The Existence of Normalized Solutions for a Nonlocal Problem in R3

<jats:p>In this paper, we study the following fractional Schrödinger equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>−</mml:mo><mml:mi>λ</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mfenced open="|" close="|"><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi></mml:math>, in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>σ</mml:mi><mml:mo>∈</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℝ</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>σ</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>4</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mfenced><mml:mi>σ</mml:mi></mml:mrow></mml:mfenced></mml:math>. By using the constrained variational method, we show the existence of solutions with prescribed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> norm for this problem.</jats:p>


Introduction
This paper concerns with the following fractional Schrödinger problem: where σ ∈ ð0, 1Þ, p ∈ ð2 + σ, 2 + ð4/3ÞσÞ, and λ ∈ ℝ. Here, the fractional Laplacian ð−ΔÞ σ in ℝ n is defined by where PV stands for the Cauchy principal value and C n,σ is a normalization constant.
In the present paper, the motivation for studying such equations comes from mathematical physics: searching for the form of standing wave ψ = e −iht u of the evolution equation leads to looking for solutions of (1). Here, i is the imaginary unit and h ∈ ℝ. This class of Schrödinger-type equations is of particular interest in fractional quantum mechanics for the study of particles on stochastic fields modelled by Lévy processes. A path integral over the Lévy flight paths and a fractional Schrödinger equation of fractional quantum mechanics are formulated by Laskin [1,2] from the idea of Feynman and Hibbs's path integrals.
On the other hand, problem (1) has attracted considerable attention in the recent period. Part of the motivation is to consider h ∈ ℝ as a fixed parameter and then to search for a solution u ∈ H σ ðℝ 3 Þ solving (1). In this direction, mainly by variational methods, many researches have been devoted to the study of the existence, multiplicity, uniqueness, regularity, and asymptotic decay properties of the solutions to fractional Schrödinger equation (1). For this information, we can refer to [3][4][5][6][7][8][9][10][11] and the references therein. Besides, some more complicated fractional equations and systems were also studied, and indeed, some interesting results were obtained. Nearly, Mingqi et al. [12] investigated a critical Schrödinger-Kirchhoff type systems driven by nonlocal integrodifferential operators and by applying the mountain pass theorem and Ekeland's variational principle; the authors obtained the existence and asymptotic behavior of solutions for this system under some suitable assumptions. Later, in [13], the same authors as in [12] studied a diffusion model of Kirchhoff-type. Under some appropriate conditions, by employing the Galerkin method, the local existence of nonnegative solutions was obtained, and then by virtue of a differential inequality technique, they proved that the local nonnegative solutions blow up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, in [14], Mingqi et al. concerned with a class of fractional Kirchhoff-type problems with the Trudinger-Moser nonlinearity. By applying minimax techniques combined with the fractional the Trudinger-Moser inequality, they found the existence of a ground state solution with positive energy and the existence of nonnegative solutions with negative energy by using Ekeland's variational principle. In [15], the three authors considered a fractional Choquard-Kirchhofftype problem involving an external magnetic potential and a critical nonlinearity and established a fractional version of the concentration-compactness principle with magnetic field, and then together with the mountain pass theorem, they verified the existence of nontrivial radial solutions in nondegenerate and degenerate cases. Furthermore, Mingqi et al. [16] concerned the Schrödinger-Kirchhoff-type problems involving the fractional p-Laplacian and critical exponent. By using the concentration-compactness principle in fractional Sobolev spaces, they showed the existence of m pairs of solutions for any m ∈ ℕ, and by applying Krasnoselskii's genus theory, they also got the existence of infinitely many solutions under some suitable conditions for the parameter. For more information on this direction, one can refer to [17][18][19][20][21][22][23][24] and the references therein.
