Semiclassical states for non-cooperative singularly perturbed fractional Schrödinger systems

We study the following non-cooperative type singularly perturbed systems involving the fractional Laplacian operator: where s ∈ (0,1), N > 2 s , and (– (cid:3) ) s is the s -Laplacian, ε > 0 is a small parameter. f and g are power-type nonlinearities having superlinear and subcritical growth at inﬁnity. The corresponding energy functional is strongly indeﬁnite, which is diﬀerent from the one of the single equation case and the one of a cooperative type. By considering some truncated problems and establishing some auxiliary results, the semiclassical solutions of the original system are obtained using “indeﬁnite functional theorem”. The concentration phenomenon is also studied. It is shown that the semiclassical solutions can concentrate around the global minima of the potential. MSC: 35Q40; 49J20


Introduction and main results
In these last years, a great deal of work has been devoted to the study of the weak solutions for the following singularly perturbed fractional Schrödinger systems (1.1) portant physical interest. For small value ε, the wave functions of (1.1) tend to concentrate as a material particle.
Recently, there has been tremendous interest in developing the fractional Laplacian problem in various fields, for instance, thin obstacle problems, optimization, population dynamics, geophysical fluid dynamics, mathematical finance, phases transitions, anomalous diffusion, crystal dislocation, ultra-relativistic limits of quantum mechanics, etc., see [1]. Different from the classical Laplace operator, the analytical methods for elliptic PDEs cannot be directly applied to (1.1) since the operator (-) s is nonlocal. In [2], Caffaralli and Silvestre gave a new formulation of the fractional Laplacian through Dirichlet-Neumann maps. This is extensively used in the recent literature since it allows to transform nonlocal problems into local ones, which enables the of use variational methods. For example, for the single nonlocal problems, this is, u = v, f = g in (1.1), there have been many results on the existence and concentration, which were studied using the idea of the s-harmonic extension [3][4][5][6][7][8][9].
In the case of the standard Laplacian operator (s = 1, local case), the existence of a solution for the Schrödinger systems has been studied, and relatively complete methods have been formed. However, for the fractional Schrödinger systems like (1.1), there are only some literature on the semiclassical states for nonlocal singularly perturbed problems; for example, see [7,[10][11][12][13].
Later, Vincenzo Ambrosio [10] applied penalization techniques, Nehari manifold arguments, and Ljusternik-Schnirelmann theory and investigated the existence, multiplicity, and concentration of positive solutions of the following nonlocal system of fractional Schrödinger equations where V , W : R N → R are positive continuous potentials, Q is a homogeneous C 2function with subcritical growth.
We point out that Manassés de Souza in [12] also considered the existence and multiplicity of solutions of the following more general nonlocal system involving the fractional Laplacian where a i (x) are continuous and unbounded potentials that may change sign, and the nonlinearities f i (x, u 1 , . . . , u m ) are continuous functions that may be unbounded in x.
In line with the above works, it is worth mentioning that the nonlinearities are cooperative type in [7,[10][11][12][13] hence the energy functions corresponding to them can be proved to have mountain pass structure, and Nehari manifold arguments can be used. However, when the nonlinearities are non-cooperative type, the corresponding energy functional is strongly indefinite; that is, the quadratic part of the energy functional has no longer a positive sign, the problems become rather complicated mathematically. In [14], Manasses de Souza established a weighted Thudinger-Morse type inequality and, as the application of this result using the Galerkin methods and a linking theorem, proved the existence of weak solutions for the following elliptic system: However, another question arises: for the following more general non-cooperative type singularly perturbed fractional systems: whether the results [7,[10][11][12][13] on the existence of semiclassical states and concentration can be obtained? Answering this question constitutes the goal of this paper.
