Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth

: In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth:


Introduction
In this paper, we continue the study started in [6] concerning the concentration phenomena for a class of fractional relativistic Schrödinger equations.More precisely, we focus on the following nonlinear fractional elliptic equation with critical growth: where ε > 0 is a small parameter, m > 0, s ∈ (0, 1), N > 2s, 2 * s := 2N N −2s is the fractional critical exponent, and V : R N → R and f : R → R are continuous functions.The operator (−∆ + m 2 ) s is defined in Fourier space as multiplication by the symbol (|k| 2 + m 2 ) s (see [28,29]), i.e., for each function u : R N → R that belongs to the Schwartz space S(R N ) of rapidly decreasing functions, we have where we denoted by the Fourier transform of u.We also recall the following alternative representation of (−∆ + m 2 ) s in terms of singular integrals (see [24,29]): where P.V. indicates the Cauchy principal value, K ν is the modified Bessel function of the third kind of index ν (see [9,23]), and When s = 1 2 , the operator √ −∆ + m 2 was considered in [43,44] for spectral problems and has a clear meaning in relativistic quantum mechanics.Indeed, the energy for the motion of a free relativistic particle of mass m and momentum p is given by: where c is the speed of the light.With the usual quantization rule p → −ı ∇, where is Planck's constant, we obtain the so-called relativistic Hamiltonian operator: The point of the subtraction of the constant mc 2 is to make sure that the spectrum of the operator H is [0, ∞), and this explains the terminology of relativistic Schrödinger operators for the operators of the form H + V (x), where V (x) is a potential (see [16]).Equations involving H arise in the study of time-dependent Schrödinger equations of the type: where Φ : R × R N → C is a wave function and f : R N × [0, ∞) → R is a nonlinear function, which describe the dynamics of systems consisting of identical spin-0 bosons whose motions are relativistic, for instance, boson stars.Physical models related to H have been widely analyzed over the past 30 years, and there exists an important literature on the spectral properties of relativistic Hamiltonians; most of it has been strongly influenced by the works of Lieb on the stability of relativistic matter (see [22,27,30,31] and references therein).
On the other hand, from a probabilistic point of view, m 2s − (−∆ + m 2 ) s is the infinitesimal generator of a Lévy process X 2s,m t called 2s-stable relativistic process having the following characteristic function: (see, for example, [16,36]).For a more detailed discussion on (−∆ + m 2 ) s , we refer the interested reader to [8].
This operator has gained tremendous popularity during the last two decades thanks to its applications in different fields, such as, among others, phase transition phenomena, crystal dislocation, population dynamics, anomalous diffusion, flame propagation, chemical reactions of liquids, conservation laws, quasi-geostrophic flows, and water waves.Moreover, the fractional Laplacian is the infinitesimal generator of a (rotationally) symmetric 2s-stable Lévy process.For a very nice introduction to (−∆) s and its applications, consult [13,20].Note that the most striking difference between the operators (−∆) s and (−∆ + m 2 ) s is that the first one is homogeneous in scaling, whereas the second one is inhomogeneous as should be clear from the presence of the Bessel function K ν in (1.2).We emphasize that in these years, several authors dealt with the existence and multiplicity of solutions for the following fractional Schrödinger equation: where ε > 0, γ ∈ {0, 1}, V : R N → R and f : R → R satisfy suitable conditions (see, for instance, [7] and references therein).As s → 1, equation (1.4) boils down to the classical nonlinear Schrödinger equation of the form: Since we cannot review the huge bibliography on this topic, we refer to [2,3,19,25,26,35,42] for some results on the existence, multiplicity, and concentration of positive solutions to (1.5) for small ε > 0. We recall that a positive solution u ε of (1.5) is said to concentrate at The interest in studying semiclassical solutions of (1.5), i.e., solutions of (1.5) with small ε > 0, is justified by the well-known fact that the transition from quantum mechanics to classical mechanics can be described by letting ε → 0. A typical feature of semiclassical solutions is that they tend to concentrate as ε → 0 around critical points of the potential V .
