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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 15, Issue 1


Volume 13 (2015)

lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

Peiluan Li
  • Corresponding author
  • School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023 Henan, China
  • Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1B 3X7, Newfoundland, Canada
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/ Youlin Shang
Published Online: 2017-05-08 | DOI: https://doi.org/10.1515/math-2017-0053


Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.

Keywords: Fractional Schrodinger equations; Variational methods; Infinitely many solutions

MSC 2010: 26A33; 35A15; 35B20

1 Introduction

In this paper, we are concerned with the existence of solutions of the following fractional Schrödinger equations (Δ)αu+V(x)u=f(x,u)+λh(x)|u|p2u,xRN,(1) where 0<α<1,2α<N,1p<2,λ>0,fC(RN×R,R),hL22p(RN). (− Δ)α is the so-called fractional Laplacian operator of order α ∈ (0,1) and can be defined pointwise for xRN by (Δ)αu(x)=12RNu(x+y)+u(xy)2u(x)|y|N+2αdy along any rapidly decaying function u of class C(RN), see Lemma 3.5 of [1]. It can also be characterized as by (− Δ)αu = ℱ− 1(|ξ|2αu), ℱ denotes the usual Fourier transform in RN. The potential V satisfies the following conditions (V0)VC(RN,R),infxRNV(x)=V0>0 and lim|x|V(x)=.

In [1], the authors proved that (− Δ)α reduces to the standard Laplacian − Δ as α → 1. When α = 1, the problem (1) reduces to the generalized integer order Schrödinger equation. Over the past decades, with the aid of different methods, for various conditions of the potential V and the nonlinear term f, the existence and multiplicity of nontrivial solutions for the classical Schrödinger equation have been extensively investigated.

Recently, a great attention has been focused on the study of problems involving the fractional Laplacian. Fractional calculus provide a powerful tool for the description of hereditary properties of various materials and memory processes (see [2, 3]). Fractional differential equations play important roles in the modeling of medical, physical, economical and technical sciences. For more details on fractional calculus theory, we refer the readers to the monographs of Kilbas et al. [4], Lakshmikantham et al. [5], Podlubny [6] and Tarasov [7]. Fractional differential equations involving the Rieman-Liouville fractional derivative or Caputo fractional derivative have received more and more attention.

The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Laskin [8, 9] as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths, where the Feynman path integral leads to the classical Schrödinger equation, and the path integral over Lévy trajectories leads to the fractional Schrödinger equation. Recently, some researchers have investigated the following fractional Schrödinger equations (2) for the different cases of the potential V (for example, V = 1 or V = u) and the nonlinear term f (for example, f(x, u) = |u|p − 1u) under the suitable assumptions. We refer the interested readers to [1020] and the references therein. (Δ)αu+V(x)u=f(x,u),xRN,(2) In [12], Felmer et al. studied the existence and regularity of solutions for the fractional Schrödinger equations (2) under the famous Ambrosetti-Rabinowitz (A-R) condition, i.e., there exists θ > 2 such that 0 < θ F(x, t) < t f(x, t), where F(x,u)=0uf(x,τ)dτ. In [17], by virtue of some nonlinear analysis techniques, the existence of weak solutions for (2) is obtained. In [18], for the case that V = 1 and f(x, u) = f(u), the authors proved that (2) has at least two nontrivial radial solutions without assuming the A-R condition by variational methods and concentration compactness principle. In [19], using variant Fountain theorems, the author proved the existence of infinitely many nontrivial high or small energy solutions of (2). In [20], the authors obtained the existence of a sequence of radial solutions for N ≥ 2, a sequence of non-radial solutions for N = 4 or N ≥ 6, and a non-radial solution for N = 5 by the variational methods.

Some researchers also investigated the related problems, see [2126].

In [27], by using the mountain pass theorem and Ekeland’s variational principle, the author showed that the generalized fractional Schrödinger equations (1) possess two solutions.

Motivated by the work above, in the present paper, we consider the generalized fractional Schrödinger equations (1). By the variational methods, we obtain the existence of infinitely many solutions without the A-R condition. The form of problem (1) is more general than (2). In (1), the nonlinearity involves a combination of superlinear or asymptotically linear terms and a sublinear perturbation. To our best knowledge, the problem (1) has received considerably less attention. We get the existence of infinitely many solutions for (1), which generalize and improve the recent results in the literature.

