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BY-NC-ND 3.0 license Open Access Published by De Gruyter January 19, 2016

Multiplicity results for elliptic Kirchhoff-type problems

  • Sami Baraket and Giovanni Molica Bisci EMAIL logo

Abstract

The aim of this paper is to establish the existence of multiple solutions for a perturbed Kirchhoff-type problem depending on two real parameters. More precisely, we show that an appropriate oscillating behaviour of the nonlinear part, even under small perturbations, ensures the existence of at least three nontrivial weak solutions. Our approach combines variational methods with properties of nonlocal fractional operators.

1 Introduction

Consider the elliptic problem

(${K_{\lambda,\mu}^{f,g}}$){-(a+bΩ|u|2𝑑x)Δu=λf(x,u)+μg(x,u)in Ω,u=0on Ω,

where a,b are two real positive constants, ΩN (with N3) is a nonempty bounded open subset with boundary Ω of class C1, f,g:Ω× are two Carathéodory functions having a suitable growth, and (λ,μ)+×0+.

In this paper, we establish sufficient conditions on the nonlinearities f and g under which the above problem possesses three nontrivial solutions for suitable values of the parameters λ and μ. The main tool in order to achieve our multiplicity result is a critical point theorem contained in [4] along with the technical approach developed in [5]; see Theorem 2.1 below.

In our setting, the presence of the term a+bΩ|u|2𝑑x implies that the equation under consideration is of nonlocal type and this fact gives rise to some mathematical difficulties. In addition, these kinds of problems have motivations from physics. For instance, the above operator appears in the Kirchhoff equation which arises in nonlinear vibrations, namely,

{utt-(a+bΩ|u|2𝑑x)Δu=h(x,u)in Ω×(0,T),u=0on Ω×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x).

Hence, problem (${K_{\lambda,\mu}^{f,g}}$) is related to the stationary counterpart of the above evolution equation. Such a hyperbolic equation is a general version of the relation

(1.1)ρ2ut2-(P0h+E2L0L|ux|2𝑑x)2ux2=0,

which was proposed by Kirchhoff in 1883 (see [11]) as an extension of the D’Alembert wave equation for free vibrations of elastic strings. The parameters in (1.1) have the following meaning: L is the length of the string, h is the area of the cross section, E is the Young modulus of the material, ρ is the mass density, and P0 is the initial tension.

Many solvability conditions for Kirchhoff-type problems are known, such as the Yang index theory and via the use of invariant sets of descent flow; see [21, 17]. Moreover, there have been several multiplicity results for Kirchhoff problems by using variational methods; see, for instance, [1, 7, 10, 13, 15, 16]. In particular, Mao and Zhang studied in [14, Theorem 1.2] the existence of multiple solutions for problem (K1,0f,0). More precisely, under suitable conditions on the nonlinearity f, combining critical point theory, the invariant set of descent flow, and minimax methods, they proved the existence of three solutions: one positive, one negative, and one sign-changing. Among others, one of their key assumptions is the condition

lim|t|0f(x,t)|t|=0

uniformly for every xΩ.

Furthermore, very recently, Ricceri established in [19] the existence of at least three weak solutions to a class of Kirchhoff-type doubly eigenvalue boundary value problem. We point out that in the paper of Ricceri no estimate of the parameters μ is given and the condition

lim supξ0supxΩ0ξf(x,t)𝑑tξ2<+

is required; see [19, Theorem 1].

For completeness, we also observe that, very recently, the nonlocal fractional counterpart of Kirchhoff-type problems has been considered. In this order of ideas, a similar variational approach to the one adopted here can be used for proving the existence of weak solutions for the (doubly) nonlocal problem

(${D_{M,f}}$){M(n×n|u(x)-u(y)|2|x-y|n+2s𝑑x𝑑y)(-Δ)su=f(x,u)in Ω,u=0in nΩ,

where Ω is a bounded domain in (n,||) with smooth boundary Ω, s(0,1) is fixed with s<n/2, and (-Δ)s is the fractional Laplace operator, which (up to normalization factors) can be defined as

(-Δ)su(x):=-nu(x+y)+u(x-y)-2u(x)|y|n+2s𝑑y,xn.