Since we are interested in positive solutions, we assume the continuous functions a(x), f , g satisfy the following conditions: there exist real numbers l 1 , l 2 > 0 and p, q > 2 such that 1 p + 1 q > N-2s N and (A 3 ) there exists δ > 0 such that 0 < (1 + δ)f (t)t ≤ f (t)t 2 for every t ∈ R and similarly for g; (A 4 ) for every μ > 0, there exists C μ > 0 such that The main result of this paper is stated as follows: 4 ) are satisfied, s ∈ (0, 1), then for all small ε > 0, (i) (Existence) the nonlocal system (1.2) admits a least energy solution (ω ε , ξ ε ); (ii) (Concentration) both functions ω ε and ξ ε attain their maximum value at some unique and common point z ε ∈ R N such that (iii) (Decay estimates) there exist constants 0 < C 1 < C 2 and large R > 0 such that for all |x| ≥ R.
A typical example of functions verifying the assumption (A 1 )-(A 4 ) is given by f (t) = l 1 |t| p-2 t, g(t) = l 2 |t| q-2 t with l 1 , l 2 > 0 and p, q > 2 such that 1 Remark 1.2 The difficulties in treating system (1.3) originate in at least five facts: (i) Although we have a variational problem, the functional I ε associated with (1.3) is strongly indefinite, compared to the single equation case and cooperative type systems, the quadratic part of the energy functional has no longer a positive sign, and so we have to recourse to the "Indefinite Functional Theorem" introduced by Benci and Rabinowitiz in [15], which is an extension of both the mountain-pass theorem and the saddle point theorem. (ii) There is a lack of compactness due to the fact that we are working in R N , in order to attain (PS) condition, we need to consider some truncated problems in Sect. 4. (iii) No uniqueness and non-degenerate results seem to be known for the autonomous system of (1.2), and thus, the Lyapunov-Schmidt reduction method can not be used. (iv) We employ the ideas in [16] and [17] to prove Theorem 1.1; however, our systems are nonlocal, a delicate analysis is needed to overcome the lack of localization. The proof is different from that of the classical case s = 1. (v) Under the natural assumption on p and q, that is p, q > 2 such that 1 p + 1 q > N-2s N , which is more general than assuming that 2 < p, q < 2 * s , the associated functional I ε may not to be well defined in the space H s (R N ) × H s (R N ), because it may happen that say p < 2 * s = 2N N-2s < q. However, as explained in Sect. 5, we only have to prove Theorem 1.1 in the case of 2 < p = q < 2 * s . In fact, given n ∈ N, we can define the truncated functions, where the coefficients are chosen so that g n is C 1 . Thus, in view of (A 2 ), we see that A n = ( l 2 p-1 +o(1))·n q-p , B n = ( l 2 (p-q) (p-1)(q-1) +o(1))·n q-1 . We show in Sect. 5 that the solutions (u ε n , v ε n ) of the corresponding system obtained using Theorem 1.1 applied to the truncated problem are such that u ε n ∞ , v ε n ∞ ≤ C for some C > 0 independent of n, and therefore they solve the original problem (1.2) if n is taken sufficiently large. Thus, in Sects. 2-4, we assume that 2 < p = q < 2 * s .
This paper is organized as follows: In Sect. 2, we review certain notations related to the fractional Laplacian and describe the appropriate functional setting, including the definition of the equivalent problems. In order to study the concentration phenomenon of semiclassical states for system (1.3), Sect. 3 is devoted to studying the autonomous systems of (1.3). We show that (PS) condition holds for the energy functional associated with (1.3) at energy levels in a suitable range in Sect. 4.1; we discuss some auxiliary problems involving appropriate truncated functions in the place of a(x) in Sect. 4.2. The proof of Theorem 1.1 is given in Sect. 4.3. In Sect. 5, we will show that the solutions are bounded in L ∞ (R N ); for this, some Liouville-type theorems need to be established. Therefore, during the proof on the existence of weak solutions for (1.3), we may assume that 2 < p = q < 2 * s .

Notations
Here, we list some notations that will be used throughout the paper.
• We denote by R N+1 is a generic constant that may vary from line to line.
• For a function U ∈ X s (R N+1 + ), we denote its trace on R N × {y = 0} as u = Tr(U).