In the present paper, inspired by the fact that physicists are often interested in normalized solutions, we look for solutions in H σ ðℝ 3 Þ having a prescribed L 2 norm to equation (1). Such types of problems were studied extensively in recent years for the classical Schrödinger equations with the standard Laplacian operator. We refer the interested reader to [25][26][27][28][29][30][31] and to the references therein. But up to our knowledge, not much is obtained for the existence of normalized solutions of equation (1) in H σ ðℝ 3 Þ with a fractional Laplacian operator. So, in this paper, the aim is to get the normalized solutions of equation (1). Here, we give the definition of prescribed ρ-L 2 norm solu- we call it a prescribed ρ-L 2 norm solution. Naturally, a prescribed ρ-L 2 norm solution u ρ ∈ H σ ℝ of (1) can be a constrained critical point of the functional Note that for any p ∈ ð2, 6/ð3 − 2σÞÞ, IðuÞ is a well-defined and C 1 functional. Set It is standard that if u ρ is a minimizer of (7), then u ρ is a solution of (1) with prescribed ρ-L 2 norm with the constraint λ ρ ∈ ℝ being the Lagrange multiplier. However, it is worth mentioning that dealing with this kind of problem, one has to face the main difficulty concerning with the lack of compactness of the minimizing sequence fu n g ⊂ B ρ . In fact, we will encounter two possible bad scenarios that u n ⇀ 0 and u n ⇀ u ≠ 0 with 0 < k uk 2 < ρ. In order to avoid the possible cases and to get that the infimum is obtained, we prove an important lemma (Lemma 6) that guarantees the compactness of minimizing sequence. As a consequence of this lemma, setting we can get our main result as follows: (1) with the constraint λ ρ ∈ ℝ. But, when p = 2 + ð4/3Þσ, m ρ has no minimizer for any ρ ∈ ðρ * 2 ,+∞Þ.
To see this, letting u ∈ B θ −1 ρ be arbitrary and considering u θ ðyÞ = θ 5/2 uðθyÞ for any θ > 0, we find u θ ∈ B ρ and Hence, Iðu θ Þ ⟶ 0 as θ ⟶ 0 and the conclusion is as follows: Finally, we give the following notations which can be used in this paper: (iv) Denote C > 0 by various positive constants which may vary from one line to another and which are not important for the analysis of the problem This paper is organized as follows: In Section 2, we will give some preliminary results which are crucial to prove 2 Advances in Mathematical Physics our main result. And then the proof of our main result is given in Section 3.
In [33], the authors have established the Pohozaev identity for the fractional Laplacian operator.
Applying the Pohozaev identity, we have the following: Proof. Define the functional energy corresponding to (1) as Then any critical point u of F λ ðuÞ satisfies the Pohozaev identity for (1) (see [33]), that is, On the other hand, if u is a critical point of IðuÞ restricted to B ρ , there is a Lagrange multiplier λ * ∈ ℝ such that So, for any ψ ∈ H σ ðℝ 3 Þ, we have Furthermore, if we now know that by (18), we find A λ * ðu * Þ = 0. As a result, Using Lemma 3, the estimate (13) leads to the following fact: Lemma 5. If p ∈ ð2 + σ, 2 + ð4/3ÞσÞ, then for any ρ > 0, functional I is bounded from below and coercive on B ρ .
(ii) We will divide it into three steps to show (ii).
Step 2. We will show that all the minimizing sequences fu n g for m ρ have a weak limit, up to translations, different from zero. Let fu n g be a minimizing sequence on B ρ for m ρ . Note that for any sequence fy n g ⊂ ℝ 3 , u n ð⋅ +y n Þ is still a minimizing sequence for m ρ . So the proof of this step can be finished if we can prove the existence of a sequence fy n g ⊂ ℝ 3 such that the weak limit of u n ð⋅ +y n Þ is different from zero.
Applying Lion's lemma, we know that if Therefore, it follows the compactness of the embedding H σ ðBð0, 1ÞÞ↪L 2 ðBð0, 1ÞÞ that the weak limit of the sequence u n ð⋅ +y n Þ is not the trivial function.
Step 3. Finally, we verify that m ρ has a minimizer for m ρ < 0. Suppose fu n g to be a minimizing sequence on B ρ for m ρ with m ρ < 0. Then by Lemma 5, fu n g is bounded in H σ ðℝ 3 Þ and L q ðℝ 3 Þ for any q ∈ ½2, 6/ð3 − 2σÞ. So there exists u ∈ H σ ðℝ 3 Þ such that u n ⇀ u in H σ ðℝ 3 Þ and then we can get If we set ν ≔ k uk 2 and ϵ n ≔ ð ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ρ 2 − v 2 p Þ/ku n − uk 2 , then by the Step 2, 0 < ν ≤ ρ. Now, we want to prove that 4 Advances in Mathematical Physics ν = ρ. To see this, we assume that 0 < ν < ρ. From (36), we find since ϵ n ⟶ 1. Noting that kϵ n ðu n − uÞk 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ 2 − ν 2 p , (37) tells that m ffiffiffiffiffiffiffiffi ffi which contradicts to (31) and (32) and then ν = ρ. Since u ∈ B ρ , we have ku n − uk 2 = oð1Þ. Hence, if we would verify that u n ⟶ u in H σ ðℝ 3 Þ, it remains to show that ku n − uk 2 σ = oð1Þ up to a subsequence. First, by assumption, there is fλ n g ⊂ ℝ such that So, which implies fλ n g is bounded and up to a subsequence; there exists λ ∈ ℝ with λ n ⟶ λ.
(iii) Now, we come to prove that if ρ n ⟶ ρ, then lim n→∞ m ρ n = m ρ . For any n ∈ ℕ + , let w n ∈ B ρ n such that Iðw n Þ < m ρ n + ð1/nÞ. Using Lemma 5, we deduce that fw n g is bounded in H σ ðℝ 3 Þ and then kw n k σ and kw n k p are bounded. So it is easy to find that On the other hand, letting fv n g ⊂ B ρ be a minimizing sequence for m ρ , we have

Proof of the Main Result
To prove our main theorem, we first give the following important results: Lemma 7. When p ∈ ð2 + σ, 2 + ð4/3ÞσÞ, there exists ρ 3 > 0 such that m ρ has no minimizer for all ρ ∈ ð0, ρ 3 Þ.
Proof. We prove it by contradiction and suppose that there exist fρ n g ⊂ ℝ + with ρ n ⟶ 0 + as n ⟶ ∞ and fu n g ⊂ B ρ n such that Iðu n Þ = m ρ n .