On the other hand, several existence and multiplicity results for fractional equations driven by (−∆+m 2 ) s , with m > 0, have been established in [5,6,14,17,37,39].In particular, in [6], the author investigated (1.1) without the presence of the critical term u 2 * s −1 and obtained the existence of solutions concentrating in a given set of local minima of V as ε → 0. He also related the number of positive solutions to the topology of the set where V attains its minimum value.We point out that, in all the aforementioned articles, only equations with subcritical nonlinearities are considered.
Motivated by the previous facts, in this paper we examine the existence of concentrating solutions to (1.1), by assuming that the potential V : R N → R is a continuous function fulfilling the following conditions: with V 0 > 0, and 0 ∈ M := {x ∈ Λ : V (x) = −V 0 }, and that the nonlinearity f : R → R is a continuous, f (t) = 0 for t ≤ 0, and satisfies the following hypotheses: , for all t ≥ 0, and lim t→∞ f (t) t is increasing in (0, ∞).The main result of this paper can be stated as follows: Theorem 1.1.Assume that (V 1 )-(V 2 ) and (f 1 )-(f 4 ) hold.Then, for every small ε > 0, there exists a solution u ε to (1.1) such that u ε has a maximum point x ε satisfying lim ε→0 dist(ε x ε , M ) = 0, and for which Moreover, for each sequence (ε n ) with ε n → 0, there exists a subsequence, still denoted by itself, such that there exist a point x 0 ∈ M with ε n x ε n → x 0 and a positive ground state solution u ∈ H s (R N ) of the limiting problem: The proof of Theorem 1.1 relies on appropriate variational techniques.Since the operator (−∆ + m 2 ) s is nonlocal, we transform (1.1) into a degenerate elliptic equation in a half-space with a nonlinear Neumann boundary condition via a variant of the extension method [15] (see [17,24,41]).Then, we adapt the penalization approach in [19], the so-called local mountain pass, by building a convenient modification of the energy functional associated with the extended problem in such a way that the corresponding modified energy functional J ε satisfies the hypotheses of the mountain pass theorem [4], and then we prove that, for ε > 0 sufficiently small, the trace of the associated mountain pass solution is, indeed, a solution to the original equation with the stated properties.The modification of the functional corresponds to a penalization outside Λ, and this is why no other global assumptions are required.With respect to [6], it is more difficult to obtain compactness for J ε due to the presence of the critical exponent.To overcome this obstacle, we first estimate from above the mountain pass level c ε of J ε , by constructing a suitable cut-off function.Roughly speaking, we choose a function, appropriately rescaled, of the type v ǫ (x, y) = ϑ(my)φ(x, y)w ǫ (x, y), with ǫ > 0, where ϑ is expressed via the Bessel function K s , φ is a smooth cut-off function, and w ǫ is the s-harmonic extension of the extremal function u ǫ for the fractional Sobolev inequality (see [18]), in such a way that the control of the quadratic term is, in some sense, reduced to the control of the term: and thus, we are able to verify that c ε < c * , where the threshold value c * depends on the best constant S * for the critical Sobolev trace inequality (see [11]), and the constants V 1 , m 2s , and θ (see Lemma 3.2).In view of this bound and by establishing a concentration-compactness principle in the spirit of Lions [32,33], we show that the modified energy functional satisfies the Palais-Smale condition in the range (0, c * ) (see Lemmas 3.1 and 3.5).Finally, we prove that, for ε > 0 small enough, the solution of the auxiliary problem is, indeed, solution of the original one by combining a Moser iteration scheme [34], a comparison argument, and some crucial properties of the Bessel kernel [9,40] (see Lemmas 5.3 and 5.4).As far as we know, this is the first time that the penalization method is used to study the concentration phenomena for a fractional relativistic Schrödinger equation with critical growth.The structure of the paper is the following.In section 2, we define some function spaces.In section 3, we focus on the modified problem.In section 4, we deal with the autonomous critical problems related to the extended modified problem.Section 5 is devoted to the proof of Theorem 1.1.In section 6, we discuss a multiplicity result for (1.1).
