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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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A multiplicity result for asymptotically linear Kirchhoff equations

Chao Ji / Fei Fang / Binlin Zhang
Published Online: 2017-03-16 | DOI: https://doi.org/10.1515/anona-2016-0240

Abstract

In this paper, we study the following Kirchhoff type equation:

-(1+bN|u|2dx)Δu+u=a(x)f(u)in N,uH1(N),

where N3, b>0 and f(s) is asymptotically linear at infinity, that is, f(s)O(s) as s+. By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on a(x).

Keywords: Kirchhoff type equations; asymptotically linear; ground state solution; variational methods.

MSC 2010: 35J60; 35J25; 35J20

1 Introduction and main result

In this paper, we study the following Kirchhoff type equations in N (N3):

-(1+bN|u|2dx)Δu+u=a(x)f(u)in N,uH1(N),(1.1)

where b>0, a(x) and f satisfy the following assumptions:

  • (A1)

    fC(,+), f(s)0 if s<0 and f(s)=o(s) as s0+.

  • (A2)

    There exists l(0,+) such that f(s)sl as s+.

  • (A3)

    a(x)>0 is a continuous function and there exists R0>0 such that

    sup{f(s)s:s>0}<inf{1a(x):|x|R0}.

Throughout this paper, we denote by H:=H1(N) the usual Sobolev space equipped with the following inner product and norm:

(u,v)=N(uv+uv)𝑑x,u=(u,u)12.

Define the energy functional Ib:H by

Ib(u)=u22+b4(N|u|2dx)2-Na(x)F(u)dx,

where F(s)=0sf(t)𝑑t. The functional Ib is well defined for each uH and belongs to C1(H,). Moreover, for any u,φH, we have

Ib(u),φ=N(uφ+uφ)dx+bN|u|2dxNuφdx-Na(x)f(u)φdx.

Clearly, the critical points of Ib are the weak solutions for problem (1.1).

In recent years, much attention has been paid to Kirchhoff type equations. Let ΩN be a bounded domain. The following Kirchhoff problem with zero boundary data:

-(a+bΩ|u|2dx)Δu=f(x,u),xΩ,u=0on Ω,(1.2)

which is related to the stationary analogue of the equation

ρ2ut2-(P0h+E2L0L|ux|2𝑑x)2ux2=0,

was proposed by Kirchhoff [10] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. In (1.2), u denotes the displacement, f(x,u) the external force and b the initial tension, while a is related to the intrinsic properties of the string, such as Young’s modulus. We would like to point out that such nonlocal problems also appear in other fields such as biological systems, where u describes a process which depends on the average of itself, for example, population density. It is worth mentioning that Fiscella and Valdinoci [5] proposed a stationary fractional Kirchhoff model, in bounded regular domains of N, which takes into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. For some recent results about stationary Kirchhoff problems involving the fractional Laplacian, we refer to [18, 20, 19, 22, 23, 25, 30, 31, 34] and the references therein. For more mathematical and physical background for problem (1.2), we refer the readers to [1, 2, 5, 24] and the references therein.

Recently, problems like type (1.2) in bounded domains have been investigated by many authors, see, for instance, [6, 7, 16, 17, 21, 26, 32, 33]. In [16], Ma and Muñoz Rivera obtained positive solutions via variational methods. In [21], Perera and Zhang obtained a nontrivial solution via the Yang index and the critical group. Zhang and Perera [33], and Mao and Zhang [17] obtained multiple and sign-changing solutions via the invariant sets of descent flow. Shuai [26] obtained one least energy sign-changing solution via a constraint variational method and the quantitative deformation lemma. He and Zou [6, 7] obtained infinitely many solutions via the local minimum method and the fountain theorems.

