Multiplicity result for non-homogeneous fractional Schrodinger--Kirchhoff-type equations in ℝn

and César E. Torres Ledesma
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  • Departmento de Mathemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n., Trujillo-Perú, Chile
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Abstract

In this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type

M(nn|u(x)-u(z)|p|x-z|n+ps𝑑z𝑑x)(-Δ)psu+V(x)|u|p-2u=f(x,u)+g(x)

in n, where (-Δ)ps is the fractional p-Laplacian operator with 0¡s¡1¡p¡, ps¡n, f : n× is a continuous function, V : n+ is a potential function and g : n is a perturbation term. Assuming that the potential V is bounded from bellow, that f(x,t) satisfies the Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, and that g(x) is sufficiently small in Lp(n), we obtain some new criterion to guarantee that the equation above has at least two non-trivial solutions.

1 Introduction

The aim of this article is to study a Schrödinger–Kirchhoff-type equation with fractional p-Laplacian in n,

M(nn|u(x)-u(z)|p|x-z|n+ps𝑑z𝑑x)(-Δ)psu+V(x)|u|p-2u=f(x,u)+g(x),

where 0<s<1<p<, ps<n, and the operator (-Δ)ps is the fractional p-Laplacian which may be defined along a function φC0(n) as

(-Δ)psu(x)=limϵ0+nB(x,ϵ)|φ(x)-φ(z)|p-2(φ(x)-φ(z))|x-z|n+ps𝑑z,

where xn, and B(x,ϵ)={yn:|x-y|<ϵ}. We invite the reader to check [16, 19, 20, 21, 25, 35] and the references therein for further details on the fractional p-Laplace operator. The function g=g(x) can be viewed as a perturbation term.

When p=2 and M=1, equation (1.1) becomes the fractional Laplacian equation in n,

(-Δ)su+V(x)u=f(x,u),

which has been study by many researchers, see, for instance, [2, 6, 7, 8, 11, 34] and the references therein for some results.

In recent years, great attention has been paid on the study of problems involving the non-local fractional Laplacian or more general integro-differential operators. This type of operators arises in a quite natural way in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, as they are the typical outcome of stochastical stabilization of Lévy processes, see [3, 4, 5, 23] and the references therein. The literature on fractional operators and their applications is very extensive, see, for example, [1, 27, 28, 2, 6, 7, 8, 15, 14, 11, 13, 12, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. For a short introduction to the fractional Laplacian and the fractional Sobolev spaces, the reader is referred to [10]. Furthermore, research has been done in recent years for the regional fractional Laplacian, where the scope of the operator is restricted to a variable region near each point. We mention the work by Guan [17] and Guan and Ma [18] where they study these operators, their relation with stochastic processes and they develop an integration by parts formula. We also mention the work by Ishii and Nakamura [21], where they studied the Dirichlet problem for regional fractional Laplacian modeled on the p-Laplacian and the recent work by Felmer and Torres [13, 12], where they considered the existence, symmetry properties and concentration phenomena of solutions of the non-linear Schrödinger equation with non-local regional diffusion. These regional operators present various interesting characteristics that make them very attractive from the point of view of mathematical theory of non-local operators.

On the other hand, Fiscella and Valdinoci in [15] first proposed a stationary Kirchhoff variational equation which models the non-local aspect of the tension arising from non-local measurements of the fractional length of the string. Indeed, problem (1.1) is a fractional version of a model, the so-called Kirchhoff equation, introduced by Kirchhoff in [22]. More precisely, Kirchhoff established a model given by the problem

ρ2ut2-(p0h+E2L0L|ux|2𝑑x)2ux2=0,

where ρ is the mass density, p0 is the initial tension, h is the area of the cross section, E is the Young modulus of the material and L is the length of the string, which extends the classical D’Alembert wave equation by considering the effects of the changes in the length of the strings during the vibrations. Note that non-local boundary problems like (1.2) can be used to model several physical and biological systems where u describes a process, which depend on the average of itself, such as the population density [9]. A parabolic version of problem (1.2) can be used to describe the growth and movement of a particular species. The movement, modeled by the integral term, is assumed to be dependent on the energy of the entire system with u being its population density. Alternatively, the movement of a particular species may be subject to the total population density within the domain (for instance, the spreading of bacteria) which gives rise to equations of the type

ut-ψ(Ωu𝑑x)Δu=h(x,u).

