Multiplicity and concentration results for some nonlinear Schr\"odinger equations with the fractional $p$-Laplacian

We consider a class of parametric Schr\"odinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.


Introduction
In the first part of this paper we focus our attention on the existence, multiplicity and concentration of positive solutions for the following fractional p-Laplacian type problem where ε > 0 is a parameter, s ∈ (0, 1), 1 < p < ∞, N > sp, W s,p (R N ) is defined as the set of the functions u : R N → R belonging to L p (R N ) such that |u(x) − u(y)| p |x − y| N +sp dxdy < ∞.
Here (−∆) s p is the fractional p-Laplacian operator which may be defined, up to a normalization constant, by setting |u(x) − u(y)| p−2 (u(x) − u(y)) |x − y| N +sp dy (x ∈ R N ) for any u ∈ C ∞ c (R N ); see [18,34] for more details and applications. Now we introduce the assumptions on the potential V and the nonlinearity f . We require that V : R N → R is a continuous function satisfying the following condition introduced by Rabinowitz [38]: and we consider both cases V ∞ < ∞ and V ∞ = ∞.
Concerning the nonlinearity f : R → R we suppose that (f 1 ) f ∈ C 0 (R, R) and f (t) = 0 for all t < 0; (f 2 ) lim |t|→0 |f (t)| |t| p−1 = 0; (f 3 ) there exists q ∈ (p, p * s ), with p * s = N p N −sp , such that lim |t|→∞ |f (t)| |t| q−1 = 0; (f 4 ) there exists ϑ > p such that 0 < ϑF (t) = ϑ t 0 f (τ ) dτ ≤ tf (t) for all t > 0; (f 5 ) the map t → f (t) t p−1 is increasing in (0, +∞). Since we deal with the multiplicity of solutions to (1.1), we recall that if Y is a given closed set of a topological space X, we denote by cat Y (Y ) the Ljusternik-Schnirelmann category of Y in X, that is the least number of closed and contractible sets in X which cover Y ; see [32] for more details.
Let us denote by Our first main result can be stated as follows: Theorem 1.1. Let N > sp, and suppose that V satisfies (V ) and f verifies (f 1 )-(f 5 ). Then, for any δ > 0 there exists ε δ > 0 such that problem (1.1) has at least cat M δ (M ) positive solutions, for any 0 < ε < ε δ . Moreover, if u ε denotes one of these solutions and x ε ∈ R N its global maximum, then lim Due to the variational nature of problem (1.1), we look for critical points of the functional defined on a suitable subspace of W s,p (R N ). Since f is only continuous, we can not apply the Nehari manifold arguments developed in [3] in which the authors considered the corresponding local problem to (1.1) under the assumptions f ∈ C 1 and (f 6 ) there exist C > 0 and σ ∈ (p, p * s ) such that f ′ (t)t 2 − (p − 1)f (t)t ≥ Ct σ for all t ≥ 0.
To overcome this difficulty, we use some variants of critical point theorems due to Szulkin and Weth [41]. As usual, the presence of the fractional p-Laplacian operator makes our analysis more delicate and intriguing. In order to obtain multiple critical points, we employ a technique introduced by Benci and Cerami [12], which consists in making precise comparisons between the category of some sublevel sets of I ε and the category of the set M . Then, after proving that the levels of compactness are strongly related to the behavior of the potential V (x) at infinity (see Proposition 3.1), we can apply Ljusternik-Schnirelmann theory to deduce a multiplicity result for problem (1.1). Finally, we study the concentration of positive solutions u ε of (1.1). More precisely, we first adapt the Moser iteration technique [36] in the fractional setting (see Lemma 3.15 in Section 3) in order to obtain L ∞ -estimates (independent of ε) of u ε 's. Then, taking into account the Hölder estimates recently established in [27] for the fractional p-Laplacian, we can also deduce C 0,α -estimates of u ε uniformly in ε. These informations allow us to infer that u ε (x) decay at zero as |x| → ∞ uniformly in ε. Moreover, we prove that our solutions have a polynomial decay; see Remark 3.3. In the second part of this work we consider the following fractional problem involving the critical Sobolev exponent: with s ∈ (0, 1), 1 < p < ∞ and N ≥ sp 2 . In order to deal with the critical growth of the nonlinearity we assume that f verifies (f 1 ), (f 2 ), (f 3 ), (f 5 ), hypothesis (f 4 ) with ϑ ∈ (p, q), and the following technical condition: (f ′ 6 ) there exist q 1 ∈ (p, p * s ) and λ > 0 such that f (t) ≥ λt q 1 −1 for any t > 0. Then we are able to obtain our second result: Theorem 1.2. Let N ≥ sp 2 , and suppose that V satisfies (V ) and f satisfies (f 1 )-(f 5 ) and (f ′ 6 ). Then, for any δ > 0 there exists ε δ > 0 such that problem (1.2) has at least cat M δ (M ) positive solutions, for any 0 < ε < ε δ . Moreover, if u ε denotes one of these solutions and x ε ∈ R N its global maximum, then lim We note that Theorem 1.2 improves and extends, in the fractional setting, Theorem 1.1 in [23] in which the author assumed f ∈ C 1 . The approach developed in this case follows the arguments used to analyze the subcritical case. Anyway, this new problem presents an extra difficulty, due to the fact that the level of non-compactness is affected by the critical growth of the nonlinearity. To overcome this hitch, we adapt some calculations performed in [35] and using the optimal asymptotic behavior of p-minimizers established in [13] we are able to prove that the functional associated to (1.2) satisfies the Palais-Smale condition at every level , and D s,p (R N ) = u ∈ L p * s (R N ) : [u] W s,p (R N ) < ∞ .
