Research Article | Open Access
Existence and Stability of Standing Waves for Nonlinear Fractional Schrödinger Equations with Hartree Type and Power Type Nonlinearities
We consider the standing wave solutions for nonlinear fractional Schrödinger equations with focusing Hartree type and power type nonlinearities. We first establish the constrained minimization problem via applying variational method. Under certain conditions, we then show the existence of standing waves. Finally, we prove that the set of minimizers for the initial value problem of this minimization problem is stable.
In the paper, we study the following nonlinear fractional Schödinger equations:where , , , , is a complex-valued function on , , and , where denotes the convolution. The fractional Laplacian is a nonlocal operator defined as where the Fourier transform is defined by Equations (1) appear in statistical physics; this problem stems from the Slater semirelativistic situation of Hartree-Fock model for particle interacting with each other via the Coulomb law; see . The power type nonlinearity reflects the exchange effect produced by the Pauli principle, and the Hartree type nonlinearity describes the Coulomb effect of particles exclusion.
The fractional Schrödinger equations have widely applied to physical and other areas and have attracted much attention of researchers, especially in fractional quantum mechanics. Laskin spread the fractional operator to quantum mechanics and formulated Schrödinger equation (see [2–4]). In [5–7], equations with fractional Laplacians have been studied recently as the Lévy processes appear widely in physics, chemistry, and biology. Equation in  can describe ground state solutions for the -critical boson star equation (when ). The authors in  considered the orbital stability of standing waves for classical nonlinear Schrödinger equations (when ). The Cauchy problems of systems of Schrödinger equations are important issues which have been studied by many researchers; see [10–13].
The existence and stability of standing waves is a very important topic of fractional Schrödinger equations. The standing waves have been raised in various fields of physics, for example, plasma physics, constructive field theory, nonlinear optics, and so on. Recently, the authors in [12–15] have been concerned with the fractional Schrödinger equations with Hartree type nonlinearity; they obtained a series of results about existence, continuity, and stability of standing waves. The results about this topic for power type nonlinearity have been studied by [16, 17]. In this paper, we are interested in considering the existence and stability of standing waves for nonlinear fractional Schrödinger equations with combined nonlinearities of Hartree type and power type.
It is well known that a standing wave for (1) is a solution of the form , where . Therefore, it is easy to see that get a standing wave of (1) is equivalent to solving the following equations for :where are complex-valued functions. In order to study the existence of solutions to (4), by the variational method, we consider the following constrained minimization problem: where and are defined byDenote the set of the minimizers of problem (5) by
We define the fractional order Sobolev space ; its norm is We always write for brevity; we denote and in the following.
The following is the main conclusion of the paper.
Theorem 1. Let and . If is a minimizing sequence of problem (5), then there exists a sequence such that contents a convergent subsequence. In particular, there exists a minimizer for problem (5), which implies is not an empty set, and we have
Theorem 2. Under the assumptions of Theorem 1, the set is -stable with respect to (1); that is, for any , there exists such that if the initial condition in (1) satisfiesThen for any , where is a solution of (1) corresponding to the initial condition .
Remark 3. By the Hermiticity of the fractional Schrödinger operator in , we obtain that the solution of (1) with initial value satisfies the following conservation laws:(1)Conservation of mass: (2)Conservation of energy: The conservations are very important to the proof of the -stability.
In the section, we will list some lemmas, which will have great effects for the following proofs.
Lemma 4. Let ; there are two properties with .
(i) The norm of is equivalent to This result follows easily from the fundamental inequalityand the definitions of and .
(ii) , which implies , where is defined by the trace interpolation (see ), and
Lemma 5. If , and , represent the Schwartz class; then the fractional Laplacian of is also expressed by the formula where means the Cauchy principal value on the integral and is some positive normalization constant.
Lemma 6 ((Hardy’s inequality)(see )). For , we havewhere the constant depends on and .
Lemma 7 (commutator estimates). If , , the Schwartz class, then where satisfying .
Lemma 8 ((fractional Rellich compactness theorem) (see )). Let and . If is a bounded domain, then every bounded sequence has a convergent subsequence in .
Lemma 9 (see ). For , , and for and , where . Moreover, for , if in , then in , and a.e. in .
