Infinitely many solutions for p harmonic equation with singular term

R E S E A R C H

Open Access

Infinitely many solutions for

p

-harmonic

equation with singular term

Huazhao Xie

_{and Jianping Wang}

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Henan University of Economics and Law, Zhengzhou, 450002, China Full list of author information is available at the end of the article

Abstract

In this paper, we study the followingp-harmonic problem involving the Hardy term:

(|

)

λ

|x|2p =f(x,u), in

∂

∂

∂

where

0≤

λ

above problem has infinitely many solutions with positive energy levels.

AMS Subject Classification: 35J60; 35J65

Keywords: infinitely many solutions;p-harmonic; Hardy term

1 Introduction

The main purpose of this paper is to show the existence of infinitely many solutions for the followingp-harmonic equation:

⎧ ⎨ ⎩

(|u|p–u) –λ|u_||p–u

x|p =f(x,u), x∈, u=∂u

∂n= , x∈∂,

(.)

whereis an open bounded domain containing the origin inRN_{, the boundary∂is} smooth.  <p<N

, ≤λ<λ= [N(p– )(N– p)/p _]p_. ∂

∂n is the outer normal derivative. Nonlinearityf(x,u) satisfies the following conditions:

(f)f(x,u) is continuous on×Rand limits subcritical growing at infinity; that is,

lim

u→∞ f(x,u)

|u|p*_–= , uniformly forx∈, (.)

wherep*_{=Np/(N– p) is the critical exponent of Sobolev’s embeddingW},p

 ()→Ls().

(f) For anyx∈,f(x,u) satisfies

lim

u→

f(x,u)u

|u|p =  and ulim→∞ f(x,u)u

|u|p = +∞. (.)

(f) DenoteG(x,t) =f(x,t)t–p

t

f(x,s)ds. For allx∈, there exists a constantM≥

such that

G(x,t)≤G(x,t) for anyM≤ |t| ≤ |t|. (.)

(f)f(x,u) is odd with respect tou.

By the Hardy-Rellich inequality (see [, ]), we know that

|u|p |x|pdx≤

 λ

|u|pdx. (.)

Obviously, for anyλ∈[,λ),

 –λ λ

|u|pdx≤

|u|p–λ|u| p

|x|p

dx≤

|u|pdx.

InW_,p(), forλ∈[,λ), we define

u=u_W,p

 ():=

|u|p–λ|u| p

|x|p

dx



p

,

this norm is equivalent to (|u|p_dx)/p_.

A weak solution of the problem (.) is a critical point of the energy functional

I(u) = 

p

|u|p–λ|u| p

|x|p

dx–

F(x,u)dx, u∈W_,p(), (.)

whereF(x,u) =_uf(x,s)ds. It is easy to check thatI(u) is a continuous even functional. Biharmonic equations can describe the static form change of a beam or the sport of a rigid body. For example, this type of equation furnishes a model for studying traveling wave in suspension bridges (see []). By using variational arguments, many authors investigated nonlinear biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions and got interesting results (see [–]).

Li and Squassina in [] considered the superlinearp-harmonic equation with Navier boundary conditions

|u|p–u=f(x,u), in, u=u=  on∂, (.)

whereis an open bounded domain inRN_{with a smooth boundary∂.p> ,N≥p+ } andf:×R→Ris a Carathéodory function such that for some positive constantC,

f(x,u)≤C  +|u|q– for allq∈,p*. (.)

By means of the Morse theory, they proved the existence of two nontrivial solutions to (.). After that, Li and Tang [] considered a more general problem than (.). They got three solutions by the three critical points theorem which was obtained in []. If

The authors in [] considered thep-harmonic equation with Dirichlet conditions and obtained the existence of a nontrivial solution. Under the condition of (.), [] proved the existence and multiplicity of weak solutions for the nonuniformly nonlinear problem

a(x,u)=f(x,u), in, u=∂u

∂n=  on∂, (.)

where|a(x,t)| ≤c(h(x) +h(x)|t|p–) withh(x)≥,h(x)≥.a(x,t) andf(x,t) are odd

with respect to the second variable. Furthermore, f(x,t) is subcritical and satisfies the Ambrosetti-Rabinowitz condition, that is, there exists a constantθ>psuch that

 <θ

t



f(x,s)ds≤f(x,t)t for anyx∈. (.)

