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Advanced Nonlinear Studies

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Volume 16, Issue 1

Issues

Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity

Jianjun Zhang
  • College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074; and Chern Institute of Mathematics, Nankai University, Tianjin 300071, P. R. China
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/ João Marcos do Ó / Marco Squassina
Published Online: 2016-01-27 | DOI: https://doi.org/10.1515/ans-2015-5024

Abstract

We consider a fractional Schrödinger–Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti–Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.

Keywords: Schrödinger–Poisson Systems; Variational Methods; Critical Growth

MSC 2010: 35B25; 35B33; 35J61

1 Introduction and Main Results

We are concerned with the fractional nonlinear Schrödinger–Poisson system

{(-Δ)su+λϕu=g(u)in 3,(-Δ)tϕ=λu2in 3,(1.1)

where λ>0 and (-Δ)α is the fractional Laplacian operator for α=s,t[0,1]. The fractional Schrödinger equation was introduced by Laskin [28] in the context of fractional quantum mechanics for the study of particles on stochastic fields modeled by Lévy processes. The operator (-Δ)α can be seen as the infinitesimal generator of Lévy stable diffusion processes (see Applebaum [3]). If λ=0, then (1.1) reduces to the nonlinear fractional scalar field equation

(-Δ)su=g(u)in 3.(1.2)

This equation is related to the standing waves of the time-dependent fractional scalar field equation

iϕt-(-Δ)sϕ+g(ϕ)=0in 3,(1.3)

which is a physically relevant generalization of the classical nonlinear Schrödinger equation. In fact, up to replacing (-Δ)α with (1-α)(-Δ)α, the operators in the above equations converge to -Δ, in a suitable sense, due to the results in Bourgain, Brezis and Mironescu [9]. Here, i is the imaginary unit and t denotes the time variable. For power-type nonlinearities, the fractional Schrödinger equation (1.3) was derived in [28] by replacing the Brownian motion in the path integral approach with the so-called Lévy flights (see, e.g., Metzler and Klafter [30]). Hence, the equation we want to study appears as a perturbation of a physically meaningful equation. Also, Frank and Lenzmann [21, 22] obtained deep results on the uniqueness and the non-degeneracy of ground states for (1.2) in the case when g(u)=|u|p-2u-u for subcritical p; see also Secchi and Squassina [34], where the soliton dynamics for (1.3) with an external potential was investigated. In [24], Giammetta studied the evolution equation associated with the one-dimensional system

{-Δu+λϕu=g(u)in ,(-Δ)tϕ=λu2in .(1.4)

In this case, the diffusion is fractional only in the Poisson equation. Our system is more general and contains this as a particular case. If 𝒦α(x)=|x|α-N, the equation

-Δu+u=(𝒦2*|u|2)u,uH1/2(3),u>0,

is studied in Frank and Lenzmann [20] and in Elgart and Schlein [19] it is shown that the dynamical evolution of boson stars is described by the nonlinear evolution equation

itψ=-Δ+m2ψ-(𝒦2*|ψ|2)ψ,m0,

for a field ψ:[0,T)×3 (see also Fröhlich, Jonsson and Lenzmann [23]). The square root of the Laplacian also appears in the semi-relativistic Schrödinger–Poisson–Slater system (see Bellazzini, Ozawa and Visciglia [6] and also the model studied in D’Avenia, Siciliano and Squassina [16]).

Observe that if we formally take s=t=1, then (1.1) reduces to the classical Schrödinger–Poisson system

{-Δu+λϕu=g(u)in 3,-Δϕ=λu2in 3,(1.5)

which describes systems of identically charged particles interacting with each other in the case when magnetic effects can be neglected (see Benci and Fortunato [7]). In recent years, the Schrödinger–Poisson system (1.5) has been widely studied by many researchers. Here, we would like to cite some related results, for example, Cerami and Vaira [11] for positive solutions, Azzollini and Pomponio [5] for ground state solutions, D’Aprile and Wei [15] for semi-classical states, and Ianni [25] for sign-changing solutions. See also Ambrosetti [2] and the references therein. In [4], Azzollini, d’Avenia and Pomponio were concerned with (1.5) under the Berestycki–Lions conditions (H2)(H4) with s=1 (see below). They proved that (1.5) admits a positive radial solution if λ>0 small enough. For the critical case, we refer to [38] and to the recent work [39] by the authors of the present work.

1.1 Main Results

In this paper, we are mainly concerned with positive solutions of (1.1). First, we consider the subcritical case with the Berestycki–Lions conditions. More precisely, we assume the following hypotheses on g.

  • (H1)

    gC1(,).