The letters c, C, C ′ , and C i will be repeatedly used to denote various positive constants whose exact values are irrelevant and can change from line to line.For x ∈ R N and R > 0, we will denote by B R (x) the ball in R N centered at x ∈ R N with radius r > 0. When x = 0, we set ).For a generic real-valued function w, we set w + := max{w, 0} and w − := min{w, 0}.

Function spaces
Let H s (R N ) be the fractional Sobolev space defined as the completion of C ∞ c (R N ) with respect to the norm s ] and compactly in L p loc (R N ) for all p ∈ [1, 2 * s ); see [1,7,8,20,29].We denote by ) with respect to the norm: .
To deal with (1.1) via variational methods, we use a variant of the extension method [15] given in [17,24,41].More precisely, for each u ∈ H s (R N ), there exists a unique function The function U is called the extension of u and fulfills the following properties: where ), and it can be expressed as: , is the constant for the (normalized) Poisson kernel with m = 0 (see [41]).
Remark 2.1.We recall (see [24]) that P s,m is the Fourier transform of k → ϑ( |k| 2 + m 2 ) and that where belongs to H 1 (R + , y 1−2s ) and solves the following ordinary differential equation: (2.7) We also have Consequently, (1.1) can be realized in a local manner through the following nonlinear boundary value problem: where V ε (x) := V (ε x).For simplicity of notation, we will drop the constant σ s from the second equation in (2.9).In order to examine (2.9), for ε > 0, we introduce the space equipped with the norm: . Clearly, ), and using (2.3) and (V 1 ), we see that Furthermore, X ε is a Hilbert space endowed with the inner product: Henceforth, with X * ε , we will denote the dual space of X ε .

Penalization argument
To study (2.9), we adapt the penalization approach in [19] (see also [6]) where χ Λ denotes the characteristic function of Λ. Set G(x, t) := t 0 g(x, τ ) dτ .By assumptions (f 1 )-(f 4 ), it is easy to prove that g is a Carathéodory function satisfying the following properties: s −1 for all x ∈ R N , t > 0, (g 3 ) (i) 0 < θG(x, t) ≤ tg(x, t) for all x ∈ Λ and t > 0, or, x ∈ Λ c and 0 < t ≤ a, (ii) 0 ≤ 2G(x, t) ≤ tg(x, t) ≤ V1 κ t 2 for all x ∈ Λ c and t > 0, (g 4 ) for each x ∈ Λ, the function t → g(x,t) t is increasing in (0, ∞), and for each x ∈ Λ c , the function t → g(x,t) t is increasing in (0, a).Let us introduce the following auxiliary problem: where g ε (x, t) := g(ε x, t).It is clear that if v ε is a positive solution of (3.1) satisfying v ε (x, 0) < a for all x ∈ Λ c ε , where Λ ε := {x ∈ R N : ε x ∈ Λ}, then v ε is a positive solution of (2.9).The energy functional associated with (3.1) is defined by: It is standard to check that J ε ∈ C 1 (X ε , R) and that its differential is given by Hence, the critical points of J ε correspond to the weak solutions of (3.1).To seek these critical points, we will apply suitable variational arguments.First, we show that J ε possesses the geometric assumptions of the mountain pass theorem [4].
In what follows, we show that J ε satisfies a local compactness condition.First, we prove the boundedness of Palais-Smale sequences of J ε .
