)) dy, |x – y|N+sp

x ∈ RN .

We may assume that the weight functions f (x) and g(x) are as follows: (H) f + = max{f , } ≡ , and f ∈ Lμq () where μq =

μ μ–(q+)

for some μ ∈ (q + , p∗s ),

with in addition f (x) ≥  a.e. in  in the case q = ; (H) g + = max{g, } ≡ , and g ∈ Lνr () where νr =

ν ν–(r+)

for some ν ∈ (r + , p∗s ).

The fractional Kirchhoff type problems have been studied by many authors in recent years; see [–] and references therein. In the subcritical case, Pucci and Saldi in [] stud© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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ied the following Kirchhoff type problem in RN : ⎧  ⎪ M( RN ⎪ ⎨ ⎪ ⎪ ⎩

|u(x)–u(y)|p |x–y|N+sp q–

= λw(x)|u|

dx dy)(–)sp u + V (x)|u|p– u

u – h(x)|u|r– u,

x ∈ , x ∈ RN \ ,

u = ,

with n > ps, s ∈ (, ), and they established the existence and multiplicity of entire solutions using variational methods and topological degree theory for the above problem with a real parameter λ under the suitable integrability assumptions of the weights V , w, and h. In [], Mishra and Sreenadh have studied the following Kirchhoff problem with sign-changing weights: ⎧  ⎨M(

RN

⎩u = ,

|u(x)–u(y)|p |x–y|N+sp

dx dy)(–)sp u = λf (x)|u|q– u + |u|α– u, x ∈ , x ∈ RN \ ,

and they obtained the multiplicity of non-negative solutions in the subcritical case α < p∗s by minimizing the energy functional over non-empty decompositions of Nehari manifold. When p = , s = , a =  and b = , problem (.) is reduced to the following semilinear elliptic equation: ⎧ ⎨–u = λf (x)uq + g(x)ur , x ∈ , ⎩u = , x ∈ ∂.

(.)

In [], Wu proved that equation (.) involving a sign-changing weight function has at least two solutions by using the Nehari manifold. Motivated by the above work, in this paper, we investigate the existence and multiplicity of solutions for a fractional Kirchhoff equation (.) and extend the main results of Wu []. This article is organized as follows. In Section , we give some notations and preliminaries. Section  is devoted to the proof that problem (.) has at least two solutions for λ sufficiently small.

2 Preliminaries For any s ∈ (, ),  < p < ∞, we define   |u(x) – u(y)|p X = u|u : RN → R is measurable, u| ∈ Lp (), and dx dy < ∞ , n+ps Q |x – y| where Q = RN \ (C  × C ) with C  = RN \ . The space X is endowed with the norm defined by

 uX = uLp () + Q

/p |u(x) – u(y)|p dx dy . |x – y|n+ps

The functional space X denotes the closure of C∞ () in X. By [], the space X is a Hilbert space with scalar product  u, v X =

Q

|u(x) – u(y)|p– (v(x) – v(y)) dx dy, |x – y|n+ps

∀u, v ∈ X ,

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and the norm

 uX =

Q

/p |u(x) – u(y)|p dx dy . |x – y|n+ps

For further details on X and X and also for their properties, we refer to [] and the references therein. Throughout this section, we denote the best Sobolev constant by Sl for the embedding of X into Ll (), which is defined as  Sl = inf

X \{}

RN

(



|u(x)–u(y)|p |x–y|N+sp

RN

dx dy > ,

p

|u|l dx) l

where l ∈ [p, p∗s ]. A function u ∈ X is a weak solution of problem (.) if



 |u(x) – u(y)|p |u(x) – u(y)|p– (u(x) – u(y))(v(x) – v(y)) M dx dy dx dy N+sp |x – y|N+sp Q |x – y| Q   = λ f (x)|u|q– uv dx + g(x)|u|r– uv dx, ∀v ∈ X . 



