Multiplicity of solutions for elliptic problems of p-Kirchhoff type with critical exponent

Abstract

In this paper we consider a class of elliptic problems of p-Kirchhoff type with critical exponent in bounded domains and new results as regards the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method.

Zhou and Song Boundary Value Problems (2015) 2015:223 DOI 10.1186/s13661-015-0489-z

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Web End = Multiplicity of solutions for elliptic problems of p-Kirchhoff type with critical exponent

Chenxing Zhou1 and Yueqiang Song2*

*Correspondence: mailto:[email protected]

Web End [email protected]

2Scientic Research Department, Changchun Normal University, Changchun, Jilin 130032, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper we consider a class of elliptic problems of p-Kirchho type with critical exponent in bounded domains and new results as regards the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method.

MSC: 35J70; 35B20Keywords: p-Kirchho-type problem; critical growth; concentration-compactness principle; variational method

1 Introduction

In this paper we deal with the existence and multiplicity of solutions to the following p-Kirchho type with critical exponent:

[g(

|u|p dx)] pu = h(x, u) + upu, x , u = , x ,

(.)

where < p < N, is a positive parameter,

RN is an open bounded domain with smooth boundary and is a positive parameter, p = Np/(N p) is the critical exponent according to the Sobolev embedding. f :

R

R, g : R+

R+ are continuous functions

that satisfy the following conditions:(G) There exists > such that g(t) for all t .

(G) There exists satised < p < p and G(t) g(t)t for all t , where

G(t) =

t g(s) ds.

(H) h(x, u) C(

R, R), h(x, u) = h(x, u) for all u

R.

(H) lim|u|

h(x,u)

|u|

p = uniformly for x . (H) lim|u|

+ h(x,u)

up/ = uniformly for x .

Much interest has grown on problems involving critical exponents, starting from the celebrated paper by Brezis and Nirenberg []. For example, Li and Zou [] obtained innitely many solutions with odd nonlinearity. Chen and Li [] obtained the existence of innitely many solutions by using the minimax procedure. For more related results, we refer the interested reader to [] and references therein.

2015 Zhou and Song. 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.

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 2 of 12

On the one hand, for the special case of problem (.), equation (.) reduces to the following Dirichlet problem of Kirchho type:

(a + b

|u| dx) u = f (x, u), x , u| = ,

RN, problem (.) is a generalization of a model introduced by Kirchho []. More precisely, Kirchho proposed a model given by the equation

ut

where

ux = , (.)

where , , h, E, L are constants, which extends the classical dAlembert wave equation, by considering the eects of the changes in the length of the strings during the vibrations. The equation (.) is related to the stationary analog of problem (.). Equation (.) received much attention only after Lions [] proposed an abstract framework to the problem. Some important and interesting results can be found; see for example []. We note that results dealing with the problem (.) with critical nonlinearity are relatively scarce.

In [], by means of a direct variational method, the authors proved the existence and multiplicity of solutions to a class of p-Kirchho-type problem with Dirichlet boundary data. In [], the author showed the existence of innite solutions to the p-Kirchho-type quasilinear elliptic equation. But they did not give any further information on the sequence of solutions. Recently, Kajikiya [] established a critical point theorem related to the symmetric mountain-pass lemma and applied to a sublinear elliptic equation. However, there are no such results on Kirchho-type problems (.).

Motivated by reasons above, the aim of this paper is to show that the existence of innitely many solutions of problem (.), and there exists a sequence of innitely many arbitrarily small solutions converging to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya [].

To the best of our knowledge, the existence and multiplicity of solutions to problem (.) has not ever been studied by variational methods. As we shall see in the present paper, problem (.) can be viewed as an elliptic equation coupled with a non-local term. The competing eect of the non-local term with the critical nonlinearity and the lack of compactness of the embedding of H,p( ) into the space Lp( ), prevents us from using the variational methods in a standard way. Some new estimates for such a Kirchho equation involving Palais-Smale sequences, which are key points to the application of this kind of theory, are needed to be re-established. We mainly follow the idea of [, ]. Let us point out that although the idea was used before for other problems, the adaptation to the procedure to our problem is not trivial at all, since the appearance of non-local term, we must consider our problem for suitable space and so we need more delicate estimates.

Our main result in this paper is the following.

