Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

R3 |u(y)|2 * μ,s + F(u(y)) |x − y|μ dy)(|u| 2*μ,s−2u + 1 2*μ,s f (u)) in R3, where ε > 0 is a small parameter, (−∆)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, 2μ,s = 6−μ 3−2s is the critical exponent in the sense ofHardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.


Introduction and the main results
In the present paper we are interested in the existence, multiplicity and concentration behavior of the semi-classical solutions of the singularly perturbed nonlocal elliptic equation ε s (−∆) s u + V(x)u = ε µ−N ( R G(u(y)) |x − y| µ dy)g (u) in R N , (1.1) solutions of the singularly perturbed equation (1.1) is known as the semi-classical problem. It was used to describe the transition between Quantum Mechanics and Classical Mechanics. Our motivation to study (1.1) mainly comes from the fact that solutions u(x) of (1.1) correspond to standing wave solutions Ψ(x, t) = e −iEt/ε u(x) of the following time-dependent fractional Schrödinger equation where i is the imaginary unit, ε is related to the Planck constant. Equations of the type (1.2) was introduced by Laskin (see [25,26]) and come from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. It also appeared in several areas such as optimization, nance, phase transitions, strati ed materials, crystal dislocation, ame propagation, conservation laws, materials science and water waves (see [11]). When s = , the equation (1.1) turns out to be the Choquard equation 3) The existence, multiplicity and concentration of solutions for (1.3) has been widely investigated. On one hand, some people have studied the classical problem, namely ε = in (1.3). When V = and G(u) = |u| q q , (1.3) covers in particular the Choquard-Pekar equation ( 1.4) The case N = , q = and µ = came from Pekar [38] in 1954 to describe the quantum mechanics of a polaron at rest. In 1976 Choquard used (1.4) to describe an electron trapped in its own hole, in a certain approximation to Hartree-Fock theory of one component plasma [27]. In this context (1.4) is also known as the nonlinear Schrödinger-Newton equation. By using critical point theory, Lions [29] obtained the existence of in nitely many radialy symmetric solutions in H (R N ) and Ackermann [1] prove the existence of in nitely many geometrically distinct weak solutions for a general case. For the properties of the ground state solutions, Ma and Zhao [30] proved that every positive solution is radially symmetric and monotone decreasing about some point for the generalized Choquard equation (1.4) with q ≥ . Later, Moroz and Van Schaftingen [32,33] eliminated this restriction and showed the regularity, positivity and radial symmetry of the ground states for the optimal range of parameters, and also derived that these solutions decay asymptotically at in nity.
On the other hand, some people have focused on the semiclassical problem, namely, ε → in (1.3). The question of the existence of semiclassical solutions for the non-local problem (1.3) has been posed in [5]. Note that if v is a solution of (1.3) for x ∈ R N , then u = v(εx + x ) veri es −∆u + V(εx + x )u = ( R N G(u(y)) |x − y| µ dy)g (u) in R N , (1.5) which means some convergence of the family of solutions to a solution u of the limit problem , which is a di cult problem that has only been fully solved in the case when N = , µ = and G(u) = |u| . Moroz and Van Schaftingen [34] used variational methods to develop a novel non-local penalization technique to show that equation (1.3) with G(u) = |u| q has a family of solutions concentrated at the local minimum of V, with V satisfying some additional assumptions at in nity. In addition, Alves and Yang [4] investigated the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. Very recently, in an interesting paper, Alves et al. [2] study (1.3) with a critical growth, they consider the critical problem with both linear potential and nonlinear potential, and showed the existence, multiplicity and concentration behavior of solutions when the linear potential has a global minimum or maximum. On the contrary, the results about fractional Choquard equation (1.1) are relatively few. Recently, d'Avenia, Siciliano and Squassina [17] studied the existence, regularity and asymptotic of the solutions for the following fractional Choquard equation where ω > , N−µ N < q < N−µ N− s . Shen, Gao and Yang [42] obtain the existence of ground states for (1.7) with general nonlinearities by using variational methods. Chen and Liu [14] studied (1.7) with nonconstant linear potential and proved the existence of ground states without any symmetry property. For critical problem, Wang and Xiang [47] obtain the existence of in nitely many nontrivial solutions and the Brezis-Nirenberg type results can be founded in [36]. For the critical Choquard equations in the sense of Hardy-Littlewood-Sobolev, Cassani and Zhang [12] developed a robust method to get the existence of ground states and qualitative properties of solutions, where they do not require the nonlinearity to enjoy monotonicity nor Ambrosetti-Rabinowitz-type conditions. For other existence results we refer to [6,8,23,24,31,48,52] and the references therein.
