一类Kirchhoff型四阶椭圆方程的无穷多个变号解

CHINESE JOURNAL

OF ENGINEERING MATHEMATICS

Vol. 35 No. 2

doi: 10.3969/j.issn.1005-3085.2018.02.007 Article ID: 1005-3085(2018)02-0206-11

Infinitely Many Sign-changing Solutions for Some Fourth-order Elliptic Equations

of Kirchhoff Type*

CHEN Jing, HAN Guo-dongt

(School of Mathematics and Information Science, Shaanxi Normal University, Xi?an 710062)

Abstract: In engineering practice, the fourth-order elliptic equation of Kirchhoff type derives

from the model of the suspension bridge. In the paper, two theorems on the ex­istence and multiplicity of sign-changing solutions for some fourth-order elliptic equations of Kirchhoff type are proved by using the descending flow invariant set method under the assumption that the nonlinear term is odd and superquadric at infinity. The main results and the proofs are different from those in literatures.

Keywords: fourth-order elliptic boundary value problems of Kirchhoff type; sign-changing

solutions; critical points

Classification: AMS(2010) 35J60; 35J65 Document code: A

CLC number: O175.4

1 Introduction

In this paper, the existence and multiplicity of sign-changing solutions for the

following fourth-order elliptic boundary value problems (BVP) of Kirchhoff type are considered

^ A2u — M(

^

\\Vu\\2dx)Au = f (x, u),

x G

— Au\\dO, — 0,

where H C RN is a bounded smooth domain, f : H x R ^ R and M : R ^ R are two continuous functions. Some papers are concerned with the existence and multiplicity of the positive solutions for BVP (1) of 1-dimension by using variational methods, fixed point theorems and mountain pass theorem in cones of ordered Banach space, see [1-4]Received: 04 Mar 2016. Biography: Chen Jing (Born in 1990), Female, Master.

Accepted: 29 July 2016. Research field: nonlinear analysis and applications.

*Foundation item: The National Natural Science Foundation of China (11101253); the Fundamental Re­search Funds for the Central Universities (GK201503016); the Science Program of Education Department of Shaanxi Province (14JK1461).

^Corresponding author: G. Han. E-mail address: gdhan@snnu.edu.cn

NO. 2 Chen Jing, Han Guodong: Infinitely Many Sign-changing Solutions for Some Elliptic Equations 207

and reference therein. The fourth-order semilinear elliptic problem of high dimension has been studied by many authors, see [5-9] and reference therein. In addition, due to the complicated and interesting properties of sign-changing solutions, more and more authors devoted to deal with the existence and multiplicity of theorems for kinds of fourth-order equations, see [10-13].

Inspired by the above references, the aim of the paper is to study the sign-changing solutions for the fourth-order elliptic boundary value problems of Kirchhoff type (1). Under some conditions on the function M and the nonlinear term f, the existence and multiplicity results will be established. The proof is based on the descending flow invariant set method.

Let A i be the positive first eigenvalue of the following second eigenvalue problem

■Av = Av, v\\dQ = 0•

x G H

Thenit is easy to seethat Ai = Ai(Ai + c) is the positive first eigenvalue of the following fourth-order eigenvalue problem

A2u — cAu = Au,

x G H

u\\ dQ — Au\\dQ — 0*

For convenience, we list some conditions as follows:(H〇):

There exist m7 > m〇 > 0 and

> 0 such that M(t) > m〇 for all t > 0 and

M(t) = m! for all t > t〇;

(Hi): (H2):

f G C(G x R,R)and f(x,0) = 0;There exist c> 0 and q such that

|f(x,t)| < c(l + \\\q-i),

V (x,t) G H x R,

where 1 < q <

(H3):

if N ^ 5 and 1 There exist ^ > 2 and M > 0 such that

for \\\ > M,

x G H;

(H4)

lim sup

t^o

where

\\f (x,t)\\

\\\

< A,

for x G H uniformly,

A — Ai(Ai + mi), mi

化（〇,十

maxM(t);

208CHINESE JOURNAL OF ENGINEERING MATHEMATICSVOL. 35

(H5): f is odd in t, that is, —f (x,t) = f (x, -t).

Remark 1 (F3).

A standard argument (see [10] for example) shows that (H3) implies

(F3): There exist ^1,^2 > 0 such that F(x,t) >

\\\^ — ^2 for all t E R.

Now, the main results on BVP (1) can be stated as follows.Theorem 1

Assume that (H〇)-(H4) hold. Then BVP (1) has at least a positive

solution, a negative solution and a sign-changing solution.

Theorem 2

Assume that (H〇)-(H5) hold. Then BVP (1) has infinitely many

sign-changing solutions.

