Multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation with potentials

We get multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation $$ (-\Delta)^su+V(\varepsilon x)u=\lambda u+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-\Delta)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\varepsilon$ is small enough.

1. Introduction 1.1.Background and motivation.In this paper, we investigate the multiplicity of normalized solutions for the fractional Schrödinger equation where 0 < s < 1, i denotes the imaginary unit and ψ(x, t) is a complex wave.A solution of (1.1) is called a standing wave solution if it has the form ψ(x, t) = e −iλt u(x) for some λ ∈ R. (−∆) s stands for the fractional Laplacian and if u is small enough, it can be computed by the following singular integral (−∆) s u = C(N, s)P.V.
Here the symbol P.V. is the Cauchy principal value and C(N, s) is a suitable positive normalizing constant.
The operator (−∆) s can be seen as the infinitesimal generators of Lévy stable diffusion processes [4], it originates from describing various phenomena in the field of applied science, such as fractional quantum mechanics, barrier problem, markov processes and phase transition phenomenon, see [13,20,30,31].In recent decades, the study of problems of fractional Schrödinger equation has attracted wide attention, see e.g.[27,28,33] and references therein.
In [2], Alves considered the following class of elliptic problems with a L 2 -subcritical nonlinear term    By using the variational approaches, the author shows that problem (1.2) admits multiple normalized solutions if ε is small enough.Particularly, the numbers of normalized solutions are at least the numbers of global maximum points of h.Moreover, for the following class of problem a similar result is also obtained for some negative and continuous potential V .
Motivated by [2], our interest is mainly focused on the fractional case with both potentials and weights.Actually, our purpose of this paper is devoted to the multiplicity of normalized solutions for the fractional Schrödinger equation where s ∈ (0, 1), a, ε > 0, λ ∈ R is an unknown parameter that appears as a Lagrange multiplier.
In the local case, when s = 1, the fractional laplace (−∆) s reduces to the local differential opterator −∆.If V (x) ≡ 0, Jeanjean's [18] exploited the mountain pass geometry to deal with existence of normalized solutions in purely L 2 -supercritical, we refer [6,14,15,21] for more results in this type of problems.In [25], they considered the related problem for q = 2 + 4  N .The multiplicity of normalized solutions for the Schrödinger equation or systems has also been extensively investigated, see [12,18,18,29].
For the non-potential case, a large body of literature is devoted to the following problem: In particular, for the case g(u) = |u| p−1 u, by assuming H 1 -precompactness of any minimizing sequences, Cazenave and Lions [7] showed the attainability of the L 2 -constraint minimization problem and orbital stability of global minimizers, it is assumed that E α < 0 for all α > 0, and then, the strict subadditivity condition: (1.5) holds.However, when dealing with the general function g, it is difficult to show (1.5) holds.Shibata [29] proved the subadditivity condition (1.5) using a scaling argument.
In addition, if V (x) ≡ 0, Ikoma and Miyamoto [16] studies the existence and nonexistence of a minimizer of the L 2 -constraint minimization problem V and f satisfy some suitable assumptions.They performed a careful analysis to exclude dichotomy and proved the precompactness of the modified minimizing sequence.When dealing with general nonlinear terms in mass subcritical cases, one can apply the subadditive inequality to prove the compactness of the minimzing sequence.
Zhong and Zou in [35] studied the existence of ground state normalized solution to Schrödinger equations with potential under different assumptions, and presented a new approach to establish the strict sub-additive inequality.Alves and Thin [3] study the existence of multiple normalized solutions to the following class of ellptic problems where ε > 0, V : R N → [0, ∞) is a continuous function, and f is a differentiable function with L 2subcritical growth.For normalized solutions of the nonlinear Schrödinger equation with potential, we also see [5,17,26] and the references therein.
In the case 0 < s < 1, few results are available.In the paper [34] the author proved some existence and asymptotic results for the fractional nonlinear Schrödinger equation.For the particular case of a combined nonlinearity of power type, namely Dinh [8] studied the existence and nonexistence of normalized solutions for the fractional Schrödinger equations By using the concentration-compactness principle, he showed a complete classification for the existence and non-existence of normalized solutions for the problem (1.7).For more results about the fractional Schrödinger equations, we can refer to [11,24] and the references therein.