Let us point out that the restriction N ≥ sp 2 is crucial to apply Lemma 2.7 in [35] to estimate the L p -norm of p-minimizers; see Lemma 4.10 and Lemma 4.11 in Section 4. When p = 2, equations (1.1) and (1.2) become fractional Schrödinger equations of the type which has been widely investigated in the last decade: see [4,6,[8][9][10]20,22,24,28,39,40] and references therein. The study of (1.3) is strongly motivated by the seeking of standing waves solutions for the time-dependent fractional Schrödinger equation namely solutions of the form ψ(x, t) = u(x)e − ıEt ε , where E is a constant. Equation (1.4) is a fundamental equation of the fractional Quantum Mechanics in the study of particles on stochastic fields modelled by Lévy processes; see [29,30] for more physical background.
In recent years there has been a surge of interest around nonlocal and fractional problems involving the fractional p-Laplacian operator, and several existence and regularity results have been established by many authors. For instance, Franzina and Palatucci [26] discussed some basic properties of the eigenfunctions of a class of nonlocal operators whose model is (−∆) s p (see also [31]). Mosconi et al. [35] used an abstract linking theorem based on the cohomological index to obtain nontrivial solutions to the Brezis-Nirenberg problem for the fractional p-Laplacian operator. Di Castro et al. [17] established interior Hölder regularity results for fractional p-minimizers (see also [27]). In [5] the first author obtained the existence of infinitely many solutions for a superlinear fractional p-Laplacian equation with sign-changing potential. Fiscella and Pucci [25] investigated the existence and asymptotic behavior of nontrivial solutions for some classes of stationary Kirchhoff problems driven by fractional integro-differential operators and involving a Hardy potential and different critical nonlinearities. Belchior et al. [11] studied the existence of ground state solutions for a fractional Choquard equation with the fractional p-Laplacian and involving subcritical nonlinearities. However, for what concerns the existence and multiplicity results for problems (1.1) and (1.2) with p = 2, the literature seems to be rather incomplete.
The goal of this paper is to consider the question related to the existence and multiplicity of positive solutions for fractional Schrödinger equations with the fractional p-Laplacian when ε → 0. More precisely, we aim to extend the results obtained in [24] and [40] in which the authors dealt with equation (1.3) and involving nonlinearities with subcritical and critical growth respectively. Unfortunately, the operator (−∆) s p is not linear when p = 2, so more technical difficulties arise in the study of our problems. For instance, we can not make use of the s-harmonic extension by Caffarelli and Silvestre [15] commonly exploited in the recent literature to apply well-known variational techniques in the study of local degenerate elliptic problems. Moreover, the arguments used in the study of (1.3) (see for example [2,4,6,7,24,40]) seem not to be trivially adaptable in our context. Indeed, some appropriate technical lemmas (see Lemma 2.2, Lemma 2.6, and Lemma 4.8) will be needed to overcome the non-Hilbertian structure of the fractional Sobolev spaces W s,p (R N ) when p = 2.
We would like to emphasize that, to our knowledge, this is the first time that the Ljusternik-Schnirelmann theory is applied to get multiple solutions for fractional Schrödinger equations in R N driven by the fractional p-Laplacian operator with p = 2, and involving nonlinearities with subcritical and critical growth. The paper is organized as follows: in Section 2 we collect some facts about the involved fractional Sobolev spaces and we provide some useful technical lemmas. In Section 3 we study the existence, multiplicity and concentration of solutions to (1.1) proving some convenient properties for the autonomous problem associated to (1.1). In Section 4 we consider critical problem (1.2) and the corresponding autonomous critical one.

Preliminaries
In this preliminary section we recall some facts about the fractional Sobolev spaces and we prove some technical lemmas which we will use later.
Let 1 ≤ r ≤ ∞ and A ⊂ R N . We denote by |u| L r (A) the L r (A)-norm of a function u : R N → R belonging to L r (A). We define D s,p (R N ) as the closure of C ∞ c (R N ) with respect to . We begin recalling the following embeddings of the fractional Sobolev spaces into Lebesgue spaces.
Theorem 2.1. [18] Let s ∈ (0, 1) and N > sp. Then there exists a sharp constant S * > 0 such that for any u ∈ D s,p (R N ) . Moreover W s,p (R N ) is continuously embedded in L q (R N ) for any q ∈ [p, p * s ] and compactly in L q loc (R N ) for any q ∈ [1, p * s ). Proceeding as in [22,39] we can prove the following compactness-Lions type result.
Then, Hölder and Sobolev inequality yield, for every n ∈ N, that τ . Now, covering R N by balls of radius R, in such a way that each point of R N is contained in at most N + 1 balls, we find Exploiting (2.1) and the boundedness of {u n } in W s,p (R N ), we obtain that u n → 0 in L τ (R N ). By using an interpolation argument, we get the thesis.
The lemma below provides a way to manipulate smooth truncations for the fractional p-Laplacian. Let us note that this result can be seen as a generalization of the second statement of Lemma 5 in [37] to the case of the space W s,p (R N ) with p = 2.
Proof. Since φ r u → u a.e. in R N as r → ∞, 0 ≤ φ ≤ 1 and u ∈ L p (R N ), it follows by the Dominated Convergence Theorem that lim r→∞ |uφ r − u| L p (R N ) = 0. In what follows, we show the first relation of limit. Let us note that Taking into account that |φ r (x) − 1| ≤ 2, |φ r (x) − 1| → 0 a.e. in R N and u ∈ W s,p (R N ), the Dominated Convergence Theorem yields B r → 0 as r → ∞.

Now, we aim to show that
A r → 0 as r → ∞.
Firstly, we observe that since R 2N can be written as We are going to estimate each integral in (2.2).