Preliminaries
In this section, we first introduce some definitions and notations. We consider the fractional Sobolev space where F u denotes the Fourier transform of u, and the fractional Laplacian (-) s : When u is assumed, in addition, sufficiently regular, we obtain the direct representation for a suitable constant C N,s and the integral is understood in a principal value sense.
The dual space H -s (R N ) is defined in the standard way, as well as the inverse operator (-) -s .
It is standard that (1.2) is equivalent to, by letting u( we now consider the problem (2.1). Since the above definition of the fractional Laplacian allows integrating by parts in the proper spaces, a natural definition of the energy solution to the problem (2.1) is the following.
holds for every functions ψ, ϕ ∈ H. Associated to the problem (2.1), we consider the energy functionals dξ . These functionals are well defined in H × H when 2 < p = q < 2 * s , and moreover, the critical points of I ε correspond to the weak solutions of (2.1).
We now include the main ingredients of a recently developed technique by Caffarelli and Silverstre [2]. Let u be a regular function in R N , we say that In [8] it is proved that . Observe that k s = 1 for s = 1 2 and k s ∼ 1 2-2s as s → 1 -. Identity (2.3) allows the formulation of nonlocal problems involving the fractional powers of the Laplacian in R N as local problems in divergence form in the half-space R N+1 + . Remarking (2.2), we introduce the function space X s (R N+1 + ) that is defined as the completion of C ∞ 0 (R N+1 + ) with respect to the norm Then it is a Hilbert space endowed with the inner product With the constant k s , we have the extension operator to be an isometry between H s (R N ) and X s (R N+1 + ); that is On the other hand, for a function U ∈ X s (R N+1 + ), we will denote its trace on R N × {y = 0} as Tr(U). This trace operator is also well defined, and it satisfies For convenience, we will use the following notation: With the above extension, we can reformulate our problem (2.1) as The energy solutions to (2.4) are functions (U, V ) ∈ X × X such that for any , ∈ X, here X := {U ∈ X s (R N+1 + ) : R N ×{y=0} a(εx)U 2 dx < +∞}. For any energy solutions (U, V ) ∈ X × X to this problem, the functions (u, v) = (U(x, 0), V (x, 0)), defined in the sense of traces, belong to the space H × H and are the energy solutions to the problem (2.1). The converse is also true. Therefore, both formulations are equivalent.
The associated energy functionals J ε : X × X → R 1 to (2.4) are given by (2.5) They are the C 2 functionals well defined over the Hilbert space E := X × X, Clearly, the critical points of J ε in E correspond to the ones of I ε in H × H.
Remark 2.2 In the sequel, in view of the above equivalence, we will see both formulations of the problem (2.1), in R N or in R N+1 + , whenever we may take some advantages. In particular, we will use the extension version when dealing with the fractional operator acting on products of functions since it is not clear how to calculate this action.
In Sect. 5, we will utilize the following Sobolev inequality on weighted spaces appeared in Theorem 1.3 of [18].

Proposition 2.3 Let be an open bounded set in R N+1
+ . Then there exists a constant C = C(N, s, ) > 0 such that holds for any function U whose support is contained in whenever the right-hand side is well-defined.
It can be observed that the following orthogonal splitting holds ). So that, denoting by Q the quadratic term of the energy functional J ε , namely We have that Q is positive definite (respectively, negative definite) in E + (respectively, in E -). Therefore, J ε are indefinite functionals; we refer to "Indefinite functional theorem" in [15] to obtain nontrivial critical points of J ε .
What as follows, we recall that the definitions of the relative Morse index and solutions having finite index.
Let E be a real Hilbert space; for a closed subspace of V ⊂ E, we denote by P V the orthogonal projection onto V and by V ⊥ the orthogonal complement of V . Following [19] and [20], we say that the closed subspaces V , W of E are commensurable if P V ⊥ P W and P W ⊥ P V are compact operators.
If V and W are commensurable, the relative dimension of W with respect to V is defined as Commensurability guarantees that both terms in the above formula are finite.
We will also borrow the definition of solutions having a finite index as defined in [21]. (2.6)