Remark 2

In [4], by using the mountain pass lemma, Wang and An obtained

that BVP (1) has at least a sign-changing solution. Their condition imposed on the nonlinear term f is as follows

f(x,t) = o(\\\) as t4

〇 uniformly for xEQ.

Obviously, (H4) is weaker than the above condition. Therefore, our results are more general than those in [4]. Besides, we established a existence theorem of infinitely many sign-changing solutions for (1).

The paper is organized as follows. In section 2, we recall some facts on the method of the invariant set of decreasing flow. The main results are proved in section 3.

2 Preliminaries

Let H be a real Hilbert space, J a C1 functional on H and Jf(u) = u — Au the

gradient operator of J at u E H. Let

K = {u E X : J7(u) = d],

H〇 = H\\K.

The following two theorems are due to [11]. For the convenience of use in section 3, we give their special cases here. See [11,14-20] for more general results and the concepts of the invariant set of descending flow.

Let ^± be two closed convex subsets of H and A E C(H, H) an operator.We shall use the following assumptions:(D): (Ai): (A2):

O = int切+ n int 切—；A(切士）c int切士；

The map A is compact, that is, A maps bounded subsets of H into precom­

pact subsets of H；

NO. 2 Chen Jing, Han Guodong: Infinitely Many Sign-changing Solutions for Some Elliptic Equations 209

(Ji): There exists a path h : [0,1] ^ H such that

h(0) G (intD+)\\D—,

h(1) G (intD-)\\D+,

and

max J(h(t)) < a〇 :=

(J2):

inf

J(u);

(2)

There exist a number ai, a sequence [Hn] of subsequences of H, and a

sequence {Rn} of positive numbers satisfying

dim Hn > n,

for

n G N,

sup

J(u) < ai < a〇,

(3)

where Bn = {u G Hn : ||u||< Rn}.

In the following, CH(S) denotes the complete invariant set of descending flow rel­ative to H expanded by the set S[17].

Theorem 3

Assume that (D), (Ai), (A2) and (Ji) hold. Then J has a critical

point in each of the four mutually disjoint sets

dCH(O)\\(D + U D—),

dCH(O) H int D +,

dCH(O) n int D—

and O.

Theorem 4 Assume that (D), (Ai), (A2) and (J2) hold, and J is an even func­

tional and satisfies PS condition on H. Then, J has a sequence of solutions {±Un} in

M = dCH(O)\\(Ch(int D+)) U (C

h

(int D—)),

such that

J(Un) 4 +⑴

as

n ^ +^.

(4)

3 Proof of the main results

In this section, we will use the abstract theorems in section 2 to prove Theorem 1

and Theorem 2.

Let H = H2(Q) n H〇i(Q) be the Hilbert space with the inner product

(u, v)=

and the deduced norm

||u||2= ,|Au|2dx+ ,|Vu|2dx.

J n J n

A function u G H is called a weak solution of BVP (1) if

AuAvdx + M^

'Vufdx)

VuVvdx =

f(x,u)vdx

(AuAv + VuVv)dx,Jn

210CHINESE JOURNAL OF ENGINEERING MATHEMATICSVOL. 35

holds for any v G H. On the other hand, we see that weak solutions of BVP (1) are critical points of the functional J : H ^ R defined by

J(u) = \\ j |Au|2dx + 1 lM( f |Vu|2d^ ^ / F(x,u)dx,

2 Jq 2 乂九 ^ Jn

where ^

IM(t) = ( M(s)ds.

Jo

Since M is continuous and f has subcritical growth, the above functional J is of class

C1 in H.

Lemma 1 dition on H.

Assume that (H〇)-(H4) hold. Then the functional J satisfies PS con­

Proof Assume that [un] C H, [un] satisfies \\ J(un)| < C and J!(un) ^ 0, there exists a number Ci > 0 such that || J7(un)|| ^ Ci.

Since f (x,t) is sub-critical by (H2), from the compactness of Sobolev embedding, to verify that J satisfies PS condition we only need to show that {un} is bounded in

H.

By (Ho), it is easy to obtain that

M(t) = f M(s)ds < f mids < mit.

o

Then, from (H2) and (H3)，we have

o

〈J(un)，un'}

/*|Aun|2dx + M^( /*|Vun|2dx、

nn

n

f (x, un)undx

In

乂 |Aun|2dx + Mf( 乂 |Vun|2dx

Wn

’

n

f (x, un)undx _ \"J(un) + \"J(un)

|Aun|2x + M^( ^ |Vun|2dx) — ^ f(x,un)undxn卞

2 乂 |Aun|2dx + 2

」

/ |Vun|2dx) — 乂 F(x,un)dx

nn

|VUn|rZ

+ ^J (un)

2 — ^2 2 — ^

|AUn^dx + *~ 2 n 2

+ /(\"F(x,un)— f(x,un)un)dx + J(un)n

<

2 — ^2

max{1，mi}||un||2 +\"J(un).