It is worth pointing out that problem (1.2) received much attention only after Lions [26] proposed an abstract framework to it. For some motivation in the physical background for the fractional Kirchhoff model, we refer to [15, Appendix A]. Also, see for example [29, 30, 31, 37] for non-degenerate Kirchhoff-type problems, and [1, 36], for degenerate Kirchhoff-type problems in this direction.

Very recently in [32], Pucci, Xiang and Zhang considered problem (1.1) under the following assumptions: For the Kirchhoff function M, suppose that:

  1. libel=($M_{0}$),leftmirgin=24ptMC(0+) satisfies inft0+M(t)a>0, where a>0 is a constant.
  2. liibel=($M_{0}$),leftmiirgiin=24ptThere exists θ[1,nn-sp) such that
    θ(t)=θ0tM(s)𝑑stM(t)for all t0+.

For the potential V, suppose that:

  1. libel=($V_{0}$),leftmirgin=24ptVC(n,) and infxnV(x)V0>0, where V0>0 is a constant.
  2. liibel=($V_{0}$),leftmiirgiin=24ptThere exists h>0 such that
    lim|y|meas({xBh(y):V(x)c})=0for any c>0.

For the nonlinearity f, suppose that:

  1. libel=($f_{0}$),leftmirgin=24ptf:n× is a Carathéodory function and there exist q, with θp<q<ps*, and a1>0 such that |f(x,t)|a1(1+|t|q-1) for a.e. xn and all t.
  2. liibel=($f_{0}$),leftmiirgiin=24ptThere exists μ>θp such that
    μF(x,t)tf(x,t)for all xn and all t.
  3. liiibel=($f_{0}$),leftmiiirgiiin=24ptf(x,t)=o(|t|p-1) as t0, uniformly for xn.
  4. livbel=($f_{0}$),leftmivrgivn=24ptinfxn,|t|=1F(x,t)>0.

Under the assumptions (M1)–(M2), (V1)–(V2), (f1)–(f4), the authors first establish a Batsch–Wang-type compact embedding theorem for the fractional Sobolev spaces. Multiplicity results are then obtained by using the Ekeland’s variational principle and the Mountain Pass Theorem. Also we mention the work by Xiang, Zhang and Ferrara [38], where they considered the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities and have obtained the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland’s variational principle.

Inspired by these previous works, in this paper we consider problem (1.1) under some weaker assumptions. More precisely, we assume that the Kirchhoff function M satisfies (M1)–(M2). Furthermore, regarding the potential V we only assume (V1) and for the functions f we suppose that:

  1. libel=($H_{0}$),leftmirgin=24ptfC(n×,) and there exists ps*>μ>θp such that
    0<μF(x,ζ)ζf(x,ζ),for all xn and all ζ{0},
    where F(x,ζ)=0ζf(x,s)𝑑s.
  2. liibel=($H_{0}$),leftmiirgiin=24ptThere exists some positive continuous function a:n with
    lim|x|a(x)=0
    such that
    |f(x,ζ)|a(x)|ζ|μ-1for all(x,ζ)n×.

Before stating our main results, we introduce some useful notations. First of all, define the Gagliardo seminorm by

[u]s,pp=nn|u(x)-u(y)|p|x-y|n+sp𝑑x𝑑y,

where u:n is a measurable function. Now, let the fractional Sobolev space be denoted as

Ws,p(n)={uLp(n):u is measurable and [u]s,pp<}

and assume that it is endowed with the norm

us,p=([u]s,pp+upp)1/p,

where the fractional critical exponent is defined by

ps*={npn-spif sp<n,if spn.

Moreover, we consider the fractional Sobolev space with potential

Xs:={uWs,p(n):nV(x)|u|p𝑑x<}

endowed with the norm

uXs=([u]s,pp+V(x)1/pupp)1/p.