Proof.By assumptions, we know that as n → ∞.Using (3.11), (g 3 ), (2.3), and (2.10), we see that, for n big enough, and (v n ) ⊂ X ε is a Palais-Smale sequence of J ε at the level c, then we may always assume that (v n ) is nonnegative.In fact, Lemma 3.3 implies that also . The next concentration-compactness principle in the spirit of Lions [32,33] will be used in the proof of the local compactness of J ε .We start by recalling some useful definitions.A sequence ) is tight if for every ξ > 0, there exists R > 0 such that From the aforementioned estimate and the tightness of ( ).Let µ and ν be two bounded nonnegative measures on R N +1 + and R N , respectively, and such that Then, there exist an at most countable set I and three families ), it holds for some constant C 0 > 0. For this purpose, we fix ϕ ∈ C ∞ c (R N +1 + ) and let K := supp(ϕ).By (3.4), we deduce that . (3.18) Now, we note that (3.13) implies that On the other hand, Since Furthermore, the Hölder inequality, the boundedness of (v n ) in X s (R N +1 + ), and (3.21) lead to Finally, taking (3.12) into account, we see that Putting together (3.18)-(3.23),we can infer that (3.17) holds with ).Moreover, there exist two bounded nonnegative measures μ and ν on R N +1 + and R N , respectively, such that Then, we are in the previous case, and we can use (3.17) to deduce that Hence, as in [32, Lemma 1.2], we can find an at most countable set I, a family of distinct points ).By the Brezis-Lieb lemma [12], we know that . The aforementioned fact combined with (3.13), (3.25), and (3.26), the boundedness of ( and the tightness of , from which .
Next, we prove the tightness of the Palais-Smale sequences of J ε .More precisely, we establish the following result.
Lemma 3.4.Let 0 < c < s N (ζS * ) N 2s and (v n ) ⊂ X ε be a Palais-Smale sequence of J ε at the level c.Then, for all ξ > 0, there exists R = R(ξ) > 0 such that which can be rewritten as: Choose R > 0 such that Λ ε ⊂ B R/2 .Thus, thanks to (g 3 )-(ii), On the other hand, by Hölder's inequality, Then, applying (2.4) to v n η R , and using ∇η R L ∞ (R N +1 + ) ≤ C/R and (3.31), we infer that Putting together (3.29)-(3.32),we arrive at ) and the definition of η R , we deduce that (3.33) implies the assertion.Remark 3.3.Differently from [3,7,19], we use to obtain (3.28).This is motivated by the fact that to estimate the quadratic terms, it is needed to apply in a careful way Inequality (2.4).
which combined with (3.33) gives Thus, At this point, we can show that the modified functional fulfills a local compactness condition.Proof.Let 0 < c < s N (ζS * ) N 2s and (v n ) ⊂ X ε be a Palais-Smale sequence at the level c, namely, In view of Remark 3.1, we can suppose that (v n ) is nonnegative.Thanks to the reflexivity of X ε and Theorem 2.1, up to a subsequence, we may assume that ) in X ε , (g 1 ), (g 2 ), (f 2 ), and (3.35), it is easy to check that J ′ ε (v), ϕ = 0 for all ϕ ∈ X ε .In particular, On the other hand, In light of (3.36) and (3.37), if we prove that then we deduce that v n ε → v ε as n → ∞, and recalling that X ε is a Hilbert space, we conclude that v n → v in X ε as n → ∞.Next, we verify that (3.38) is valid.By virtue of Lemma 3.4, fixed ξ > 0, there exists R = R(ξ) > 0 such that (3.28) is true.By (g 2 ), (f 1 ), (f 2 ), and (3.34), we see that On the other hand, because Thus, (3.39) and (3.40) yield Now, it follows from the definition of g that Since B R ∩ Λ c ε is bounded, we can use the aforementioned estimate, (f 1 ), (f 2 ), (3.35), and the dominated convergence theorem to infer that, as n → ∞, At this point, we aim to show that In fact, if we assume that (3.43) holds, then we can exploit (g 2 ), (f 1 ), (f 2 ), (3.35), and the dominated convergence theorem again to obtain ), we may suppose that where µ and ν are two bounded nonnegative measures on R N +1 + and R N , respectively.Thus, applying Proposition 3.1, we can find an at most countable index set I and sequences i , for all i ∈ I. (3.45) Let us show that , and so or equivalently, Since f has subcritical growth and ψ ρ (•, 0) has compact support, we can use (3.35) to see that and Now, we prove that From the Hölder inequality, .