Associated with equation (.), we consider the energy functional Jλ,M in X λ  ˆ p Jλ,M (u) = M uX – p q+

 f |u|

q+



 dx – r+

 g|u|r+ dx, 

t ˆ where M(t) =  M(μ) dμ. It is easy to see that the solutions of equation (.) are the critical points of the energy functional Jλ,M . The Nehari manifold for Jλ,M is defined as      Nλ,M () = u ∈ X \ {} : Jλ,M (u), u =     p p q+ r+ = u ∈ X \ {}|M uX uX – λ f |u| dx – g|u| dx =  . 



The Nehari manifold Nλ,M () is closely linked to the behavior of functions of the form hλ,M : t → Jλ,M (tu) for t > , named fibering maps []. If u ∈ X , we have   t q+  ˆ p t r+ p hλ,M (t) = M t uX – λ f |u|q+ dx – g|u|r+ dx, p q+  r+    p p p  p– q q+ r hλ,M (t) = t M t uX uX – λt f |u| dx – t g|u|r+ dx, 



and p p p p hλ,M (t) = (p – )t p– M t p uX uX + pt p– M t p uX uX   – qλt q– f |u|q+ dx – rt r– g|u|r+ dx. 



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Obviously, p p thλ,M (t) = M t p uX tuX – λ



  = Jλ,M (tu), tu ,

 f |tu|q+ dx –

g|tu|r+ dx





which implies that for u ∈ X \ {} and t > , hλ,M (t) =  if and only if tu ∈ Nλ,M (), i.e., positive critical points of hλ,M correspond to points on the Nehari manifold. In particular, hλ,M () =  if and only if u ∈ Nλ,M (). Hence, we define   + Nλ,M () = u ∈ Nλ,M () : hu,M () >  ,    Nλ,M () = u ∈ Nλ,M () : hu,M () =  ,   – Nλ,M () = u ∈ Nλ,M () : hu,M () <  . For each u ∈ Nλ,M (), we have p p p p hλ,M () = (p – )M uX uX + pM uX uX   – qλ f |u|q+ dx – r g|u|r+ dx 



p p p p = (p – r – )M uX uX + pM uX uX – λ(q – r) p p p p = (p – q – )M uX uX + pM uX uX – (r – q)

 f |u|q+ dx (.)





g|u|r+ dx. (.) 

 (), then hλ,M () = , and Let M(t) = a + bt p– , where a > , b ≥  and p > . If u ∈ Nλ,M we have by (.) and (.)

a(p – r

p – )uX



p + b p – r –  uX – λ(q – r)



p p a(p – q – )uX + b p – q –  uX – (r – q)





f |u|q+ dx = ,

(.)



g|u|r+ dx = .

(.)



⎧ ⎨> p – , b = , (H)  < q < , p >  + q and p∗s –  > r ⎩> p – , b = . For convenience, we let

Lemma . If (H) and (H) hold, then the energy functional Jλ,M is coercive and bounded below on Nλ,M (). Proof For u ∈ Nλ,M (), we have by the Hölder and Sobolev inequalities  

    p p – uX + b  – uX p r+ p r+ 

  – –λ f |u|q+ dx q+ r+   



    p p – uX + b  – uX =a p r+ p r+

Jλ,M (u) = a

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 r–q –λ f |u|q+ dx (q + )(r + ) 

 

    p p ≥a – uX + b  – uX p r+ p r+ r–q q+ f Lμq Sμq+ uX , –λ (q + )(r + ) where μq =

μ , μ–(q+)

μ ∈ (q + , p∗s ). Thus Jλ,M is coercive and bounded below on Nλ,M (). 

Lemma . Let (H)-(H) hold. There exists λ >  such that for any λ ∈ (, λ ), we have  Nλ,M () = ∅.  Proof If not, that is, Nλ,M () = ∅ for each λ > , then by (.) and the Hölder and Sobolev  () inequalities, we have for u ∈ Nλ,M



p p p a(r – p + )u X ≤ a(r – p + )u X + b r – p +  u X  = λ(r – q) f |u |q+ dx, 

which implies that λ(r – q) a(r – p + )

p

u X ≤

 f |u |q+ dx 

λ(r – q) q+ f Lμq Sμq+ u X a(r – p + )

≤ and so

u X ≤

λ(r – q) f Lμq Sμq+ a(r – p + )

  p–q–

.