It seems that the only works concerning the concentration behavior of solutions are due to [13,51]. Assuming the global condition on V: which was rstly introduced by Rabinowitz [39] in the study of the nonlinear Schrödinger equations. By using the method of Nehari manifold developed by Szulkin and Weth [46], authors in [13,51] obtained the multiplicity and concentration of positive solutions for the following fractional Choquard equation ε s (−∆) s u + V(x)u = ε µ− ( R |u(y)| * µ,s + F(u(y)) |x − y| µ dy)(|u| * µ,s − u + * µ,s f (u)) in R , (1.8) where ε > , < µ < , F is the primitive function of f . Di erent to [13,51], in this paper, we are devote to establishing the existence and concentration of positive solutions for the fractional Choquard equation (1.8) when the potential function satis es the following local conditions [18]: Without loss of generality, we may assume that M = {x ∈ Ω : To go on studying the problem (1.8), the following Hardy-Littlewood-Sobolev inequality [28] is the starting point.
f ∈ L t (R ) and h ∈ L r (R ). There exists a sharp constant C(t, µ, r), independent of f , h such that In particular, if t = r = −µ , then In this case there is equality in (1.9) if and only if f ≡ Ch and Notice that, by the Hardy-Littlewood-Sobolev inequality, the integral Thus, −µ is called the lower critical exponent and * µ,s := −µ − s is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the fractional Laplace operator.
For the nonlinearity term, we assume that the continuous function f vanishes in (−∞, ) and satis es: Note that there is no (AR) type assumption on f . Then it is di cult to show that the functional satis es the (PS) condition even for the autonomous case, which is necessary to use Ljusternik-Schnirelmann category theory. We shall investigate the (PS) sequence carefully and restore the compactness for (PS) sequence via some compactness Lemmas.
In order to describe the multiplicity, we rst recall that, if Y is a closed subset of a topological space X, the Ljusternik-Schnirelmann category cat X Y is the least number of closed and contractible sets in X which cover Y. Then we state our main result as follows.  [18], we will proposed some control conditions on the non-local term ( |x| µ (|u| * µ,s + F(u))) , which need some regularity (see Lemma 2.7 and 2.8), where we introduce the assumption < µ < s.
We shall use the method of Nehari manifold, concentration compactness principle and category theory to prove the main results. There are some di culties in proving our theorems. The rst di culty is that the nonlinearity f is only continuous, we can not use standard arguments on the Nehari manifold. To overcome the nondi erentiability of the Nehari manifold, we shall use some variants of critical point theorems from Szulkin and Weth [46]. The second one is the lack of compactness of the embedding of H s (R ) into the space L * s (R ). We shall borrow the idea in [2,12] to deal with the di culties brought by the critical exponent. However, we require some new estimates, which are complicated because of the appearance of fractional Laplacian and the convolution-type nonlinearity. Moreover, the potential V satis es (V ) and (V ) instead of the global condition. Since we have no information on the potential V at in nity, we adapt the truncation trick explored in [18]. It consists in making a suitable modi cation on the nonlinearity, solving a modi ed problem and then check that, for ε small enough, the solutions of the modi ed problem are indeed solutions of the original one. It is worthwhile to remark that in the arguments developed in [18], one of the key points is the existence of estimates involving the L ∞ -norm of the modi ed problem. But for the critical nonlocal problem (1.8), this kind of estimates are more delicate.
