Solutions of stationary Kirchhoff equations involving nonlocal operators with critical nonlinearity in R N ∗

Abstract. In this paper, we consider the existence and multiplicity of solutions for fractional Schrödinger equations with critical nonlinearity in R. We use the fractional version of Lions’ second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PSc) condition holds locally. Under suitable assumptions, we prove that it has at least one solution and, for any m ∈ N, it has at least m pairs of solutions. Moreover, these solutions can converge to zero in some Sobolev space as ε → 0.

The fractional Laplacian operator (−∆) s (up to normalization constants) may be defined as where P.V. stands for the principal value.It may be viewed as the infinitesimal generators of a Lévy stable diffusion processes [1].This operator arises in the description of various phenomena in the applied sciences, such as phase transitions, materials science, conservation laws, minimal surfaces, water waves, optimization, plasma physics, and so on; see [21] and references therein for more detailed introduction.
In these last years, a great deal of work has been devoted to the study of semiclassical standing waves for the fractional nonlinear Schrödinger equation of the form where ε is a small positive constant, which corresponds to the Planck constant, (−∆) s , 0 < s < 1, is the fractional Laplacian, P (x) is a potential function.Problem (2) models naturally many physical problems, such as phase transition, conservation laws, especially in fractional quantum mechanics, etc.; see [14].It was introduced by Laskin [16,17] as a fundamental equation of fractional quantum mechanics in the study of particles on stochastic fields modeled by Lévy process.We refer to [21] for more physical background.
To obtain standing waves of the fractional nonlinear Schrödinger equation (2), we set ψ(x, t) = e −iEt/ε u(x) for some function u ∈ H s (R N ), and let V (x) = P (x) − E. Then problem (2) is reduced to the following equation: In quantum mechanics, when ε tends to zero, the existence and multiplicity of solutions to (3) is of particular importance.
In the nonlocal case, that is, when s ∈ (0, 1), the nonlocal model has attracted much attentions recently.For the case of a bounded domain, Ricceri [24] established a theorem tailor-made for a class of nonlocal problems involving nonlinearities with bounded primitive.In [8], Molica Bisci and Repovš studied a class of nonlocal fractional Laplacian equations depending on two real parameters and obtained the existence of three weak solutions by exploiting the result established by Ricceri in [24].For more related results, we refer the readers to [3,4,6,7,11,13,18] and the references therein.For the whole space R N were also studied by a number of authors.Felmer et al. [14] studied the existence and regularity of positive solution when f has subcritical growth and satisfies the Ambrosetti-Rabinowitz condition.Secchi [25] obtained the existence of ground state solutions of (3) when V (x) → ∞ as |x| → ∞ and Ambrosetti-Rabinowitz condition holds.In [29], the authors obtained the existence of infinitely many weak solutions for (3) by variant fountain theorem when f has subcritical growth.For the case of critical growth, Shang and Zhang [26] studied the existence and multiplicity of solutions for the critical fractional Schrödinger equation Based on variational methods, they showed that problem (4) has a nonnegative ground state solution for all sufficiently large λ and small ε.Moreover, Shen and Gao in [28] obtained the existence of nontrivial solutions for problem (4) under various assumptions on f and potential function V (x), in which the authors assumed the well-known Ambrosetti-Rabinowitz condition.See also recent papers [2,22,25,27] on the fractional Schrödinger equations (4).In [32], the fractional Schrödinger equations with a critical nonlinearity considered by using fractional version of concentration-compactness principle and radially decreasing rearrangements, they obtained the existence of a ground state solutions.However, there are no such results on Kirchhoff type problems (1).The interest in studying problems like problem (1) relies not only on mathematical purposes but also on their significance in real models.For example, in the Appendix of paper [15], the authors construct a stationary Kirchhoff variational problem, which models, as a special significant case, the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string.
In this paper, inspired by [12,31], we consider the existence and multiplicity of standing wave solutions of the fractional Schrödinger equation (1).To the best of our knowledge, the existence and multiplicity of standing wave solutions to problem (1) on R N has not ever been studied by variational methods.To prove all the results, we mainly follow the ideas in [12,31].Our proofs are based on variational methods.Let us point out that although the idea was used before for other problems, the adaptation to the procedure to our problem is not trivial at all.Because the appearance of non-local term and the function g, we must reconsider this problem and need more delicate estimates.
Our main result is the following.
Remark 1.We should point out that Theorem 1 is different from the previous results of [12,31] in two main directions: (i) g(t) ≡ C.There exist many functions g(t) satisfying condition (g 1 )-(g 2 ), for example, g(t) = a + bt, a, b > 0, and Σ = 1/2.(ii) Other potentials V (x) guaranteeing compactness of the embedding from E → H s (R N ) can also be used in this paper.(iii) We use the fractional version of Lions' second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PS c ) condition holds, which is different from methods used in [12].(iv) The method are employed to establish the existence and multiplicity of standing wave solutions for problems (1), which is different from methods used in [31].
This paper is organized as follows: Section 2 is devoted to preliminary.In Section 3, we introduce the variational framework and restate the problem in a equivalent form by replacing ε −2s with λ.Furthermore, we describe the corresponding main results (Theorem 3).In Section 4, we prove the behaviors of the bounded (PS) sequences and then show that the energy functional satisfies (PS c ) by using the fractional version of concentrationcompactness principle.In Section 4, we give behaviors of (PS c ) sequences and its consequences.In Section 4, we verify the geometry of the mountain pass theorem and estimate the minimax value.At last, we give the proof of the main results.