NO. 2 Chen Jing, Han Guodong: Infinitely Many Sign-changing Solutions for Some Elliptic Equations 211

Therefore, according to Cauchy-Schwartz inequality, we know that

n — 2

-^—max[1,m1}^un^2 < /iJ(un) - (J'(un),un)

引 J(u„)| + ||7>„)|||卜„|| S nC + Ci||u„||.

Then, the above inequality shows that {un} is bounded in H.From (Hi), (H3) and (H4), there exists m > 0 such that

uf (x, u) + mu2 > 0,

for u = 0. (5)

In order to prove Theorem 1 and Theorem 2, we need to construct two convex subsets

of H and an operator Am satisfying the assumptions (D) and (Ai). We begin by transforming BVP (1) into the following equivalent problem

A2u — M( fQ lVul2dXj Au + mu = fm(x, u), u| OQ — Aul〇Q — 0,

where m > 0 is as in (5) and fm(x, u) — f (x, u) + mu for all (t, u) E Q x R. The space is still the Hilbert space H with the deduced norm

||u||m — / |Au|2dx + 乂 |Vu|2dx + m 乂 |u|2dx.

J Q,

J Q,

J Q

x E Q,

(6)

In addition, we see that weak solutions of BVP (6) are critical points of the functional

Jm : H ^ R defined by

Jm(u) —1 , |Au|2dx + 1^( , |Vu|2dx) — ,Fm(x,u)dx,

where

Fm(x,t)

t

(f (x, s) + mSj ds

It is easy to see that Jm satisfies PS condition. We will use the notation P :— {u E H : u(x) > 0 a.e. x E Q}. For m > 0 we consider the operator Am : H ^ H defined by

Am(u) — (A2 — M(乂 |V • |2dx)A • +m) (f (x,u) + mu),

The distance in H with respect to || • ||m is denoted by distm.

Lemma 2 such that

Am(D±) C int (D±),

where

for u E H.

Assume that (H〇)-(H5) hold. Then, there exist m > 0 and

> 0

for

e E (0,£〇],

(7)

— {u E H : distm(u, 士P) < e}.

(8)

212CHINESE JOURNAL OF ENGINEERING MATHEMATICSVOL. 35

Proof By (H2) and (H4), there exist 5 > 0 and c> 0 such that

\\f (x, t)+ mtl < (A + m — S)\\~t\\ + c\\\q-1,

t E R.

(9)

For u E H, we denote v = Amu, u+(x) = max{0,u(x)} and u-(x) = min{0,u(x)}. Obviously, ||u—||2 = inf ||u — w||2. Therefore

wEPllv—lim =〈v—,v—〉=d〈(AmU) —v〈Amu- ,v-) = (f(x,u) + mu-)v-dx.Q

Since v+ E P, v = v+ + v—, so

distm(v,P).||v—||m<||v—Iim< / (f(x,u) + mu)v—dx.

Next, we show that there exists

> 0 such that

distm(v,P) < distm(u,P),

0 < distm(u,P) <

distm(v, —P) < distm(u, —P),

0 < distm(u, —P) < ^〇.

According to Sobolev imbedding theorem and (9), we have

llv—|

=

(f (x, u) + mu—) v—dxJq

< (A + m — 5) \\u—\\\\v—\\dx + ^ \\u—\\q—1\\v—\\dxQ

Q

< (A + m — 5)llu—y

v—lb +c|u—liq—q Ilvq—1

1q—1

&

—yv—llm+秦

llq)”v—llm.

Thus

distm(v,P) <

A + m — 5|u—||cl(||u—||q)

q—1

\\/A + m

2+<

A + m — 5A + inf ( inf ||u q—1m w.,,r Dep

||u — w|| + c2\\ wEP

— w||)

A + m — 5A + distm(u,P)+c2(distm(u,P))q \\

mTherefore, there exists > 0 such that

distm(v,P) < distm(u,P),

0 < distm(u, P) < ^〇.

Similarly,

distm(v, —P) < distm(u, —P),

0 < distm(u, —P) < ^〇.

(10)(11)

NO. 2 Chen Jing, Han Guodong: Infinitely Many Sign-changing Solutions for Some Elliptic Equations 213

The proof is completed.

Proof of Theorem 1

We only need to verify that all the conditions of Theorem

3 hold. From Lemma 1 and Lemma 2, Jm satisfies PS condition on H, and it is easy to see that Am and

satisfy (D) and (Ai). Now we need to show that (Ji) holds.