This paper is organized as follows. In section 2, besides describing the functional setting to study problem (1.8), we give some preliminary Lemmas which will be used later. In section 3, in uenced by the work [18] and [45], we introduce a modi ed functional and show it satis es the Palais-Smale condition. In section 4, we study the autonomous problem associated. This study allows us to show that the modi ed problem has multiple solutions. Finally, we show the critical point of the modi ed functional which satis es the original problem, and investigate its concentration behavior, which completes the proof Theorem 1.1.

Variational settings and preliminary results
Throughout this paper, we denote | · |r the usual norm of the space L r (R ), ≤ r < ∞, Br(x) denotes the open ball with center at x and radius r, C or C i (i = , , · · · ) denote some positive constants may change from line to line. and → mean the weak and strong convergence. Let E be a Hilbert space, the Fréchet derivative of a functional Φ at u, Φ (u), is an element of the dual space E * and we shall denote Φ (u) evaluated at v ∈ E by Φ (u), v .

. The functional space setting
Firstly, fractional Sobolev spaces are the convenient setting for our problem, so we will give some skrtchs of the fractional order Sobolev spaces and the complete introduction can be found in [19]. We recall that, for any s ∈ ( , ), the fractional Sobolev space H s (R ) = W s, (R ) is de ned as follows: where F denotes the Fourier transform. We also de ne the homogeneous fractional Sobolev space D s, (R ) as the completion of C ∞ (R ) with respect to the norm The embedding D s, (R ) → L * s (R ) is continuous and for any s ∈ ( , ), there exists a best constant Ss > such that According to [16], Ss is attained by where C ∈ R, b > and a ∈ R are xed parameters. We use S H,L to denote the best constant de ned by . (2. 2) The fractional Laplacian, (−∆) s u, of a smooth function u : R → R, is de ned by Also (−∆) s u can be equivalently represented [19] as Also, by the Plancherel formular in Fourier analysis, we have For convenience, we will omit the normalization constant in the following. As a consequence, the norms on H s (R ) de ned below which is a Hilbert space equipped with the inner product We denote · Hε by · ε in the sequel for convenience. For the reader's convenience, we review some useful result for this class of fractional Sobolev spaces: |un| dx = , then un → in L r (R ) for any < r < * s .
If, in addition, φ ≡ in a neighbourhood of the origin, then uφr → u in D s, (R ) as r → +∞.
(2.5) Therefore, the Hardy-Littlewood-Sobolev inequality implies that and R R G(u(y))g(u(x))u(x) |x − y| µ dydx ≤ C(|u| q q r + |u| q q r + |u| * µ,s * s ). (2.7) It is clear that problem (2.3) is the Euler-Lagrange equations of the functional I : Hε → R de ned by From (2.6) we know that I(u) is well de ned on Hε and belongs to C , with its derivative given by for all u, v ∈ Hε. Hence the critical points of I in Hε are weak solutions of problem (2.3). In the following, we will consider critical points of I using variational methods. Firstly, we give the following Lemma, whose simple proof is omit.