  • (H2)

    -<lim infτ0g(τ)τlim supτ0g(τ)τ=-m<0.

  • (H3)

    lim supτg(τ)τ2s-10, where 2s=63-2s.

  • (H4)

    There exists ξ>0 such that G(ξ):=0ξg(τ)dτ>0.

Our first result is the following theorem.

Suppose that g satisfies (H1)(H4) and 2t+4s3. Then, the following hold.

  • (i)

    There exists λ0>0 such that, for every λ(0,λ0) , ( 1.1 ) admits a nontrivial positive radial solution (uλ,ϕλ).

  • (ii)

    Along a subsequence, (uλ,ϕλ) converges to (u,0) in Hs(3)×𝒟t,2(3) as λ0 , where u is a radial ground state solution of ( 1.2 ).

The hypotheses (H2)(H4) are the so-called Berestycki–Lions conditions, which were introduced in Berestycki and Lions [8] for the derivation of the ground state of (1.2) with s=1. Under (H1)(H4), Chang and Wang [12] proved the existence of ground state solutions to (1.2) for s(0,1). The hypothesis (H1) is only used to get the better regularity of solutions to (1.2), which guarantees the Pohožaev identity. By the Pohožaev identity, (H4) is necessary.

The hypothesis 2t+4s3 is just used to guarantee that the Poisson equation (-Δ)tϕ=λu2 makes sense, due to the fact that 𝒟t,2(3)L2t(3). For details, see Section 2 below.

In the variational approach to the study of elliptic problems, the Palais–Smale condition ((PS) condition for short) plays a crucial role. To verify the (PS) condition, the so-called Ambrosetti–Rabinowitz condition

μ0τf(ξ)dξτf(τ),τ{0},μ>2,(AR)

has been frequently used in the literature. The main role of ((AR)) is to guarantee the boundedness of the (PS) sequence in some suitable Sobolev space. More recently, Pucci, Xiang and Zhang [32] considered fractional p-Laplacian equations of Schrödinger–Kirchhoff type

M(NN|u(x)-u(y)|p|x-y|N+psdxdy)(-Δ)psu+V(x)|u|p-2u=f(x,u)+g(x).(1.6)

With the use of ((AR)), they established the existence of multiple solutions to (1.6) via the Ekeland variational principle and the mountain pass theorem. In fact, ((AR)) is a technical assumption. Many mathematicians have tried to remove or weaken it. In [8], Berestycki and Lions considered the autonomous scalar field equation. Without using ((AR)), they proved the existence of ground state solutions by the constraint variational method. However, it is not easy to use the idea in [8] in order to deal directly with non-autonomous problems. In [26], Jeanjean introduced a monotonicity trick to overcome the difficulty due to the lack of ((AR)) in the non-autonomous case. In [39], without ((AR)), the authors of the present work considered the existence and the concentration of positive solutions to (1.1) in the critical case for s=t=1. It is natural to wonder if similar results can hold for the critical fractional case. This is just our second goal of the present paper. In the critical case, we assume the following hypotheses on g.

  • (H2)’

    limτ0g(τ)τ=-a<0.

  • (H3)’

    limτg(τ)τ2s-1=b>0.

  • (H4)’

    There exists μ>0 and q<2s such that g(τ)-bτ2s-1+aτμτq-1 for all τ>0.

Our second result is the following theorem.

Suppose that g satisfies (H1) and (H2)’(H4)’ . Then, the following hold.

  • (i)

    The limit problem ( 1.2 ) admits a ground state solution if max{2s-2,2}<q<2s.

  • (ii)

    If 2t+4s3 , then there exists λ0>0 such that, for every λ(0,λ0) , ( 1.1 ) admits a nontrivial positive radial solution (uλ,ϕλ) if max{2s-2,2}<q<2s.

  • (iii)

    Along a subsequence, (uλ,ϕλ) converges to (u,0) in Hs(3)×𝒟t,2(3) as λ0 , where u is a radial ground state solution of ( 1.2 ).

In the case s=1, the hypotheses (H2)’(H4)’ were introduced in Zhang and Zou [40] (see also Alves, Souto and Montenegro [1]) to obtain the ground state of the scalar field equation -Δu=g(u) in N. In [36], Shang and Zhang considered the fractional problem (1.2) in the critical case (see also Shang, Zhang and Yang [37]). With the help of the monotonicity of τg(τ)/τ, the ground state solutions were obtained by using the Nehari approach. To the best of our knowledge, there are few results in the literature about the ground states of the critical fractional problem (1.2) with a general nonlinearity, particularly without the Ambrosetti–Rabinowitz condition and the monotonicity of g(τ)/τ. Theorem 1.4 seems to be the first result in this direction.