(.)

Similarly, we obtain by (.) and the Hölder and Sobolev inequalities p

u X ≤

r–q gLνr Sνr+ u r+ X , a(p – q + )

which implies that

u X ≥

a(p – q + ) –(r+) g– Lνr Sν r–q

  r–p+

.

(.)

But (.) contradicts (.) if λ is sufficiently small. Hence, we conclude that there exists  () = ∅ for λ ∈ (, λ ).  λ >  such that Nλ,M Let cλ =

inf

u∈Nλ,M ()

Jλ,M (u).

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+ – From Lemma ., for λ ∈ (, λ ), we write Nλ,M () = Nλ,M () ∪ Nλ,M () and define

c+λ =

inf

+ () u∈Nλ,M

Jλ,M (u) and c–λ =

Jλ,M (u).

inf

– () u∈Nλ,M

 + Lemma . (i) If u ∈ Nλ,M (), then  f |u|q+ dx > .  – (), then  g|u|r+ dx > . (ii) If u ∈ Nλ,M The proof is immediate from (.) and (.). Define the function ku : R+ → R as follows: ku (t) = t

p–q–



M t

p

p p uX uX

 –t

r–q

g|u|r+ dx

t > .

(.)



Obviously, tu ∈ Nλ,M () if and only if ku (t) = λ



q+ dx.  f |u|

Moreover,

p p p p ku (t) = (p – q – )t p–q– M t p uX uX + pt p–q– M t p uX uX  – (r – q)t r–q– g|u|r+ dx,

(.)

 + – which implies that t q ku (t) = hλ,M (t) for tu ∈ Nλ,M (). That is, u ∈ Nλ,M () (or Nλ,M ()) if  and only if ku (t) >  (or < ). Set

 p–q–

r–p+ a(r – p + ) a(p – q – ) A= r+ ν r–q (r – q)gL r Sν  –q–  pr–p+

a(p – q – ) b(r – p + ) + . r–q (r – q)gLνr Sνr+

(.)

Lemma . Assume that (H)-(H) hold. Let λ =

A q+ . f Lμq Sμ

Then, for each u ∈ X \ {}

and λ ∈ (, λ ), we have:  () If  f |u|q+ dx ≤ , then there exists a unique t – = t – (u) > tmax (u) such that – () and t – u ∈ Nλ,M Jλ,M t – u = sup Jλ,M (tu) > .

(.)

t≥

 () If  f |u|q+ dx > , then there exists a unique  < t + = t + (u) < tmax (u) < t – such that + – (), t – u ∈ Nλ,M () and t + u ∈ Nλ,M Jλ,M t + u =

inf

≤t≤tmax (u)

Jλ,M (tu),

Jλ,M t – u = sup Jλ,M (tu).

(.)

t≥

Proof From (.) and (.), we have p

 –q–

ku (t) = at p–q– uX + bt p

p

uX – t r–q

 g|u|r+ dx

t ≥ ,



and   

 p p ku (t) = t –q– a(p – q – )t p– uX + b p – q –  t p – uX – (r – q)t r g|u|r+ dx , 

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which implies that ku () = , ku (t) → –∞ as t → ∞, limt→+ ku (t) >  and limt→∞ ku (t) < . Thus there exists a unique tmax (u) := tmax >  such that ku (t) is increasing on (, tmax ), decreasing on (tmax , ∞) and ku (tmax ) = . Moreover, tmax is the root of p p– a(p – q – )tmax uX



p – p r + b p – q –  tmax uX – (r – q)tmax

 g|u|r+ dx = .

(.)



From (.), we obtain

tmax ≥

p

a(p – q – )uX  (r – q)  g|u|r+ dx

  r–p+

 ≥ uX



a(p – q – ) (r – q)gLνr Sνr+

  r–p+

:= t∗ .