In addition, (f ) and ( (2.12) (ii). Assume that un u in H s (R ). For any v ∈ C ∞ (R ) with support Ω, by Lemma 2.1 we may assume that un(x) → u(x) a.e. in R and un → u in L p (Ω), p < * s . We rst check that if un u in H s (R ), then as n → ∞. By the Hardy-Littlewood-Sobolev inequality, we have and it is a linear bounded operator from L r (R ) to L µ (R ). Choosing hn(y) := |un(y)| * µ,s ∈ L r (R ), we have Therefore, by Hölder inequality we can prove the sequence as n → ∞. Then (2.13) follows for every v ∈ H s (R ) ⊂ L * s (R ). By (f ) we have |f (u)| ≤ C( + |u| q − ) for all u ∈ R + , and then f (un) → f (u) in L q r q − (Ω). Then Hardy-Littlewood-Sobolev inequality implies that Since F(un) is bounded in L r (R ), we may assume that F(un) Moreover, as (2.12) we get which combining with (2.14) we have from our argument above it is then easy to prove Hence, for any v ∈ C ∞ (R ), , we conclude that (2.15) holds for any v ∈ H s (R ), and so I (un)

. Regularity of solutions and Pohožaev identity
The assumption (f ) is too weak for the standard bootstrap method as in [4,15,32]. Therefore, in order to prove regularity of solutions of (2.3) we shall rely on a nonlocal version of the Brezis-Kato estimate. Note that a special case of the regularity result of Brezis and Kato [10, , then u ∈ L p (R N ) for every p ≥ . Similar to [33,42], we extent this result to the fractional Choquard equation with critical growth. For the convenience of readers, we give here a short proof. We rst have the following useful inequality.
Lemma 2.6. [33] Let p, q, r, t ∈ [ , +∞) and λ ∈ [ , ] such that If θ ∈ ( , ) satis es and Applying Lemma 2.6, we have the following result, which is a nonlocal counterpart of the estimate [10, Lemma By the Sobolev inequality, we have thus prove that for every u ∈ H s (R N ), The conclusion follows by choosing H * and K * such that Now, we have the following result, which is a nonlocal Brezis-Kato type regularity estimate.
Proof. By Lemma 2.7 with θ = , there exists λ > such that for every φ ∈ H s (R N ), is bilinear and coercive. Therefore, applying the Lax-Milgram theorem [9, Corollary 5.8], there exists a unique solution u k ∈ H s (R N ) satis es where u ∈ H s (R N ) is the given solution of (2.16). Moreover, we can prove that the sequences {u k } k∈N converges weakly to u in H s (R N ) as k → ∞.
For µ > , we de ne the truncation u k,µ by For p ≥ , we have |u k,µ | p− u k,µ ∈ H s (R N ), so we can take it as a test function in (2.17), we have By Lemma 2.7 with θ = , there exists C > such that We have thus In view of the Sobolev estimate, we have proved the inequality By iterating over p a nite number of times we cover the range

The penalized problem
In this section, we will adapt for our case an argument explored by the penalization method introduction by del Pino and Felmer [18] to overcome the lack of compactness. Let K > to be determined later, and take a > to be the unique number such that G(a) a = V K where V is given by (V ). We de nẽ and where χ is characteristic function of set Ω. From hypotheses (f )−(f ) we get that H is a Carathéodory function and satis es the following properties: Moreover, in order to nd positive solutions, we shall henceforth consider H(x, u) = for all u ≤ . It is easy to check that if u is a positive solution of the equation and therefore u is also a solution of problem (1.8).
In view of this argument above, we shall deal with in the following with the penalized problem and we will look for solutions uε of problem (3.1) verifying The energy functional associated with (3.1) is which is of C class and whose derivative is given by Hence the critical points of Jε in Hε are weak solutions of problem (3.1). Now, we denote the Nehari manifold associated to Jε by Obviously, Nε contains all nontrivial critical points of Iε. But we do not know whether Nε is of class C under our assumptions and therefore we cannot use minimax theorems directly on Nε. To overcome this di culty, we will adopt a technique developed in [45,46] to show that Nε is still a topological manifold, naturally homeomorphic to the unit sphere of Hε, and then we can consider a new minimax characterization of the corresponding critical value for Iε. For this we denote by H + ε the subset of Hε given by where Sε is the unit sphere of Hε.
Proof. Suppose by contradiction there are a sequence {un} ⊂ Hε\H + ε and u ∈ H + ε such that un → u in Hε.
But, this contradicts the fact that u ∈ H + ε . Therefore H + ε is open.