Without loss generality, from now on, we assume that a=b=μ=1.

We conclude by fixing some notation that we will use throughout the paper. We define the norm

up:=(3|u|pdx)1/p,p[1,),

the value

2α:=63-2α,α(0,1),

and we let u^=(u) denote the Fourier transform of u.

In the rest of the paper, we use the perturbation approach to prove Theorem 1.1 and Theorem 1.4. Similar arguments can also be found in [39]. The paper is organized as follows. In Section 2, we introduce the functional framework and some preliminary results. In Section 3, we construct the min-max level. In Section 4, we use a perturbation argument to complete the proof of Theorem 1.1 and we give the proof of Theorem 1.4.

2 Preliminaries and Functional Setting

2.1 Fractional-Order Sobolev Spaces

The fractional Laplacian (-Δ)α with α(0,1) of a function ϕ:3 is defined by

((-Δ)αϕ)(ξ)=|ξ|2α(ϕ)(ξ),ξ3,

where is the Fourier transform, i.e.,

(ϕ)(ξ)=1(2π)3/23exp(-2πiξx)ϕ(x)dx,

where i is the imaginary unit. If ϕ is smooth enough, it can be computed by the singular integral

(-Δ)αϕ(x)=cαP.V.3ϕ(x)-ϕ(y)|x-y|3+2αdy,x3,

where cα is a normalization constant and P.V. stands for the principal value.

For any α(0,1), we consider the fractional-order Sobolev space

Hα(3)={uL2(3):3|ξ|2α|u^|2dξ<}

endowed with the norm

uα=(3(1+|ξ|2α)|u^|2dξ)1/2,uHα(3),

and with the inner product

(u,v)=3(1+|ξ|2α)u^v^¯dξ,u,vHα(3).

It is easy to see that the inner products

u,v3(1+|ξ|2α)u^v^¯dξandu,v3(uv+(-Δ)α/2u(-Δ)α/2v)dx

on Hs(3) are equivalent (see [36]). The homogeneous Sobolev space 𝒟α,2(3) is defined by

𝒟α,2(3)={uL2α(3):|ξ|αu^L2(3)},

which is the completion of C0(3) under the norm

u𝒟α,22=(-Δ)α/2u22=3|ξ|2α|u^|2dξ,u𝒟α,2(3),

and the inner product

(u,v)𝒟α,2=3(-Δ)α/2u(-Δ)α/2vdx,u,v𝒟α,2(3).

For a further introduction on fractional-order Sobolev spaces, we refer the interested reader to Di Nezza, Palatucci and Valdinoci [17]. Let

Hrs(3)={uH3(3):u(x)=u(|x|)}.

Now, we introduce the following Sobolev embedding theorems.

(Lions [29])

For any α(0,1), Hα(3) is continuously embedded into Lq(3) for q[2,2α] and compactly embedded into Llocq(3) for q[1,2α). Moreover, Hrα(3) is compactly embedded into Lq(3) for q(2,2α).

(Cotsiolis and Tavoularis [14], Di Nezza, Palatucci, and Valdinoci [17])

For any α(0,1), 𝒟α,2(3) is continuously embedded into L2α(3), i.e., there exists Sα>0 such that

(3|u|2αdx)2/2αSα3|(-Δ)α/2u|2dx,u𝒟α,2(3).

2.2 The Variational Setting

Now, we study the variational setting of (1.1). By Lemma 2.1, for 2t+4s3, we have

Hs(3)L12/(3+2t)(3).

Then, for uHs(3), by Lemma 2.2, the linear operator P:𝒟t,2(3) defined by

P(v)=3u2vu12/(3+2t)2v2tCus2v𝒟t,2

is well defined on 𝒟t,2(3) and is continuous. Thus, it follows from the Lax–Milgram theorem that there exists a unique ϕut𝒟t,2(3) such that (-Δ)tϕut=λu2. Moreover, for x3, we have

ϕut(x):=λct3u2(y)|x-y|3-2tdy,(2.1)

where we have set

ct=Γ(32-2t)π3/222tΓ(t).

Formula (2.1) is called the t-Riesz potential. Substituting (2.1) into (1.1), we can rewrite (1.1) in the equivalent form

(-Δ)su+λϕutu=g(u),uHs(3).(2.2)

We define the energy functional Γλ:Hs(3) by

Γλ(u)=123|(-Δ)s/2u|2dx+λ43ϕutu2dx-3G(u)dx

with

G(τ)=0τg(ζ)dζ.