The aim of this paper is to obtain infinitely many solutions for the problem (.) when p<N. The main difficulty lies in the fact that the embeddingsW_,p()→Lp*_{() and}

W_,p()→Lp(,|x|–pdx) are not compact. Furthermore, the assumption (f) is not the

usual subcritical growth (.). The condition (f) is weaker than the A-R condition (.).

We use the concentration compactness principle (see [, ]) to overcome those difficul-ties. The following theorem is our main result.

Theorem . Assume f(x,u)satisfies(f)-(f),then the problem(.)possesses infinitely many weak solutions and the corresponding critical values are positive.

This paper proceeds as follows. In the next section, we prove the energy functionalI(u) satisfies the Palais-Smale condition. In Section , by using the symmetric mountain pass theorem (see []), we get the main result of this paper. Throughout the paper, denote

Sp,λ:= inf u∈W_,p()\{}

up up_p*

.

We useuq= (|u|q_dx)/q_{to denote the norm ofL}q_(),C,C

istand for universal con-stants. We omitdxandin the integrals if there is no other indication.

2 Palais-Smale condition

To show the (PS) sequence{un}of the variational functionalI(u) is compact inW_,p(), we first prove the boundedness of{un}by the analytic argument which has been used in []. Then, using the concentration compactness principle, we get the compactness of{un}.

Lemma . The condition(f)implies that for anyε> ,there exists a positive constant Cεsuch that

F(x,u)≤ε|u|p*+Cε|u|, (x,u)∈×R. (.)

(f)implies that F(x,u) =o(up)as u→.Furthermore,for any|u| ≥M> ,there exists a small positive constantθ

Lemma . Under the conditions(f), (f)and(f),if the sequence{un} ∈W,p()satisfies

I(un)→c, I(un)→ as n→+∞, (.)

then the sequence{un}is bounded in W_,p().

Proof We prove this lemma by contradiction. Without loss of generality, we assume that

un →+∞ asn→+∞. (.)

Setwn=un/un, thenwn=  for anyn∈N. Asn→+∞, there existsw∈W,p() such

that

wn w weakly inW,p(),

wn→w strongly inLs() for any ≤s<p*,

wn→w a.e.x∈.

Ifw ≡, we denote={x∈:w= }, set\is nonempty. (.) implies that

unp–

f(x,un)undx=o().

Therefore,

 –o() =

f(x,un)un unp dx=



f(x,un)un unp dx+

\

f(x,un)un

unp dx. (.)

Asn→+∞,|un|p_{dx≤ un →. So,|un| →+∞forx∈\}

. It follows from (f)

that

lim

un→∞

f(x,un)un unp =_unlim_→∞

f(x,un)un |un|p |wn|

p_{= +∞.}

Since|\|> , then

\

f(x,un)un

unp dx→+∞ asn→+∞. (.)

Using (.) and (.), we derive that



f(x,un)un unp dx=

 unp

(|un|>M)

· · ·+

(|un|≤M)

· · ·

≥– 

unp

(|un|≤M)

f(x,un)undx

≥– C

unp M+M p*_|

|

From (.) and (.), we get

f(x,un)un

unp dx→+∞ asn→+∞. (.)

This contradicts (.).

Ifw≡, thenwn→ strongly inL(). SinceI(un)→c, we know that

c←  pun

p_–

F(x,un)dx≥ 

pun

p_–ε

|un|p*dx–Cε

|un|dx. (.)

In (.), chooseε= , then



pun

p_{≤ un}p*

p*+Cun+c≤Cunp

*

p*+c.

From (.), it follows thatunp*→+∞asn→+∞. Denote

An=M

un

unp* ≥

MS/_p,pλ, (.)

whereMis any fixed positive constant. It is clear that

An un=

M

unp* → (asn→+∞).

Therefore, fornlarge enough,An/un ∈(, ).

For everyn∈N, we define a sequence oftnas follows:

I(tnun) = max t∈[,]I(tun).

We claim that

I(tnun)→+∞ (asn→+∞). (.)

In fact, by (.)-(.) and the definition oftn, we have

I(tnun)≥I

An unun

≥ 

pA

p

n–ε

An unun

p

*

dx–CεAn

|wn|dx

= 

pA

p n–εMp

*

–CεAn

|wn|dx.