From de nition of S + ε and Lemma 3.1 it follows that S + ε is a incomplete C , -manifold of codimension 1, modeled on Hε and contained in the open H + ε . Hence, In the rest of this section, we show some Lemmas related to the function Jε and the set H + ε . First, we show the functional Jε satisfying the Mountain Pass geometry.
Proof. (i). For any u ∈ Hε\{ }, it follows from (g ) and the Hardy-Littlewood-Sobolev inequality that Hence, Therefore, we can choose positive constants α, ρ such that (ii). Fix a positive function u ∈ H + ε with supp(u ) ⊂ Ωε, and we set where Since H(εx, u ) = F(u ) and by using Lemma 2.4, we deduce that Taking e = tu with t su ciently large, we can see (ii) holds.
(A ) First of all we observe thatmε, mε and m − ε are well de ned. In fact, by (A ), for each u ∈ H + ε , there exists a unique τu > such that τu u ∈ Nε, hence there is a uniquemε(u) = τu u ∈ Nε. On the other hand, if u ∈ Nε then u ∈ H + ε . Therefore, m − ε (u) = u u ε ∈ S + ε , is well de ned and it is a continuous function. Since we conclude that mε is a bijection.
By Lemma 2.5 and passing to the limit as n → ∞, it follows that which means that τ u ∈ Nε and τu = τ . This provesmε(un) →mε(u) in H + ε . So,mε, mε are continuous functions and (A ) is proved. Now we de ne the functionsΨ ε : H + ε → R and Ψε : S + ε → R, byΨε(u) = Iε(mε(u)) and Ψε :=Ψε| S + ε . The next result is a direct consequence of Lemma 3.3. The details can be seen in the relevant material from [46]. For the convenience of the reader, here we do a sketch of the proof.  Proof. (B ) Let u ∈ H + ε and v ∈ Hε. From de nition ofΨε and tu and the mean value theorem, we obtain where |h| is small enough and θ ∈ ( , ). Similarly, where ς ∈ ( , ). Since the mapping u → τu is continuous according to Lemma 3.3, we see combining these two inequalities that Since Jε ∈ C , it follows that the Gâteaux derivative ofΨε is bounded linear in v and continuous on u. From [50] we know thatΨε ∈ C (H + ε , R) and The item (B ) is proved. (B ) The item (B ) is a direct consequence of the item (B ). (B ) We rst note that Hε = Tu S + ε ⊕ Ru for every u ∈ S + ε and the linear projection P : Moreover, by (B ) we have where w = mε(u). Since w ∈ Nε, we conclude that Hence, from (3.5) and (3.7) we have Since w ∈ Nε, we have w ≥ γ > . Therefore, the inequality in (3.8) together with Jε(w) = Ψε(u) imply the item (B ). (B ) It follow from (3.8) that Ψ ε (u) = if and only if J ε (w) = . The remainder follows from de nition of Ψε.
As in [46], using the mountain pass theorem without the (PS) condition, we get the existence of a (PS)c ε sequence {un} ⊂ Hε with If vn is vanishing, i.e.
If un ε → , then it follows from (3.11) and (3.12) that cε = , which is impossible. Then un ε and by virtue of (3.12) we get which is a contradiction. Therefore, {un} is non-vanishing. Proof. By Lemma 3.5, we can have {un} is bounded in Hε. Therefore, we may assume that un u in Hε and un → u in L r loc (R ) for any r ∈ [ , * s ). Fix R > and let ψ R ∈ C ∞ (R ) be such that By Lemma 2.8, taking For n ≥ n and ε > xed, take R > big enough such that Ωε ⊂ B R/ . Then we have Now, we note that the Hölder inequality and the boundedness of {un} imply that Therefore, it is enough to prove that to conclude our result. Let us note that R × R can be written as (3.14) Now, we estimate each integral in (3.14).