Obviously, the critical points of Γλ are the weak solutions of (2.2).

  • (i)

    We call (u,ϕ)Hs(3)×𝒟t,2(3) a weak solution of (1.1) if u is a weak solution of (2.2).

  • (ii)

    We call uHs(3) a weak solution of (2.2) if

    3((-Δ)s/2u(-Δ)s/2v+λϕutuv)dx=3g(v)vdxfor all vHs(3).

Setting

T(u):=143ϕutu2dx,

we summarize some properties of ϕut and T(u) which will be used later.

If t,s(0,1) and 2t+4s3, then, for any uHs(3), the following hold.

  • (i)

    uϕut:Hs(3)𝒟t,2(3) is continuous and maps bounded sets into bounded sets.

  • (ii)

    ϕut(x)0, x3 , and T(u)cλus4 for some c>0.

  • (iii)

    T(u(/τ))=τ3+2tT(u) for any τ>0 and uHs(3).

  • (iv)

    If unu weakly in Hs(3) , then ϕunϕu weakly in 𝒟t,2(3).

  • (v)

    If unu weakly in Hs(3) , then T(un)=T(u)+T(un-u)+o(1).

  • (vi)

    If u is a radial function, so is ϕut.

Proof.

The proof is similar to that in [33], so we omit the details here. ∎

3 The Subcritical Case

3.1 The Modified Problem

It follows from Lemma 2.4 that Γλ is well defined on Hs(3) and is of class C1. Since we are concerned with positive solutions of (2.2), similarly to [8] (see also [12]), we modify our problem first. Without loss of generality, we assume that

0<ξ=inf{τ(0,):G(τ)>0},

where ξ is given in (H4). Let

τ0=inf{τ>ξ:g(τ)=0}[ξ,]

and define a function g~: by

g~(τ)={g(τ)for τ[0,τ0],0for ττ0,

and g~(τ)=0 for τ0. If uHs(3) is a solution of (2.2), where g is replaced by g~, then, by the maximum principle (see Cabré and Sire [10]), we get that u is positive and u(x)τ0 for any x3, i.e., u is a solution of the original problem (2.2) with g. Thus, from now on, we can replace g by g~, but still use the same notation g. In addition, for τ>0, let

g1(τ)=max{g(τ)+mτ,0}andg2(τ)=g1(τ)-g(τ).

Then, we have g2(τ)mτ for τ0,

limτ0g1(τ)τ=0andlimτ+g1(τ)τ2s-1=0,(3.1)

and, for any ε>0, there exists Cε>0 such that

g1(τ)εg2(τ)+Cετ2s-1,τ0.(3.2)

Let

Gi(u)=0ugi(τ)dτ,i=1,2.

Then, by (3.1) and (3.2), for any ε>0, there exists Cε>0 such that

G1(τ)εG2(τ)+Cε|τ|2s,τ.(3.3)

3.2 The Limit Problem

In the following, we will find solutions of (2.2) by seeking critical points of Γλ. If λ=0, (2.2) becomes

(-Δ)su=g(u),uHs(3),(3.4)

which is referred to as the limit problem of (2.2). We define an energy functional for the limit problem (3.4) by

L(u)=123|(-Δ)s/2u|2dx-3G(u)dx,uHs(3).

In [12], Chang and Wang proved that, with the same assumptions on g as in Theorem 1.1, there exists a positive ground state solution UHr3(3) of (3.4). Moreover, each such solution U of (3.4) satisfies the Pohožaev identity

3-2s23|(-Δ)s/2U|2dx=33G(U)dx.(3.5)

Let S be the set of positive radial ground state solutions U of (3.4). Then, S and we have the following compactness result which plays a crucial role in the proof of Theorem 1.1.

Under the assumptions in Theorem 1.1, S is compact in Hrs(3).

As shown in Cho and Ozawa [13], for general s(0,1), we do not have a similar radial lemma in Hrs(3). So the Strauss compactness lemma (see [8]) is not applicable here. Before we prove Proposition 3.1, we begin with the following compactness lemma which is a special case of [12, Lemma 2.4.]

(Chang and Wang [12])

Assume that QC(,) satisfies

limτ0Q(τ)τ2=lim|τ|Q(τ)|τ|2s=0

and there exists a bounded sequence {un}n=1Hrs(3) for some vL1(3) with

limnQ(un(x))=v(x)a.e. x3.

Then, up to a subsequence, we have Q(un)v strongly in L1(3) as n.

Proof of Proposition 3.1.

Let {un}n=1S and denote by E the least energy of (3.4). Then, for any n, un satisfies L(un)=E and the Pohožaev identity (3.5), which implies that

E=s33|(-Δ)s/2un|2dxand3G(un)dx=3-2s2sE.