(.)

Hence, we have by (.), (.), and the Hölder and Sobolev inequalities ku (tmax ) =

p–q– tmax

   p p p(p–) r–p+ r+ auX + btmax uX – tmax g|u| dx 



=

b(r – p + ) p –q– a(r – p + ) p–q– p p tmax uX + tmax uX r–q r–q

a(r – p + ) p–q– b(r – p + ) p –q– p p t∗ t∗ uX + uX r–q r–q

 p–q– r–p+ a(p – q – ) a(r – p + ) q+ ≥ uX r+ r–q (r – q)gLνr Sν ≥

 –q–

 pr–p+ b(r – p + ) a(p – q – ) q+ + uX r–q (r – q)gLνr Sνr+

q+

= AuX .

(.)

  Case ():  f |u|q+ dx ≤ . Then ku (t) = λ  f |u|q+ dx has unique solution t – > tmax and ku (t – ) < . On the other hand, we have  p

 p a(p – q – )t – uX + b p – q –  t – uX – (r – q) 





 r+ g t – u dx



 p–q–

p –q– +q p p a(p – q – ) t – uX + b p – q –  t – uX = t– r–q– – (r – q) t – +q  – ku t <  = t–



 r+

g|u|

dx



and   – –  Jλ,M t u , t u p p q+ p p = a t – uX + b t – uX – λ t –    – – q+ q+ ku t – λ f |u| dx = . = t 

 

r+ f |u|q+ dx – t –

 g|u|r+ dx 

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– Hence, t – u ∈ Nλ,M () or t – = . For t > tmax , we obtain



p p a(p – q – )tuX + b p – q –  tuX – (r – q)

 g|tu|r+ dx < , 

d Jλ,M (tu) < , dt    d  p p Jλ,M (tu) = at p– uX + bt p – uX – λt q f |u|q+ dx – t r g|u|r+ dx = , dt   for t = t – . Thus, Jλ,M (u) = supt≥ Jλ,M (tu). Furthermore, we have a b   r+ p p Jλ,M (u) ≥ Jλ,M (tu) ≥ t p uX +  t p uX – t p p r+

 g|u|r+ dx,

t ≥ .



Let a b   r+ p p t hu (t) = t p uX +  t p uX – p p r+

 g|u|r+ dx,

t ≥ .



Similar to the argument in the function ku (t), we see that hu (t) achieves its maximum at tm ≥ ( 

p   r–p+ r+ g|u| dx

auX 

)

. Thus, we have

p  r–p+

aur+ ap(r +  – p) + b(r +  – p ) X  Jλ,M (u) ≥ hu (tm ) ≥ > . r+ dx p (r + )  g|u|

Case ():



q+ dx > .  f |u|

By (.) and



q+

ku () =  < λ 

f |u|q+ dx ≤ λf Lμq Sμq+ uX q+

q+

< λ f Lμq Sμq+ uX = AuX ≤ ku (tmax ),

for λ ∈ (, λ ).

Then there exist t + and t – such that  < t + < tmax < t – , ku t + = λ



f |u|q+ dx = ku t – . 

Moreover, we have ku (t + ) >  and ku (t – ) < . Thus, there are two multiples of u lying in + – Nλ,M (), that is, t + u ∈ Nλ,M () and t – u ∈ Nλ,M (), and Jλ,M (t – u) ≥ Jλ,M (tu) ≥ Jλ,M (t + u) for each t ∈ [t + , t – ] and Jλ,M (t + u) ≤ Jλ,M (tu) for each t ∈ [, t + ]. Hence, t – =  and

Jλ,M (u) = sup Jλ,M (tu), t≥

Jλ,M t + u =

inf

≤t≤tmax

Jλ,M (tu).

Lemma . If (H) holds, then we have cλ ≤ c+λ < . + , we get Proof For u ∈ Nλ,M

 (r – q)λ 



p p f |u|q+ dx > a(r – p + )uX + b r – p +  uX .