In light of Lemmas 3.1 and 3.5, we can apply the mountain pass theorem [4] to infer the next existence result for (3.1).Theorem 3.1.For all ε > 0, there exists a nonnegative function , we can argue as in the proof of Lemma 5.1 to prove that there exist (z n ) ⊂ R N and r, β > 0 such that On the other hand, reasoning as in Remark 3.4, we have that for all R > 0 such that In view of the aforementioned estimates, we deduce that the sequence (z n ) is bounded in R N .Now, because (v n ) is bounded in X ε , up to a subsequence, we may assume that there exists Then, it is easy to check that v ε is a critical point of J ε .Moreover, we can see that v ε ≡ 0. In fact, due to the boundedness of (z n ), we can find k > 0 such that B r (z n ) ⊂ B k for all n ∈ N. Consequently, which implies that v ε ≡ 0. Finally, to verify that J ε (v ε ) = c ε , it suffices to use (g 3 ) and Fatou's lemma to get

Autonomous critical problems
) and introduce the following autonomous problem related to (1.1): The extended problem associated with (4.1) is given by: and the corresponding energy functional ) endowed with the norm: To check that • Yµ is a norm equivalent to ) , one can argue as at pag. 5671 in [6] (observe that ).We also note that, by (2.3) Obviously, Y µ is a Hilbert space with the inner product v, w Yµ := Denote by M µ the Nehari manifold associated with L µ , i.e., As in the previous section, it is easy to verify that L µ has a mountain pass geometry [4].Thus, invoking a variant of the mountain pass theorem without the Palais-Smale condition (see [45, Theorem 2.9]), we can find a Palais-Smale sequence (v n ) ⊂ Y µ at the mountain pass level d µ of L µ given by: where which implies the boundedness of (v n ) in Y µ .As in [35,45], by assumptions on f , we can see that Arguing as in the proof of Lemma 3.2, it is easy to prove that Our claim is to establish the existence of a ground state solution for (4.2).We start by recalling a vanishing Lions-type result.
Proof.Assume that (b) does not occur.Therefore, for all R > 0, it holds By Lemma 4.1, we know that v n (•, 0) → 0 in L r (R N ) for all r ∈ (2, 2 * s ).This fact and (f 1 )-(f 2 ) imply that Exploiting L ′ µ (v n ), v n = o n (1) and (4.5), we see that we may assume that there exists ℓ ≥ 0 such that Suppose by contradiction that ℓ > 0. By virtue of L µ (v n ) = d µ + o n (1), (4.5), and (4.6), we have On the other hand, by Theorem 2.1 and (4.3), Yµ , and passing to the limit as n → ∞, we arrive at Taking (4.7) into account, we obtain that , which is impossible in view of (4.4).Then, ℓ = 0, and this completes the proof.Now we are ready to provide the main result of this section.