IfAn→+∞asn→+∞, then (.) follows from the above estimate. IfAnis bounded for anyn, that is, there exists a constantK>  such thatAn≤K, then

I(tnun) ≥ 

pA

p n–εMp

*

–CεAn

|wn|dx≥ pSp,λM

p_–εMp*_–C εK

|wn|dx

→ 

pSp,λM

where we use the fact thatwn→ asn→+∞. The positive constantεis arbitrary,

thus

lim

n→+∞I(tnun)≥ 

pSp,λM

p _{for any positive constantM.}

(.) follows from the above inequality. From (.), we know that

pc+o() =pI(un) –

I(un),un

= f(x,un)un–pF(x,un)

dx. (.)

By the definition oftn, we obtain _dtdI(tun)|t=tn=  andtn∈[, ], then|tnun| ≤ |un|. The

condition (f) and (.) derive that

f(x,un)un–pF(x,un)

dx ≥ f(x,tnun)tnun–pF(x,tnun)

dx

= pI(tnun) –

I(tnun),tnun

= pI(tnun) –tn

d dtI(tun)

t=tn

→+∞.

Which is a contradiction to (.). Therefore, the sequence{un}is bounded inW_,p().

Under the conditions of Lemma ., we knowun ≤C. Therefore, there exists a sub-sequence, still denoted by{un}, and someu∈W_,p() such that

un u weakly inW,p(),

un u weakly inLp ,|x|–pdx

,

un→u strongly inLs() for any ≤s<p*,

un→u a.e.x∈.

Obviously,uis a weak solution of (.). Now we prove thatu ≡. According to the con-centration compactness principle (see [, ]), there exists a subsequence, still denoted by{un}, at most countable setJ, a set of different points{xj}j∈J⊂\ {}and two positive number sequences{μj}j_∈J∪{},{νj}j∈J∪{}such that

|un|p dμ≥ |u|p+ j∈J

μjδxj+μδ weakly in the sense of measure,

|un|p* dν=|u|p*+ j∈J

νjδxj+νδ weakly in the sense of measure,

|un|p

|x|p dγ= |u|p

|x|p+γδ weakly in the sense of measure,

Sp,λν p p*

 ≤μ–λγ, and Sp,ν p p*

j ≤μj forj∈J,

Lemma . Assume f(x,u)satisfies(f),the sequence{un}is bounded in W,p()and sat-isfies(.).Then the set J={x,x, . . . ,xj} ⊂is finite,μ–λγ= .

Proof We first prove thatμj=  for anyxj∈J.

Letε>  be small enough such that  /∈Bε(xj) andBε(xi)∩Bε(xj) =∅fori=j.φj(x)∈[, ] is a cutting-off function inC∞().φj(x)≡ for|x–xj|<ε/,φj(x)≡ for|x–xj| ≥ε, and |φj| ≤/ε. Since

I(un),unφj

=

|un|pφjdx+ 

|un|p–un∇un∇φjdx

+

un|un|p–unφjdx

–

λ|un| p

|x|pφjdx–

f(x,un)unφjdx, (.)

we deduce that

lim

ε→n→+∞lim

|un|p_φ

jdx=lim ε→

φjdμ≥lim ε→

|u|p_φ

jdx+μj=μj,

lim

ε→n→+∞lim

|_|u_xn|_|ppφjdx

≤lim

ε→n→+∞lim

Bε(xj)

|un|p

(|xj|–ε)p|φj|dx= ,

lim

ε→n→+∞lim

|un|p*_φ

jdx=lim ε→

|u|p*_φ

jdx+νj=νj, (.)

lim

ε→n→+∞lim

|un|φjdx=lim ε→

|u|φjdx= . (.)

By Hölder’s inequality,

lim

ε→n→+∞lim

|un|p–un∇un∇φjdx

≤lim

ε→n→+∞lim un

p–

Lp_(B

ε(xj))∇unLNp/(N–p)(Bε(xj))∇φjLN(Bε(xj))

≤Clim

ε→un

p Lp_(B

ε(xj))= ,

lim

ε→n→+∞lim

|un|p–ununφjdx

≤lim

ε→n→+∞lim un p–

Lp_(B

ε(xj))unLp*_(Bε(x

j))φjLN/(Bε(xj))

≤Clim

ε→unLp*_(B ε(xj))= .

Using (f), (.) and (.), we have

lim

ε→n→+∞lim f(x,un)unφjdx≤εlim→n→+∞lim

ε

|un|p*φjdx+Cε

|un|φjdx

= .

By the above estimates, (.) becomes

 =lim

ε→n→+∞lim

I(un),unφj

≥μj. (.)

If the concentration is at the origin, letφ(x)∈[, ] be a cutting-off function inC∞().