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Thus, we have  a(r – p + ) b(r – p + ) λ(r – q) p p uX + u – f |u|q+ dx X p(r + ) p (r + ) (q + )(r + )      a(r – p + )   b(r – p + )   p p < – uX + uX < , –  r+ p q+ r+ p q+

Jλ,M (u) =

which implies that cλ ≤ c+λ < .



3 Main results Using the idea of Ni-Takagi [], we have the following. Lemma . For each u ∈ Nλ,M (), there exist >  and a differentiable function ξ : B(; ) ⊂ X → R+ such that ξ () = , the function ξ (v)(u – v) ∈ Nλ,M () and    ξ (), v =

W p a(p – q – )uX

+ b(p

p

– q – )uX – (r – q)



r+ dx  g|u|

,

(.)

for all v ∈ X , where 

|u(x) – u(y)|p– (u(x) – u(y))(v(x) – v(y)) dx dy |x – y|N+sp Q   |u(x) – u(y)|p – (u(x) – u(y))(v(x) – v(y))  + bp dx dy |x – y|N+sp Q   – (q + )λ f |u|q– uv dx – (r + ) g|u|r– uv dx.

W = ap



(.)



Proof For u ∈ Nλ,M (), we define a function F : R × X → R by

   ξ (u – w) , ξ (u – w) Fu (ξ , w) = Jλ,M p p = ξ p M ξ p u – wX u – wX   – ξ q+ λ f |u – w|q+ dx – ξ r+ g|u – w|r+ dx 

= aξ

p



p u – wX



+ bξ

p

p u – wX



f |u – w|q+ dx – ξ r+

– ξ q+ λ 

g|u – w|r+ dx. 

 Then Fu (, ) = Jλ,M (u), u =  and

  d p p Fu (, ) = apuX + bp uX – (q + )λ f |u|q+ dx – (r + ) g|u|r+ dx dξ   

p p = a(p – q – )uX + b p – q –  uX – (r – q) g|u|r+ dx = . 

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From the implicit function theorem, we know that there exist >  and a differentiable function ξ : B(; ) ⊂ X → R such that ξ () = ,    ξ (), v =

W p a(p – q – )uX

p

+ b(p – q – )uX – (r – q)



r+ dx  g|u|

,

where W is as in (.), and

Fu ξ (v), v =  for all v ∈ B(; ) which is equivalent to  

 Jλ,M ξ (v)(u – v) , ξ (v)(u – v) = 

for all v ∈ B(; ),

which implies that ξ (v)(u – v) ∈ Nλ,M ().



Similar to the argument in Lemma ., we can obtain the following lemma. – Lemma . For each u ∈ Nλ,M (), there exist >  and a differentiable function ξ – : – + – () and B(; ) ⊂ X → R such that ξ () = , the function ξ – (v)(u – v) ∈ Nλ,M



  ξ – (), v =

W p a(p – q – )uX

+ b(p

p

– q – )uX – (r – q)



r+ dx  g|u|

,

for all v ∈ X , where W is as in (.). Let (H) p <  + (r–)q . r Moreover, we let p∗ =

(p – )r –q r–

and

λ =

a(p – q – )(r – p + ) (r – q)(p – q – )

×



 q+

f Lμq Sμ



a(p – q – ) r–q 





(p–q–) (p–q––p∗ )(r–)

(p–q–) (r–)(p–q––p∗ )

gLνr Sνr+

.

Remark . By (H) we know that p∗ < . Lemma . Assume that (H)-(H) hold. Let  = min{λ , λ , λ }, then for λ ∈ (,  ): (i) There exists a minimizing sequence {un } ⊂ Nλ,M () such that

Jλ,M (un ) = cλ + o(),

 Jλ,M (un ) = o() in (X )∗ .

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– (ii) There exists a minimizing sequence {un } ⊂ Nλ,M () such that

Jλ,M (un ) = c–λ + o(),

 Jλ,M (un ) = o() in (X )∗ .