We say that uXs is a weak solution of problem (1.1) if

M([u]s,pp)2n|u(x)-u(z)|p-2(u(x)-u(z))(φ(x)-φ(z))|x-z|n+ps𝑑z𝑑x+nV(x)|u(x)|p-2u(x)φ(x)𝑑x
=nf(x,u)φ(x)𝑑x+ng(x)φ(x)𝑑x

for any φXs.

From here on we set p=p/(p-1), the Hölder conjugate of p. Our main result in this paper is the following theorem.

Theorem 1.1.

Let (M1)–(M2), (V1), (H1)–(H2) hold and suppose that gLp(Rn) and g0. Then there exists a constant δ0>0 such that problem (1.1) has at least two non-trivial solutions in Xs, provided that gLpδ0.

Remark 1.1.

We note that condition (V2) is used to establish some compact embedding theorems to guarantee that the (PS) condition holds, which is the essential step to obtain the existence of weak solutions of (1.1) via the Mountain Pass Theorem. In the present paper, we assume that V(x) is bounded from below but we could not obtain some compact embedding theorem. Therefore, one difficulty is to adapt some new technique to overcome this difficulty and test that the (PS) condition is verified, see Lemmas 3.1 and 3.2 below.

The remaining part of this paper is organized as follows. Some preliminary results are presented in Section 2. In Section3, we are devoted to accomplishing the proof of Theorem 1.1.

2 Preliminaries

To study the fractional problem (1.1), the so-called fractional Sobolev spaces Ws,p(n) with 0<s<1 are expedient. If 1<p<, as usual, the norm is defined through

us,pp=nn|u(x)-u(y)|p|x-y|n+sp𝑑x𝑑y+n|u(x)|p𝑑x.

We recall the Sobolev embedding theorem.

Theorem 2.1 ([10]).

Let s(0,1) and p[1,) be such that sp<n. Then there exists a positive constant C=C(n,p,s) such that

uLps*pCnn|u(x)-u(y)|p|x-y|n+sp𝑑x𝑑y,

where ps*=npn-sp is the so-called “fractional critical exponent”. Consequently, the space Ws,p(Rn) is continuously embedded in Lq(Rn) for any q[p,ps*]. Moreover, the embedding Ws,p(Rn)Llocq(Rn) is compact for q[p,ps*).

Consider now the space Xs defined by

Xs:={uWs,p(n):nV(x)|u|p𝑑x<},

endowed with the norm

uXs=(n×n|u(x)+u(y)|p|x-y|n+sp𝑑x𝑑y+nV(x)|u(x)|p𝑑x)1/p.

Remark 2.1.

From (V1), Theorem 2.1 and the Hölder inequality, we have XsLq(n) for pqps* and XsLlocq(n) compactly for q[p,ps*). Moreover, there exists 𝔖q such that

uq𝔖quXs.

Now we introduce some more notations and necessary definitions. Let be a real Banach space. If I is a continuously Fréchet-differentiable functional defined on , we write IC1(,). Recall that IC1(,) is said to satisfy the (PS) condition if every sequence {un}n for which {I(un)}n is bounded and I(un)0 as n possesses a convergent subsequence in .

Moreover, let Br be the open ball in with radius r and centered at 0, and let Br denote its boundary. Under the conditions of Theorem 1.1, we obtain the existence of the first weak solution of (1.1) using the following well-known Mountain Pass Theorem, see [33].

Lemma 2.1.

[33, Theorem 2.2] Let B be a real Banach space and let IC1(B,R) satisfy the (PS) condition. Suppose that I(0)=0 and

  1. libel=($i_{0}$),libelwidth=*there are constants ρ, η>0 such that I|Bρη,
  2. liibel=($ii_{0}$),liibelwiidth=*there is an eB¯ρ such that I(e)0.

Then I possesses a critical value cη. Moreover, c can be characterized as

c=infgΓmaxs[0,1]I(g(s)),

where

Γ={gC([0,1],):g(0)=0,g(1)=e}.

As far as the second solution is concerned, we obtain it using the minimizing method, which is contained in a small ball centered at 0.