Theorem . Suppose that (G)-(G), (H)-(H) hold. There then exists > such that, for any (, ), problem (.) has a sequence of non-trivial solutions {un} and un

as n .

dx

(.)

h +

E L

L

u

x

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 3 of 12

2 Preliminary lemmas

We consider the energy functional J : W,p( )

R dened by

J(u) =

pG

|u|p dx,

where W,p( ) is the Sobolev space endowed with the norm u p =

|u|p dx. Standard arguments [] show that a critical point of J is a weak solution of problem (.). We try to use a new version of the symmetric mountain-pass lemma due to Kajikiya []. But since the functional J(u) is not bounded from below, we could not use the theory directly. So we follow [] to consider a truncated functional of J(u). Denote by J : E E the derivative

operator of J in the weak sense. Then

J (u), v

= g

u p

H(x, u) dx p

u p

dx

|u|puv dx

|u|pu v

h(x, u)v dx, u, v W,p( ).

To use variational methods, we give some results related to the Palais-Smale compactness condition. Recall that a sequence (un) is a Palais-Smale sequence of J at the level c, if J(un) c and J (un) .

We recall the second concentration-compactness principle of Lions [, ].

Lemma . [, ] Let {un} be a weakly convergent sequence to u in W,p( ) such that |un|p and |un| in the sense of measures. Then, for some at most countable index

set I,(i) = |u|p +

jI xjj, j > ,(ii) |u|p +

jI xjj, j > ,(iii) j Sp/p

j ,

where S is the best Sobolev constant, i.e. S = inf{

RN|u|p dx :

RN|u|p dx = }, xj

RN, xj

are Dirac measures at xj and j, j are constants.

Under assumptions (H) and (H), we have

h(x, s)s = o

|s|p

, H(x, s) = o

|s|p

,

which means that, for all > , there exist a(), b() > such that

h(x,

s)s

a() + |s|p, (.)

H(x,

s)

b() + |s|p. (.)

Hence,

H(x, s)

p h(x, s)s c(

) + |s|p (.)

for some c() > .

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 4 of 12

Lemma . Suppose that (G)-(G), (H)-(H) hold. Then, for any > , the functional J satises the local (PS)c condition in

c

,p ppp (S)N/p c

p p pp

| |

in the following sense: if

J(un) c <

p p

pp (

S)N/p c

p p pp

| |

and J (un) for some sequence in W,p( ), then {un} contains a subsequence converging

strongly in W,p( ).

Proof Let {un} be a sequence in W,p( ) such that

J(un) =

pG

un p

H(x, un) dx p

|un|p dx = c + o(), (.)

J (un), v

= g

un p

|un|pun v

dx

|un|punv dx

h(x, un)v dx = o() un . (.)

By (.) and (.), we have

c + o() un = J(un)

p

J (un), un

=

pG

un p

p g

un p

un p + p p

|

un|p dx

H(x, un) dx +p

h(x, un)un dx

p p

pp

|un|p dx

H(x, un) dx

p

+ h(x, un)un dx,

i.e.

p p

pp

|un|p dx

H(x, un) p h(x, un)un dx + c + o() un .

Then by (.), we have

p ppp

|

un|p dx c()| | + c + o() un .

Setting = p

ppp , we get

|un|p dx M + o() vn , (.)

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 5 of 12

where o() and M is a some positive number. On the other hand, by (.) and (.),

we have

c + o() un = J(un)

=

pG

un p

H(x, un) dx p

|un|p dx

p un p

b()| |

p +

|

un|p dx. (.)

Therefore, the inequalities (.) and (.) imply that {un} is bounded in W,p( ). Hence,

up to a subsequence, we may assume that

un u weakly in W,p( ),

un u a.e. in ,un u in Ls( ), s < p, (.)

|un|p

weak-sense of measures

,

|un|p

weak-sense of measures

,

where and are a nonnegative bounded measures on . Then, by the concentration-compactness principle due to Lions [, ], there exists some at most countable index set I such that

= |u|p +

jIjxj, j > ,

|u|p +

jIjxj, j > ,

Sp/p

j j,

where xj is the Dirac measure mass at xj . Let (x) C such that ,

(x) =

if |x| < , if |x| ,

(.)

and || .

For > and j I, denote j(x) = ((x xj)/). Since J (un) and (jun) is bounded, J (un), jun as n ; that is,

g

un p

|un|pj dx

= g

un p

un|un|punj dx

+

h(x, un)unj dx + |un|pj dx + on(). (.)