Proof By the Ekeland variational principle [] and Lemma ., there exists a minimizing sequence {un } ⊂ Nλ,M () such that

Jλ,M (un ) < cλ +

 n

(.)

and  Jλ,M (un ) < Jλ,M (w) + w – un X n

∀w ∈ Nλ,M ().

(.)

Let n large enough, by Lemma ., we obtain

Jλ,M (un ) =

a(r – p + ) b(r – p + ) λ(r – q) p p un X + un X –  p(r + ) p (r + ) (q + )(r + )

< cλ +

 f |un |q+ dx 

 cλ < , n 

which implies that q+

f Lμq Sμq+ un X ≥

 f |un |q+ dx > – 

(q + )(r + ) cλ > . λ(r – q) 

(.)

This implies un =  and by using (.), (.), and the Hölder inequality, we get   q+  (q + )(r + ) cλ –(q+) f – S un X > – Lμq μ λ(r – q) 

(.)

and  un X <

λp(r – q)(r + ) f Lμq Sμq+ a(q + )(r + )(r – p + )

  p–q–

.

(.)

In the following, we will prove that    J (un ) λ,M (X

)

∗

→  as n → ∞.

By using Lemma . with un we get the functions ξn : B(; n ) → R+ for some n > , such that ξn (w)(un – w) ∈ Nλ,M (). For fixed n ∈ N, we choose  < ρ < n . Let u ∈ X with u =  ρu . Set ηρ = ξn (wρ )(un – wρ ), since ηρ ∈ Nλ,M (), we deduce from (.) that and let wρ = u X 

 Jλ,M (ηρ ) – Jλ,M (un ) ≥ – ηρ – un X n

∀w ∈ Nλ,M (),

and by the mean value theorem, we obtain 

   Jλ,M (un ), ηρ – un + o ηρ – un X ≥ – ηρ – un X . n

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Hence,  

    Jλ,M (un ), –wρ + ξn (wρ ) –  Jλ,M (un ), un – wρ

 ≥ – ηρ – un X + o ηρ – un X . n

(.)

By ξn (wρ )(un – wρ ) ∈ Nλ,M () and (.) it follows that   (un ), –ρ Jλ,M

 

  u  (un ) – Jλ,M (ηρ ), un – wρ + ξn (wρ ) –  Jλ,M uX

 ≥ – ηρ – un X + o ηρ – un X . n Thus, 

 Jλ,M (un ),



u   ηρ – un X + o ηρ – un X ≤ uX nρ ρ +

 (ξn (wρ ) – )    Jλ,M (un ) – Jλ,M (ηρ ), un – wρ . ρ

(.)

Since     ηρ – un X ≤ ρ ξn (wρ ) + ξn (wρ ) – un X and lim

n→∞

 |ξn (wρ ) – |  ≤ ξn (), ρ

taking the limit ρ →  in (.), we obtain 

 Jλ,M (un ),

  u C   + ξn () ≤ uX n

for some constant C > , independent of ρ. In the following, we will show that ξn () is uniformly bounded in n. From (.), (.), and the Hölder inequality, we obtain for some κ >    ξn (), v ≤

κvX p a(p – q – )un X

+ b(p

p

– q – )un X – (r – q)



r+ dx  g|un |

.

We only need to prove that     

 a(p – q – )un p + b p – q –  un p – (r – q) g|un |r+ dx > c X X  

(.)



for some c >  and n large enough. If (.) is fails, then there exists a subsequence {un } such that

p p a(p – q – )un X + b p – q –  un X – (r – q)

 g|un |r+ dx = o(). 

(.)

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Combining (.) with (.), we may find a suitable constant d >  such that  g|un |r+ dx ≥ d

for n sufficiently large.

(.)



By (.) and un ∈ Nλ,M (), we have  f |un |q+ dx

λ 

p = aun X

p + bun X

 g|un |r+ dx

– 

 



 p p – a p – q –  u  + b p – q –  u  g|un |r+ dx n n X X p – q –   

 p p ≥  a(p – q – )un X + b p – q –  un X – g|un |r+ dx p –q–    r–q =  g|un |r+ dx – g|un |r+ dx + o() p –q–     r–p + =  g|un |r+ dx + o(). p –q–  =

(.)