3 Proof of Theorem 1.1

The aim of this section is to establish the proof of Theorem 1.1. For this purpose, we are going to establish the corresponding variational framework to obtain solutions of (1.1). To this end, define the functional I:Xs by

I(u)=J(u)-H(u),

where

J(u)=1p(([u]s,pp)+V(x)1/puLpp),
H(u)=nF(x,u)+g(x)u(x)dx.

If (M1) and (V1) hold, then J:Xs is of class C1(Xs) and

J(u),v=M([u]s,pp)n×n|u(x)-u(z)|p-2(u(x)-u(z))(v(x)-v(z))|x-z|n+ps𝑑z𝑑x
+nV(x)|u(x)|p-2u(x)v(x)𝑑x

for all u,vXs. Moreover, J is weakly lower semi-continuous in Xs, see [32].

Now we prove our key lemma.

Lemma 3.1.

Under the conditions of Theorem 1.1, Φ is compact, i.e., Φ(uk)Φ(u) if uku in Xs, where Φ:XsR is defined by

Φ(u)=nF(x,u(x))𝑑x.

Proof.

From (H2) it is obvious that

f(x,u)=o(|u|p-1)as |u|0,a.e. xn

and

f(x,u)=o(|u|ps*-1)as |u|,a.e. xn.

Therefore, by (3.3) and (3.4), for any ϵ>0, there exists Cϵ such that

|f(x,u)|ϵ|u|p-1+Cϵ|u|ps*-1for all u and a.e. xn.

Now assume that unu in Xs. Then, there is some constant K>0 such that

1𝔖qukqukXsKand1𝔖ququXsK

for k. Furthermore, by Remark 2.1, given R>0 up to a subsequence, we may assume that:

  1. (i)uku in Lq(B(0,R)) for q[p,ps*) as k,
  2. (ii)uk(x)u(x) a.e. in n as k,
  3. (iii)there is ψLq(B(0,R)) such that |uk(x)|ψ(x) a.e in n and for all k.

Thus, by the previous inequality

|f(x,uk)|aϕμ-1Lμ(B(0,R))

and

limk|f(x,uk(x))-f(x,u(x))|=0a.e. in B(0,R).

Hence, by the Lebesgue Dominated Convergence Theorem we have

limkB(0,R)|f(x,uk)-f(x,u)|μ𝑑x=0,

where μ+μ=μμ.

On the other hand, from (H2), for any ϵ>0 there exists R>0 such that

|f(x,u)|ϵ|u|μ-1and|f(x,uk)|ϵ|uk|μ-1for all |x|>R.

Consequently, in view of (2.1), (3.6)–(3.8), for k large enough, we have

|Φ(uk)-Φ(u),v|n|f(x,uk)-f(x,u)||v|𝑑x
B(0,R)|f(x,uk)-f(x,u)||v|𝑑x+B(0,R)c|f(x,uk)||v|𝑑x+B(0,R)c|f(x,u)||v|𝑑x
ϵvLμ+ϵB(0,R)c|uk|μ-1|v|𝑑x+ϵB(0,R)c|u|μ-1|v|𝑑x
ϵ𝔖μvXs+ϵB(0,R)cl(μ-1μ|uk|μ+1μ|v|μr)𝑑x+ϵB(0,R)cl(μ-1μ|u|μ+1μ|v|μr)𝑑x
ϵ𝔖μvXs+ϵμ-1μB(0,R)c(|uk|μ+|u|μ)𝑑x+ϵ2μB(0,R)c|v|μ𝑑x.

Here, we apply the Young inequality

abapp+bqq,a,b>0,p,q>1and1p+1q=1.

Consequently, we obtain that

Φ(uk)-Φ(u)=supvXs=1|n(f(x,uk)-f(x,u),v)dx|
ϵ𝔖μ+2ϵ𝔖μμμ[(μ-1)Kμ+1],

which yields that Φ(uk)Φ(u) as uku, so Φ is compact. ∎

Remark 3.1.

Under the conditions of Lemma 3.1, by the Hölder inequality we get

limkng(x)uk(x)𝑑x=ng(x)u(x)𝑑x.

Therefore, H is weakly continuous in Xs.

To prove Theorem 1.1 we first consider some lemmas.