By using 0 ≤ φ ≤ 1, |∇φ| ≤ 2 and by applying the Mean Value Theorem, we can see that Regarding the integral on X 3 r we obtain At this point, by using the Mean Value Theorem and noting that if (x, y) ∈ (R N \ B 2r (0)) × B 2r (0) and |x − y| ≤ r then |x| ≤ 3r, we get (2.6) Let us observe that for any K > 4 it holds Let us note that if (x, y) ∈ (R N \ B Kr (0)) × B r (0), then |x − y| ≥ |x| − |y| ≥ |x| 2 + K 2 r − 2r > |x| 2 , and by using Hölder inequality we can see that Therefore, taking into account (2.7) and (2.8), we get Putting together (2.2)-(2.6) and (2.9), we obtain Now, fixed ε > 0, we introduce the following fractional Sobolev space In view of assumption (V ) and Theorem 2.1, it is easy to check that the following result holds.
Lemma 2.3. The space W ε is continuously embedded in W s,p (R N ). Therefore, W ε is continuously embedded in L r (R N ) for any r ∈ [p, p * s ] and compactly embedded in L r loc (R N ) for any r ∈ [1, p * s ). Moreover, when V is coercive, we get the following compactness lemma.
Then W ε is compactly embedded in L r (R N ) for any r ∈ [p, p * s ). Proof. We argue as in [42]. Let r = p. By using Lemma 2.3 we know that W ε ⊂ L p (R N ). Let {u n } be a sequence such that u n ⇀ 0 in W ε . Then, u n ⇀ 0 in W s,p (R N ). Let us define M := sup n∈N u n ε < ∞. (2.10) Since V is coercive, for any η > 0 there exists R = R η > 0 such that Hence, for any n ≥ n 0 , by using (2.10)-(2.12), we have Therefore, u n → 0 in L p (R N ).
For r > p, using the conclusion of r = p, interpolation inequality and Theorem 2.1, we can see that , which yields the conclusion as required.
The next two results are technical lemmas which will be very useful in this work; their proofs are obtained following the arguments developed by Brezis and Lieb [14].
Proof. From the Brezis-Lieb Lemma [14] we know that if r ∈ (1, ∞) and {g n } ⊂ L r (R k ) is a bounded sequence such that g n → g a.e. in R k , then we have and taking in (2.13), we can see that Lemma 2.6. Let w ∈ D s,p (R N ) and {z n } ⊂ D s,p (R N ) be a sequence such that z n → 0 a.e. and [z n ] W s,p (R N ) ≤ C for any n ∈ N. Then we have and p ′ = p p−1 is the conjugate exponent of p.
Proof. Firstly we consider the case p ≥ 2. We resemble some ideas in Lemma 3 in [1]. By using the Mean Value Theorem, Young inequality and p ≥ 2, we can see that for fixed ε > 0 there exists C ε > 0 such that (2.14) Taking N+sp p in (2.14), we obtain |(z n (x) + w(x)) − (z n (y) + w(y))| p−2 ((z n (x) + w(x)) − (z n (y) + w(y))) Let us define the function H ε,n : R 2N → R + by setting We can see that H ε,n → 0 a.e. in R 2N as n → ∞, and By using the Dominated Convergence Theorem, we have From the definition of H ε,n , we deduce that so we obtain and by the arbitrariness of ε we get the thesis. Now we deal with the case 1 < p < 2. By using Lemma 3.1 in [33], we know that we can conclude the proof in view of the Dominated Convergence Theorem.
Lemma 2.7. Let {u n } be a sequence such that u n ⇀ u in W ε , and v n := u n − u. Then we have and sup Proof. We begin proving (2.15). Let us note that In view of (f 2 ) and (f 3 ), for any δ > 0 there exists C δ > 0 such that By using (2.17) with δ = 1 and (a + b) r ≤ C r (a r + b r ) for any a, b ≥ 0 and r ≥ 1, we can see that By applying the Young inequality ab ≤ ηa r + C η b r ′ with 1 r + 1 r ′ = 1 and η > 0 to the first and third term on the right hand side of (2.18), we can deduce that Then G η,n → 0 a.e. in R N as n → ∞, and 0 ≤ G η,n ≤ C ′ η (|u| p + |u| p * s ) ∈ L 1 (R N ). As a consequence of the Dominated Convergence Theorem, we get On the other hand, from the definition of G η,n , it follows that The arbitrariness of η ends the proof of (2.15). Now we prove (2.16). Arguing as in Lemma 5.7 in [19] we can find a subsequence {u n j } such that for all η > 0 there exists r η > 0 satisfying In view of Lemma 2.2 we can see that By using (2.20) we obtain On the other hand, from (2.19) and the definition ofũ j , it follows that Putting together (2.24) and (2.26), we can deduce that (2.23) holds true. Finally we verify that Noting that |Cã j ∩ W δ j | ≤ |W δ j | → 0 as j → ∞, we can argue as before to infer that there exists j 0 ∈ N such that In view of (2.20), we can find j 1 ∈ N such that j 1 ≥ j 0 and Taking into account (2.29), (2.31) and the boundedness of {v n j } we can see that for all j ≥ j 1 Now, we recall the following inequalities for all a, b ∈ R Assume p ∈ (1, 2]. By using |g(t)| ≤ C(1 + |t| q−p ), Hölder inequality and (2.31) we have When p > 2, we can deduce that From the above estimates, and using (2.30) and uniformly in w ε ≤ 1, which together with (2.28) yields (2.27).

Subcritical case
3.1. Functional setting in the subcritical case. After a change of variable, we are led to consider the following problem Weak solutions of (P ε ) can be obtained as critical points of the functional On the other hand, hypothesis (f 5 ) implies that By using Lemma 2.3, it is easy to see that I ε ∈ C 1 (W ε , R) and its differential I ′ ε is given by for any u, ϕ ∈ W ε . Now, let us introduce the Nehari manifold associated to I ε , that is Let us note that I ε possesses a mountain pass geometry.
Lemma 3.1. The functional I ε satisfies the following conditions: Choosing ξ ∈ (0, V 0 ), there exist α, ρ > 0 such that Since f is only continuous, the next results are very important because they allow us to overcome the non-differentiability of N ε . We begin proving some properties for the functional I ε .