The autonomous system
In order to investigate the semiclassical states and their concentration phenomenon of the noncooperative type system (2.1), we firstly give some results on the autonomous system as follows: where λ > 0 is any constant. In view of hypothesis 2 < p = q < 2 * s , we work with the space E := X s (R N+1 + ) × X s (R N+1 + ). So, we consider the functional J λ : E → R 1 defined by for any , ∈ X s (R N+1 + ). So, the critical points of J λ satisfy the equations and k s for any , ∈ X s (R N+1 + ). Equations (3.2)-(3.3) are the weak formulation of (3.1). The following Lemma will play a significant role in the sequel, whose proof is similar to [22], Lemma 2.1, so we omit it.
Pproof of Theorem 3.1 By the assumptions (A 1 )-(A 3 ), it is easy to check that the energy function J λ possesses the linking structure; that is, , for some small r > 0, ρ > 0; moreover, if r > 0 is sufficiently large and e = (e 1 , e 2 ) ∈ E, e 1 > 0, e 2 > 0, then Then, according to "Indefinite functional theorem" [15], J λ has a (PS) c sequence {(U n , V n )} ⊂ E, where 0 < ρ ≤ c ≤ sup E -⊕R + e J λ , using Lemma 3.2, (U n , V n ) are bounded in E and may assume (U n , V n ) (U, V ) as n → ∞, then clearly J λ (U, V ) = 0. Next we need to show that there exists a non-trivial critical point. For this purpose, by concentration compact principle [17], it is possible to find a sequence {x n } ⊂ R N and some constants R > 0 and β > 0 such that Indeed, assuming the contrary, we have But then, for large n and some constants a > 0 and C 1 , C 2 > 0, we have proving a contradiction, since c > 0. Now we define U n (x, y) = U n (x + x n , y), V n (x, y) = V n (x + x n , y), then ( U n , V n ) (U 0 , V 0 ) = (0, 0) is a non-trivial critical point of J λ .

The noncooperative singularly perturbed system 4.1 For the original problem (2.1)
Now we temporarily come back to our original problem (2.1). Note that using the similar arguments as Sect. 3, J ε also possess the linking structure, and thus there exist (PS) c ε sequences with 0 < c ε ≤ sup E -⊕R + e J ε , e = (e 1 , e 2 ) ∈ E, e 1 > 0, e 2 > 0. (4.1) On the other hand, according to the assumption (A 0 ), we may fix a ∈ R such that We denote by J 0 and c 0 the energy functional defined in (2.5) with a(0) in place of a(εx) and least energy, respectively. Let (U 0 , V 0 ) be a ground-state for J 0 , it is easy to check that and uniformly for bounded , ∈ X. Applying Theorem 3.1 and Lemma 3.2 to the functionals J ε , we deduce that which complies with (4.1) to conclude that 0 < c ε < c(a).

Some auxiliary problems
To apply "Indefinite functional theorem" to get the positive solutions of (2.1), we only need to show that (PS) c ε condition holds for 0 < c ε < c(a). For this purpose, in this subsection, we will study some auxiliary problems and establish several auxiliary results employing the ideas from [16]. Consider the nonlocal system as follows: where b(x) ∈ C(R N ), b(x) ≥ b > 0 for any x ∈ R N and lim |x|→∞ b(x) = b ∞ ∈ R. Page 12 of 24 The associated energy functional to the extension problem of (4.3) is defined by We denote by J ∞ the corresponding functional with b ∞ in place of b(x). Of course, here we work in the space E := X × X, X = {U ∈ X s (R N+1 + ) : R N ×{y=0} b(x)U 2 dx < +∞}. Now we prove the following Lemma.
Indeed, one has and for any bounded functions , ∈ X.
Now we compute the third term of (4.5) and (4.6), respectively, Similarly, uniformly for any bounded and in X.
Combining these estimates with (4.4)-(4.6) gives and , a similar conclusion as (4.7) and (4.8) holds for J ∞ (U n , V n ) and J ∞ (U n , V n ); if lim inf n→∞ J ∞ (U n , V n ) > 0, we deduce from which contradicts with the assumption c < c(b ∞ ). Consequently, which turns to be lim inf n→∞ U n X + V n X = 0.
Therefore, U n → U, V n → V as n → ∞; that is the (PS) c condition holds for 0 < c < c(b ∞ ). Applying "Indefinite functional theorem", we can derive that there exists a nontrivial critical point (U, V ) for the functional J b . We set By the standard arguments, the infimum is actually a minimum, and it follows that J b admits a ground-state critical level c b . Finally, we compare the least energy levels of J b with the ones of J¯b; this estimate is crucial for the proof of Theorem 1.1.
Assume that c(b) > c b . For t ∈ [0, 1], let b t (x) := (1t)b(x) + tb and denote by c , c t the corresponding ground-state level and linking level, respectively.
It follows from the assumption c(b) > c b and the fact (1t)b ∞ + tb ≥b and Lemma 3.2 that for every t ∈ [0, 1]. This is a contradiction for t = 1, since c 1 = c(b), which ends the proof.
Next, utilizing Lemma 4.1, we prove an important auxiliary result that is directly applied to the proof of Theorem 1.1.  (1), (4.10) and (1), (4.11) uniformly for all bounded functions and in X. From (4.9)-(4.11), we can employ Lemma 4.1 to get J b n admitting a ground state critical level c b n > 0 and This completes the proof.