φ(x)≡ for|x|<ε/,φ(x)≡ for|x| ≥ε, and|φ| ≤/ε. Chooseε>  small enough

such thatxj∈/Bε() for allj∈J. Then

lim

ε→n→+∞lim

|un|pφdx=lim

ε→

φdμ≥lim

ε→

|un|pφdx+μ=μ, (.)

lim

ε→n→+∞lim

|_|u_xn|_|ppφdx

=lim

ε→

Bε(xj)

|u|p

|x|p|φ|dx+γ=γ. (.)

Similarly, we can get

lim

ε→n→+∞lim

|un|p–un∇un∇φ+|un|p–ununφj

dx= , (.)

lim

ε→n→+∞lim f(x,un)unφdx= . (.)

From equalities (.)-(.), we get

 =lim

ε→n→+∞lim

I(un),unφ

≥μ–λγ. (.)

On the other hand,μ–λγ≥Sp,λν p p*

 ≥, thusμ–λγ= . The proof is complete.

Lemma . Assume f(x,u)satisfies(f),the sequence{un}satisfying(.)is bounded in W_,p().Then there exist some u∈W_,p()and a subsequence,still denoted by{un},such that

un→u strongly in Lp

*

(),

∇un→ ∇u strongly in L

Np N–p_(),

where=\j_i=B(xi),xi∈J={x,x, . . . ,xj} ∪ {}.

Proof Lemma . implies thatJis finite. Chooseϕ∈C∞ () withϕ(xi) =  for anyxi∈

J∪ {}. Then

|ϕun|p*dx=

|ϕu|p*dx+ j

i=

νiϕp

*

(xi) +νϕp

*

()

=

|ϕu|p*dx. (.)

Sinceϕun→ϕua.e. in, we get that

un→u strongly inLp

*

().

Similarly,

∇un→ ∇u strongly inL

Np

By using the Lebesgue decomposition theorem, we have

dμ–λdγ =|u|p–λ|u| p

|x|p +dσ, (.)

where (|u|p–λ|u|p|x|–p)⊥dσ. Thendσ≥_j∈Jμjδxj+ (μ–λγ)δ.

Lemma . Assume f(x,u)satisfies(f)-(f),the sequence{un} ∈W,p()satisfies(.). Then there exist u∈W_,p()and a subsequence,still denoted by{un},such that un

con-verges to u strongly in W_,p().

Proof We first prove that

σ– j∈J

μjδxj– (μ–λγ)δ

() = . (.)

Claim for any closed setF⊂\J,J=J∪ {},σ(F) = . In fact, denoter=dist(F,J) > . Then, by using a finite covering theorem, there exist finite open ballsBr/(zi), withzi∈F,

i= , . . . ,m, such thatF⊂m_i=Br/(zi)⊂\J. Let(t) be a smooth cutting-off function. ≤(t)≤ for anyt∈[,∞).(x)≡ for ≤t≤/, and≡ fort≥. Denoteηε,y= (|x–y|/ε), then

RN|∇

ηε,y|Ndx=

 /

(t)Ndt=C, (.)

RN|

ηε,y| N

 _dx:=

 /

(t)N _dt=C

. (.)

The sequence{ηε,yun}is still bounded inW,p(). Define

η(x) = m

i=

ηr/,zi(x), (.)

thenη(x)∈C_∞(\J) and ≤η(x)≤m. Furthermore,

F⊂

m

i= Br

(zi)⊂sptη(x)⊂r, (.)

wheresptη(x) denotes the support ofη(x). Asn→+∞, (.) implies

←I(un) –I(u), (un–u)η

=

η |un|p–|un|p–unu–|u|p–uun+|u|p

dx

–λ

η |x|p |un|

p_–|un|p–_u

nu–|u|p–uun+|u|p

dx

+ 

∇η∇(un–u)|un|p–un–|u|p–u

dx

+

η(un–u) |un|p–un–|u|p–u

–

η(un–u)f(x,un) –f(x,u)

dx

= I+II+III+IV+V.

Asn→+∞,

I+II=

η |un|p–|un|p–unu–|u|p–uun+|u|p

dx –λ η |x|p |un|

p_–|un|p–_u

nu–|u|p–uun+|u|p

dx = η

|un|p–|u|p–λ|un| p_–|u|p

|x|p

dx+o()

=

ηdσ+o(),

where we have used (.). Lemma . implies thatun ≤M. Hölder’s inequality, (.)-(.) and Lemma . deduce that

|III|= 

∇η∇(un–u) |un|p–un–|u|p–u

dx

≤C

|un|p+|u|pdx∇ηN

spt(η)

_∇(un–u)

Np N–p_dx

N–p Np

≤C(m,M,p)C

r



_∇(un–u)

Np N–p_dx

N–p Np

→ asn→+∞,

|IV|=

(un–u)η |un|p–un–|u|p–u

dx

≤C(m,M,p)C

r



|un–u|p

*

dx



p*

→ asn→+∞.