(iii) Without loss of generality, we may assume that u ε = 1 for each u ∈ K. For u n ∈ K, after passing to a subsequence, we obtain that u n → u ∈ S ε . Then, by using (f 4 ) and Fatou's Lemma, we can see that Under the assumptions of Lemma 3.2, for ε > 0 we have: By using (f 5 ), it is easy to verify the uniqueness of a such t u .
(ii) By using (3.1) and Lemma 2.3 we can see that for any u ∈ N ε From the proof of (ii), we can see that Since W is compact, we can find u ∈ W such that u n → u in W ε and u n → u a.e. in R N . By using Lemma 3.2-(iii), we can deduce that I ε (t un u n ) → −∞ as n → ∞, which gives a contradiction because (f 4 ) implies that (iv) Let us define the mapsm ε : W ε \ {0} → N ε and m ε : S ε → N ε by settinĝ In view of (i)-(iii) and Proposition 3.1 in [41] we can deduce that m ε is a homeomorphism between S ε and N ε and the inverse of m ε is given by m −1 ε (u) = u u ε . Therefore N ε is a regular manifold diffeomorphic to S ε .
(v) For ε > 0, t > 0 and u ∈ W ε \ {0}, we can see that (3.2) yields so we can find ρ > 0 such that I ε (tu) ≥ ρ > 0 for t > 0 small enough. On the other hand, by using (i)-(iii), we know (see [41]) that Now we introduce the following functionalsΨ ε : wherem ε (u) = t u u is given in (3.4). As in [41] we have the following result: Under the assumptions of Lemma 3.2, we have that for ε > 0: Moreover the corresponding critical values coincide and We conclude this section proving the following useful result.
Proof. By using assumption (f 4 ) we have and being ϑ > p we get the thesis.
3.2. Autonomous subcritical problem. Let us consider the autonomous problem associated to The corresponding functional is given by Clearly, J µ ∈ C 1 (X µ , R) and its differential J ′ µ is given by for any u, ϕ ∈ X µ . Let us define the Nehari manifold associated to J µ , that is Arguing as in the previous section and using (3.6), it is easy to prove the following lemma.
Lemma 3.6. Under the assumptions of Lemma 3.2, for µ > 0 we have: Now we define the following functionalsΨ µ : X µ \ {0} → R and Ψ µ : S µ → R by settinĝ Then we have the following result: Lemma 3.7. Under the assumptions of Lemma 3.2, we have that for µ > 0: Moreover the corresponding critical values coincide and Lemma 3.8. Let {u n } ⊂ N µ be a minimizing sequence for J µ . Then, {u n } is bounded and there exist a sequence {y n } ⊂ R N and constants R, β > 0 such that Proof. Arguing as in the proof of Lemma 3.5, we can see that {u n } is bounded in X µ . Now, in order to prove the latter conclusion of this lemma, we argue by contradiction. Assume that for any R > 0 it holds lim Since {u n } is bounded in X µ , from Lemma 2.1 it follows that Fix ξ ∈ (0, µ). By using J ′ µ (u n ), u n = 0, (3.1) and the fact that {u n } is bounded in X µ , we have . In view of (3.8), we can conclude that u n → 0 in X µ . Now, we prove the following useful compactness result for the autonomous problem. Proof. From (v) of Lemma 3.6, we know that c µ > 0 for each µ > 0. Moreover, if u ∈ N µ verifies J µ (u) = c µ , then m −1 µ (u) is a minimizer of Ψ µ and it is a critical point of Ψ µ . In view of Lemma 3.7, we can see that u is a critical point of J µ . Now we show that there exists a minimizer of J µ | Nµ . By applying Ekeland's variational principle [21] there exists a sequence {ν n } ⊂ S µ such that Ψ µ (ν n ) → c µ and Ψ ′ µ (ν n ) → 0 as n → ∞. Let u n = m µ (ν n ) ∈ N µ . Then, thanks to Lemma 3.7, J µ (u n ) → c µ and J ′ µ (u n ) → 0 as n → ∞. Therefore, arguing as in Lemma 3.5, {u n } is bounded in X µ and u n ⇀ u in W s,p (R N ). It is easy to check that J ′ µ (u) = 0. Now, we prove that J µ (u) = c µ . Assume u ≡ 0 and we aim to show that In fact, once proved the previous limit, we can use Lemma 2.5 to deduce that u n → u in X µ , and recalling that J µ (u n ) → c µ , we obtain the thesis. Now, we prove (3.9). Let us observe that Fatou's Lemma yields Let us note that Recalling that lim sup n→∞ (a n + b n ) ≥ lim sup n→∞ a n + lim inf n→∞ b n and ϑ > p, we can see that Fatou's Lemma, (3.11), (3.12) and J ′ µ (u) = 0 produce which gives a contradiction. Finally, we consider the case u = 0. Arguing as in the proof of Lemma 3.8, we can find a sequence {y n } ⊂ R N and constants R, β > 0 such that Set v n := u n (· + y n ). Then, by using the invariance by translations of R N , it is clear that {v n } is a (P S) cµ for J µ , {v n } ⊂ N µ and v n ⇀ v = 0 in W s,p (R N ). Thus, we can proceed as above to deduce that {v n } converges strongly in W s,p (R N ).
Remark 3.2. Let us observe that the ground state obtained in Lemma 3.9 is positive. Indeed, which implies that u − = 0, that is u ≥ 0. Arguing as in Lemma 3.15 below, we can see that u ∈ L ∞ (R N ) ∩ C 0 (R N ), and by applying the maximum principle [16] we deduce that u > 0 in R N .

3.3.
Existence result for (1.1). In this section we focus on the existence of a solution to (1.1) provided that ε is sufficiently small. Let us begin proving the following useful lemma.