To state our results, we make the following assumptions.

(H1) Let fC(RN × R, R) . There exist constants a1,a20,q[2,2N+4αN)[2,2α) with a12S22+a2qSqq<12, such that |f(x,u)|a1|u|+a2|u|q1,(x,u)RN×R, where 2α=2NN2α with 2α < N, Sr is the best constant for the embedding of XLq(RN), the details of Sr and the definition of X will be given in the section 2.

(H2) lim|u|F(x,u)|u|2= uniformly in xRN and there exists r0 > 0 such that F(x, u) ≥ 0, for any xRN, uR and |u| ≥ r0, where F(x,u)=0uf(x,τ)dτ.

(H3)2F(x, u) < uf(x, s), ∀(x, u) ∈ RN × R.

The main results are as follows.

Theorem 1.1

Assume the hypotheses (V0), (H1)-(H3) hold. Suppose that F(x, − u) = F(x, u) for all (x, u) ∈ RN×RandhL22p(RN), then there exists a constant λ0 > 0, for λ ∈ (0, λ0), such that (1) possesses infinitely many solutions.

Remark 1.2

A-R condition plays an important role in variational methods, which could guarantee boundedness of the sequence. The problem in bounded domain with A-R condition or weaker conditions was studied in many works.

However, in the unbounded domain RN, there are few papers that considered the problem without A-R condition. Furthermore, there are very few papers which replaced A-R condition by (H3) in RN. Moreover, the form of (1) is more general. Hence, our results can be viewed as extension to the related results of the fractional Schrödinger equation.

Remark 1.3

We can identify (− Δ)α with − Δ when α = 1. Hence, Theorem 1.1 is also valid for α = 1.

The rest of this paper is organized as follows. In Section 2, some definitions and lemmas which are essential to prove our main results are stated. In Section 3, we give the main results. Finally, one example is offered to demonstrate the application of our main results.

2 Preliminaries

At first, we present the necessary definitions for the fractional calculus theory and several lemmas which will be used further in this paper.

In order to establish a variational structure which enables us to reduce the existence of solutions of problem (1) to one of finding critical points of corresponding functional, it is necessary to construct appropriate function spaces.

Let us recall that for any fixed tRN and 1< q ≤ ∞, u=maxtRN|u(t)|,uLq(RN)=(RNu(s)qds)1q.

Throughout this paper, we denote by ∥uq the Lq-norm, for 1 < q ≤ ∞.

Let H=Hα(RN):={uL2(RN):RNRN|u(x)u(z)|2|xz|N+2αdxdz<+} with the inner product and the norm u,vH=RNRN[u(x)u(z)][v(x)v(z)]|xz|N+2αdxdz+RNu(x)v(x)dx,uH=u,uH12, while [u]Hα=(RNRN|u(x)u(z)|2|xz|N+2αdxdz)12 is the Gagliardo(semi) norm. The space Hα(RN) can also be described by means of the Fourier transform, which can be denoted by Hα(RN):={uL2(RN):RN(1+|ζ|2)α(|Fu(ζ)|2)dζ<+}, and the norm is defined as uH=(RN(1+|ξ|2)α(|Fu(ξ)|2)dξ)12.

In the following, we introduce the definition of Schwartz function δ(is dense in Hα(RN)), that is, the rapidly decreasing C function on RN. If u ∈ Δ, the fractional Laplacian (− δ)α acts on u as (Δ)αu(x)=C(N,α)P.V.RNu(x)u(y)|xy|N+2αdy=C(N,α)limε0+RNB(0,ε)u(x)u(y)|xy|N+2αdy where the symbol P.V. represents the principal value integrals, the constant C(N, α) depends only on the space dimension N and the order α, and it is explicitly given by the formula 1C(N,α)=RN1cosζ|ζ|N+2αdζ.

In [1], the authors show that for all u ∈ δ, (Δ)α=F1(|ζ|2αFu) and [u]Hα=(2C(N,α)RN|ζ|2α|Fu|2dζ)12.