The proof of Theorem 1.1
In this subsection, we prove Theorem 1.1 divided into three results. First of all, we give the existence result. Proof Recall that there exists a (PS) c ε sequence {(U n , V n )} ⊂ E for J ε , assume that (U n , V n ) (U, V ) ∈ E, and for sufficiently small ε > 0, 0 < c ε < c(a). And there exists a constant R 0 > 0 such that for some fixed ε > 0, a(εx) ≥ a for all |x| ≥ R 0 .
According to "Indefinite functional theorem" [15], we only need to show that for fixed ε > 0, the (PS) c ε condition holds for J ε at critical levels 0 < c ε < c(a).
Introduce the following truncated function Observe that a(εx)U n V n dx + o n (1).
Using the similar arguments as Lemma 4.1, it is easy to check that If lim inf n→∞ J b (U n , V n ) > 0, we can derive applying Lemma 4.2 that which is a contradiction. As a result, lim inf n→∞ J b (U n , V n ) ≤ 0. This gives U n → 0, V n → 0 in X. Next, we show c ε → c 0 as ε → 0. By (4.2), we obtain that lim sup ε→0 c ε ≤ c 0 . On the other hand, let (U ε , V ε ) be the ground states for J ε , then (U ε , V ε ) are bounded in E. In particular, we have and for any , ∈ X, By Lemma 3.2, we conclude that Hence, c 0 ≤ lim inf ε→0 c ε . This completes the proof.
From now, we consider the positive functions U ε > 0, V ε > 0 given by Theorem 1.
Next, we study the concentration behavior of this family of solution (u ε , v ε ) as ε → 0. Proof Since u ε > 0, v ε > 0, for each small ε > 0, there exist x ε , y ε ∈ R N such that, respectively, We split the proof into several steps.
Step 1. {εx ε } and {εy ε } are bounded. Suppose by contradiction that there exists a subsequence {ε j x ε j } such that |ε j x ε j | → +∞ as ε j → 0. Define the functions u j (x) = u ε j (x + x ε j ) and v j (x) = v ε j (x + x ε j ). Observe that these functions satisfy the following system: here a j (x) = a(ε j x + ε j x ε j ). Denote by I j the corresponding energy functional, Recall from Theorem 1.1(i) that Let b j (x) := max{a, a j (x)} and denote by I j the corresponding energy functional defined as in (4.12) with a j (x) replaced by b j (x). By the assumption |ε j x ε j | → +∞, it holds that ( u j , v j ) is a (PS) sequence for I j and Since b j (x) ≥ a for every x ∈ R N , it follows then from Lemma 4.2 that which contradicts with (4.13). Hence {εx ε } is bounded, the same is {εy ε }. Furthermore, we have that there is a subsequence { u j n } and { v j n } such that for any η > 0, there exists r η > 0 satisfying f ( u j n ) u j n + g( v j n ) v j n dx ≤ η, (4.14) for all r ≥ r η .
x ε = y ε as ε → 0 and the maximum point of u ε (and of v ε as well) is unique if ε > 0 is sufficiently small.
Let us prove that there exists C > 0 such that, for every subsequence {ε j } and every large j, For this purpose, let u j = u ε j (x + y ε j ), v j = v ε j (x + y ε j ), we can follow Step 1 to assume (u j , v j ) (u, v) = (0, 0); it follows from (4.14) that there is a subsequence {u j n } and {v j n } such that, for any η > 0 there exists R > 0 such that, for every large n, Denoting w j n := y ε jnx ε jn , the above inequality reads as So, if |w j n | → +∞, we conclude from (4.14) that for every η > 0, for every large n, which conclude that R N (f (u)u + g(v)v) dx = 0, and thus u = v = 0. This is a contradiction.
Now, according to (4.15), , combining with the fact that both u j (x) and u j (x) have the same limit function u (and similarly for v j (x) and v j (x)), we can derive u(x) = u(x + w 0 ) (and similarly for v). u has 0 and w 0 as maximum points, so that w 0 = 0. This establishes Similarly, if z ε ∈ R N is also a maximum point of u ε , then the preceding arguments yield that z ε = x ε (= y ε ) for small values of ε.
Indeed, assume by contradiction that there exist δ > 0 and ξ ε ∈ R N with |ξ ε | → ∞ such Assume that u ε u in H, we obtain, as ε → 0, which is a contradiction completing the proof. Define Then (ω ε , ξ ε ) is a solution to (1.2) for all small ε > 0. Since z ε is a maximum point of ω ε and ξ ε , we have lim ε→0 a(z ε ) = a(0). Now, we study the decay behavior of this family of solution (ω ε , ξ ε ).