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 6 of 12

By (.) and Vitalis theorem, we see that

lim

n

u

nj

p

dx =

uj

p

dx.

Hence, by Hlders inequality we obtain

lim sup

n

un|un|punj dx lim sup

n

|

un|p dx (p)/p u

nj

p

dx

/p

C

B(xj,) |

u|p

j

p

dx

/p

C

B(xj,) j

N

dx

/N

B(xj,) |

u|p dx /p

C

B(xj,) |

u|p dx /p

as . (.)

Since j has compact support, letting n in (.) we deduce from (.) and (.)

that

j d C B(xj,) |u|p dx /p +

B(xj,) f (x, u)u dx + j d.

Letting , we obtain j j. Therefore,

(S)N/p j. (.)

We will prove that this inequality is not possible. Let us assume that (S)N/p j for some j I. From (G) we see that

G

un p

g

un p

un p for all n.

Since

c = J(un)

p

J (un), un

+ on(),

it follows that

c = lim

n

J(un) p

J (un), un

p p

|

un|p dx

H(x, un)

p h(x, un)un

dx

p ppp

j|un|p dx c()| |.

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 7 of 12

Letting = p

ppp and n , we obtain

c

p p

pp

jJj(xj)j c

p p pp

| |

p p

pp (

S)N/p c

p p pp

| |.

This is impossible. Then I = , and hence un u in Lp( ).

Then, using (.) and the fact that un u in Lp( ), we have

lim

n

h(x, un)(un u) dx = (.)

and

lim

n

|un|pun(un u) dx = . (.)

From J (un), un u = on(), we deduce that

J (un), un u

= g

un p

|un|pun(un u) dx

h(x, un)(un u) dx |un|pun(un u) dx = on().

This, (.), and (.) imply

lim

n g u

n p

|un|pun(un u) dx = .

Since un is bounded and g is continuous, up to subsequence, there is t such that

g

un p

g

tp as n ,

and so

lim

n

|un|pun(un u) dx = .

Thus by the (S+) property, un u strongly in W,p( ). The proof is complete.

3 Existence of a sequence of arbitrarily small solutions

In this section, we prove the existence of innitely many solutions of (.) which tend to zero. Let X be a Banach space and denote

:=

A X \ {} : A is closed in X and symmetric with respect to the origin

.

For A , we dene genus (A) as

(A) := inf

m N : C

A, Rm \ {}, (x) = (x)

.

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 8 of 12

If there is no mapping as above for any m N, then (A) = +. Let k denote the family

of closed symmetric subsets A of X such that /

A and (A) k. We list some properties

of the genus (see [, ]).

Proposition . Let A and B be closed symmetric subsets of X which do not contain the origin. Then the following hold.

() If there exists an odd continuous mapping from A to B, then (A) (B).

() If there is an odd homeomorphism from A to B, then (A) = (B). () If (B) < , then (A \ B) (A) (B).

() Then n-dimensional sphere Sn has a genus of n + by the Borsuk-Ulam theorem. () If A is compact, then (A) < + and there exists > such that U(A) and

(U(A)) = (A), where U(A) = {x X : x A }.

The following version of the symmetric mountain-pass lemma is due to Kajikiya [].

Lemma . Let E be an innite-dimensional space and J C(E, R) and suppose the fol

lowing conditions hold.

(C) J(u) is even, bounded from below, J() = and J(u) satises the local Palais-Smale condition, i.e. for some c > , in the case when every sequence {uk} in E satisfying

limk J(uk) = c < c and limk J (uk) E = has a convergent subsequence.

(C) For each k N, there exists an Ak k such that supuAk J(u) < . Then either (R) or (R) below holds.

(R) There exists a sequence {uk} such that J (uk) = , J (uk) < , and {uk} converges to zero.

(R) There exist two sequences {uk} and {vk} such that J (uk) = , J(uk) < , uk = ,

limk uk = , J (vk) = , J(vk) < , limk vk = , and {vk} converges to a non-zero

limit.

Remark . From Lemma . we have a sequence {uk} of critical points such that J(uk)

, uk = and limk uk = .

In order to get innitely many solutions we need some lemmas. Let =

p , from (.)

we have

J(u) :=

pG

p

p

L u p L u p L,

where L, L, L are some positive constants.