Lemma 3.10. Let {u n } ⊂ N ε be a sequence such that I ε (u n ) → c and u n ⇀ 0 in W ε . Then, one of the following alternatives occurs (a) u n → 0 in W ε ; (b) there are a sequence {y n } ⊂ R N and constants R, β > 0 such that Proof. Assume that (b) does not hold true. Then, for any R > 0 it holds Since {u n } is bounded in W ε , from Lemma 2.1 it follows that u n → 0 in L t (R N ) for any t ∈ (p, p * s ). Now, we can argue as in the proof of Lemma 3.8 to deduce that u n ε → 0 as n → ∞.
In order to get a compactness result for I ε , we need to prove the following auxiliary lemma.
Lemma 3.11. Assume that V ∞ < ∞ and let {v n } ⊂ N ε be a sequence such that Proof. Let {t n } ⊂ (0, +∞) be such that {t n v n } ⊂ N V∞ . Claim 1: We aim to prove that lim sup n→∞ t n ≤ 1.
Assume by contradiction that there exist δ > 0 and a subsequence, still denoted by {t n }, such that t n ≥ 1 + δ for any n ∈ N. (3.14) Since In view of t n v n ∈ N V∞ , we also have Putting together (3.15) and (3.16) we obtain By hypothesis (V ) we can see that, given ζ > 0 there exists R = R(ζ) > 0 such that Now, taking into account v n → 0 in L p (B R (0)) and the boundedness of {v n } in W ε , we can infer that Thus, Since v n → 0 in W ε , we can apply Lemma 3.10 to deduce the existence of a sequence {y n } ⊂ R N , and two positive numbersR, β such that Let us considerv n = v n (x + y n ). From condition (V ) and the boundedness of {v n } in W ε , we can see that v n Taking into account that W s,p (R N ) is a reflexive Banach space, we may assume thatv n ⇀v in W s,p (R N ). By (3.19) there exists Ω ⊂ R N with positive measure and such thatv > 0 in Ω. By using (3.14), assumption (f 5 ) and (3.18) we can infer Letting the limit as n → ∞ and by applying Fatou's Lemma we obtain ζC for any ζ > 0, and this is a contradiction. Now, we distinguish the following cases: Case 1: Assume that lim sup n→∞ t n = 1. Thus there exists {t n } such that t n → 1. Recalling that Now, by using condition (V ), v n → 0 in L p (B R (0)), t n → 1, (3.17), and On the other hand, since {v n } is bounded in W ε , we can see that Hence, putting together (3.21), (3.22) and (3.23), we obtain At this point, we show that Indeed, by using the Mean Value Theorem and (3.1) we have and taking into account the boundedness of {v n } in W ε we get the thesis. Now, putting together (3.20), (3.24) and (3.25) we can infer that and passing to the limit as ζ → 0 we get d ≥ c V∞ . Case 2: Assume that lim sup n→∞ t n = t 0 < 1. Then there is a subsequence, still denoted by {t n }, such that t n → t 0 (< 1) and t n < 1 for any n ∈ N. Let us observe that Exploiting the facts that t n v n ∈ N V∞ , (3.3) and (3.26), we obtain Taking the limit as n → ∞ we get d ≥ c V∞ .
At this point we are able to prove the following compactness result.
Proof. It is easy to see that {u n } is bounded in W ε . Then, up to a subsequence, we may assume that u n ⇀ u in W ε , u n → u in L q loc (R N ) for any q ∈ [p, p * s ), u n → u a.e. in R N .
Now, we prove that I ′ ε (v n ) = o n (1). By using Lemma 2.6 with z n = v n and w = u we get Arguing as in the proof of Lemma 3.3 in [33], we can see that Hence, by using Hölder inequality, for any ϕ ∈ W ε such that ϕ ε ≤ 1, it holds and in view of (2.16) of Lemma 2.7, (3.29), (3.30), I ′ ε (u n ) = 0 and I ′ ε (u) = 0 we obtain the thesis. Now, we note that by using (f 4 ) we can see that Since I ′ ε (v n ), v n = o n (1) and applying (3.32) we can infer that v n p ε = o n (1), which yields u n → u in W ε .
We end this section giving the proof of the existence of a positive solution to (P ε ) whenever ε > 0 is small enough.
Proof. From (v) of Lemma 3.3, we know that c ε ≥ ρ > 0 for each ε > 0. Moreover, if u ε ∈ N ε verifies I ε (u) = c ε , then m −1 ε (u) is a minimizer of Ψ ε and it is a critical point of Ψ ε . In view of Lemma 3.4 we can see that u is a critical point of I ε . Now we show that there exists a minimizer of I ε | Nε . By applying Ekeland's variational principle [21] there exists a sequence {v n } ⊂ S ε such that Ψ ε (v n ) → c ε and Ψ ′ ε (v n ) → 0 as n → ∞. Let u n = m ε (v n ) ∈ N ε . Then, from Lemma 3.4 we deduce that I ε (u n ) → c ε , I ′ ε (u n ), u n = 0 and I ′ ε (u n ) → 0 as n → ∞. Therefore, {u n } is a Palais-Smale sequence for I ε at level c ε . It is standard to check that {u n } is bounded in W ε and we denote by u its weak limit. It is easy to verify that I ′ ε (u) = 0. Let us consider V ∞ = ∞. By using Lemma 2.3 we have I ε (u) = c ε and I ′ ε (u) = 0. Now, we deal with the case V ∞ < ∞. In view of Proposition 3.1 it is enough to show that c ε < c V∞ for small ε. Without loss of generality, we may suppose that Let µ ∈ R such that µ ∈ (V 0 , V ∞ ). Clearly c V 0 < c µ < c V∞ . By Lemma 3.9, it follows that there exists a positive ground state w ∈ W s,p (R N ) to the autonomous problem (P µ ).