Theorem 1.1(iii)
There exist 0 < C 1 ≤ C 2 and R > 1 such that for all small ε > 0, Proof Firstly, we use the following Claims, according to [25].
By u ε (x), v ε (x) → 0 as |x| → ∞ uniformly for all small ε and the condition (A 0 ), (A 1 ), we conclude that there is a large R 1 > 0 such that Moreover, by the continuity of solution (u ε , v ε ) and w 2 , for all small ε > 0, there exists C > 0 such that Therefore, Using comparison arguments, we get Hence, by rescaling, it follows that there exists C 2 > 0 such that On the other hand, by the continuity of (u ε , v ε ) and w 1 , there exist constants C 2 , C 3 > 0 such that, respectively, By the similar comparison arguments, we conclude the second inequality, which ends the proof of Theorem 1.1.

The case p = q
In Sect. 3 and Sect. 4, we have proved Theorem 1.1(i)(ii)(iii) except that we have worked with a truncated problem as explained in Remark 1.2. The full statement of Theorem 1.1(i)(ii)(iii) will be established once we prove uniform bounds in L ∞ of the solutions constructed so far. So, in this section, let us suppose that p, q > 2 are such that 1 p + 1 q > N-2s N with say, 2 < p < 2 * s and p < q. We only show that the weak solutions to the modified problem of (3.1) are bounded uniformly in L ∞ . In the same way, the weak solutions to the modified problem of (2.1) are ones of the original system (2.1) for large value of n.
Given n ∈ N, we can define the truncated functions, where the coefficients are chosen in such a way that g n is C 1 . Thus, in view of (A 2 ), we see that A n = ( l 2 p-1 + o(1)) · n q-p , B n = ( l 2 (p-q) (p-1)(q-1) + o(1)) · n q-1 . The energy functionals associated to the modified problem of (3.1) are given by where G n is the primitive of g n . They are C 2 functionals defined over the Hilbert space E. The critical points of J n correspond to weak solutions of the modified problem in R N . In particular, u n and v n are solutions to the problem (3.1) for large values of n.
The proof of Theorem 5.2 is based on the following simple fact, whose proof is the same as Lemma 1.2 in [26].

(5.5)
We conclude by combing (5.3)-(5.5), which together lead to u ∈ L p (R N ). Thanks to R N (-) s uv dx = R N g ∞ (v)v dx = R N u p dx < +∞ and the condition (a), v ∈ L q (R N ) also. Making use of the well-known Pohozăve-Rellich type identity, By the condition (c), we deduce that Since 1 p + 1 q > N-2s N , this implies that u = v = 0 and concludes the proof of Proposition 5.4.
Once Proposition 5.4 is settled, we may use the classical blow-up argument to give the proof of Theorem 5.2 that is similar to [27], we omit it.