Obviously, {(-Δ)s/2un2} is bounded. It follows from Lemma 2.2 that {un2s} is bounded. By (3.3), as we can see in [8], {un2} is bounded, which yields that {un} is bounded in Hrs(3). Without loss of generality, we can assume that there exists u0Hrs(3) such that unu0 weakly in Hrs(3), strongly in Lq(3) for q(2,2s), and un(x)u0(x) a.e. x3.

In the following, we adopt some ideas from [8] to prove that unu0 strongly in Hrs(3). For uHs(3), let

J(u)=s33|(-Δ)s/2u|2dxandV(u)=3G(u)dx.

Then, we know that un is a minimizer of the constrained minimizing problem

inf{J(u):uHrs(3),V(u)=3-2s2sE}.

By (3.1) and Lemma 3.2 we get that

limn3G1(un)=3G1(u0).

Then, by Fatou’s Lemma,

V(u0)3-2s2sE,

which implies that u00. Meanwhile, it is easy to see that J(u0)E. Similarly to [8], we know that u0 satisfies

J(u0)=EandV(u0)=3-2s2sE,

which yields that

limn3G2(un)=3G2(u0).

By Fatou’s Lemma, we know that un2u02 as n. Thus, unu0 strongly in Hrs(3). ∎

3.3 The Min-Max Level

Take US and let

Uτ(x)=U(xτ),τ>0.

Then, by the definition of U^=(U), we know that U^(/τ)=τ3U^(t) and

3|(-Δ)s/2Uτ|2dx=3|ξ|2s|U^(ξτ)|2=τ3-2s3|(-Δ)s/2U|2dx.

By the Pohožaev identity, we have

L(Uτ)=(τ3-2s2-3-2s6τ3)3|(-Δ)s/2U|2.

Thus, there exists τ0>1 such that L(Uτ)<-2 for ττ0. Set

Dλmaxτ[0,τ0]Γλ(Uτ).

By virtue of Lemma 2.4, we have Γλ(Uτ)=L(Uτ)+O(λ). Note that since maxτ[0,τ0]L(Uτ)=E, we get that DλE as λ0+.

Moreover, similarly to [39], we can prove the following lemma, which is crucial in defining the uniformly bounded set of the mountain paths (see below).

There exist λ1>0 and 𝒞0>0 such that, for any 0<λ<λ1, we have

Γλ(Uτ0)<-2,Uτs𝒞0for all τ(0,τ0],us𝒞0for all uS.

Now, for any λ(0,λ1), we define a min-max value Cλ as

Cλ=infγΥλmaxτ[0,τ0]Γλ(γ(τ)),

where

Υλ={γC([0,τ0],Hrs(3)):γ(0)=0,γ(τ0)=Uτ0,γ(τ)s𝒞0+1,τ[0,τ0]}.

Obviously, for τ>0, we have

Uτs2=τ3-2s(-Δ)s/2U22+τ3U22.

Then, we can define U00 so UτΥλ. Moreover, we have

lim supλ0+Cλlimλ0+Dλ=E.

We have limλ0+Cλ=E.

Proof.

It suffices to prove that

lim infλ0+CλE.

Now, we give the mountain pass value

b=infγΥmaxτ[0,1]L(γ(τ)),

where

Υ={γC([0,1],Hrs(3)):γ(0)=0,γ(1)<0}.

It follows from [12, Lemma 3.2] that L satisfies the mountain pass geometry. As we can see in Jeanjean and Tanaka [27], b agrees with the least energy level of (3.4), i.e., b=E. Note that ϕut(x)0 for x3. Then, γ~()=γ(τ0)Υ for any γΥλ and it follows that Cλb, which concludes the proof. ∎

3.4 Proof of Theorem 1.1

Now, for α,d>0, define

Γλα:={uHrs(3):Γλ(u)α}

and

Sd={uHrs(3):infvSu-vsd}.

In the following, we will find a solution uSd of (2.2) for sufficiently small λ>0 and some 0<d<1. The following proposition is crucial for obtaining a suitable (PS) sequence for Γλ and plays a key role in our proof.

Let {λi}i=1 be such that limiλi=0 and {uλi}Sd with

limiΓλi(uλi)E𝑎𝑛𝑑limiΓλi(uλi)=0.

Then, for d small enough, there is u0S, up to a subsequence, such that uλiu0 in Hrs(3).

Proof.

For convenience, we write λ for λi. Since uλSd and S is compact, we know that {uλ} is bounded in Hrs(3). Then, by Lemma 2.4, we see that

limiL(uλ)EandlimiL(uλ)=0.