Furthermore, from the Plancherel formula in Fourier analysis, we can easily see that [u]Hα2=2C(N,α)(Δ)α222.

Hence, the norms on Hα(RN) defined below are all equivalent: uH=(RNRN|u(x)u(z)|2|xz|N+2αdxdz+RN|u(x)|2dx)12, uH=(RN(1+|ξ|2)α(|Fu(ξ)|2)dξ)12,uH=(RN|u(x)|2dx+(Δ)α222)12, uH=(RN|u(x)|2dx+RN|ξ|2α|Fu(ξ)|2dξ)12.

In order to investigate the problem (1), we define the following space X={uL2(RN):RNRN|u(x)u(z)|2|xz|N+2αdxdz+RNV(x)u(x)2dx<+} with the inner product and the norm u,v=RNRN|u(x)u(z)|2|xz|N+2αdxdz+RNV(x)u(x)v(x)dx,u2=u,u.

It is easy to see that X is a Hilbert space with the inner product 〈u, v〉, XH and XLq(RN) for all q[2,2α) with the embeddings being continuous.

Lemma 2.1

([27]). Assume that the condition (V0) holds. Then there exists a constant c0 > 0 such that RNRN|u(x)u(z)|2|xz|N+2αdxdz+RNV(x)u(x)2dxc0uH2,uH.(3)

Lemma 2.2

([27]). Assume that the condition (V0) holds. Then X is compactly embedded in XLq(RN) for all q[2,2α).

Remark 2.3

Lemma 2.2 implies that Srur ≤ ∥u∥, where Sr > 0 is the best constant for the embedding of XLq(RN).

We consider the functional φ: XR, defined by φ(u)=12u2RNF(x,u)dxλpRNh(x)|u|pdx.(4)

Then φ is continuously differentiable under the assumption (H1), and φ(u),v=RNRN[u(x)u(z)][v(x)v(z)]|xz|N+2αdxdz+RNV(x)u(x)v(x)dxRNf(x,u(x))v(x)dxλRNh(x)|u(x)|p2u(x)v(x)dx,(5) for all u, vX Hence, the critical point of φ is the solution of problem (1). Next, we only need to consider the critical point of φ.

Definition 2.4

Let (E, ∥•∥) be a Banach space and φC1(E, R). We say φ satisfies (Cerami)C condition, if any sequence {um} ⊂ E for which φ(um) → c andφ′(um)∥(1+∥um∥) → 0 as m → 0 posses a convergent subsequence, and {um} is called a(Cerami)c sequence.

From now on, we will denote (Cerami)C condition by the abbreviation (C)C condition.

Lemma 2.5

([28] Theorem 9.12]). Let (E, ∥•∥) be an infinite dimensional Banach space, X = YZ with dim Y < ∞. Let φC1(X, R) be an even functional, which satisfies the (C)C condition and φ(0) = 0. In addition, if φ satisfies:

  1. There exist a, ρ > 0 such that φ|BρZa where Bρ = {uX :∥u∥ < ρ},

  2. For any finite dimensional subspace WX, there exists R = R(W) such that φ(u) ≤ 0 on WBR(W).

Then, φ possesses an unbounded sequence of critical values.

3 Main results

Without loss of generality, if we take a subsequence {um}, we also use the same notation {um}.

Lemma 3.1

Assume that (V0), (H1)-(H3) hold. Then there exists a constant λ0 > 0, for any λ ∈ (0, λ0), φ satisfies the (C)c condition.


Let {um} ⊂ X be a (C)c sequence, that is φ(un)c,φ(un)(1+un)0, as n,(6) which also implies φ(un),un0, as n.(7)

First, we prove that {um} is bounded.

We argue it by contradiction. If {um} is unbounded in X, then there exists a subsequence {um} with ∥un∥ → ∞, as n → ∞. From (4)-(5), (H3) and the Hölder inequality, one has φ(un)φ(un),un2=RNF(x,un)dx(1p12)λRNh(x)|un|pdx+12RNf(x,un)undx(1p12)λRNh(x)|un|pdx(1p12)λ0h22pun2p.