Equations of type (1.1) in the whole space N (N3), but with the nonlinear term a(x)f(u) being replaced by a more general nonlinear term f(x,u), have also been studied extensively, see, for example, [8, 9, 15, 13, 12, 28, 35] and the references therein. More recently, the researchers paid their attention on asymptotically linear Kirchhoff equations. In [11], for a special type of a Kirchhoff equation with asymptotically linear term, in which the nonlocal term is like 1+bN(|u|2+V(x)u2)𝑑x (this makes the functional contain a term like 12N(|u|2+V(x)u2)𝑑x+b4(N(|u|2+V(x)u2)𝑑x)2), Li and Sun only needed to verify the (PS) condition in order to apply the mountain pass theorem to obtain the existence and multiplicity of solutions. Moreover, we noticed that the potential V(x) in [11] was assumed to be radially symmetric, in order to consider the problem in a radial function Sobolev space in which the compact Sobolev embedding holds, and thus being easy to verify the (PS) condition. In some cases, this symmetric condition may be replaced by other compact conditions which make the compact embedding holds; here we just quote [3]. In [29], Wu and Liu studied the existence and multiplicity of nontrivial solutions for a Kirchhoff equation in 3 with asymptotically linear term via Morse theory and local linking. To overcome the loss of compactness, the usual strategy is to restrict the functional to a subspace of H1(3), which embeds compactly into L2(3) with certain assumptions on radially symmetric functions. In this paper, we will study directly problem (1.1) in H, not in any subspaces. To overcome the loss of compactness, motivated by [14] in which Liu, Wang and Zhou studied an asymptotically linear Schrödinger equation, we will make a careful prior estimate for the Cerami sequence (defined later). Unlike the problem in [11], we not only show the convergence of the Cerami sequence, but also show that N|un|2dxN|u|2dx as n, where {un} is a Cerami sequence. This makes the study of our problem more difficult. Through a careful observation, we show that this limit holds only by assuming that (A1), (A2) and (A3) hold, without any more assumptions.

Now let us state the main result of this paper.

Theorem 1.1.

Assume that (A1), (A2) and (A3) hold. If l>μ with

μ=inf{N(|u|2+u2)dx:uH with Na(x)u2dx=1},

then there exists b~>0 such that for any b(0,b~), problem (1.1) admits at least two nontrivial nonnegative solutions, in which one is a mountain pass type solution and the other is a ground state solution.

We use the following notation:

  • C, C1, C2, etc. will denote positive constants whose exact values are not essential.

  • , is the duality pairing between H-1 and H, where H-1 denotes the dual space of H.

  • p is the norm of the space Lp(N).

  • 2*=2NN-2 if N3.

  • D1,2(N):={uL2*(N):uL2(N)}.

  • BR(x) denotes the open ball centered at x having radius R.

2 Preliminary lemmas

To prove Theorem 1.1, we use a variant version of the mountain pass theorem, which allows us to find a so-called Cerami type (PS) sequence. The properties of this kind of (PS) sequence are very helpful in showing the boundedness of the sequence in the asymptotically linear case.

Theorem 2.1 ([4]).

Let E be a real Banach space with its dual space E*, and suppose that IC1(E,R) satisfies

max{I(0),I(e)}μ<ηinfu=ρI(u)

for some μ<η, ρ>0 and eE with e>ρ. Let cη be characterized by

c=infγΓmax0τ1I(γ(τ)),

where Γ={γC([0,1],E):γ(0)=0,γ(1)=e} is the set of continuous paths joining 0 and e. Then there exists a sequence {un}E such that

I(un)cη𝑎𝑛𝑑(1+un)I(un)E*0as n.

This kind of sequence is usually called a Cerami sequence.

Lemma 2.1.

Assume that (A1), (A2) and (A3) hold. Then there exist ρ>0, η>0 such that

inf{I(u):uH with u=ρ}>η.

Proof.

By (A1) and (A2), for any ϵ>0, there exists Cϵ>0 such that

f(s)ϵ|s|+Cϵ|s|2*-1for all x,(2.1)

and then

F(s)ϵ2|s|2+Cϵ2*|s|2*for all x.(2.2)

Moreover, by (A1), (A2) and (A3), there exists C1>0 such that

a(x)C1for all xN.(2.3)

So, from (2.2), (2.3) and the Sobolev inequality, for any uH, we have

|Na(x)F(u)dx|ϵC12N|u|2dx+CϵC12*N|u|2*dxϵC12u2+CϵC22*u2*.

Thus, one has

Ib(u)=u22+b4(N|u|2dx)2-Na(x)F(u)dx1-ϵC12u2-CϵC22*u2*,

thanks to the fact that b4(N|u|2dx)2 is nonnegative. Fixing ϵ(0,C1-1) and letting u=ρ>0 be small enough, there exists η>0 such that the desired conclusion holds. ∎

Lemma 2.2.

Assume that (A1), (A2) and (A3) hold. If l>μ, then there is b~>0 such that for any b(0,b~), there exists eH with e>ρ such that Ib(e)<0.