Lemma 3.2.

Under the conditions of Theorem 1.1, the functional I satisfies the (PS) condition.

Proof.

Although the proof of this lemma is just the repetition of the process of [32, Lemma 6], we give the details for convenience of the reader.

Assume that {uk}kXs is a sequence such that {I(uk)}k is bounded and I(uk)0 as k. Then there exists a constant C>0 such that

|I(uk)|CandI(uk)(Xs)*C

for every k, where (Xs)* is the dual space of Xs.

Firstly, we show that {uk}k is bounded. In fact, in view of (M1)–(M2), (H1), (3.10) and using also that μ>θpp>1, we obtain

C+CμukXsI(uk)-1μI(uk)uk
=1p([uk]s,pp)-1μM([uk]s,pp)[uk]s,pp+(1p-1μ)ukXsp
-n[F(x,uk(x))-1μf(x,uk(x))uk(x)]𝑑x-μ-1μng(x)uk𝑑x
(1pθ-1μ)M([uk]s,pp)[uk]s,pp+(1p-1μ)ukXsp-μ-1μ𝔖pgLpukXs
min{(1θp-1μ)a,1p-1μ}ukXsp-𝔎ukXs.

where 𝔎=𝔖pgLp. Hence, {uk}k is bounded in Xs. Then, the sequence {uk}k has a subsequence, again denoted by {uk}k, and there exists uXs such that

ukuweakly inXs

which implies that

I(uk)-I(u),uk-u0as k.

Furthermore, according to Lemma 3.1, we have

H(uk)H(u)as k.

Let φXs be fixed and denote by Bφ the linear functional on Xs defined by

Bφ(v)=n×n|φ(x)-φ(z)|p-2(φ(x)-φ(z))|x-z|n+ps(v(x)-v(z))𝑑z𝑑x

for all vXs. By the Hölder inequality, Bφ is continuous and hence

limk(M([uk]s,pp)-M([u]s,pp))Bu(uk-u)=0,

since {M([uk]s,pp)-M([u]s,pp)}k is bounded in .

Therefore, for k large enough we get

o(1)=I(uk)-I(u),uk-u
=M([uk]s,pp)Buk(uk-u)-M([uk]s,pp)Bu(uk-u+(M([uk]s,pp)-M([u]s,pp))Bu(uk-u))
+nV(x)(|uk|p-2uk-|u|p-2u)(uk-u)𝑑x-n(f(x,uk)-f(x,u))(uk-u)𝑑x+o(1),

that is

limk(M([uk]s,pp)[Buk(uk-u)-Bu(uk-u)]+nV(x)(|uk|p-2uk-|u|p-2u)(uk-u)dx=0.

Note that

M([uk]s,pp)[Buk(uk-u)-Bu(uk-u)]0andV(x)(|uk|p-2uk-|u|p-2u)(uk-u)0

for all k by convexity. Considering also (M1) and (V1), we have in particular

limk[Buk(uk-u)-Bu(uk-u)]0andlimkV(x)(|uk|p-2uk-|u|p-2u)(uk-u)0.

Now, by the well-know Simon inequalities [32], for p2, (M1) and (3.14) as k, one has

[uk-u]s,ppcp[Buk(uk-u)-Bu(uk-u)]=o(1).

Similarly, by (V1) and (3.14) as k, we get

V(x)1/p(uk-u)Lppo(1).

In conclusion, uk-uXs0 as k, as required.

In the case 1<p<2, since ukuweakly inXs there exists K>0 such that uks,ppK for all k. Now by the Simon inequality, the Hölder inequality and (3.14) as k, we have

[uk-u]s,ppC[Buk(uk-u)-Bu(uk-u)]p/2=o(1),

where C=2CpKp(2-p)/2. Furthermore, by the Hölder inequality and (3.14) as k

V(x)1/p(uk-u)LppL(nV(x)(|uk|p-2uk-|u|p-2u)(uk-u)𝑑x)p/2,

where L=2CpKp(2-p)/p. Hence, uk-uXs0 as k also in the second case. Therefore, I satisfies the (PS) condition, as stated. ∎

Lemma 3.3.