Let η r ∈ C ∞ c (R N ) be a cut-off function such that η r = 1 in B r (0) and η r = 0 in B c 2r (0). Let us define w r (x) := η r (x)w(x), and take t r > 0 such that J µ (t r w r ) = max t≥0 J µ (tw r ). Now we prove that there exists r sufficiently large for which J µ (t r w r ) < c V∞ . Assume by contradiction J µ (t r w r ) ≥ c V∞ for any r > 0. Taking into account w r → w in W s,p (R N ) as r → ∞ in view of Lemma 2.2, t r w r and w belong to N µ and by using assumption (f 5 ), we have that t r → 1. Therefore, which leads to a contradiction being c V∞ > c µ . Hence, there exists r > 0 such that J µ (τ (t r w r )) and J µ (t r w r ) < c V∞ . (3.33) Now, condition (V ) implies that there exists ε 0 > 0 such that Therefore, by using (3.33) and (3.34), we deduce that for all ε ∈ (0, ε 0 ) which implies that c ε < c V∞ for any ε > 0 sufficiently small.

3.4.
Multiple solutions for (1.1). This section is devoted to the study of the multiplicity of solutions to (1.1). We begin proving the following result which will be needed to implement the barycenter machinery. Proof. Since I ′ εn (u n ), u n = 0 and I εn (u n ) → c V 0 , we know that {u n } is bounded in W ε . From c V 0 > 0, we can infer that u n εn → 0. Therefore, as in the proof of Lemma 3.10, we can find a sequence {ỹ n } ⊂ R N and constants R, β > 0 such that Let us define v n (x) := u n (x +ỹ n ). In view of the boundedness of {u n } and (3.35) we may assume that v n ⇀ v in W s,p (R N ) for some v = 0. Let {t n } ⊂ (0, +∞) be such that w n = t n v n ∈ N V 0 , and we set y n := ε nỹn . Thus, by using the change of variables z → x +ỹ n , V (x) ≥ V 0 and the invariance by translation, we can see that Then we can infer J V 0 (w n ) → c V 0 . This fact and {w n } ⊂ N V 0 imply that there exists K > 0 such that w n V 0 ≤ K for all n ∈ N. Moreover, we can prove that the sequence {t n } is bounded. In fact, v n → 0 in W s,p (R N ), so there exists α > 0 such that v n V 0 ≥ α. Consequently, for all n ∈ N, we have |t n |α ≤ t n v n V 0 = w n V 0 ≤ K, which yields |t n | ≤ K α for all n ∈ N. Therefore, up to a subsequence, we may suppose that t n → t 0 ≥ 0. Let us show that t 0 > 0. Otherwise, if t 0 = 0, from the boundedness of {v n }, we get w n = t n v n → 0 in W s,p (R N ), that is J V 0 (w n ) → 0 in contrast with the fact c V 0 > 0. Thus t 0 > 0, and up to a subsequence, we may assume that w n ⇀ w : From Lemma 3.9, we deduce that w n → w in W s,p (R N ), that is v n → v in W s,p (R N ). Now, we show that {y n } has a subsequence such that y n → y ∈ M . Assume by contradiction that {y n } is not bounded, that is there exists a subsequence, still denoted by {y n }, such that |y n | → +∞. Firstly, we deal with the case V ∞ = ∞. By using {u n } ⊂ N εn and a change of variable, we can see that By applying Fatou's Lemma and v n → v in W s,p (R N ), we deduce that which gives a contradiction. Let us consider the case V ∞ < ∞. Taking into account w n → w strongly in W s,p (R N ), condition (V ) and using the change of variable z = x +ỹ n , we have which is an absurd. Thus {y n } is bounded and, up to a subsequence, we may assume that y n → y. If y / ∈ M , then V 0 < V (y) and we can argue as in (3.36) to get a contradiction. Therefore, we can conclude that y ∈ M .
At this point, we introduce a subset N ε of N ε by taking a function h : R + → R + such that h(ε) → 0 as ε → 0, and setting Fixed y ∈ M , from Lemma 3.12 we deduce that h(ε) = |I ε (Φ ε (y)) − c V 0 | → 0 as ε → 0. Hence Φ ε (y) ∈ N ε , and N ε = ∅ for any ε > 0. Moreover, we have the following lemma. Thus, recalling that {u n } ⊂ N εn ⊂ N εn , we deduce that which implies that I εn (u n ) → c V 0 . By using Proposition 3.2, there exists {ỹ n } ⊂ R N such that y n = ε nỹn ∈ M δ for n sufficiently large. Thus Since u n (·+ỹ n ) converges strongly in W s,p (R N ) and ε n z + y n → y ∈ M , we can infer that β εn (u n ) = y n + o n (1), that is (3.45) holds. Now we show that (P ε ) admits at least cat M δ (M ) positive solutions. In order to achieve our aim, we recall the following result for critical points involving Ljusternik-Schnirelmann category. For more details one can see [32].
Theorem 3.2. Let U be a C 1,1 complete Riemannian manifold (modelled on a Hilbert space). Assume that h ∈ C 1 (U, R) bounded from below and satisfies −∞ < inf U h < d < k < ∞. Moreover, suppose that h satisfies Palais-Smale condition on the sublevel {u ∈ U : h(u) ≤ k} and that d is not a critical level for h. Then Since N ε is not a C 1 submanifold of W ε , we can not directly apply Theorem 3.2. Fortunately, from Lemma 3.3, we know that the mapping m ε is a homeomorphism between N ε and S ε , and S ε is a C 1 submanifold of W ε . So we can apply Theorem 3.2 to Ψ ε (u) = I ε (m ε (u))| Sε = I ε (m ε (u)), where Ψ ε is given in Lemma 3.4.
3.5. Concentration of solutions to (1.1). Let us prove the following result which will play a fundamental role to study the behavior of maximum points of solutions to (1.1).
Then v n ∈ L ∞ (R N ) and there exists C > 0 such that |v n | L ∞ (R N ) ≤ C for all n ∈ N. Moreover, lim |x|→∞ v n (x) = 0 uniformly in n ∈ N.