Proof.

According to the definition of μ and l>μ, there exists ϕH, with ϕ0, such that

Na(x)ϕ2𝑑x=1andμN(|ϕ|2+ϕ2)𝑑x<l.

By (A2) and Fatou’s Lemma, we have

limt+I0(tϕ)t2=ϕ22-limt+Na(x)F(tϕ)(tϕ)2ϕ2𝑑x12(ϕ2-l)<0,

by taking e=t0ϕ with t0 large enough so that I0(e)=I0(t0ϕ)<0 and e=t0ϕ>ρ. Since

Ib(e)=I0(e)+b4(N|e|2dx)2

is continuous and increasing in b0 and I0(e)<0, there exists b~>0 sufficiently small such that Ib(e)<0 for all b(0,b~). ∎

By means of Lemmas 2.12.2 and Theorem 2.1, there exists a sequence {un}H such that

Ib(un)cηand(1+un)Ib(un)H-10as n.(2.4)

In order to get the existence of a nontrivial nonnegative solution, we first show that this sequence is bounded.

Lemma 2.3.

Let (A1), (A2) and (A3) hold, and let l>μ. Then, for any b(0,b~), where b~ given by Lemma 2.2, the sequence {un} defined in (2.4) is bounded in H.

Proof.

Assume on the contrary that un+ as n. Define ωn=unun. Clearly, {ωn} is bounded in H and there exists ωH such that, going if necessary to a subsequence,

ωnωin H,ωnωin Lloc2(N),ωnω a.e. in N as n.(2.5)

We claim that ω0. Assume on the contrary that ω0. By (A3), there exists a constant θ(0,1) such that

sup{f(s)s:s>0}<θinf{1a(x):|x|R0}.

For any n, this yields

|x|R0a(x)f(un)unωn2𝑑xθ|x|R0ωn2𝑑xθωn2=θ<1.(2.6)

Since the embedding H1(BR0(0))L2(BR0(0)) is compact, we have ωnω in L2(BR0(0)). According to [27, Lemma A.1], going if necessary to a subsequence, there exists gL2(BR0(0)) such that

|ωn|g(x)a.e. in BR0(0).(2.7)

By (A1) and (A2), there exists C>0 such that

f(t)tCfor all tR.(2.8)

By (2.3), (2.7) and (2.8), for any n, we have

0a(x)f(un)unωn2Ca(x)ωn2CC1ωn2CC1g2a.e. in BR0(0).(2.9)

Noting that ωnω0 a.e. in N, we obtain

a(x)f(un)unωn20a.e. in N.(2.10)

It follows, from (2.9), (2.10) and the dominated convergence theorem, that

limn|x|<R0a(x)f(un)unωn2𝑑x=0.(2.11)

Thus, by (2.6) and (2.11), we get

lim supnNa(x)f(un)unωn2𝑑x<1.(2.12)

Since un+ as n, it follows from (2.4) that

o(1)=Ib(un),unun2=1+b(N|un|2dx)2un2-Na(x)f(un)unωn2𝑑x1-Na(x)f(un)unωn2𝑑x.

Therefore,

Na(x)f(un)unωn2𝑑x+o(1)1,

which contradicts (2.12), so ω0.

On the other hand, since un+ as n, it follows form (2.4) that

Ib(un),unun4=o(1),

that is,

o(1)=1un2+b(N|ωn|2dx)2-Na(x)f(un)unωn2𝑑xun2.(2.13)

From (2.3) and (2.8), one has

Na(x)f(un)unωn2𝑑xun2=o(1).(2.14)

From (2.13) and (2.14), it is clear that

b(N|ωn|2dx)2=o(1).

Fatou’s Lemma yields

0=lim infnb(N|ωn|2dx)2b(N|ω|2dx)20,

that is,

(N|ω|2dx)2=0.(2.15)

Since the embedding HD1,2(N) is continuous, ω also belongs to D1,2(N). According to the definition of the norm of D1,2(N) and (2.15), ω0. This is a contradiction. So, the sequence {un} is bounded in H. ∎

To prove that the Cerami sequence {un} in (2.4) converges to a nonzero critical point of Ib, we need the following compactness lemma.

Lemma 2.4.