Let us note that, if (x, y) ∈ (R \B kR ) × B R , then Therefore, taking into account ≤ ψ R ≤ , |∇ψ R | ≤ C R and applying Hölder inequality, we can see (3.16) Now, x ε ∈ ( , ), and we note that By using the de nition of ψ R , ε ∈ ( , ) and ψ R ≤ , we have (3.20) Putting together (3.14),(3.15),(3.16) and (3.20), we can infer Since {un} is bounded in Hε, we may assume that un → u in L loc (R ) for some u ∈ Hε. Then, taking the limit as n → ∞ in (3.21), we have where in the last passage we use Hölder inequality.
Since u ∈ L * s (R ), k > and ε ∈ ( , ), we obtain lim sup Choosing ε = k , we get which complete our proof. Proof. Since {un} is bounded in Hε, we may assume Let us prove that un → u in Hε as n → ∞. Setting ωn = un − u ε , we have

The autonomous problem
Since we are interestd in giving a multiplicity result for the modi ed problem, we start by considering the limit problem associated to (1.8), namely, the problem which has the following associated functional The functional I is well de ned on the Hilbert space H = H s (R ) with the inner product   As in the previous section, we have the following variational characterization of the in mum of I over N : The next Lemma allows us to assume that the weak limit of a (PS)c sequence is non-trivial. Thus, by (f ) we have Recalling that I (un)un → , we get un = on( ).
Therefore the conclusion follows.
In particular, we consider the following family of functions Uε de ned as for ε > and x ∈ R , the minimizer of Ss (see, [41]), which satis es Then, by a simple calculation, we know is the unique minimizer for S H,L that satis es Let φ ∈ C ∞ (R , [ , ]) and small δ > be such that φ ≡ in B δ ( ) and ϕ ≡ in R \B δ ( ). For any ε > , de ne the best function by uε = φUε.
Similar to [22,Lemma 1.2], we can easily draw the following conclusion.

Lemma 4.4. The constant S H,L de ned in (2.2) is achieved if and only if
where C > is a xed constant, a ∈ R and b > are parameters. Thus if and only if Then, by the de nition of S H,L , we get In addition, if q < * µ,s , then there holds Proof. For the proof of (4.5) and (4.6), we can see that in [41]. So we only need to estimate (4.7) and (4.8).

Lemma 4.7. Suppose that (f ) − (f ) hold. Then the number c V satis es that
Proof. By the de nition of c V , it su ces to prove that there exists v ∈ N such that (4.12) By Lemma 4.1, there exists τε > such that τε uε ∈ N . We claim that for ε > small enough, there exist A and A independent of ε such that < A ≤ τε ≤ A < ∞. (4.13) Indeed, note that N is bounded away from , we have that τε ≥ A > using (4.5) and (4.6 (4.14) For I , we set and consider the function θ : [ , ∞) → R de ned by For I , given A > , we invoke (f ) to obtain R = R(A ) > such that, for x ∈ R , t ≥ R, By (4.6) and (4.16), we need to estimate I in three cases. Since the argument is similar, we only consider the case that < s. For |x| < ε < δ, noting that φ ≡ in B δ ( ), by the de nition of uε and (4.13), we get a constant β > such that Then we can choose ε > such that τε uε ≥ R, for |x| < ε, < ε < ε . It follows from (4.16) that for |x| < ε, < ε < ε . Then for any < ε < ε , by (4.8) we get Note that F(u) ≥ , (4.6) and (4.13), we have Inserting (4.15) and (4.17) into (4.14), we get . (4.18) Observe that −µ > − s for < s, and A > is arbitrary, we choose large enough A such that C + C − C A < . Then for small ε > we have v := τε uε satis es (4.12). |un| dx ≥ δ.