Proof. For any L > 0 and β > 1, let us consider the function where v L,n = min{v n , L}. Let us observe that, since γ is an increasing function, it holds Define the functions Fix a, b ∈ R such that a > b. Then, from the above definitions and applying Jensen inequality we get In similar fashion, we can prove that the above inequality is true for any a ≤ b. Thus we can infer that In particular, by (3.47) it follows that as test-function in (3.46), in view of (3.48) we have Since Γ(v n ) ≥ 1 β v n v β−1 L,n , from the Sobolev inequality we can deduce that On the other hand, from assumptions (f 2 )-(f 3 ), we know that for any ξ > 0 there exists C ξ > 0 such that Choosing ξ ∈ (0, V 0 ), and using (3.50) and (3.51), we can see that (3.49) yields where w L,n := v n v β−1 L,n . Now, we take β = p * s p and fix R > 0. Observing that 0 ≤ v L,n ≤ v n , we can deduce that Since v n → v in W s,p (R N ), we can see that for any R sufficiently large Putting together (3.52), (3.53) and (3.55) we get and taking the limit as L → ∞, we obtain v n ∈ L (p * s ) 2 p (R N ). Now, using 0 ≤ v L,n ≤ v n and by passing to the limit as L → ∞ in (3.52), we have from which we deduce that .

Let us define
. By using an iteration argument, we can find C 0 > 0 independent of m such that Taking the limit as m → ∞ we get |v n | L ∞ (R N ) ≤ K for all n ∈ N. Moreover, by using Corollary 5.5 in [27], we can deduce that v n ∈ C 0,α (R N ) for some α > 0 (independent of n) and [v n ] C 0,α (R N ) ≤ C, with C independent of n. Since v n → v in W s,p (R N ), we can infer that lim |x|→∞ v n (x) = 0 uniformly in n ∈ N. Remark 3.3. We can also provide a more precise estimate on the decay of v n at infinity. Indeed, by using (f 2 ) and lim |x|→∞ v n (x) = 0, we can see that there exists By using Theorem A.4 in [13], we know that Γ(x) = |x| − N−sp p−1 is a weak solution to for all r > 0. In view of the continuity of v n and Γ, there exists C 1 > 0 such that w n (x) = v n (x) − C 1 Γ(x) ≤ 0 for all |x| = R (with R larger if necessary). Taking φ = max{w n , 0} ∈ W s,p 0 (B c R (0)) as test function in (3.56) and using (3.57) withΓ = C 1 Γ, we can deduce that Therefore, if we prove that To achieve our purpose, we first note that for all a, b ∈ R it holds Taking b = v n (x) − v n (y) and a = Γ(x) − Γ(y) we can see that where I(x, y) ≥ 0 stands for the integral. Since we can infer that (|b| p−2 b − |a| p−2 a)(φ(x) − φ(y)) ≥ 0, that is (3.59) holds true. As a consequence, we can conclude that v n (x) ≤ C|x| Proof. Assume by contradiction that |v n | L ∞ (R N ) → 0 as n → ∞. By using (f 2 ), there exists n 0 ∈ N such that < V 0 2 for all n ≥ n 0 . Therefore, in view of (f 5 ) we can see that which is impossible. Now, we end this section studying the behavior of maximum points of solutions to (1.1). If u εn is a solution to (P εn ), then v n (x) = u εn (x +ỹ n ) is a solution to (3.46). Moreover, up to subsequence, v n → v in W s,p (R N ) and y n = ε nỹn → y ∈ M in view of Proposition 3.2. If p n denotes a global maximum point of v n , we can use Lemma 3.15 and Lemma 3.16 to see that p n ∈ B R (0) for some R > 0. As a consequence, the point of maximum of u εn is of the type z εn = p n +ỹ n and then ε n z εn = ε n p n + ε nỹn → y because {p n } is bounded. This fact and the continuity of V yield V (ε n z εn ) → V (y) = V 0 as n → ∞.

Critical case
4.1. Functional setting in the critical case. In this section we deal with critical problem (1.2). Since many calculations are adaptations to that presented in the early sections, we will emphasize only the differences between the subcritical and the critical case. By using a change of variable we consider the following problem The functional associated to (P * ε ) is given by which is well defined on W ε . Let us introduce the Nehari manifold associated to I ε , that is Arguing as in Section 3.1 we can prove that the following lemmas hold true.
Lemma 4.1. The functional I ε satisfies the following conditions: (i) there exist α, ρ > 0 such that I ε (u) ≥ α with u ε = ρ; (ii) there exists e ∈ W ε with e ε > ρ such that I ε (e) < 0.  (i) for all u ∈ S ε , there exists a unique t u > 0 such that t u u ∈ N ε . Moreover, m ε (u) = t u u is the unique maximum of I ε on W ε , where S ε = {u ∈ W ε : u ε = 1}. (ii) The set N ε is bounded away from 0. Furthermore N ε is closed in W ε .
(iii) There exists α > 0 such that t u ≥ α for each u ∈ S ε and, for each compact subset W ⊂ S ε , there exists C W > 0 such that t u ≤ C W for all u ∈ W . (iv) For each u ∈ N ε , m −1 ε (u) = u u ε ∈ N ε . In particular, N ε is a regular manifold diffeomorphic to the sphere in W ε .