Assume that (A1), (A2) and (A3) hold. Then, for any ϵ>0, there exists R(ϵ)>R0 and n(ϵ) such that

|x|R(|un|2+un2)𝑑xϵ

for all RR(ϵ) and nn(ϵ).

Proof.

Let ξR:N[0,1] be a smooth function such that

ξR(x)={0,0|x|R,1,|x|2R,(2.16)

and, for some constant C0>0 (independent of R),

|ξR(x)|C0Rfor all xN.

Then, for all n and RR0, we have

N|(unξR)|2dx2N|un|2ξR2dx+N|un|2|ξR|2dx2N|un|2dx+2C02R2N|un|2dx2(1+C02R2)un22(1+C02R02)un2.

This implies that

unξR2(2+C02R02)12un(2.17)

for all n and RR0. By (2.4), I(un)H-1un0 as n. So, for any ϵ>0, there exists n(ϵ)>0 such that

I(un)H-1unϵ2(2+C02R02)12(2.18)

for all nn(ϵ). Hence, it follows from (2.17) and (2.18) that

|I(un),unξR|I(un)H-1unξRϵ(2.19)

for all RR0 and nn(ϵ). Note that

I(un),unξR=N|un|2ξRdx+Nun2ξRdx+bN|un|2dxN|un|2ξRdx+NununξRdx+bN|un|2dxNununξRdx-Na(x)f(un)unξRdx.(2.20)

For any ϵ>0, there exists R(ϵ)>R0 such that

1R24ϵ2C02for all RR(ϵ).(2.21)

By (2.21) and Young’s inequality, for all n and RR(ϵ), we get

N|ununξR|dxϵN|un|2dx+14ϵ|x|2R|un|2C02R2dxϵN|un|2dx+ϵ|x|2R|un|2dxϵun2.(2.22)

By (A1), (A2), (A3) and (2.16), there exists η1(0,1) such that for all n and RR0,

N|a(x)f(un)unξR|dxη1Nun2ξRdx.(2.23)

In virtue of the fact that bN|un|2dxN|un|2ξRdx is nonnegative, together with (2.20), (2.22) and (2.23), for all n and RR(ϵ)R0, we have

I(un),unξRN|un|2ξRdx+(1-η1)Nun2ξRdx-ϵun2-ϵbun4.(2.24)

Since the sequence {un} is bounded in H, it follows from (2.19) and (2.24) that there exists C3>0 such that for all RR(ϵ) and nn(ϵ),

N|un|2ξRdx+(1-η1)Nun2ξRdxC3ϵ.

From η1(0,1) and (2.16), the desired conclusion easily follows. ∎

3 Proof of Theorem 1.1

Now we are in a position to give the proof of Theorem 1.1.

Proof of Theorem 1.1.

By Lemma 2.3, the sequence {un} defined in (2.4) is bounded in H. Since H is a reflexive space, going if necessary to a subsequence, unu in H for some uH. In order to prove the theorem, we need to show that N|un|2dxN|u|2dx and that the sequence {un} has a strong convergence subsequence in H, that is, unu as n. Note that, by (2.4),

Ib(un),un=N(|un|2+un2)dx+b(N|un|2dx)2-Na(x)f(un)undx=o(1)(3.1)

and

Ib(un),u=N(unu+unu)dx+bN|un|2dxNunudx-Na(x)f(un)udx=o(1).(3.2)

Since unu in H, we have

N(unu+unu)𝑑x=N(|u|2+u2)𝑑x+o(1).(3.3)

Since the embedding HD1,2(N) is continuous, unu in D1,2(N), and thus

Nunudx=N|u|2dx+o(1).(3.4)

To show that

N|un|2dxN|u|2dxandunuas n,

we first prove that

Na(x)f(un)un𝑑x=Na(x)f(un)u𝑑x+o(1).(3.5)

For any ϵ>0, by Lemma 2.4, (A3) and Hölder’s inequality, for n large enough, we have

|x|R(ϵ)a(x)f(un)un𝑑x-|x|R(ϵ)a(x)f(un)u𝑑x|x|R(ϵ)(a12|un-u|)(a12|f(un)|)𝑑x(|x|R(ϵ)a|un-u|2𝑑x)12(|x|R(ϵ)a|f(un)|2𝑑x)12(|x|R(ϵ)a|un-u|2dx)12(|x|R(ϵ)|un|2dx)12C4ϵ.