Up to s subsequence, there exists u ∈ H s (R ) such that As Lemma 2.5, we have I (u) = . Since I and I are both invariant by translation, without lost of generality, we can assume that {yn} is bounded. Note that un → u in L loc (R ). Then u ̸ ≡ . So u ∈ N . Then where we used Fatou Lemma and Lemma 2.4. Therefore, I (u) = c V , which means that u is a ground state solution for (4.1). Next we prove that the solution u is positive, using u − = max{−u, } as a test function in (4.1) we obtain On the other hand, Thus, it follows from (4.19) that u − = and u ≥ . Rewriting the equation (4.1) in the form of By Lemma 2.8, we know u ∈ L p (R ) for all p ∈ [ , ( −µ)( − s) ). Using the growth assumption (f ) and the higher integrability of u, for some C > we have (4.20) which is nite since the various exponents live within the range [ , ( −µ)( − s) ). Thus, By the Moser iteration, similar arguments developed in Lemma 6.1 below, we can get u ∈ L ∞ (R ) and lim |x|→+∞ u(x) = uniformly in n. Then, by regularity theory [43], there exists α ∈ ( , ) such that u ∈ C ,α loc (R ). Therefore, if u(x ) = for some x ∈ R , we have that (−∆) s u(x ) = and by [19,Lemma 3.2], we have yielding u ≡ , a contradiction. Therefore, u is a positive solution of the equation (4.1) and the proof is completed.
The next result is a compactness result on autonomous problem which we will use later.
From Theorem 1.1 in [21], for λ = k , there exist a sequence {ṽ k } ⊂ S + such that In particular, for any u ∈ S + we have Hence, similar the proof for Theorem 3.1 in [21], we have that there exists λ k ∈ R such that where g (u) = u − . Which means that Therefore, we can conclude there is a sequence {ṽn} ⊂ S + such that {ṽn} is a (PS)c V sequence for Ψ on S + and un −ṽn = on( ). Now the remainder of the proof follows from Lemma 4.2, Theorem 4.1 and arguing as in the proof of Lemma 3.8.

Solutions for the penalized problem
In this section, we shall prove the existence and multiplicity of solutions. We begin showing the existence of the positive ground-state solution for the penalized problem (3.1). Then, we know that there exists a (PS) sequence at cε, i.e. J ε (un) → and Jε(un) → cε . Therefore, by Lemma 3.7, the existence of ground state solution uε is guaranteed. Moreover, similarly to the proof in Theorem 4.1, we know that uε(x) > in R .
Next, we will relate the number of positive solutions of (3.1) to the topology of the set M. For this, we consider δ > such that M δ ⊂ Ω and by Theorem 4.1, we can choose w ∈ N with I (w) = c V . Let η be a smooth nonincreasing cut-o function de ned in [ , +∞) such that Then for small ε > , one has Ψε,y ∈ Hε\{ } for all y ∈ M. In fact, using the change of variable z = x − y ε , one has Moreover, using the change of variable x = x − y ε , z = z − y ε , we have where ηε(x) = η(|εx|). By Lemma 2.3, we see that ηε w ∈ D s, (R ) as ε → , and hence Ψε,y ∈ D s, (R ) for ε > small. Hence Ψε,y ∈ Hε. Now we proof Ψε,y ≠ . In fact, as ε → . Then Ψε,y ≠ for small ε > . Therefore, there exists unique τε > such that max τ≥ Iε(τΨε,y) = Iε(τε Ψε,y) and τε Ψε,y ∈ Nε .
By construction, Φε(y) has a compact support for any y ∈ M and Φε is a continuous map. Taking into account that w is a ground state solution to (4.1) and using (f ), we deduce that T = . It follows from (5.4), we have lim n→+∞ Jε n (Φε n (yn)) = J (w) = c V , which is a contradiction with (5.1). This completes the proof.
For ∀x ∈ R xed, since εn x + yn → y ∈ M δ , we have that the sequence {yn} satis es (5.11). This completes the proof.