Moreover, we have by (.) and (.)

p p p a(p – q – )un X ≤ a(p – q – )un X + b p – q –  un X  = (r – q) g|un |r+ dx + o() 

≤λ



(p – q – )(r – q) r – p + 

 f |un |q+ dx + o() 

(p – q – )(r – q) q+ ≤λ f Lμq Sμq+ un X + o(), r – p +  which implies that

un X ≤ λ

(p – q – )(r – q) f Lμq Sμq+ a(p – q – )(r – p + )

  p–q–

+ o().

(.)

Let

Iλ,M (u) = K(p, q, r) 

pr

uX

  r–

r+ dx  g|un |

 f |u|q+ dx,

–λ 

where

K(p, q, r) =

a(p – q – ) r–q

r  r–

r – p +  . p – q – 

From (.), it is easy to see that p

un X ≤

r–q a(p – q – )

 g|un |r+ dx. 

(.)

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Thus,  r

r–q r  r+ r  a(p – q – ) r– r – p +  ( a(p–q–) ) (  g|un | dx) r–  Iλ,M (un ) ≤ r+ dx r–q p – q –   g|un |  r – p +  –  g|un |r+ dx + o() p –q– 

= o().

(.)

But, by (.), (.), and λ ∈  ,

Iλ,M (un ) ≥ K(p, q, r)

  r–

pr

un X gLνr Sνr+ un r+ X

q+

– λf Lμq Sμq+ un X

 r+

q+ p∗ q+ –r = un X K(p, q, r)gL–r νr Sν un X – λf Lμq Sμ 

≥

q+ un X

   r+ –r K(p, q, r)gL–r λ νr Sν

– λf Lμq Sμq+

 Jλ,M (un ),

p∗  p–q–

,

which contradicts (.), where p∗ = Hence, we obtain 

(p – q – )(r – q) f Lμq Sμq+ a(p – q – )(r – p + )

(p–)r r–

– q < .

 u C ≤ . uX n

This completes the proof of (i). Similarly, we can prove (ii) by using Lemma ..



Theorem . Assume that (H)-(H) hold. For each  < λ <  ( is as in Lemma .), + the functional Jλ,M has a minimizer u+λ in Nλ,M () satisfying: + + () Jλ,M (uλ ) = cλ = cλ ; () u+λ is a solution of (.). Proof By Lemma .(i), there exists a minimizing sequence {un } ⊂ Nλ,M () for Jλ,M on Nλ,M () such that

Jλ,M (un ) = cλ + o(),

 Jλ,M (un ) = o() in (X )∗ .

From Lemma . and the compact embedding theorem, we see that there exist a subsequence {un } and u+λ ∈ X such that un  u+λ

weakly in X

un → u+λ

strongly in Lη () for  < η < p∗s .

and (.)

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In the following we will prove that Hölder inequality we can obtain 



+ q+ dx  f |uλ |

= . In fact, if not, by (.) and the

 f |un |q+ dx →



f |u+λ |q+ dx =  

as n → ∞. Hence, 

p

p

aun X + bun X =

g|un |r+ dx + o() 

and

Jλ,M (un ) = a

 

    p p – un X + b  – un X + o(), p r+ p r+

which contradicts Jλ,M (un ) → cλ <  as n → ∞. Furthermore,      +  o() = Jλ,M uλ , φ + o() (un ), φ = Jλ,M

for all φ ∈ X .