This and the compactness of the embedding H1(N)Lloc2(N) imply (3.5). Since {un} is bounded in D1,2(N), we assume that N|un|2dxλ0. If λ=0, by Fatou’s Lemma,

N|un|2dxN|u|2dx=0=λasn,(3.6)

so u0 in H. From (3.1)–(3.6), it is easy to see that

un2u2=0as n.(3.7)

But (3.6) and (3.7) invoke a contradiction with (2.4) in which c>0. So λ>0, and by Fatou’s Lemma we have

λN|u|2dx.

If λ=N|u|2dx, from (3.1)–(3.5), it is also easy to see that unu in H as n, hence the proof is completed.

If λ>N|u|2dx, from (3.1)–(3.5), we obtain

o(1)=N(|un|2+un2)dx-N(unu+unu)dx+b(N|un|2dx)2-bN|un|2dxNunudx+Na(x)f(un)udx-Na(x)f(un)undx=N(|un|2+un2)dx-N(|u|2+u2)dx+bλ(λ-N|u|2dx)+o(1).

By Fatou’s Lemma, it is easy to see from our assumption that

0bλ(λ-N|u|2dx)=C5>0,

which is impossible. So,

N|un|2dxN|u|2dx>0as n.

Moreover, we get unu as n. Now we show that the solution u is nonnegative. Multiplying equation (1.1) by u- and integrating over N, where u-=min{u(x),0}, we find

u-2+bN|u|2dxN|u-|2dx=0.

Hence, u-=0 and u is a nonnegative solution of problem (1.1).

To get a ground state solution, we denote by K the nontrivial critical set of Ib. Set

m:=inf{Ib(u):uK}.

It is easy to see that K is nonempty. For any uK, we have

0=Ib(u),u=u2+b(N|u|2dx)2-Na(x)f(u)udxu2-Na(x)f(u)udx.

Now we choose ϵ(0,C1-1) as in the proof of Lemma 2.1 and use (2.1), (2.3) and the Sobolev embedding theorem to get

|Na(x)f(u)u𝑑x|N(ϵC1u2+C1Cϵ|u|2*)𝑑xϵC1u2+C2Cϵu2*.

Therefore, for any uK, we have

0u2-ϵC1u2-C2Cϵu2*.(3.8)

We recall that u0 whenever uK, and (3.8) implies

u(1-ϵC1C2Cϵ)12*-2>0for all uK.(3.9)

Hence, any limit point of a sequence in K is different from zero.

We claim that Ib is bounded from below on K, i.e., there exists M>0 such that Ib(u)-M for all uK. Otherwise, there exists {un}K such that

Ib(un)<-nfor all n.(3.10)

It follows from (2.3) that

Ib(un)14un2-CCϵun2*.

This and (3.10) imply that un+ as n. Let ωn=unun. There exists ωH such that (2.5) holds. Note that Ib(un)=0 for unK. As in the proof of Lemma 2.3, we obtain that un+ is impossible. Then Ib is bounded from below on K. So m-M. Let {u¯n}K be such that Ib(u¯n)m as n. Then (2.4) holds for the sequence {u¯n} and m. Following almost the same procedures as in the proofs of Lemmas 2.3 and 2.4, and using the above arguments, we can show that {u¯n} is bounded in H and, going if necessary to a subsequence, u¯nu¯, where u¯H{0} and N|u¯n|2dxN|u¯|2dx as n. There is only one difference in showing that (3.6) does not hold. Based on the aforementioned discussions, we know that the possible critical value c>0. Here we do not know if m>0, but (3.9) holds, and so λ>0. Moreover, Ib(u¯)=m and Ib(u¯)=0. Therefore, u¯H{0} is a ground state solution of problem (1.1). Finally, we can also show that the ground state solution u¯ is nonnegative. ∎

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About the article

Received: 2016-11-08

Revised: 2016-12-11

Accepted: 2017-01-20

Published Online: 2017-03-16


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11301181

Funding Source: Natural Science Foundation of Heilongjiang Province

Award identifier / Grant number: A201306

The first author is supported by NSFC (No. 11301181) and China Postdoctoral Science Foundation, the second author is supported by Young Teachers Foundation of BTBU (No. QNJJ2016-15). The third author was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667) and Doctoral Research Foundation of Heilongjiang Institute of Technology (No. 2013BJ15).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 267–277, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0240.

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