Under the conditions of Theorem 1.1, there exist ρ>0 and α>0 such that

infuXs=ρI(u)>α.

Proof.

By (3.5), for any ϵ>0, there exists Cϵ such that

|F(x,u)|ϵ|u|p+Cϵ|u|ps*for all (x,u)n×.

Thus, using the Hölder inequality, we have

I(u)=1p(([u]s,pp)+V(x)1/puLpp)-nF(x,u(x))𝑑x-ng(x)u𝑑x
min{1,a}puXsp-ϵuLpp-CϵuLps*ps*-gLpuLp
min{1,a}puXsp-ϵ𝔖ppuXsp-Cϵ𝔖ps*ps*uXsps*-𝔖pgLpuXs
=uXs[(min{1,a}p-ϵ𝔖pp)uXsp-1-Cϵ𝔖ps*ps*uXsps*-1-𝔖pgLp].

Taking ϵ=min{1,a}p𝔖pp and setting

η(t)=min{1,a}2ptp-1-Cϵ𝔖ps*ps*tps*-1for all t0+,

we see that there exists ρ>0 such that maxt0+η(t)=η(ρ)>0, since ps*>p>1 by (H1). Taking δ0:=η(ρ)2𝔖pp, we obtain that

I(u)α=ρη(ρ)2>0

for all uXs with uXs=ρ and for all gLp(n) with gLpδ0. ∎

Lemma 3.4.

Under the conditions of Theorem 1.1, let ρ>0 be defined as in Lemma 3.3. Then, there exists an eXs with eXs>ρ such that I(e)<0.

Proof.

First, we note that by (H1),

lim|u|F(x,u)|u|θp=.

Fix φC0(n) with φXs=1. Then, for any ϵ>0 we have

limssupp(φ)F(x,sφ)sθp𝑑x1ϵsupp(φ)|φ(x)|θp𝑑x.

Since ϵ is arbitrary, by (3.17) we obtain

limssupp(φ)F(x,sφ)sθp𝑑x=.

Furthermore, by assumption (M2), we also get that

(ξ)(1)ξθfor all ξ1.

Consequently, by (3.18) and (3.19), we have as s,

I(sφ)sθp=1psθp(([sφ]s,pp)+spV(x)1/pφLpp)-nF(x,sφ)sθp𝑑x-1sθp-1ngφ𝑑x
1psθp((1)sθp[φ]s,pθp+spV(x)1/pφLpp)-supp(φ)F(x,sφ)sθp𝑑x-1sθp-1ngφ𝑑x
max{1,(1)}p-supp(φ)F(x,sφ)sθp𝑑x-1sθp-1ngφ𝑑x-

Hence, if s0 is big enough and e=s0φ, one gets I(e)<0. ∎

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

Proof of Theorem 1.1.

Since gLp(n) and g0, we can choose a function χC0(n)Xs such that

ng(x)χ(x)𝑑x>0.

Then, by (H1) we note that F(x,t)0 a.e. and

I(σχ)1p([σχ]s,pp)+σppnV(x)|χ|p𝑑x-σng(x)χ(x)𝑑x
1p(supξ[0,ρp](ξ))σp[χ]s,pp+σppnV(x)|χ|p𝑑x-σng(x)χ(x)𝑑x<0,

for σ>0 small enough, where ρ is given in Lemma 3.3. Thus, we get

c1=inf{I(u):uB¯ρ}<0,

where Bρ={uXs:uXs<ρ}. By the Ekeland variational principle and Lemma 3.3, there exists a sequence {uk}kBρ such that

c1I(uk)c1+1kandI(v)I(uk)-1kv-ukXs

for all vB¯ρ. Then, a standard procedure gives that {uk}k is a bounded (PS) sequence of I. Therefore, Lemmas 3.2 and 3.3 imply that there exists a function u1Bρ such that

I(u1)=0andI(u1)=c1<0.

On the other hand, by Lemmas 3.3 and 3.4 and the Mountain Pass Theorem, there exists a sequence {uk}kXs such that, as k,

I(uk)c2>0andI(uk)0.