Thus, u+λ ∈ Nλ,M () is a nonzero solution of (.) and Jλ,M (u+λ ) ≥ cλ . Next, we will prove that Jλ,M (u+λ ) = cλ . Since    q+  r+ a  p b  p λ  Jλ,M u+λ = u+λ X +  u+λ X – f u+λ  dx – g u+λ  dx   p p q+  r+ 



     a  a u+ p + b – b u+ p = – λ X λ X p r+ p r +  

 q+ λ λ – + f u+λ  dx r+ q+   



a b b a p p – un X + un X – ≤ lim inf  n→∞ p r+ p r+

  λ λ + f |un |q+ dx – r+ q+  = lim inf Jλ,M (un ) = cλ . n→∞

+ – Hence, Jλ,M (u+λ ) = cλ . Moreover, we have u+λ ∈ Nλ,M (). In fact, if u+λ ∈ Nλ,M (), by + – + – + + – + Lemma ., there are unique t and t such that t uλ ∈ Nλ,M () and t uλ ∈ Nλ,M (), + – we have tλ < tλ = . Since



d Jλ,M tλ+ u+λ =  dt

and



d Jλ,M tλ+ u+λ > ,  dt

there exists tλ+ < t ∗ ≤ tλ– such that Jλ,M (tλ+ u+λ ) < Jλ,M (t ∗ u+λ ). By Lemma ., we get





Jλ,M tλ+ u+λ < Jλ,M t ∗ u+λ ≤ Jλ,M tλ– u+λ = Jλ,M u+λ , + which is a contradiction. Since Jλ,M (u+λ ) = Jλ,M (|u+λ |) and |u+λ | ∈ Nλ,M (), we see that u+λ is a solution of (.) by Lemma .. 

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Similarly, we can obtain the theorem of existence of a local minimum for Jλ,M on – Nλ,M () as follows. Theorem . Assume that (H)-(H) hold. For each  < λ <  ( is as in Lemma .), – () satisfying: the functional Jλ,M has a minimizer u–λ in Nλ,M – – () Jλ,M (uλ ) = cλ ; () u–λ is a solution of (.). Finally, we give the main result of this paper as follows. Theorem . Suppose that the conditions (H)-(H) hold. Then there exists  >  such that for λ ∈ (,  ), (.) has at least two solutions. Proof From Theorems ., ., we see that (.) has two solutions u+λ and u–λ such that + + – – (), u–λ ∈ Nλ,M (). Since Nλ,M () ∩ Nλ,M () = ∅, we see that u+λ and u–λ are u+λ ∈ Nλ,M different.  Remark . Obviously, if p = , then (H) and (H) hold. Moreover, if p = , s = , a = , and b = , then Theorem . is in agreement with Theorem . in [].

Competing interests The author declares that he has no competing interests. Author’s contributions All results belong to CB. Acknowledgements This work is supported by Natural Science Foundation of China (11571136 and 11271364). Received: 5 August 2016 Accepted: 11 November 2016 References 1. Wu, TF: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mt. J. Math. 39, 995-1011 (2009) 2. Autuori, G, Fiscella, A, Pucci, P: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699-714 (2015) 3. 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) 4. Fiscella, A, Valdinoci, E: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156-170 (2014) 5. Pucci, P, Saldi, S: Critical stationary Kirchhoff equations in RN involving nonlocal operators. Rev. Mat. Iberoam. 32, 1-22 (2016) 6. Pucci, P, Xiang, M, Zhang, B: Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN . Calc. Var. Partial Differ. Equ. 54(3), 2785-2806 (2015) 7. Mishra, PK, Sreenadh, K: Existence and multiplicity results for fractional p-Kirchhoff equation with sign changing nonlinearities. Adv. Pure Appl. Math. (2015). doi:10.1515/apam-2015-0018 8. Di Nezza, E, Palatucci, G, Valdinoci, E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521-573 (2012) 9. Drabek, P, Pohozaev, SI: Positive solutions for the p-Laplacian: application of the fibering method. Proc. R. Soc. Edinb. A 127, 703-726 (1997) 10. Ni, WM, Takagi, I: On the shape of least energy solution to a Neumann problem. Commun. Pure Appl. Math. 44, 819-851 (1991) 11. Ekeland, I: On the variational principle. J. Math. Anal. Appl. 17, 324-353 (1974)