Then, (4.1) has a ground state solution.Proof.Since L µ has a mountain pass geometry [4], we can find a Palais-Smale sequence (v n ) ⊂ Y µ at the level d µ .Hence, (v n ) is bounded in Y µ , and so, up to a subsequence, we may suppose that there exists v ∈ Y µ such that v n ⇀ v in Y µ .Using the growth assumptions on f and the density of ), it is standard to verify that L ′ µ (v), ϕ = 0 for all ϕ ∈ Y µ .If v ≡ 0, we can apply Lemma 4.2 to deduce that for some sequence (z n ) ⊂ R N , vn (x, y) := v n (x + z n , y) is a bounded Palais-Smale sequence at the level d µ and vn ⇀ v ≡ 0 in Y µ .Thus, v ∈ M µ .Moreover, by (f 3 ) and Fatou's lemma, and so L µ (v) = d µ .When v ≡ 0, as before, we can prove that v is a ground state solution to (4.2).Consequently, for each We conclude this section by establishing an important relation between c ε and d The numbers c ε and d V (0) verify the following inequality: Proof.In light of Theorem 4.1, there exists a ground state solution w to (4.2 On the other hand, by the definition of c ε , we have for some t ε > 0. Using w ∈ M V (0) and (f 4 ), we deduce that t ε → 1 as ε → 0. Observe that Then, since V ε (x) is bounded on the support of w ε (•, 0) and V ε (x) → V (0) as ε → 0, we can exploit the dominated convergence theorem, (4.4), (4.8), and (4.9) to reach the desired conclusion.

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1.Let us recall that, by Theorem 3.1, for all ε > 0 there exists a nonnegative mountain pass solution v ε to (3.1).We begin with a useful result.Lemma 5.1.There exist r, β, ε * > 0 and (y ε ) ⊂ R N such that Proof.On account of (3.51) and the growth conditions on f , we can find α > 0, independent of ε > 0, such that v ε 2 ε ≥ α, for all ε > 0. (5.1) Let (ε n ) ⊂ (0, ∞) be such that ε n → 0. Assume, by contradiction, that there exists r > 0 such that By Lemma 4.1, we know that v ε n (•, 0) → 0 in L r (R N ) for all r ∈ (2, 2 * s ).Hence, (3.51) and the growth assumptions on f yield

This implies that
and (5.3) Because of J ′ εn (v ε n ), v ε n = 0 and (5.3), we obtain It is clear that ℓ > 0; otherwise, v ε n ε n → 0, and this is impossible due to (5.1) (alternatively, one can observe that which is a contradiction since Remark 4.1 ensures that c εn ≥ d −V1 > 0 for all n ∈ N).From (5.4), we derive that On the other hand, noting that by the definitions of S * and ζ, we see that ) \ {0} such that Moreover, there exists x 0 ∈ Λ such that Proof.Hereafter, we denote by (y n ) and (v n ), the sequences (y ε n ) and (v ε n ), respectively.Exploiting (3.51), Lemma 4.3 and (2.10), we can argue as in the proof of Lemma 3.3 to deduce that (w n ) is bounded in ).Hence, up to a subsequence, there exists w ∈ X s (R and Br w 2 (x, 0) dx ≥ β > 0, (5.8)where we have used Lemma 5.1.Next, we will show that (ε n y n ) is bounded in R N .To this end, it suffices to prove that dist(ε n y n , Λ) → 0 as n → ∞. (5.9) In fact, if (5.9) does not hold, there exist δ > 0 and a subsequence of (ε n y n ), still denoted by itself, such that dist(ε n y n , Λ) ≥ δ, for all n ∈ N.