Main results.
In what follows, we assume f ∈ C 1 (R N , R) is odd, continuous and satisfies the following assumptions on f .
Moreover, h and V satisfy the following assumptions. ( The problem (1.3) is variational and the associated energy functional is given by (1.8) It is easy to know that I ε ∈ C 1 (H s (R N ), R) and The solutions to (1.3) can be characterized as critical points of the function I ε (u) constrained on the sphere (1.9) Now, we are ready to state the main result of this paper.
Theorem 1.1.Suppose (A 1 ), (A 2 ), (f 1 ) − (f 3 ) hold, then there exists ε 1 > 0 such that problem (1.3) admits at least k couples (u j , λ j ) ∈ H s (R N ) × R of weak solutions for ε ∈ (0, ε 1 ) with The paper is organized as follows.In Section 2, we study the autonomous problem and give some useful results which will be used later.Section 3 is devoted to the non-autonomous problem.In Section 4, the proof of Theorem 1.1 is given.

The autonomous problem
In this section, we focus on the existence of normalized solution for the autonomous problem (2.1) where s ∈ (0, 1), a, µ > 0, η ≤ 0 and λ ∈ R is an unknown parameter that appears as a Lagrange multiplier.With the assumptions (f 1 ) − (f 3 ), it is standard to show that the solutions to (2.1) can be characterized as critical points of the function as follows (2.2) restricted to the sphere S a given in (1.9).Meanwhile, set Theorem 2.1.Suppose that f satisfies the conditions (f 1 ) − (f 3 ).Then, problem (2.1) has a couple (u, λ) solution, where u is positive, radial and λ < η.
The proof of Theorem 2.1 is standard.For the sake of convenience, we give the details.Before the proof, some lemmas are given below.Lemma 2.2.Assume u is a solution to (2.1), then u ∈ S a ∩ P , where Proof.Let u be a solution (2.1), then we get In addition, one can show that u satisfies the Pohozeav identity Combining with (2.3), we obtain that Proof.(i) According the assumptions (f 1 ) − (f 2 ), there exists C > 0 such that By the fractional Gagliardo-Nirenberg-Sobolev inequality [10], (2.5) , for some positive constant C(s, N, α) > 0.Then, (2.4) and (2.5) give that (ii) Since u ∈ S a , the conclusion immediately follows from (2.6).
The lemma above guarantees that is well defined.Now we study the properties of the function J defined in (2.1) restrict to S a and prove Theorem 2.1.
Proof.According (f 1 ), lim t→0 qF (t) t q = c > 0 and then there exists ζ > 0 such that In fact, taking u ∈ S a ∩ L ∞ (R N ) as a fixed nonnegative function, we define u(e τ x), for all x ∈ R N and all τ ∈ R, then τ * u ∈ S a .Moreover, for τ < 0 and |τ | large enough, we have It follows that Hence, we obtain and then E a < 0. In particular, we have E a < ηa 2 .The proof is complete.
In the following, we adopt some idea introduced in [35] to get the sub-additive inequality.
Lemma 2.5.For µ > 0, η ≤ 0 and let a, b > 0, then Proof.(i).For any ε > 0 small, there exist Since u and v have compact support, by using parallel translation, we can take R large enough satisfying Then u + ṽ ∈ S b and Noting that |x − y| ≥ R large enough, we have Here we used the fact Υ b−a < 0. Then by (2.9) and the arbitrariness of ε, we obtain that E b ≤ E a for any b > a > 0.
(ii).We prove the following two claims.
For ε > 0, by the definition of E a , there exists u ∈ S a such that (2.10) and u t (x) = u( x t ), we get (2.11) lim Then, by using (i), we have J(u t ) ≥ E a−h .In addition, by (2.10) and (2.11), we obtain lim Since ε is arbitrary, the claim holds.