Next we prove our multiplicity result by presenting a relation between the topology of M the number of solutions of the modi ed problem (3.1), we will apply the Ljusternik-Schnirelmann abstract result in [44,46]. By Lemma 2.8, we know u ∈ L p (R ) for all p ∈ [ , ( −µ)( − s) ). Using the growth assumption (f ) and the higher integrability of u, for some C > we have R G(u(y)) |x − y| µ ∞ ≤ C |u| * µ,s + |u| q + |u| q
We are now ready to prove the main result of the paper. Proof of Theorem . . We x a small δ > such that M δ ⊂ Ω. We rst claim that there exists someε δ > such that for any ε ∈ ( ,ε δ ) and any solution uε ∈Ñε of the problem (3.1), there holds uε L ∞ (R \Ωε) < a. (6.10) In order to prove the claim we argue by contradiction. So, suppose that for some sequence εn → + we can obtain un ∈Ñε n such that I εn (un) = and un L ∞ (R \Ωε) ≥ a. (6.11) As in the proof of Lemma 5.3, we have that Jε n (un) → c V and we can obtain a sequence {ỹn} ∈ R such that εnỹn → y ∈ M.
If we take r > such that Br(y ) ⊂ B r (y ) ⊂ Ω we have that B r εn ( y εn ) = εn Br(y ) ⊂ Ωε n .
Moreover, for any z ∈ B r εn (ỹn), there holds |z − y εn | ≤ ||z −ỹn| + |ỹn − y εn | < εn (r + on( )) < r εn , for n large. For these values of n we have that B r εn (ỹn) ⊂ Ωε n , that is, R \Ωε n ⊂ R \B r εn (ỹn). On the other hand, it follows from Lemma 6.1 that there is R > such that un(x) < a for |x| ≥ R and ∀ n ∈ N, from where it follows that vn(x −ỹn) < a for x ∈ B c R (ỹn) and n ∈ N.
Thus, there exists n ∈ N such that for any n ≥ n and r εn > R, there holds R \Ωε n ⊂ R \B r εn (ỹn) ⊂ R \B R (ỹn).
Then, there holds un(x) < a ∀ x ∈ R \Ωε n , which contradicts to (6.11) and the claim holds true. Letε δ given by Theorem 5.2 and let ε δ := min{ε δ ,ε δ }. We will prove the theorem for this choice of ε δ . Let ε ∈ ( , ε δ ) be xed. By using Theorem 5.2 we get cat M δ (M) nontrivial solutions of problem (3.1). If u ∈ Hε is one of these solutions, we have that u ∈Ñ ε, and we can use (6.10) and the de nition of g to conclude that H(·, u) = G(u). Hence, u is also a solution of the problem (2.1). An easy calculation shows that ω(x) = u( x ε ) is a solution of the original problem (1.8). Then, (1.8) has at least cat M δ (M) positive solutions. Now we consider εn → + and take a sequence un ∈ Hε n of positive solutions of the problem (3.1) as above. In order to study the behavior of the maximum points of un, we rst notice that, by the de nition of H and (h ), (h ), there exists < γ < a such that H(εn x, u)u ≤ V K u , for all x ∈ R , u ≤ γ. (6.12) Using a similar discussion above, we obtain R > and {ỹn} ⊂ R such that un L ∞ (B c R (ỹn)) < γ. (6.13) Up to a subsequence, we may assume that un L ∞ (B R (ỹn)) ≥ γ. (6.14) Indeed, if this is not the case, we have un L ∞ < γ, and therefore it follows from J εn (un) = and (6.12) that un εn ≤ V K R u n dx. (6.15) The above expression implies that un εn → as n → ∞, which leads to a contradiction. Thus, (6.14) holds. By using (6.13) and (6.14) we conclude that the maximum points pn ∈ R of un belongs to B R (ỹn). Hence, pn =ỹn + qn for some qn ∈ B R ( ). Recalling that the associated solution of (1.8) is of the form ωn(x) = un( x ε ), we conclude that the maximum point ηε of vn is ηε := εnỹn + εn qn. Since {qn} ⊂ B R ( ) is bounded and εnỹn → y ∈ M, we obtain lim n→∞ V(ηε) = V(y ) = V .
Thus, the proof of Theorem 1.1 is completed.