Furthermore, f:Ω¯× and M:[0,+)[0,+) are suitable continuous maps.

We explicitly observe that Theorem 3.1 below and the above cited results are mutually independent; see Remark 3.4 and Example 3.5. Finally, we emphasize that our novel approach adopted in the present paper yields the existence of at least three nontrivial solutions for the unperturbed problem as well; see Remark 3.3.

The structure of the paper is as follows. In Section 2, we recall our abstract framework. Then, Section 3 is devoted to the main theorem and, finally, we give some consequences and applications of the presented result.

2 Abstract framework

Let us put, as usual, 2*:=2N/(N-2). The space H01(Ω) indicates the closure of C0(Ω) in the Sobolev space W1,2(Ω) with respect to the norm

u:=(Ω|u(x)|2𝑑x)1/2.

From the Sobolev embedding theorem (see, for instance, [18, Proposition B.7]), for every q[1,2*], there exists a positive constant cq such that

(2.1)uLq(Ω)cqu,uH01(Ω),

and, in particular, the embedding H01(Ω)Lq(Ω) is compact for every q[1,2*[. It follows that

(2.2)cqmeas(Ω)(2*-q)/(2*q)N(N-2)π(N!2Γ(N/2+1))1/N,

where meas(Ω) denotes the Lebesgue measure of the set Ω; see, for instance, [20].

From now on, we assume that there exist four nonnegative constants a1, a2, b1, b2 and q,p]1,2*[ such that

|f(x,t)|a1+a2|t|q-1and|g(x,t)|b1+b2|t|p-1

for every (x,t)Ω×.

Moreover, let us fix (λ,μ)+×0+. A function u:Ω is said to be a weak solution of problem (${K_{\lambda,\mu}^{f,g}}$) if uH01(Ω) and

(a+bu2)Ωu(x)v(x)𝑑x-λΩf(x,u(x))v(x)𝑑x-μΩg(x,u(x))v(x)𝑑x=0

for every vH01(Ω). They are also classical solutions if f is locally Lipschitz continuous in Ω×; see, for instance, [14].

From a variational standpoint, the weak solutions of (Kλ,μf,g) in H01(Ω) are exactly the critical points of the C1-functional given by

Jλ,μ(u):=a2u2+b4u4-λΩ(0u(x)f(x,t)𝑑t)𝑑x-μΩ(0u(x)g(x,t)𝑑t)𝑑x

for every uH01(Ω). Moreover, its Gâteaux derivative is given by

Jλ(u)(v)=(a+bu2)Ωu(x)v(x)𝑑x-λΩf(x,u(x))v(x)𝑑x-μΩg(x,u(x))v(x)𝑑x

for every u,vH01(Ω).

Finally, we recall the following critical point theorem, obtained in [4].

Theorem 2.1

Let X be a reflexive real Banach space, let Φ:XR be a coercive, continuously Ga^teaux differentiable, and sequentially weakly lower semicontinuous functional whose Ga^teaux derivative admits a continuous inverse on X*, and let Ψ:XR be a continuously Ga^teaux differentiable functional whose Ga^teaux derivative is compact such that

Φ(0)=Ψ(0)=0.

Assume that there exist r>0 and x¯X with r<Φ(x¯) such that

  1. there holds

    supΦ(x)rΨ(x)r<Ψ(x¯)Φ(x¯);
  2. for each

    λΛr:=]Φ(x¯)Ψ(x¯),rsupΦ(x)rΨ(x)[,

    the functional Jλ:=Φ-λΨ is coercive.

Then, for each λΛr, the functional Jλ has at least three distinct critical points in X.

In the next section, adopting the technical approach developed in [5] and using the above abstract framework, we obtain Theorem 3.1.

3 Existence of three weak solutions

Put

F(x,ξ):=0ξf(x,t)𝑑tandG(x,ξ):=0ξg(x,t)𝑑t

for every (x,ξ)Ω×. Moreover, set

K:=(2N-12N-1)1/2πN/4D(N-2)/2Γ(1+N/2)[aΓ(1+N/2)+bπN/2DN-2(2N-12N-1)]1/2,

where D:=supxΩdist(x,Ω).