Consequently, we can find R > 0 such that B R (ε n y n ) ⊂ Λ c for all n ∈ N. Since w ≥ 0, it follows from the definition of X s (R N +1 + ) that there exists a nonnegative sequence ( g εn (x + y n , w n (x, 0))ψ j (x, 0) dx. (5.10) Note that, by the properties of g ε , which combined with (V 1 ) and (5.10) yields Recalling that ψ j has compact support and exploiting ε n → 0, (5.7), Theorem 2.1, the growth assumptions on f , we have that, as n → ∞, Hence, for all j ∈ N, and letting j → ∞, we obtain Using (2.3) and κ > V1 m 2s −V1 , we arrive at which contradicts (5.8).By virtue of (5.9), there exist a subsequence of (ε n y n ), still denoted by itself, and x 0 ∈ Λ such that ε n y n → x 0 as n → ∞.Next, we claim that x 0 ∈ Λ.By (g 2 ) and (5.10), we have that and taking the limit as n → ∞, it follows from (5.7), Theorem 2.1 and the continuity of V that Letting j → ∞, we find Hence, there is t 1 ∈ (0, 1) such that t 1 w ∈ M V (x0) .Thus, by Lemma 4.3, we see that ∈ ∂Λ, and we can infer that x 0 ∈ Λ.Now, we aim to show that w n → w in X s (R N +1 + ) as n → ∞.For this purpose, for all n ∈ N and x ∈ R N , we set Λn := Λ − ε n ỹn ε n and Let us define the following functions for x ∈ R N and n ∈ N: In view of (f 3 ), (g 3 ), (V 1 ), and our choice of κ, the aforementioned functions are nonnegative in R N .Furthermore, using the following relations of limits, as n → ∞, we deduce that, as n → ∞, The aforementioned inequalities yield (5.11) and and we obtain (5.12) Combining (5.11) and (5.12), and recalling that X s (R N +1 + ) is a Hilbert space, we conclude that Now we use a Moser iteration argument [34] to establish a fundamental L ∞ -estimate.
Lemma 5.3.Let (w n ) be the sequence defined as in Lemma 5.2.Then, (w n (•, 0)) ⊂ L ∞ (R N ), and there exists C > 0 such that Proof.It suffices to argue as in the proof of [6, Lemma 4.1].However, for the reader's convenience, we provide a different proof here.First, we observe that w n is a weak solution to (5.13) For β > 1 and T > 0, we consider the following function: Note that H : R → R is convex, nondecreasing, and Lipschitz continuous with Lipschitz constant βT Testing (5.13) with ϕ n , we can write (5.14) By (V 1 ), (3.2), and ϕ n ≥ 0, we see that where we have used ϕ n (x, 0) ≤ w n (x, 0)(H ′ (w n (x, 0))) 2 .Then, from (5.14), we derive that On the other hand, using Theorem 2.1 and w n , ϕ n ≥ 0, we obtain (5.16) Combining (5.15) and (5.16), we find (x, 0)H 2 (w n (x, 0)) dx, (5.17 where C > 0 is independent of β and T .We stress that the last integral in (5.17) is well defined for every T > 0 in the definition of H. Now we choose β in (5.17) such that 2β − 1 = 2 * s , and we name it β 1 , i.e., Let R > 0 to be fixed later.Concerning the last integral in (5.17), applying the Hölder inequality with exponents r := , where C is the constant appearing in (5.17).This together with (5.17 Now, we suppose β > β 1 .Thus, using H(w n (x, 0)) ≤ w β n (x, 0) on the right-hand side of (5.17) and passing to the limit as T → ∞, we deduce that ) for all n ∈ N.
This completes the proof of Theorem 1.1.
6. Final comments: multiple concentrating solutions to (1.1) As in [6], if we suppose that the continuous potential V : R N → R satisfies the following conditions: (V ′ 1 ) there exists V 0 ∈ (0, m 2s ) such that −V 0 := inf x∈R N V (x), (V ′ Moreover, if u ε denotes one of these solutions and x ε is a global maximum point of u ε , then we have Since the proof of Theorem 6.1 is similar to the one of [6, Theorem 1.2], we only point out the main differences.For the proof of the critical version of [6, Lemma 5.1], we only need to replace in formula (67) in [6] weakly in the sense of measures (3.12) and |v n (•, 0)| 2 * s ⇀ ν weakly in the sense of measures.(3.13) weakly in the sense of measures, (3.24)|w n (•, 0)| 2 *s ⇀ ν weakly in the sense of measures.(3.25)
weakly in the sense of measures, v 2 * s n (•, 0) ⇀ ν weakly in the sense of measures,(3.44)

.51) Remark 3 . 5 .
It is possible to give an alternative proof of Theorem 3.1 without using Proposition 3.1.We outline the details.Let (v n ) ⊂ X ε be a Palais-Smale sequence at the mountain pass level c ε .By Lemma 3.3, we know that