Lemma 10. Suppose is a minimizing sequence for the problem (5) satisfyingThen we have in , where .
Proof. For , let ; by the Hölder inequality and Lemma 9, we deduce thatwhere . Let ; from (22), we haveNext we divide the domain. Covering by countable balls in such a way that every point of belongs to at most balls, by (23) and Lemma 4, we getSince for some , applying (24), then we obtain in . For every , we have such that Since we have divided the domain, we know thatIf in some , , then there exists at most balls such that where only depends on . Then by Hölder and Hardy’s inequality, we can get If we take to infinity, we find the second part can also be bounded by , and we finish the proof of the lemma.
Theorem 11. Let , , . If , then there exists a global solution to the Cauchy problem of nonlinear fractional Schrödinger equations (1) with the initial date .
Remark 12. The authors in  concerned with a class of systems of fractional nonlinear Schrödinger equations. By Faedo-Galërkin method, they have obtained the existence and uniqueness of the global solution to the periodic boundary value problem. In the proof of Theorem in , let the period ; we can get Theorem 11. In [10–13], many researchers have also studied the Cauchy problem of Schrödinger equations. Here we will prove Theorem 11 by the semigroup method.
Proof of Theorem 11. Let By Stone’s Theorem , is the infinitesimal generator of a group of unitary operator , , on . Therefore, (1) can be rewritten in the form of the integral equation By Banach Fixed Point Theorem, for , we know that there exist local weak solutions to (1) with initial data . We can use the standard contraction mapping argument to prove the local existence of solutions; since the argument has been considered by many authors (see [10, 12, 13, 23]), we omit it. Next we will give the uniform estimates of local weak solutions. By the conservation laws, we haveBy the Gagliardo-Nirenberg inequality, we obtainwhere satisfies ; that is, . Applying the Hardy’s inequality and the Gagliardo-Nirenberg inequality, we haveSince implies that , , using Young inequality and combining (31) and (32), we have From the above inequality and (30), for , This completes the proof of Theorem 11.
3. Main Results
Lemma 13. For every , we have
Proof. For given , , setting , . By Lemma 5, we obtain Since , , we have sufficient small such that . Therefore, .
By Hardy’s inequality, we can deduce thatSobolev’s inequality and Young’s inequality imply thatwhere is a sufficiently small positive constant. Using Hölder inequality and the Sobolev inequality, we havewhere , . From Young inequality and , for , from (38), we can getHence, for , , and sufficiently small , we have Hence, .
To solve the constrained minimization problem (5), the most difficult problem is the lack of compactness of the minimizing sequences . However, there are two scenarios and they are impossible for the problem: vanishing ; dichotomy and .
In order to rule out the above two cases and to show that the infimum is achieved, we apply the concentration compactness principle in [24, 25]. At first, we introduce the Lévy concentration function is nondecreasing on . By Helly’s selection theorem, we find a convergent subsequence of denoted again by such that where is a nondecreasing function. We know that , so there exists , , such that
Lemma 14. Every minimizing sequence for problems is bounded in , and for sufficiently large , there exists a constant such that
Proof. At first, from (37) and (39), we obtain Since is a minimizing sequence, by , we get the result.
We prove the second part of Lemma 14 with an argument by contradiction. Suppose does not exist; then there exists a subsequence of such thatUsing the definition of energy, we deduce that which contradicts The proof of Lemma 14 is completed.
Lemma 15. Vanishing does not occur, that is, , for any .
Proof. We prove the lemma with an argument by contradiction. If , then there exist a positive and a subsequence of a minimizing sequence such that As is also a minimizing sequence, according to Lemma 10, we can find which contradicts Lemma 14.
By Lemma 15, we exclude the possibility of vanishing. We prove dichotomy will not occur yet by the next two Lemmas.
Lemma 16. Let ; then .
Lemma 17. Suppose , then .
Proof. According to the boundedness of in , the definition of , we can get for , such thatand for all , we have Then there exists such that, for , we obtain Next we let such thatWe define and for , , where is a smooth cutoff function satisfying , for , for and . With this notation, we writeIt follows easily from (53) thatWe can get the conclusion if