It follows from [12, Lemma 3.3] that there is u0Hrs(3), up to a subsequence, such that uλu0 strongly in Hrs(3). Obviously, 0Sd for d small. This implies that u00, L(u0)E, and L(u0)=0. Thus, L(u0)=E, i.e., u0S, which completes the proof. ∎

By Proposition 3.5, for small d(0,1), there exist ω>0, λ0>0 such that

Γλ(u)sω,uΓλDλ(SdSd/2),λ(0,λ0).(3.6)

Similarly to [39], we have the following proposition.

There exists α>0 such that, for small λ>0,

Γλ(γ(τ))Cλ-αimplies thatγ(τ)Sd/2,

where γ(τ)=U(/τ) for τ(0,τ0].

Proof.

From Lemma 2.4 and the Pohožaev identity, we have

Γλ(γ(τ))=(τ3-2s2-3-2s6τ3)3|(-Δ)s/2U|2+λτ3+2tT(U).

Then,

limλ0+maxτ[0,τ0]Γλ(γ(τ))=maxτ[0,τ0](τ3-2s2-3-2s6τ3)3|(-Δ)s/2U|2=E

and the conclusion follows. ∎

Similarly as in [39], thanks to (3.6) and Proposition 3.6, we can prove the following proposition, which assures the existence of a bounded (PS) sequence for Γλ.

For λ>0 small enough, there exists {un}nΓλDλSd such that Γλ(un)0 as n.

Proof of Theorem 1.1.

It follows from Proposition 3.7 that there exists λ0>0 such that, for λ(0,λ0), there exists {un}ΓλDλSd with Γλ(un)0 as n. Noting that S is compact in Hrs(3), we get that {un} is bounded in Hrs(3). Assume that unuλ weakly in Hrs(3). Then Γλ(uλ)=0. It follows from the compactness of S that uλSd and un-uλs3d for n large. So, uλ0 for small d>0. By Lemma 2.4, we have

Γλ(un)=Γλ(uλ)+Γλ(un-uλ)+o(1).

Noting that

G2(τ)m2τ2for any τ,

it follows from (3.3) that, for some C>0,

Γλ(un-uλ123(|(-Δ)s/2(un-uλ)|2+m4|un-uλ|2)dx-C3|un-uλ|2sdx.

Then, by Lemma 2.2, for small d>0, it is easy to verify that Γλ(un-uλ)0 for large n. So uλΓλDλSd with Γλ(uλ)=0. Thus, uλ is a nontrivial solution of (2.2). Finally, by Proposition 3.5, we can get the asymptotic behavior of uλ as λ0+. ∎

4 The Critical Case

In this section, we consider the Schrödinger–Poisson system (1.1) in the critical case. First, we establish the existence of ground state solutions to the fractional scalar field equation (1.2) with a general critical nonlinear term. Then, by a perturbation argument, we seek solutions of (1.1) in some neighborhood of the ground states of (1.2).

4.1 The Limit Problem

In this subsection, we use the constraint variational approach to seek ground state solutions of (1.2). A similar argument also can be found in [8, 40, 18]. Let

T(u)=123|(-Δ)s/2u|2dxandV(u)=3G(u)dx.

We recall that U is called a ground state solution of (1.2) if and only if I(U)=m0, where

m0:=inf{I(u):uHs(3){0} is a solution of (1.2)}

and

I(u)=T(u)-V(u).

The existence of ground states is reduced to looking at the constraint minimization problem

M:=inf{T(u):V(u)=1,uHs(3)}(4.1)

and eventually removing the Lagrange multiplier by some appropriate scaling. Now, we state the main result in this subsection.

Let s(0,1) and assume that (H2)’(H4)’ hold along with

  • (H0)

    gC(,) and g is odd, i.e., g(-τ)=-g(τ) for τ.

Then, (1.2) admits a positive ground state solution.

Since we are concerned with positive solutions of (1.2), (H0) can be replaced by

  • (H0)’

    gC(+,).

Moreover, similarly to Theorem 4.1, a similar result in N for N>2s can be also obtained.

Proof of Theorem 4.1.

The proof follows the lines of that in [40]. For completeness, we give the details here.

Step 1. Let M be given by (4.1) and let Ss be the Sobolev best constant in Lemma 2.2 for s(0,1). Then, we claim that

0<M<12(2s)(3-2s)/3Ss.