In view of the proof of Lemma 3.2, we know that there exists a critical point u2Xs of I. Moreover,

I(u2)=c2>0=I(0).

Thus, u20 and u2u1. ∎

References

  • [1]

    G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125 (2015), 699–714.

  • [2]

    X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.

  • [3]

    L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations. The Abel Symposium 2010 (Oslo 2010), Abel Symp. 7, Springer, Berlin (2012), 37–52.

  • [4]

    L. Caffarelli, J. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111–1144.

  • [5]

    L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) no. 7–9, 1245–1260.

  • [6]

    X. Chang and Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494.

  • [7]

    X. Chang and Z. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations 256 (2014), 2965–2992.

  • [8]

    M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53 (2012), Article ID 043507.

  • [9]

    M. Chipot and J. Rodrigues, On a class of non-local non-linear elliptic problems, RAIRO Modél. Math. Anal. Num. 26 (1992), no. 3, 447–467.

  • [10]

    E. Di Nezza, G. Patalluci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

  • [11]

    P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2002), no. 6, 1237–1262.

  • [12]

    P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrödinger equation, Comm. Pure Appl. Anal. 13 (2007), no. 6, 2395–2406.

  • [13]

    P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var. Partial Differential Equations 54 (2015), 75–98.

  • [14]

    M. Ferrara, G. Molica Bisci and B. Zhang, Existence of weak solutions for non-local fractional problems via Morse theory, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 8, 2483–2499.

  • [15]

    A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170.

  • [16]

    G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373–386.

  • [17]

    Q.-Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys. 266 (2006), 289–329.

  • [18]

    Q.-Y. Guan and Z. M. Ma, Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Related Fields 134 (2006), 649–694.

  • [19]

    A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. (2014), 10.1515/acv-2014-0024.

  • [20]

    A. Iannizzotto and M. Squassina, Weyl-type laws for fractional p-eigenvalue problems, Asymptot. Anal. 88 (2014), 233–245.

  • [21]

    H. Ishii and G. Nakamura, A class of integral equations and approximation of p-Laplace equations, Calc. Var. Partial Differential Equations 37 (2010), 485–522.

  • [22]

    G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

  • [23]

    N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), 298–305.

  • [24]

    N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002), no. 5, Article ID 056108.

  • [25]

    E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795–826.

  • [26]

    J. Lions, On some questions in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Rio de Janeiro 1977), North-Holland, Amsterdam (1978), 284–346.

  • [27]

    G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett. 27 (2014), 53–58.

  • [28]

    G. Molica Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Anal. Appl. (Singap.) 13 (2015), no. 4, 371–394.

  • [29]

    N. Nyamoradi, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type, Math. Commun. 18 (2013), 489–502.

  • [30]

    N. Nyamoradi and N. Chung, Existence of solutions to nonlocal Kirchhoff equations of elliptic type via genus theory, Electron. J. Differ. Equ. (2014), Paper No. 86.

  • [31]

    P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in n {\mathbb{R}^{n}} involving nonlocal operators, Rev. Mat. Iberoam., to appear.

  • [32]

    P. Pucci, M. Xiang and B. Zhang, Multiple solutions for non-homogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in n {\mathbb{R}^{n}}, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785–2806.

  • [33]

    P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986.

  • [34]

    S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in n {\mathbb{R}^{n}}, J. Math. Phys. 54 (2013), Article ID 031501.

  • [35]

    C. Torres, Existence and symmetry result for fractional p-Laplacian in n {\mathbb{R}^{n}}, preprint (2015).

  • [36]

    M. Xiang and B. Zhang, Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition, Complex Var. Elliptic Equ. 60 (2015), no. 9, 1277–1287.

  • [37]

    M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), 1021–1041.

  • [38]

    M. Xiang, B. Zhang and M. Ferrara, Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities, Proc. R. Soc. A (2015), 10.1098/rspa.2015.0034.

  • [39]

    B. Zhang and M. Ferrara, Multiplicity of solutions for a class of superlinear non-local fractional equations, Complex Var. Elliptic Equ. 60 (2015), 583–595.