Then we have the following result: Lemma 4.4. Under the assumptions of Lemma 4.2, we have that for ε > 0: (i) Ψ ε ∈ C 1 (S ε , R), and Finally, it is easy to prove that

Autonomous critical problem. Let us consider the following autonomous critical problem
with N ≥ sp 2 . The functional associated to the above problem is defined as , and the Nehari manifold associated to J µ is given by It is standard to check that J µ has a mountain pass geometry. Moreover we have the following useful results: Lemma 4.6. Under the assumptions of Lemma 4.2, for µ > 0 we have: (i) for all u ∈ S µ , there exists a unique t u > 0 such that t u u ∈ N µ . Moreover, m µ (u) = t u u is the unique maximum of J µ on W ε , where S µ = {u ∈ X µ : u µ = 1}. (ii) The set N µ is bounded away from 0. Furthermore N µ is closed in X µ . (iii) There exists α > 0 such that t u ≥ α for each u ∈ S µ and, for each compact subset W ⊂ S µ , there exists C W > 0 such that t u ≤ C W for all u ∈ W . (iv) N µ is a regular manifold diffeomorphic to the sphere in X µ .
(v) c µ = inf Nµ J µ > 0 and J µ is bounded below on N µ by some positive constant.
Then we obtain the following result: (i) Ψ µ ∈ C 1 (S µ , R), and In order to obtain the existence of a nontrivial solution to the autonomous critical problem, we need to prove the following fundamental result.
In particular c µ < s N S N sp * .
Before giving the proof of the above lemma, we recall some facts which will be crucial to estimate the mountain pass level c µ . For any ε > 0, let us define where U ∈ D s,p (R N ) is a solution to As showed in [13], we know that U ∈ L ∞ (R N ) ∩ C 0 (R N ) is a positive, radially symmetric and decreasing function with We also have the following interesting estimates: [13] There exist constants c 1 , c 2 > 0 and θ > 1 such that for all r ≥ 1, and Let θ be the universal constant in Lemma 4.9 that depends only on N, p and s. For ε, δ > 0, set Let us observe that g ε,δ and G ε,δ are nondecreasing and absolutely continuous functions. Now, we consider the radially symmetric nonincreasing function which, in view of the definition of G ε,δ , satisfies We recall the following useful estimates established in Lemma 2.7 in [35]: Lemma 4.10. There exists C = C(N, p, s) > 0 such that for any ε ≤ δ 2 the following estimates hold In what follows, we prove an upper bound for the L p -norm of u ε,δ : Lemma 4.11. There exists a constant C = C(N, p, s) > 0 such that for any ε ≤ δ Proof. Firstly, we consider the case N > sp 2 . Let us observe that from the definition of u ε,δ it follows that u p ε,δ dx =: I + II.

(4.4)
Now we estimate the two integrals on the right hand side of (4.4). By using a change of variable, Lemma 4.9 and the fact that ε ≤ δ 2 , we can infer that where C is a positive constant. Since U ε is radially nonincreasing, for any δ ≤ r ≤ θδ, we have By using the definition of U ε , δ ε ≥ 2 and Lemma 4.9 we obtain (4.6) Putting together (4.4)-(4.6) we get the thesis. Let us consider the case N = sp 2 . Then, we can see that Therefore, being log( δ ε ) ≥ log(2) if ε ≤ δ 2 , we can conclude that Thus, recalling that for C, D > 0 it holds for all t ≥ 0, and using (4.8), we can see that Now, in view of the following elementary inequality and gathering the estimates in Lemma 4.10 and Lemma 4.11, we get Hence, if N > sp 2 , we deduce that q 1 > p > N (p−1) N −sp and by using Lemma 4.12, we have provided that ε > 0 is sufficiently small. When N = sp 2 , we get q 1 > p = N (p−1) N −sp , and in view of Lemma 4.12 we obtain Observing that q 1 > p yields lim ε→0 ε sp 2 −s(p−1)q 1 ε sp (1 + log(1/ ε)) = ∞, we again get the conclusion for ε small enough. Now, we prove the following lemma.
Proof. It is easy to check that {u n } is bounded in X µ . Now, we assume that for any R > 0 it holds From the boundedness of {u n } and Lemma 2.1 it follows that u n → 0 in L r (R N ) for any r ∈ (p, p * s ).
(4.11) By using (3.1), (3.2) and (4.11) we deduce that Since V (x) ≥ V 0 and {u n } is bounded in X µ , we can pass to the limit as ξ → 0 in (4.12) and (4.13) to see that R N f (u n )u n dx = o n (1) and R N Moreover, we use Lemma 4.13 instead of Lemma 3.8.

4.3.
Existence result for the critical case. Arguing as in Lemma 4.13 we can prove the "critical" version of Lemma 3.10.
Lemma 4.15. Let d < s N S N sp * and let {u n } ⊂ N ε be a sequence such that I ε (u n ) → d and u n ⇀ 0 in W ε . Then, one of the following alternatives occurs (a) u n → 0 in W ε ; (b) there are a sequence {y n } ⊂ R N and constants R, β > 0 such that The next result can be obtained following the lines of the proof of Lemma 3.11.  Proof. Since I ε (u n ) → c and I ′ ε (u n ) = 0, we can see that {u n } is bounded in W ε and, up to a subsequence, we may assume that u n ⇀ u in W ε . Clearly I ′ ε (u) = 0. Now, let v n = u n − u. By using the Brezis-Lieb Lemma [14] and Lemma 3.3 in [33], we know that Since {v n } is bounded in W ε , we may assume that v n p ε → ℓ and |v n | p * s L p * s (R N ) → ℓ, for some ℓ ≥ 0. Let us show that ℓ = 0. If by contradiction ℓ > 0, by using the fact that I ε (v n ) = d + o n (1), we get Taking the limit as n → ∞ we have that s N ℓ = d, that is ℓ = d N s . Therefore we get a contradiction. Hence, ℓ = 0 and u n → u in W ε .
Finally we have the existence result for problem (1.2) for ε > 0 small enough.  has a subsequence which converges in W s,p (R N ). Moreover, up to a subsequence, {y n } := {ε nỹn } is such that y n → y ∈ M .
For any δ > 0, let ρ > 0 be such that M δ ⊂ B ρ (0). Let χ : R N → R N be defined as Let us consider the barycenter map β ε : N ε → R N given by Arguing as in the proof of Lemma 3.13 we can prove the following result.