Claim 2: lim Actually, we consider the case By Lemma 2.3, we know {u n } is bounded in H s (R N ).Morever, we have Hence, we get u n ∈ S a .On the other hand, Thus, we obtain that lim Moreover, E a−h ≥ E a ≥ E a+h holds due to (i).Hence, we get lim We complete the proof of (ii).(iii).Firstly, we prove that For any ε > 0, we take u ∈ S a ∩ P such that , by the assumption, we have |ũ| 2 2 = νa and Then, we get that Since u ∈ P , we know Obviously, if ξ > 0 small, it follows that Then by (2.12) and (f 3 ), we obtain that Therefore, for any θ ∈ (1, 1 + ξ), we have Then, it is easy to see that Since the arbitrariness of ε, we get and if E a is attained, we can take u as a minimizer in the above step, then we have Furthermore, following the proof of (i), since E a is nonincreasing, if E a < 0, for any b ∈ (a, +∞), we can get some uniform ξ > 0 satisfying Now, for any a > 0 with E a < 0 and θ > 1, we take ξ > 0 such that Then, we may choose k 0 ∈ (0, ξ) and n ∈ N such that and so Then, if E a is attained, we get that E θa < θE a for any θ > 1.For 0 < b ≤ a, we obtain that and then E a+b < E a + E b .The proof is complete.
The next compactness lemma on S a is useful in the study of the autonomous problem as well as non-autonomous problem.
Lemma 2.6.Let {u n } ⊂ S a be a minimizing sequence with respect to E a .Then, for some subsequence, one of the following alternatives holds: Proof.By Lemma 2.3, we know J is coercive on S a , the sequence {u n } is bounded, so u n ⇀ u in H s (R N ) for some subsequence.Now we consider the following three possibilities.
(1) If u ≡ 0 and |u| 2 2 = b = a, we must have b ∈ (0, a).Set v n = u n − u, by the Brézis-Lieb Lemma [32], Since F is a C 1 function and has a subcritical growth in the Sobolev sense, then it follows that (2.15) 2 , and by using , we obtain that d n ∈ (0, a) for n large enough and Letting n → +∞, by Lemma 2.5, we find that which is a contradiction.This possibility can not exist. ( Then, by (2.4) and (2.5), we have that 4s Hence, we get R N F (u n − u)dx → 0. From (2.15), we obtain that Since u ∈ S a , we infer that E a = J(u), then ||u n || 2 → ||u|| 2 , where || || denotes the usual norm in H s (R N ).Thus u n → u in H s (R N ), which implies that (i) occurs.
(3) If u ≡ 0, that is, u n ⇀ 0 in H s (R N ).We claim that there exists β > 0 such that Indeed, otherwise by [9, Lemma 2.2], we have which contradicts the Lemma 2.4.Hence, from this case, (2.16) holds and |y n | → +∞, then we consider ũn (x) = u(x + y n ), obviously {ũ n } ⊂ S a and it is also a minimizing sequence with respect to J a .It is observed that there exists ũ ∈ H s (R N )\{0} such that ũn (x) ⇀ ũ in H s (R N ).Following as in the first two possibilities of the proof, we infer that ũn (x) → ũ in H s (R N ), which implies that (ii) occurs.This proves the lemma.
In what follows, we begin to prove Theorem 2.1.
Proof of Theorem 2.1.By Lemma 2.3, Lemma 2.4, there exists a bounded minimizing sequence {u n } ⊂ S a satisfying J(u n ) → E a .Then applying Lemma 2.6, there exists u ∈ S a such that J(u) = E a .By the Lagrange multiplier, there exists λ ∈ R such that (2.17) where Φ(u) : Therefore, from (2.17), we have By Lemma 2.2, we can get Furthermore, according to the condition (f 3 ) and the claim 3, we must have λ < η.
Next, we will prove that u can be chosen to be positive.Obviously, we have J(u) = J(|u|).Moreover, since u ∈ S a shows that |u| ∈ S a , we infer that which implies that J(|u|) = E a , we can replace u by |u|.Furthermore, if u * denotes the Symmetrization radial decreasing rearrangement of u (see [1,Section 9]), we observe that then u * ∈ S a and J(u * ) = E a , it follows that we can replace u by u * .Similarly as in [23], one can show that u(x) > 0 for any x ∈ R.This completes the proof.