Simple calculations show that there exists x0Ω such that B(x0,D)Ω, where B(x0,D) denotes the open ball of center x0 and radius D. Finally, for every q[1,2*[, let

Aq:=cqqq(2a)q/2,

where cq is given by (2.1).

Our result reads as follows.

Theorem 3.1

Let N4 and let f:Ω×RR be a Carathéodory function such that

  1. there exist two nonnegative constants a1,a2 and q]1,2*[ such that

    |f(x,t)|a1+a2|t|q-1.

Assume that

  1. F(x,ξ)0 for every (x,ξ)Ω×+;

  2. there exist two positive constants γ and δ with γ<Kmin{1,δ2} such that

    A1a1γ+Aqa2γq-2<1K2B(x0,D/2)F(x,δ)𝑑xmax{1,δ4}.

Then, for each parameter

λΛ(γ,δ):=]K2max{1,δ4}B(x0,D/2)F(x,δ)𝑑x,1A1a1/γ+Aqa2γq-2[

and for every Carathéodory function g:Ω×RR satisfying that

  1. there are two nonnegative constants b1,b2 and p]1,2*[ such that

    |g(x,t)|b1+b2|t|p-1;
  2. G(x,ξ)0 for every (x,ξ)Ω×+,

there exists a positive constant

δλ,g:=1-λ(A1a1/γ+Aqa2γq-2)A1b1/γ+Apb2γp-2

such that, for each μ[0,δλ,g[, problem (${K_{\lambda,\mu}^{f,g}}$) has at least three weak solutions in H01(Ω).

Proof.

Fix λ, g, and μ as in the theorem’s statement and take X=H01(Ω) endowed with the usual norm

u:=(Ω|u(x)|2𝑑x)1/2

for each uX. Moreover, set

Φ(u):=a2u2+b4u4

and

Ψ(u):=Ω(F(x,u(x))+μλG(x,u(x)))𝑑x

for every uX.

Since the critical points of the functional Jλ:=Φ-λΨ on X are exactly the weak solutions of (${K_{\lambda,\mu}^{f,g}}$), our aim is to apply Theorem 2.1 to Φ and Ψ. To this end, we take into account that the regularity assumptions of Theorem 2.1 on Φ and Ψ are satisfied; see, for instance, [19]. Hence, we will verify (a1) and (a2). Owing to (f1) and (g1), one has

F(x,ξ)a1|ξ|+a2|ξ|qqandG(x,ξ)b1|ξ|+b2|ξ|pp

for every (x,ξ)Ω×.

Let η]0,+[ and consider the function

χ(η):=supuΦ-1(]-,η])Ψ(u)η.

Taking into account the above inequalities for F and G, it follows that

Ψ(u)a1uL1(Ω)+a2quLq(Ω)q+μλ(b1uL1(Ω)+b2puLp(Ω)p).

Then, for every uX such that Φ(u)η, due to (2.1), we get

Ψ(u)(2ηac1a1+2q/2cqqa2qaq/2ηq/2)+μλ(2ηac1b1+2p/2cppb2pap/2ηp/2).

Hence,

(3.1)supuΦ-1(]-,η])Ψ(u)(2ηac1a1+2q/2cqqa2qaq/2ηq/2)+μλ(2ηac1b1+2p/2cppb2pap/2ηp/2).

From (3.1), one has

(3.2)χ(η)(2aηc1a1+2q/2cqqa2qaq/2ηq/2-1)+μλ(2aηc1b1+2p/2cppb2pap/2ηp/2-1)

for every η>0. Then, by using our notation, the previous inequality can be written as

χ(η)(A1a1η+Aqaqη(q-2)/2)+μλ(A1b1η+Apbpη(p-2)/2)

for every η>0.

Hence, let uδH01(Ω) given by

uδ(x):={0if xΩB(x0,D),2δD(D-|x-x0|)if xB(x0,D)B(x0,D/2),δif xB(x0,D/2).