Preliminaries
For the convenience of the reader, in this part we recall some definitions and basic properties of fractional Sobolev spaces H s (R N ).For a deeper treatment on these spaces and their applications to fractional Laplacian problems of elliptic type, we refer to [9,21] and references therein.
For any s ∈ (0, 1), the fractional Sobolev space H s (R N ) is defined by , and H s (R N ) is endowed with the norm As it is well known, H s (R N ) turns out to be a Hilbert space with scalar product By Proposition 3.6 in [21], we have Thus, and the embedding

An equivalent variational problem
We set λ = ε −2s and rewrite (1) in the following form: for λ → ∞. https://www.mii.vu.lt/NA We introduce the space which is a reflexive Banach space under the scalar product The norm induced by the product By assumption (V ), we know that the embedding E → H s (R N ) is continuous.Note that the norm • E is equivalent to the one • λ defined by In the following, we denote by |•| s the norm in L s (R N ) and by • E the norm in H s (R N ).Note that the norm • E is equivalent to the • λ for each λ > 0.
The energy functional J λ : E → R associated with problem (7) is well defined.Define the Nahari manifold Under the assumptions, it is easy to check that as arguments [23,30] J λ ∈ C 1 (E, R) and its critical points are solutions of (7).
We say that u ∈ E is a weak solution of (7) if where v ∈ E.

Behaviors of (PS) sequences
We recall the fractional version of concentration-compactness principle in the fractional Sobolev space [31,32], which due to Lions [19,20].
If, in addition, Ω is bounded, then there exist a positive measure µ ∈ M(R N ) with supp µ ⊆ Ω and positive numbers {µ j } j∈I such that where S is the best Sobolev constant, i.e.