  • [40]

    B. Zhang and M. Ferrara, Two weak solutions for perturbed non-local fractional equation, Appl. Anal. 94 (2015), 891–902.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125 (2015), 699–714.

  • [2]

    X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.

  • [3]

    L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations. The Abel Symposium 2010 (Oslo 2010), Abel Symp. 7, Springer, Berlin (2012), 37–52.

  • [4]

    L. Caffarelli, J. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111–1144.

  • [5]

    L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) no. 7–9, 1245–1260.

  • [6]

    X. Chang and Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494.

  • [7]

    X. Chang and Z. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations 256 (2014), 2965–2992.

  • [8]

    M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53 (2012), Article ID 043507.

  • [9]

    M. Chipot and J. Rodrigues, On a class of non-local non-linear elliptic problems, RAIRO Modél. Math. Anal. Num. 26 (1992), no. 3, 447–467.

  • [10]

    E. Di Nezza, G. Patalluci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

  • [11]

    P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2002), no. 6, 1237–1262.

  • [12]

    P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrödinger equation, Comm. Pure Appl. Anal. 13 (2007), no. 6, 2395–2406.

  • [13]

    P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var. Partial Differential Equations 54 (2015), 75–98.

  • [14]

    M. Ferrara, G. Molica Bisci and B. Zhang, Existence of weak solutions for non-local fractional problems via Morse theory, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 8, 2483–2499.

  • [15]

    A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170.

  • [16]

    G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373–386.

  • [17]

    Q.-Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys. 266 (2006), 289–329.

  • [18]

    Q.-Y. Guan and Z. M. Ma, Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Related Fields 134 (2006), 649–694.

  • [19]

    A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. (2014), 10.1515/acv-2014-0024.

  • [20]

    A. Iannizzotto and M. Squassina, Weyl-type laws for fractional p-eigenvalue problems, Asymptot. Anal. 88 (2014), 233–245.

  • [21]

    H. Ishii and G. Nakamura, A class of integral equations and approximation of p-Laplace equations, Calc. Var. Partial Differential Equations 37 (2010), 485–522.

  • [22]

    G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

  • [23]

    N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), 298–305.

  • [24]

    N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002), no. 5, Article ID 056108.

  • [25]

    E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795–826.

  • [26]

    J. Lions, On some questions in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Rio de Janeiro 1977), North-Holland, Amsterdam (1978), 284–346.

  • [27]

    G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett. 27 (2014), 53–58.

  • [28]

    G. Molica Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Anal. Appl. (Singap.) 13 (2015), no. 4, 371–394.

  • [29]

    N. Nyamoradi, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type, Math. Commun. 18 (2013), 489–502.

  • [30]

    N. Nyamoradi and N. Chung, Existence of solutions to nonlocal Kirchhoff equations of elliptic type via genus theory, Electron. J. Differ. Equ. (2014), Paper No. 86.

  • [31]

    P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in n {\mathbb{R}^{n}} involving nonlocal operators, Rev. Mat. Iberoam., to appear.

  • [32]

    P. Pucci, M. Xiang and B. Zhang, Multiple solutions for non-homogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in n {\mathbb{R}^{n}}, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785–2806.

  • [33]

    P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986.

  • [34]

    S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in n {\mathbb{R}^{n}}, J. Math. Phys. 54 (2013), Article ID 031501.

  • [35]

    C. Torres, Existence and symmetry result for fractional p-Laplacian in n {\mathbb{R}^{n}}, preprint (2015).

  • [36]

    M. Xiang and B. Zhang, Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition, Complex Var. Elliptic Equ. 60 (2015), no. 9, 1277–1287.

  • [37]

    M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), 1021–1041.

  • [38]

    M. Xiang, B. Zhang and M. Ferrara, Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities, Proc. R. Soc. A (2015), 10.1098/rspa.2015.0034.

  • [39]

    B. Zhang and M. Ferrara, Multiplicity of solutions for a class of superlinear non-local fractional equations, Complex Var. Elliptic Equ. 60 (2015), 583–595.

  • [40]

    B. Zhang and M. Ferrara, Two weak solutions for perturbed non-local fractional equation, Appl. Anal. 94 (2015), 891–902.

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