Moreover, Φ(uδ)>γ2, indeed

Φ(uδ)=a2uδ2+b4uδ4
=a2Ω|uδ(x)|2𝑑x+b4(Ω|uδ(x)|2𝑑x)2
=a2B(x0,D)B(x0,D/2)(2δ)2D2𝑑x+b4(B(x0,D)B(x0,D/2)(2δ)2D2𝑑x)2
=a2(2δ)2D2[meas(B(x0,D))-meas(B(x0,D/2))]+b4(2δ)4D4[meas(B(x0,D))-meas(B(x0,D/2))]2
=2δ2πN/2(DN-(D/2)N)D4(Γ(1+N/2))2[aD2Γ(1+N/2)+2bδ2πN/2(DN-DN2N)].

Therefore, one has

(3.3)min{1,δ4}K2Φ(uδ)max{1,δ4}K2.

Now, since γ<Kmin{1,δ2}, it follows that Φ(uδ)>γ2.

Then,

(3.4)χ(γ2)=supuΦ-1(]-,γ2])Ψ(u)γ2(A1a1γ+Aqa2γq-2)+μλ(A1b1γ+Apb2γp-2).

Moreover, since μ<δλ,g, then

μ<1-λ(A1a1/γ+Aqa2γq-2)A1b1/γ+Apb2γp-2,

which means that

(3.5)(A1a1γ+Aqa2γq-2)+μλ(A1b1γ+Apb2γp-2)<1λ.

On the other hand, the conditions (f2) and (g2) yield

Ψ(uδ)=ΩF(x,uδ(x))𝑑x+μλΩG(x,uδ(x))𝑑xB(x0,D/2)F(x,δ)𝑑x.

Hence, from (3.3) and bearing in mind the above inequality, it follows that

(3.6)Ψ(uδ)Φ(uδ)1K2B(x0,D/2)F(x,δ)𝑑xmax{1,δ4}.

So, owing to the choice of the parameter λ, that is,

1λ<1K2B(x0,D/2)F(x,δ)𝑑xmax{1,δ4},

one has

(3.7)Ψ(uδ)Φ(uδ)>1λ.

Then, from (3.4), (3.5), and (3.7), it follows that the condition (a1) of Theorem 2.1 holds. Now, observe that from conditions (f1) and (g1), bearing in mind that N4, it automatically follows that

lim sup|ξ|supxΩG(x,ξ)ξ40andlim sup|ξ|supxΩF(x,ξ)ξ40.

Moreover, we can find l>0 such that

lim sup|ξ|supxΩG(x,ξ)ξ4<landμl<b4c44.

Therefore, there exists a function hlL1(Ω) such that

G(x,ξ)lξ4+hl(x)

for each (x,ξ)Ω×. Now, fix

0<ε<b-4μlc444λc44.

Furthermore, there exists a function hεL1(Ω) such that

F(x,ξ)εξ4+hε(x)

for each (x,ξ)Ω×. Then, for each uX,

Φ(u)-λΨ(u)(b4-λεc44-μlc44)u4-(λhεL1(Ω)+μhlL1(Ω)).

This leads to the coercivity of Jλ and the condition (a2) of Theorem 2.1 is verified. Hence, Theorem 2.1 assures the existence of three critical points for the functional Jλ and the proof is complete. ∎

Remark 3.2

With the usual notation, let us put

Bq:=cqqq(4b)q/4,

where q[1,2*[. In Theorem 3.1, instead of condition (f3), assume that

  1. there exist two positive constants γ and δ with γ<K1/2min{1,δ} such that

    B1a1γ3+Bqa2γq-4<1K2B(x0,D/2)F(x,δ)𝑑xmax{1,δ4}.

In this setting, our result guarantees that, for each parameter

λΛ(γ,δ):=]K2max{1,δ4}B(x0,D/2)F(x,δ)𝑑x,1B1a1/γ3+Bqa2γq-4[

and for every Carathéodory function g:Ω× satisfying (g1) and (g2), there exists a positive constant

δλ,g:=1-λ(B1a1/γ3+Bqa2γq-4)B1b1/γ3+Bpb2γp-4

such that, for each μ[0,δλ,g[, problem (${K_{\lambda,\mu}^{f,g}}$) has at least three weak solutions in H01(Ω).