First, we prove that {uHs(3):V(u)=1}. By [14, 35], Ss can be achieved by

Uε(x)=κε-(3-2s)/2(μ2+|xεSs1/2s|2)-(3-2s)/2

for any ε>0, where κ, μ>0 are fixed constants. Let φC0(3) be a cut-off function with support B2 such that φ1 on B1 and 0φ1 on B2, where Br:={x3:|x|<r}. Let ψε(x)=φ(x)Uε(x). From [35], it follows that

3|ψε|2s=Ss3/2s+O(ε3)and3|(-Δ)s/2ψε|2=Ss3/2s+O(ε3-2s).(4.2)

Letting

vε=ψεψε2s,

we have

(-Δ)s/2vε22Ss+O(ε3-2s).

Letting

Γε:=1qvεqq-12vε22,

by (H4) we have

V(vε)12s+Γε.

In the following, we will show that

limε0Γεε3-2s=+.(4.3)

By max{2s-2,2}<q<2s, we know that (3-2s)q>3. Then, it is easy to see that there exist C1(s),C2(s)>0 such that

vεqq1ψε2sqB1|Uε|qC1(s)ε3-(3-2s)q/201/(εSs1/(2s))r2(μ2+r2)(3-2s)q/2dr=O(ε3-(3-2s)q/2)

and

vε221ψε2s2B2|Uε|2C2(s)ε2s02/(εSs1/(2s))r2(μ2+r2)3-2sdr={O(ε2s),for s<34,O(ε2sln1ε),for s=34,O(ε3-2s),for s>34.

Then, we obtain that

ΓεO(ε3-(3-2s)q/2)for s(0,1).

Noting that max{2s-2,2}<q<2s, it is easy to verify that (4.3) is true. Thus, it follows that V(vε)>0 for small ε>0. By a scaling, we get that {uHs(3):V(u)=1}.

Next, obviously, M(0,+). For small ε>0, we have V(vε)>0 and

MT(vε)(V(vε))2/2s12(-Δ)s/2vε22(12s+Γε)2/2s12(2s)2/2sSs1+O(εN-2s)(1+2sΓε)2/2s.

If p1, then (1+t)p1+p(1+t)1+pt for all t-1. From (4.3), it follows that

(1+O(εN-2s))2s/2-12s2(1+O(εN-2s))1+2s/2O(εN-2s)<2sΓε

for small ε>0, which yields 1+O(εN-2s)<(1+2sΓε)2/2s. Then,

M<12(2s)(3-2s)/3Ss.

Step 2. Here, we show that M can be achieved. Noting that g is odd and using the fractional Pólya–Szegő inequality (see Park [31]), without loss of generality, we can assume that there exists a positive minimizing sequence {un}Hrs(3) such that V(un)=1 and T(un)M as n. By Lemma 2.2, it is easy to see that {un} is bounded in Hrs(3). By Lemma 2.1 we can assume that unu0 weakly in Hs(3), strongly in Lq(3), and a.e. in 3. Setting vn=un-u0, we have T(un)=T(vn)+T(u0)+o(1) and

un2s2s=vn2s2s+u02s2s+o(1)andun22=vn22+u022+o(1),

where o(1)0 as n. Letting f(s)=g(s)-s2s-1+s, it follows from Lemma 3.2 that

3F(un)=3F(u0)+3F(vn)+o(1).

So, V(un)=V(vn)+V(u0)+o(1).

Next, we prove that u0 is the minimizer for M. Setting Sn=T(vn), S0=T(u0), V(vn)=λn, and V(u0)=λ0, we have λn=1-λ0+o(1) and Sn=M-S0+o(1). Under a scale change, we get that

T(u)M(V(u))(3-2s)/3(4.4)

for all uHs(3) and V(u)0. By (4.4) we have λ0[0,1]. If λ0(0,1), then, again by (4.4), we have

M=limn(S0+Sn)limnM((λ0)(3-2s)/3+(λn)(3-2s)/3)=M((λ0)(3-2s)/3+(1-λ0)(3-2s)/3)>M(λ0+1-λ0)=M,

which is a contradiction. On the other hand, if λ0=0, then S0=0, which implies that u0=0. Then,

lim supnvn2s2(2s)(3-2s)/3

and

M=12limn(-Δ)s/2vn2212(2s)(3-2s)/3lim infn(-Δ)s/2vn22vn2s212(2s)(3-2s)/3Ss,

which is again a contradiction. Then, we conclude that λ0=1, i.e., M is achieved by u0.

Finally, letting U()=u0(/σ0), where

σ0=(3-2s3M)1/2,

we have that U is a ground state solution of (1.2). ∎

Furthermore, similarly to Chang and Wang [12], if we additionally assume that gC1(,), then U satisfies the Pohožaev identity

3-2s23|(-Δ)s/2U|2dx=33G(U)dx.