By (6)-(7), we can easily get that ∥un2 is bounded.

Also from (4)-(5), we have φ(un),un=un2RNf(x,un)undxλRNh(x)|un|pdx.

Then, it follows from (H1) that un2φ(un),un+RNf(x,un)undx+λ0h22pun2pφ(un),un+a1un22+a2unqq+λ0h22pun2p.(8)

By the Fractional Gagliardo-Nirenberg inequality ([29], corollary 2.3]) and the definition of the norm in X, we know that uLq(RN)ζsqusquL2(RN)1sq,(9) where ζ=2απα2(Γ(N)Γ(N/2))αNΓ(Nα2)/Γ(N+α2) and s(12αN)+qs2=1.

For 0<s=(q2)N2α<q[2,2N+4αN)[2,2α), it is easy to see that 0<s<2.(10)

Then it follows (8)-(10) that 1=un||2||un||2φ(un),unun2+a1un22un2+a2unqqun2+λ0h||22pun2p||un2φ(un),unun2+a1un22un2+a2ζsuns||un2qs||un||2+λ0h||22pun2p||un20, as n.

This is a contradiction. Hence, we know that {un} is bounded in X.

Next we prove φ satisfies the (C)c condition.

For any {un} ⊂ X being a (C)c sequence, from the boundedness of {un}, we know there exists a weakly convergent subsequence {un} such that unu weakly in X. From Lemma 2.2, we can obtain that unu strongly in Lq(RN) for q[2,2N+4αN).

Then we prove that unu in X.

From (4)-(5), we have unu2=φ(un)φ(u),unu+RN[f(x,un(x))f(x,u(x))](unu)dx+λRNh(x)|unu|pdx.(11)

It is easy to see that φ(un)φ(u),unu0, as n.(12)

Based on the fact that hL22p(RN),unu in Lq(RN) and Hölder inequality, one has λRNh(x)|unu|pdxλ0h22pun2p0, as n.(13)

From (H1) and the Hölder inequality, we can obtain RN[f(x,un(x))f(x,u(x))](unu)dxRN|f(x,un(x))f(x,u(x))||unu|dxRNa1(u+un)+a2(|u|q1+|un|q1)|unu|dxa1(un2+u2)unu2+a2(unqq1+uqq1)unuq0, as n.(14)

It follows from (11)-(14) that ∥unu2 → 0, which shows that unu in X. Hence, φ satisfies the (C)c condition.

We complete the proof of Lemma 3.1.□

Let {ej} be a total orthonormal basis of X. We define Xj:=span{ej},Yk:=j=1kXj and Zk=j=k+1Xj,¯kN.

Clearly, X = YkZk with dim Yk < ∞.

Lemma 3.2

Assume that (V0), (H1)-(H3) hold. Then there exist constants a, ρ > 0 such that φ|BρZka, where Bρ = {uX :∥u∥ < ρ}.


From (H1), (4) and Hölder inequality, for any uZk,q[2,2N+4αN), we have φ(u)=12u2RNF(x,u)dxλpRNh(x)|u|pdx12u2a12un22a2qunqqλ0h22pun2p12u2a12S22un2a2qSqqunqλ0h22pun2p.

We denote c=12(a12S22+a2qSqq), then (H1) implies c > 0. Let αk=supuZk,||u||=1ur, for r[1,2α).

By a similar proof to the Lemma 3.2 of [19], we can deduce that αk → 0 as k → ∞. Assume 0 < ρ≤ 1, then we have φ|BρZkcu2λ0h22pαkpup=up[cu2pλ0h22pαkp].

Then it is easy to see that we can choose constant ρ ∈ (0,1] such that φ|BρZk > 0, as k →∞, which implies the conclusion of Lemma 3.2.□

Lemma 3.3

Assume that (V0), (H1)-(H3) hold. Then for any finite dimensional subspace WX, there exists R = R(W) such that φ(u) ≤ 0 on WBR(W).


First, we prove that for any finite dimensional subspace WX and ∥u∥ →∞, uW, there holds φ(u) → − ∞.