Remark 3.3

We explicitly observe that if f(x,0)0 in a set Ω0Ω with positive Lebesgue measure, then, for every Carathéodory function g:Ω× satisfying (g1) and (g2), Theorem 3.1 ensures the existence of at least three nontrivial weak solutions for problem (${K_{\lambda,\mu}^{f,g}}$). The attained result is also new for the unperturbed case, that is, for μ=0.

Now, if f and g are locally Lipschitz continuous, our result guarantees the existence of at least three classical solutions for problem (${K_{\lambda,\mu}^{f,g}}$). Moreover, if in addition f and g are nonnegative functions, the strong maximum principle (see [9, Theorem 8.19]) guarantees that the (nontrivial) solutions are positive.

Indeed, let u0 be a solution of (${K_{\lambda,\mu}^{f,g}}$). Arguing by contradiction, assume that the set

A={xΩ:u0(x)<0}

is of positive measure. Put v¯(x)=min{0,u0(x)} for all xΩ. Clearly, v¯H01(Ω) and one has

(a+bu02)Ωu0(x)v¯(x)𝑑x=λΩf(u0(x))v¯(x)𝑑x+μΩg(u0(x))v¯(x)𝑑x,

that is,

(a+bu02)A|u0(x)|2𝑑x=λAf(x,u0(x))u0(x)𝑑x+μAg(x,u0(x))u0(x)𝑑x.

Therefore, it follows that

0(a+bu02)A|u0(x)|2𝑑x0.

Hence, u0=0 in A and this is absurd. Then, u0 is nonnegative in Ω. From the strong maximum principle one has that either u00 or u0>0 in Ω.

Remark 3.4

After a careful analysis of the proof of Theorem 3.1, one can see that it also works for N=3, just requiring that the conditions (f1) and (g1) hold true for some constants q,p]1,4[. Moreover, for the sake of simplicity, let us consider the case μ=0 and take f: to be a continuous function, nonnegative in ]0,+[, such that

|f(t)|a2|t|q-1

for all t and for some q]2,4[.

Clearly, the above growth condition is a particular case of condition (f1) and implies that f(0)=0. In this setting, Theorem 3.1 ensures the existence of at least three (two nontrivial) solutions for every λ sufficiently large. In [19, Theorem 2], Ricceri obtained an analogous existence result for N4 and q]1,2*[. In that paper, it was proposed to investigate if Theorem 2 also holds for N=3. Anello, in [2], gave a negative answer to this question for every q]4,6[. We just point out that our approach produces a positive answer to the proposed problem whenever q]2,4[. Finally, as already pointed out in Section 1, [19, Theorem 1] and its consequences deal with elliptic Kirchhoff problems for which the assumption

lim supξ0supxΩF(x,ξ)ξ2<+

is required.

The cited result and Theorem 3.1 are mutually independent as the following example shows.

Example 3.5

Let N=3 and assume that K>1. Furthermore, fix q]4,6[ and consider the continuous and positive function h: defined by

h(t):={1+|t|q-1if tδ,(1+r2)(1+rq-1)1+t2if t>δ,

where δ is a fixed constant such that

δ>max{1,(qK2meas(B(x0,D/2))(A1+Aq))1/(q-4)}.

Clearly, the function h satisfies the conditions (f1) and (f2) of Theorem 3.1. Moreover, (f3) is also verified taking into account the choice of δ. Since δ2>1, one has 1<Kmin{1,δ2}. Furthermore, setting

H(ξ):=0ξh(t)𝑑t

for every ξ, it follows that

H(δ)δ4=δq-4q+1δ3>K2meas(B(x0,D/2))(A1+Aq).

Then, for each parameter λ belonging to

]K2δ4meas(B(x0,D/2))H(δ),1A1+Aq[

and for every μ such that

0μ<1-λ(A1+Aq)2A1,

the perturbed problem

(${{K}_{\lambda,\mu}^{h}}$){-(a+bΩ|u|2𝑑x)Δu=λh(u)+μ(|sinu|+1)in Ω,u=0on Ω

has at least three (positive) classical solutions in H01(Ω).

Funding statement: This project was funded by the National Plan of Sciences, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, award no. 12-MAT2880-02.

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Received: 2015-12-1
Accepted: 2015-12-20
Published Online: 2016-1-19
Published in Print: 2017-2-1

© 2017 by De Gruyter

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