By (.), for anyε> , we have

|V| =

f(x,un) –f(x,u)

(un–u)ηdx

≤

f(x,un)+f(x,u)|un–u|ηdx ≤

Cε+ε |un|p

*_–

+|u|p*–|un–u|ηdx

≤ Cε

|un–u|ηdx+ε unp

*_–

p* +u

p*–

p*

spt(η)

|un–u|p

*

dx



p*

≤ Cε

|un–u|dx+εC(M)

Therefore, asn→+∞,

≤σ(F)≤σ spt(η)≤

ηdσ≤ |III|+|IV|+|V|= . (.)

Since the setFis arbitrary, (.) is obtained.

Lemma . means thatμj=  for anyj∈Jandμ–λγ= , soσ() = , which implies

that asn→+∞,

|un|p–λ|un| p

|x|p

dx→

|u|p–λ|u| p

|x|p

dx. (.)

Together withun→ua.e. in, we complete the proof.

3 The proof of the main result

In this section, by using the following symmetric mountain pass theorem (see []), we give the proof of Theorem ..

Lemma .(Symmetric mountain pass theorem) Assume functional I satisfies the follow-ing conditions:

() I∈C_(W,p

 (),R)is even and satisfies the Palais-Smale condition.

() There exists a finite dimensional subspaceX∈W_,p()such thatI|X∩∂Br≥a> . () There exists a sequence of the finite dimensional subspace{Xj},dim(Xj) =jandrj> 

such that

I(u)≤ for anyu∈Xj\Brj,j= , , . . . .

Then I has infinitely many different critical points,and the corresponding energy values are positive.

Proof of Theorem . We know the energy functional I(u) is continuous and even. Lemma . implies thatI(u) satisfies the PS condition inW_,p(). In any finite dimen-sional subspaceX⊂W_,p(), all norms are equivalent. For anyu∈X, sinceF(x,u) =o(up₎ asu→, there exists a constantq>psuch that

I(u) = 

p

|u|p–λ|u| p

|x|p

dx–

F(x,u)dx

≥ 

pu

p_–Cuq_{>  forusmall enough. (.)}

Thus, there existsr>  such thatI|X∩∂Br ≥a> . On the other hand, by using (.), we

have

I(u) = 

p

|u|p–λ|u| p

|x|p

dx–

F(x,u)dx

≤ 

pu

p_–C

{x∈:|u|>M}

|u|p+θ_dx–C

≤ 

pu

p_–Cup+θ

There exists a sequence of the finite dimensional subspace{Xj},dim(Xj) =jandrj>  such that

I(u)≤ for anyu∈Xj\Brj,j= , , . . . .

Finally, all the assumptions of Lemma . are satisfied. Hence, the problem (.) possesses infinitely many weak solutions, and the corresponding critical values are positive.

Remark . Under the same conditions of Theorem ., we can also prove that the fol-lowingp-harmonic type equation with Navier boundary conditions:

⎧ ⎨ ⎩

(|u|p–_{u) –λ}|u|p–_u

|x|p =f(x,u), x∈,

u=u= , x∈∂,

(.)

has infinitely many solutions{un} ∈W,p_()∩W,p

 (), wherecontaining the origin

is an open bounded domain inRN_{,∂is smooth.  <p<} N

, ≤λ<λ= [N(p– )(N–

p)/p_]p_.

Remark . All the results obtained above obviously hold if we choosef(x,u) =|u|q–_u+

|u|r–uwithp<q<r<p*.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HX carried out the existence of infinitely many solutions for thep-harmonic equation studies and drafted the manuscript. JW participated in its coordination. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Henan University of Economics and Law, Zhengzhou, 450002, China.}2_{College of}

Information and Management Science, Henan Agricultural University, Zhengzhou, 450002, China.

Acknowledgements

The authors are thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper. This research is supported by the Education Department of Henan Province (12B110002) and NSF of China (61143002).

Received: 19 May 2012 Accepted: 13 November 2012 Published: 6 January 2013 References

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doi:10.1186/1029-242X-2013-9