On the contrary, we assume that for some sequence {un} ⊂ W with ∥un∥ →∞, there exists a positive constant M > 0 such that φ(un)M.(15)

Let vn=un||un||, then ∥vn∥ = 1. From the boundedness of vn, there exits a weakly convergent subsequence vn such that vnv weakly in X. Then for the finite dimensional subspace WX, we know that vnv strongly in W. By the equivalence of finite dimensional spaces, we can get that vnv a.e. in RN, which also implies ∥v∥ = 1. Let V : = {xRN:v(x) ≠ 0}, then we know the measure of the set V is positive, i.e., meas(V) > 0. Hence, for xV, from un = ∥unvn we can deduce |un|, as n.(16)

For any 0 < x1 < x2, we denote Ωn(x1, x2) = {xRN:x1 ≤ |un| < x2}, which implies Ωn(x1, x2) ⊂ V. Then for n large enough, it easy to see that χΩn(r0,∞)(x) = 1, where χΩn(r0,∞) is the characteristic function on Ωn(r0,∞), r0 > 0 is given in (H2). Hence, we have χΩn(r0,)(x)vnv, as n, for a.e.xΩn(r0,).(17)

On one hand, from (4), (15) and Remark 2.3, for any λ ∈ (0, λ0), we have limn+RNF(x,un)dxun2=limn+12un2λpRNh(x)|un|pdxφ(un)un2limn+12un2+λ0S2ph||22pup+Mun||2=12.(18)

On the other hand, (H2) shows there exists r0 > 0 such that F(x, u) ≥ 0, for |u| ≥ r0. From (16), we know that F(x,un)0,fornlargeenough.(19)

Then it follows from (17), (19), (H2) and Fatou’s Lemma that limnRNF(x,un)dxun2limnVF(x,un)dxun2=limnVF(x,un)vn2dxun2liminfnVF(x,un)vn2un2dxVliminfnF(x,un)vn2un2dxΩn(r0,)liminfnF(x,un)vn2un2dx=Ωn(r0,)liminfnF(x,un)vn2|un|2χΩn(r0,)2(x)vn2dx=,

which is a contradiction with (18). Therefore φ(u) → −∞ for ‖u‖ → ∞, uW. Hence, we can easily choose R = R(W) such that φ(u) ≤ 0 on W \ BR(W). Then we complete the proof.□

Proof of Theorem 1.1

Let Y = Yk, Z = Zk, then X = YZ with dim Y < ∞. From the condition that F(x,u)=F(x,u)andhL22p(RN), we know φ is even and φ(0) = 0. Lemma 3.1-3.3 imply that φ satisfies other conditions of Lemma 2.5. Consequently, we can deduce that φ possesses an unbounded sequence of critical values, which are the solutions of the fractional Schrödinger equation (1).□

Finally, we give one example to illustrate the usefulness of our main result. Consider the following fractional Schrödinger equations.

Example 3.4


Obviously, α=1/2,N=2,p=1,λ=1,f(x,u)=8S8/38/3(sin2x)u5/39 is continuous, V(x) = 1 + x2 and h(x)=ln(1+|sinx2|)e|x|(1+x2) is a L2 integrable function.

First, we can see that 2α=1<N=2,F(x,u)=S8/38/3(sin2x)u8/33, then we have |f(x,u)|=|8S8/38/3(sin2x)u5/39|S8/38/3|u|831,

with 83=q[2,3)[2,2α=4),a1=0,a2=S8/38/3anda12S22+a2qSqq=38<12, which shows that (H1) of Theorem 1.1 holds.

From the fact that 2F(x,u)=2S8/38/3(sin2x)u8/33<8S8/38/3(sin2x)u8/39=uf(x,u), we can verify (H3) of Theorem 1.1 is also satisfied.

It is also easy to check that the hypotheses (V0), (H2) and other conditions of Theorem 1.1 hold. Then all the conditions in Theorem 1.1 are satisfied. In virtue of Theorem 1.1, we conclude that (20) possesses infinitely many solutions.


The authors thank the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper.We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contribute to the perfection of the paper.


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About the article

Received: 2017-02-13

Accepted: 2017-03-22

Published Online: 2017-05-08

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 578–586, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0053.

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© 2017 Li and Shang. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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