Similarly to [27, 40], U is also a mountain pass solution.

Let S1 be the set of positive radial ground state solutions U of (1.2). Then, as in Step 2 in the proof of Theorem 4.1, we have the following compactness result.

Under the assumptions of Theorem 4.1, S1 is compact in Hrs(3).

4.2 Proof of Theorem 1.4

In the following, we are ready to prove Theorem 1.4. Similarly to Section 3, take US1 and let

Uτ(x)=U(xτ),τ>0.

Then, there exists τ1>1 such that I(Uτ)<-2 for ττ1. Setting

Dλ1maxτ[0,τ1]Γλ(Uτ),

there exist λ2>0 and 𝒞1>0 such that, for any 0<λ<λ2,

Υλ={γC([0,τ1],Hrs(3)):γ(0)=0,γ(τ1)=Uτ1,γ(τ)s𝒞1+1,τ[0,τ1]}.

Then, for any λ(0,λ1), we define a min-max value Cλ1 as

Cλ1=infγΥλmaxτ[0,τ1]Γλ(γ(τ)).

Similarly to Section 3, we have the following proposition.

We have limλ0+Cλ1=limλ0+Dλ1=m, where m is the least energy of (1.2).

Now for α,d>0, define

Γλα:={uHrs(3):Γλ(u)α}

and

S1d={uHrs(3):infvS1u-vsd}.

Similarly to Section 3, for small λ>0 and some 0<d<1, we will find a solution uS1d of (2.2) in the critical case. Also, similarly to [39], we can get the following compactness result, which can yield the gradient estimate of Γλ.

Let {λi}i=1 be such that limiλi=0 and {uλi}S1d with

limiΓλi(uλi)m𝑎𝑛𝑑limiΓλi(uλi)=0.

Then, for d small enough, there is u1S1, up to a subsequence, such that uλiu1 in Hrs(3).

Proof.

For convenience, we write λ for λi. Since uλS1d and S1 is compact, we know that {uλ} is bounded in Hrs(3). Moreover, up to a subsequence, there exists u1S1d such that uλu1 weakly in Hs(3), a.e. in 3, and uλ-u1s3d for i large. Then, by Lemma 2.4, we see that

limiI(uλ)mandlimiI(uλ)=0.

Then I(u1)=0. Obviously, u00 if d small. So, I(u1)m. Meanwhile, thanks to Lemma 3.2, we have

I(uλ)=I(u1)+I(uλ-u1)+o(1)

and

I(uλ-u1)=12uλ-u1s2-12suλ-u12s2s+o(1)o(1).

Then, by Lemma 2.2, for d small enough, uλu1 strongly in Hrs(3). ∎

By Proposition 4.6, for small d(0,1), there exist ω1>0, λ2(0,λ1) such that

Γλ(u)sω1,uΓλDλ1(S1dS1d/2),λ(0,λ2).(4.5)

Similarly to Section 3, we have the following proposition.

There exists α1>0 such that, for small λ>0,

Γλ(γ(τ))Cλ1-α1implies thatγ(τ)S1d/2,

where γ(τ)=U(/τ) for τ(0,τ1].

Proof of Theorem 1.4.

With the help of (4.5) and Proposition 4.7, similarly to [39], for λ>0 small enough, there exists {un}nΓλDλ1S1d such that Γλ(un)0 as n. As above, there exists uλS1d with uλ0 for small d>0. Moreover, up to a subsequence, unuλ weakly in Hrs(3), a.e. in 3, and un-uλs3d for n large. Furthermore, Γλ(uλ)=0. By Lemma 2.4, we have

Γλ(un)=Γλ(uλ)+Γλ(un-uλ)+o(1).

By (H2)’(H3)’, for some C>0, we have

Γλ(un-uλ)123(|(-Δ)s/2(un-uλ)|2+12|un-uλ|2)dx-C3|un-uλ|2sdx.

Then, by Lemma 2.2, lim infnΓλ(un-uλ)0 for small d>0. So, uλΓλDλ1S1d with Γλ(uλ)=0. Thus, uλ is a nontrivial solution of (2.2). The asymptotic behavior of uλ follows from Proposition 4.6. ∎

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About the article

Received: 2015-07-31

Accepted: 2015-09-12

Published Online: 2016-01-27

Published in Print: 2016-02-01


Research partially supported by INCTmat/MCT/Brazil. The first author was partially supported by CAPES/Brazil and CPSF (grant no. 2013M530868). The second author was supported by CNPq and CAPES/Brazil.


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 15–30, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5024.

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