S = inf
x j ∈ R N , δ xj are Dirac measures at x j , and µ j , ν j are constants.
https://www.mii.vu.lt/NARemark 2. In the case Ω = R N , the above principle of concentration-compactness does not provide any information about the possible loss of mass at infinity.The following result expresses this fact in quantitative terms.
Lemma 2. (See [31,32]. µ weakly- * in M(R N ), and define The quantities ν ∞ and µ ∞ exist and satisfy We recall that a C 1 functional J on Banach space X is said to satisfy the Palais-Smale condition at level c ((PS c ) in short) if every sequence {u n } ⊂ X satisfying lim n→∞ J λ (u n ) = c and lim n→∞ J λ (u n ) X * = 0 has a convergent subsequence.Lemma 3. Suppose that (V ) and (H) hold.Then any (PS c ) sequence {u n } is bounded in E and c 0.
By ( 11), (12), and condition (h 3 ), we have Therefore, inequality (13) imply that {u n } is bounded in E. Taking the limit in ( 13), we show that c 0. This completes the proof of Lemma 3.
The main result in this section is the following compactness result.
Proof.Let {u n } be a (PS c ) sequence.By Lemma 3, {u n } is bounded in E. Hence, up to a subsequence, we may assume that Moreover, by Prokhorov's theorem (see [7, Thm.8.6.2])there exist µ, ν ∈ M(R N ) such that where µ and ν are a nonnegative bounded measures on R N .It follows from Lemma 1 that Similarly to the proof of Lemma 3.4 in [31], we have where K > 4. As {u n } is bounded in H s (R N ), it follows from ( 14) and ( 15) Nonlinear Anal.Model.Control, 22(5):614-635 as n → ∞ and as ε → 0. Note that the Hölder's inequality implies Similarly to the proof of Lemma 3.4 in [31], we have In the following, we just give a sketch of the proof for the reader's convenience.
On the one hand, we have On the other hand, we have https://www.mii.vu.lt/NANote that u n u weakly in E. By Theorem 1, we obtain It follows from the definition of φ ε and Since φ ε has compact support, letting n → ∞ in ( 16), we deduce from ( 17), (18), and that α 0 µ {x j } λν j .
Combing this with Lemma 1, we obtain ν j α 0 λ −1 Sν . This result implies that either (I) To obtain the possible concentration of mass at infinity, similarly, we define a cut off Notice that It is easy to verify that lim sup Similarly to the proof of Lemma 3.4 in [31], we have It follows from the definition of φ ε that Letting R → ∞ in (20), we obtain By Lemma 2, we obtain ν ∞ α 0 λ −1 Sν Next, we claim that (II) and (IV) cannot occur.If case (IV) holds for some j ∈ I, then by using Lemma 2 and condition (h 3 ), we have that where 2s) .This is impossible.Consequently, ν j = 0 for all j ∈ J. Similarly, we can prove that (II) cannot occur for each j.Thus, As Thus, from the weak lower semicontinuity of the norm, conditon (g 1 ), and Brezis-Lieb lemma [10] we have Here we use J λ (u) = 0. Thus, we prove that {u n } strongly converges to u in E. This completes the proof of Lemma 4.

Proof of Theorem 3
In the following, we always consider λ 1.By assumptions (V ), (G), and (H), one can see that J λ (u) has mountain pass geometry.Lemma 5. Assume (V ), (G), and (H) hold.There exist α λ , ρ λ > 0 such that J where c s is the embedding constant of (8).It follows that Since 2 * s > 2, we know that the conclusion of Lemma 5 holds.This completes the proof of Lemma 5. Lemma 6.Under the assumption of Lemma 5, for any finite dimensional subspace F ⊂ E, Proof.On the one hand, by integrating (g 2 ), we obtain for all t t 0 > 0. Using conditions (V ) and (h 1 )-(h 3 ), we can get for all u ∈ F .Since all norms in a finite-dimensional space are equivalent and 2/Σ < 2 * s , 2 < 2 * s .This completes the proof of Lemma 6.
Since J λ (u) does not satisfy condition (PS c ) for all c > 0, in the following, we will find a special finite-dimensional subspaces by which we construct sufficiently small minimax levels.
Recall that assumption (V ) implies that there exists x 0 ∈ R N such that V (x 0 ) = min x∈R N V (x) = 0. Without loss of generality, we assume from now on that x 0 = 0.
Observe that, by (h 3 ), Define the function I λ ∈ C 1 (E, R) by Then J λ (u) I λ (u) for all u ∈ E, and it suffices to construct small minimax levels for I λ .
Observe that for each u = m * i=1 c i f i λ ∈ H m * λζ , we have for some constant C > 0, As before, we can obtain the following: for all λ Λ m * ζ and some constant C > 0.
Using this estimate we have the following.where Γ m * is the set of all odd homeomorphisms ι ∈ C(E, E), and ρ λ is the number from Lemma 5. Then j is a version of Benci's pseudoindex [5].Let It follows from Lemma 4 that J λ satisfies (PS c λ ) condition at all levels c < Σ 0 λ 1−N/2s .By the usual critical point theory, all c λi are critical levels, and J λ has at least m * pairs of nontrivial critical points satisfying α λ J λ (u λ ) κλ 1−N/(2s) .
Hence, problem (7) has at least m * pairs of solutions.In the end, as in the proof of Theorem 3(i), we see that these solutions satisfy estimates ( 9) and ( 10).

Lemma 1 .
(See[31,32].)Let Ω ⊆ R N be an open subset, and let {u n } be a weakly convergent sequence to u in H s (R N ) weakly converging to u as n → ∞ and such that|u n | 2 * s ν and |(−∆) s/2 u n | 2µ in the sense of measures.Then, either u n → u in L 2 * s loc (R N ) or there exists a (at most countable) set of distinct points {x j } j∈I ⊆ Ω and positive numbers {ν j } j∈I such that