Let Q(t) = Ltp Ltp L. Then

J(u) Q

u

.

u p

H(x, u) dx p

|u|p dx

|u|p dx p +

|

u|p dx b()| |

=

|u|p dx p

|u|p dx b p

| |

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 9 of 12

Furthermore, there exists := pLNL ( pLpL )(Np)/p such that for (, ), Q(t) attains its

positive maximum, that is, there exists

R =

pL pL

(Np)/p

such that

e = Q(R) = max

t Q(t) > .

Therefore, for e (, e), we may nd R < R such that Q(R) = e. Now we dene

(t) =

, t R,

LtpLeLtp , t R,

C, (t) [, ], R t R.

Then it is easy to see (t) [, ] and (t) is C. Let (u) = ( u ) and consider the

perturbation of J(u):

G(u) :=

pG

u p

p

(u)

|u|p dx (u)

H(x, u) dx. (.)

Then

G(u) L u p L(v) u p L = Q

u

,

where Q(t) = Ltp L(t)tp L and

Q(t) =

Q(t), t R,

e, t R.

From the above arguments, we have the following.

Lemma . Let G(u) is dened as in (.). Then:(i) G C(W,p( ),

R) and G is even and bounded from below.(ii) If G(u) < e, then Q( u ) < e, consequently, u < R and J(u) = G(u).

(iii) There exists such that, for (, ), G satises a local (PS)c condition for

c < e

, min

e, p ppp (S)N/p c

p p pp

| | .

Lemma . Suppose that (G)-(G), (H) hold. Then, for any k

N, there exists = (k) >

such that ({u W,p( ) : G(u) (k)} \ {}) k.

Proof First of all, by (H) of Theorem ., for any xed u W,p( ), u = , we have

H(x, u) M()(u)

p

with M() as .

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 10 of 12

On the other hand, by integrating (G), we obtain

G(t)

G(t)

t/ t/ = Ct/ for all t t > . (.)

Second, given any k N, let Ek be a k-dimensional subspace of W,p( ). There then exists

a positive constant such that

u |u|p/ for all u Ek.

Therefore, for any u Ek with u = and small enough, by (.) and (H) we have

G(u) =

pG

u p

p

(u)

|u|p dx (u)

H(x, u) dx

C p

p

M()

p/

p

Cp M() p/

p

= (k) < ,

since lim|| M() = +. That is,

u Ek : u =

u W,p( ) : G(u) (k) \ {}.

This completes the proof.

Now we give the proof of Theorem . as follows.

Proof of Theorem . Recall that

k =

A W,p( ) \ {} : A is closed and A = A, (A) k

and dene

ck = inf

A k

sup

uA

G(u).

By Lemma .(i) and Lemma ., we know that < ck < . Therefore, assumptions (C)

and (C) of Lemma . are satised. This means that G has a sequence of solutions {un}

converging to zero. Hence, Theorem . follows by Lemma .(ii).

4 A special case of problem (1.1)

We consider the following special case of problem (.):

+

|u|p dx pu = f (x, u) + |u|pu in ,

u = on ,

(.)

Zhou and Song Boundary Value Problems (2015) 2015:223 Page 11 of 12

where is a bounded smooth domain of RN, < p < N < p, and are positive constants.

Set g(t) = + t. Then g(t) and

G(t) =

g(s) ds =

k

t +

t

(

+ t)t = g(t)t,

where = /. Hence the conditions (G) and (G) are satised.

For this case, a typical example of a function satisfying the conditions (F)-(F) is given by

f (x, t) =

i= ai(x)|t|qit,

where k , < qi < p , and ai(x) C( ). In view of Theorem ., we have the following

corollary.

Corollary . Suppose that (F)-(F) hold. There then exists > such that for any

(, ), problem (.) has a sequence of non-trivial solutions {un} and un as n .

Competing interests

The authors declare that they have no competing interests.

Authors contributions

CZ carried out the theoretical studies, and participated in the sequence alignment and drafted the manuscript.

YS participated in the design of the study and performed the statistical analysis. All authors read and approved the nal manuscript.

Author details

1College of Mathematics, Changchun Normal University, Changchun, Jilin 130032, P.R. China. 2Scientic Research Department, Changchun Normal University, Changchun, Jilin 130032, P.R. China.

Acknowledgements

The authors are supported by NSFC (Grant No. 11301038).

Received: 21 June 2015 Accepted: 19 November 2015

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