The non-autonomous problem
In this section, we first give some properties of the functional I ε (u) given by (1.8) restricted to the sphere S a , and then prove Theorem 1.1.Define the following energy functionals Moreover, denoted by E ε,a , E a i ,a and E ∞,a the following real numbers The next two lemmas establish some crucial relations involving the levels E ε,a , E ∞,a and E a i ,a .For any α, β ∈ R, set where Proof.The proof is standard and we omit the details.
Lemma 3.2.lim sup Proof.By the proof of the Theorem 2.1, choose u 0 ∈ S a such that Then u ∈ S a for all ε > 0, we have Letting ε → 0 + , by the Lebesgue dominated convergence theorem, we deduce Noting that E ∞,a can be achieved, due to 0 < h ∞ < h(a i ) and V (a i ) < 0, we have It completes the proof.
Hence by Lemma 3.2, there exists ε 1 > 0 satisfying E ε,a < E ∞,a for all ε ∈ (0, ε 1 ), In the following, we always assume that ε ∈ (0, ε 1 ).The next three lemmas will be used to prove the (P S) c condition for I ε restricts to S a at some levels.
Proof.We argue by contradiction and assume that δ = 0, then up to a subsequence, we have u n → 0 in L l (R N ) for all l ∈ (2, 2N N −2s ), by the Lebesgue dominated convergence theorem and (f 1 )-(f 2 ), we infer that which is a contradiction.
So if u ≡ 0, there exists {y n } satisfying |y n | → ∞, let ũn = u n (x + y n ), obviously {ũ n } ⊂ S a , we have which is absurd, because c < E ∞,a < 0. This proves the lemma.
Lemma 3.5.Let {u n } ⊂ S a be a (P S) c sequence of I ε restricted to S a with c < E ∞,a < 0 and let Proof.Setting the functional Φ : It follows that S a = Φ −1 ({a/2}).Then, by Willem [32, Proposition 5.12], there exists {λ n } ⊂ R such that By the boundedness of {u n } in H s (R N ), we know {λ n } is a bounded sequence, thus there exists λ ε such that λ n → λ ε as n → +∞.Then, together with (3.3), we get , for all ε ∈ (0, ε 1 ).
From (3.4), we get which combined with (3.5) to give for some constant C 3 > 0 that does not depend on ε ∈ (0, ε 1 ).If ), we know that there exists C 0 > 0 independent of ε such that Then, by the fractional Gagliardo-Nirenberg-sobolev inequality, , for some positive constant C(s, N, α) > 0. We have Clearly also, for K > 0 is a suitable constant independent of ε satisfying the condition This together with (3.8) and (3.9) gives that there exists β > 0 independent of ε ∈ (0, ε 1 ) such that lim inf n→+∞ |v n | 2 2 ≥ β. we get desired result.
Next we will give the compactness lemma.
Then, for each ε ∈ (0, ε 1 ), the functional I ε satisfies the (P S) c condition restricts to S a if c < E a i ,a + ρ 0 .
Proof.Let {u n } be a (P S) c sequence for I ε restricts to S a and c < E a i ,a + ρ 0 .It follows that c < E ∞,a < 0, since which combing with (3.10), we obtain that Letting n → ∞, by the inequation (2.13), we have which is a contradiction, because c < E a i ,a + β a (E ∞,a − E a i ,a ).Therefore, we can obtain In what follows, let us fix ρ, r > 0 satisfying: (1) B ρ(a i ) ∩ B ρ(a j ) for i = j and i, j ∈ {1, . . .k}. ( We set the function , where χ : R N → R N denotes the characteristic function, that is, The next two lemmas will be useful to get important (P S) sequences for I ε restricted to S a .Lemma 3.7.For ε ∈ (0, ε 1 ), there exist δ 1 > 0 such that if u ∈ S a and I ε (u) ≤ E a i ,a + δ 1 , then , ∀ε ∈ (0, ε 1 ).
Proof.If the lemma does not occur, there must be δ n → 0, ε n → 0 and {u n } ⊂ S a such that (3.12) According to Lemma 2.6, we have one of the following two cases: For (i): By Lebesgue dominated convergence theorem, for n large enough, that contradicts (3.12).For (ii): We will study the following two case: (I) which contradicts E a i ,a < E ∞,a in Lemma 3.2.
If (II) holds, by (3.13), we obtain that and then E h(y)V (y),a ≤ I h(y)V (y) (v) ≤ E a i ,a .By Lemma 3.1, we must have h(y) = h(a i ) and V (y) = V (a i ).Namely, y = a i for some i = 1, 2, . . ., k. Hence for n large enough, That contradicts (3.12).The proof is complete.
n } is a (P S) β i ε for I ε restricts to S a .Since β i ε < E a i ,a + ρ 0 , it follows from Lemma 3.6, there exists u i such that u i n → u i in H s (R N ).Then, we get and B ρ(a i ) ∩ B ρ(a j ) = ∅ for i = j, which implies that u i = u j for i = j while 1 ≤ i, j ≤ k, we can get I ε has at least k nontrivial critical points for any ε ∈ (0, ε 1 ).Therefore, we obtain the theorem.

Lemma 3 . 4 .
Under the assumption of Lemma 3.3, assume u n ⇀ u in H s (R N ), then u ≡ 0. Proof.By Lemma 3.3, we have that lim inf n→∞ sup y∈R N Br(y)

I
ε (û i ε ) = I a i (u) = E a i ,a .

4 .
Proof of Theorem 1.1 Proof.By Lemma 3.8, for each i ∈ {1, 2, ..., k}, we can use the Ekeland's variational principle to find a sequence {u i n } ⊂ S a satisfyingI ε (u i n ) → β i ε and I ε (w) ≥ I ε (u i n ) −Recalling Lemma 3.8, β i ε < βi ε , and so u i n ∈ θ i ε \∂θ i ε for n large enough.Let w ∈ T u i n S a , there exists δ > 0 such that the path γ : (−δ, δ) → S a defined by