This implies that

(12)

It follows from (7) that Am has no fixed point on d

K n (D+ n D-) = {

〇},

where 0 denotes the function w(w)三 0 for w G Q. Then according to (7) and [11]

inf _ Jm(u) = Jm(0) = 0.

ueD+nD-(13)

Define a path h : [0,1] ^ H such that

h(t) = Rei cos

+ Re2 sin (14)

Letei G Hand ||ei|| = 1beaneigenfunctioncorrespondingtoBVP(1)suchthatei > 0 and choose e2 G H, so that ei and e2 are orthogonal. Then h(0) = Rei, h(1) = —Rei. Since

dist(士Rei,干P) = ||Rei ||,

(15)

it is easy to see that when R is large enough, so we have h(0) G (/(D+))\\De—,h(1) G(/(D-))\\D+. As

Jm(h(t))

2 乂 |Ah(t)|2dw + 2」

|Vh(t)|2d^ ^ /Q

E^{x,h(t))dx

m i2

s 2 乂 |Ah(t)|2dx + 2分(丄 |vh(t)|2dx)—^i

n

|h(t)^dx + ^2|Q|

2

h(t)|2

< max{2,mi} \"h(t)iim—\"i 丄1h(t)rdx+内叫

^max{ 1 mi} f

=----------1-------||Rei cos nt + Re2 sinnt||m — \"i / |Rei cosnt + Re2 sinnt|Mdx + \"2|Q|

2 Jn< max{1,mi}R2(||ei||m + Ihlim) —\"iRM / |eicosnt + e2sinnt|Mdx + \"2|Q|2 Jn

< \"3R2 — \"where

aR^ +

\"2|Q|,

max{1, mi}

\"3

2

(iieiiim+ihiim),

\"4

|ei cosnt + e2 sinnt|Mdx.

214CHINESE JOURNAL OF ENGINEERING MATHEMATICSVOL. 35

Since ^ > 2

lim max

R4+⑴圯[0，1]

Thus,

v v n

hit)) = —^.

v

(16)

when R is sufficiently large

max Jm (h(t)) < 0 =

inf

J(u).

拓[0，1]

(17)

ueD+nD-Therefore, all the conditions of Theorem 1 are satisfied. This completes the proof.

Proof of Theorem 2 We only need to verify that all the conditions of Theorem

4 hold. From Lemma 1 and Lemma 2, Jm satisfies PS condition on H, and it is easy to see that Am and 纪 satisfy (D) and (Ai). According to (H5)，J is an even functional. Next, we show that (J2) holds.

Let Hn = span such that

u{ei

, ••• ,en}, since u E Hn, so there exist ai,a2, ••• ,an

= aiei + d2^2 + ••• + ^n^n-It follows from F(x,t) > ^i\\\^ — ^2, ^ > 2 that

Jm(u) = ^ / |Au|2dx + 1 IM f \\Vu\\2dx ^ / Fm(x,u)dx

2 Jq 2 Jn Jn

< max{2，mi}|iu\"m —

丄(\"iif—叱一一^|卜||2

< max{21,mi}||u|im — \"i||uK+\"2\\Q\\

< max{21,mi}|u|m—\"iiiuiim+\"2\\Q\\.

Since ^ > 2

lim Jm(u)=—⑴,

||u||4 +⑴

This implies that

sup

ueHn\\Bn u G Hn.

Jm (u) < ai < a〇 = 0 = inf Jm(u),

ueD+nD-where Bn = {u G Hn : ||u|| < Rn}. Now, all the conditions of Theorem 1 are satisfied. So BVP (1) has many infinitely many sign-changing solutions in

^

= dCH(〇)\\(CH( int (切+))) UC

h

( int (切—)).

This completes the proof.

NO. 2 Chen Jing, Han Guodong: Infinitely Many Sign-changing Solutions for Some Elliptic Equations 215

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216CHINESE JOURNAL OF ENGINEERING MATHEMATICSVOL. 35

一类Kirchhoff型四阶椭圆方程的无穷多个变号解

陈晶，韩国栋

(陕西师范大学数学与信息科学学院，西安710062)

摘要：四阶Kirchhoff型椭圆问题来源于工程实际中的悬索桥模型.本文应用下降流不变集方 法研究了一类四阶Kirchhoff型椭圆边值问题，在非线性项是奇函数且无穷远处超二次的条件 下，证明了关于变号解存在性与多重性的两个定理.主要结果及其证明方法均不同于文献中的 结果.

关键词：四阶Kirchhoff型椭圆边值问题；变号解；临界点