Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent

where (−∆)p is the fractional p–Laplacian with 0 < s < 1 < p < N/s, ps = Np/(N − ps) is the critical fractional Sobolev exponent, λ > 0 is a real parameter, 1 < θ ≤ ps/p, and f : RN × R → R is a Carathéodory function satisfying superlinear growth conditions. For θ ∈ (1, ps/p), by using the concentration compactness principle in fractional Sobolev spaces, we show that if f (x, t) is odd with respect to t, for any m ∈ N+ there exists a Λm > 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λm]. For θ = ps/p, by using Krasnoselskii’s genus theory, we get the existence of in nitely many solutions for the above problem for λ large enough. The main features, as well as the main di culties, of this paper are the facts that the Kirchho function is zero at zero and the potential function satis es the critical frequency infx∈R V(x) = 0. In particular, we also consider that the Kirchho term satis es the critical assumption and the nonlinear term satis es critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p–Laplacian case.


Introduction
In this article we concern with existence and multiplicity of solutions for critical Kirchho -type problems involving the fractional p-Laplacian. More precisely, we consider where p * s = Np/(N−sp), N > sp with s ∈ ( , ), (−∆) s p is the fractional p-Laplacian which (up to normalization factors) may be de ned for any x ∈ R N as |x − y| N+ps dy for any φ ∈ C ∞ (R N ), where B δ (x) denotes the ball in R N centered at x with radius δ. For a simple introduction about the fractional p-Laplacian, we refer to [1] and the references therein. Furthermore, we always assume M, V and f satisfy the following assumptions: (M) M ∈ C(R, R) and there exist θ ∈ ( , p * s /p) and < m ≤ m such that m t θ− ≤ M(t) ≤ m t θ− for all t ∈ R + ; (V) V ∈ C(R N , R), V(x ) = min x∈R N V(x) = and there exists a constant h > such that the Lebesgue measure of set V h = {x ∈ R N : V(x) < h} is nite; there is ϱ > such that lim |y|→∞ meas({x ∈ Bϱ(y) : V(x) < c}) = for any c ∈ R + ; (f ) f : R N × R → R is a Carathéodory function and there exists q ∈ (θp, p * s ) such that for any ε > there exists Cε > and |f (x, ξ )| ≤ θpε|ξ | θp− + qCε|ξ | q− for a.e. x ∈ R N and all ξ ∈ R; (f ) There exists q > m θp/m such that where F(x, ξ ) = ξ f (x, τ)dτ, m and m are the numbers given in (M); (f ) There exists q ∈ (θp, p * s ) such that F(x, ξ ) ≥ a |ξ | q for a.e. x ∈ R N and all ξ ∈ R. Note that condition (V ), which is weaker than the coercivity assumption: V(x) → ∞ as |x| → ∞, was rst introduced by Bartsch and Wang in [2] to conquer the lack of compactness.
In the last few years, great attention has been paid to the study of non-local fractional Laplacian problems involving critical nonlinearities. It is worth mentioning that the semilinear Laplace equation of elliptic type involving critical exponent was investigated in the crucial paper of Brézis and Nirenberg [3]. After that, many researchers dedicated to the study of several kinds of elliptic equations with critical growth in bounded domains or in the whole space. For example, by variational techniques, Servadei and Valdinoci [4] showed a Brézis-Nirenberg type result for non-local fractional Laplacian in bounded domains with homogeneous Dirichlet boundary datum, see also [5] for further discussions. In [6], Ros-Oton and Serra considered nonexistence results for nonlocal equations involving critical and supercritical nonlinearities. Autuori and Pucci [7] obtained a multiplicity result for fractional Laplacian problems in R N by using the mountain pass theorem and the direct method in variational methods, in which one of two superlinear nonlinearities could be critical or even supercritical.
Indeed, the interest in the study of partial di erential equations involving the non-local fractional Laplacian goes beyond the mathematical curiosity. This type of non-local operator comes to real world with many di erent applications in a quite natural way, such as nance, ultra-relativistic limits of quantum mechanics, materials science, water waves, phase transition phenomena, anomalous di usion, soft thin lms, minimal surfaces and game theory, see for example [1,8,9] and the references therein. The literature on fractional Laplace operators and their applications is quite large and interesting, here we just list a few, see [10][11][12] and the references therein. For the basic properties of fractional Sobolev spaces and the study of fractional Laplacian based on variational methods, we refer the readers to [1,13]. It is worth pointing out that one of the reasons that forced the rapid expansion of the fractional Laplacian results has been the nonlinear fractional Schrödinger equation, which was proposed by Laskin [14,15] as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths.
In the last decade, the existence and multiplicity of solutions for the Kirchho -type elliptic equations with critical exponents have attracted much interest of many scholars. For instance, we refer to [16][17][18] for the setting of bounded domains; we collect also some articles, see [19][20][21] for the context set in the whole space. In particular, Fiscella and Valdinoci [22] proposed a stationary Kirchho -type equation which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. More precisely, they considered a model as follows: where M(γ) = α + βγ for all γ ≥ , here α > , β ≥ . Note that M is this type, problem (1.2) is called nondegenerate if α > and β ≥ , while it is named degenerate if α = and β > , see [23] for some physical motivation about degenerate Kirchho problems. For more details about the physical background of the fractional Kirchho model, we refer to [22,Appendix A]. Afterwards, the fractional Kirchho -type problems have been extensively investigated, for example, we refer to [24][25][26][27] for some recent results about non-degenerate Kirchho -type problems. In the following, let us recall some existence results about degenerate Kirchho -type fractional p-Laplacian problems. By using the mountain pass theorem and Ekeland's variational principle, Xiang et al. [28] obtained the existence of two solutions for a nonhomogeneous Kirchho type problem driven by the fractional p-Laplacian, where the nonlinearity is convex-concave, see [26] for related results obtained by the same methods. In [29], Mingqi et al. investigated the existence of in nitely many solutions for Kirchho type fractional p-Laplacian problems, in which the symmetric mountain pass theorem is applied to study the suplinear case and the Krasnoselskii's genus theory is used to consider the sublinear case. In [23], Pucci et al. studied the existence and multiplicity of entire solutions for a class of fractional p-Laplacian problems of Kirchho type via variational methods and topological degree theory. In [30], Mingqi et al. considered the multiplicity of solutions for a class of quasilinear Kirchho system involving the fractional p-Laplacian by using the Nehari manifold method and the symmetric mountain pass theorem. Evidently, the above works did not involve the critical case. For the critical case, with the help of Kajikiya's new version of the symmetric mountain pass lemma, the existence of in nitely many solutions for a critical problem similar to (1.1) is proved in [31], see [32,33] for more related results.
However, there are few results in the available literature on problems like problem (1.1). In particular, there are no result on the multiplicity of solutions for problem (1.1). There is no doubt that we encounter serious di culties because of the lack of compactness and of the nonlocal nature of the fractional p-Laplacian. To overcome the loss of compactness, Xiang et al. [34] extended the concentration compactness principle of Lions [35] to the setting of fractional p-Laplacian in R N , and used it to get the existence of solutions for the following critical p-Kirchho problem where a ≥ , b > , θ > , λ > is a parameter and f ∈ L p * s p * s − (R N ). In [36], Fiscella and Pucci studied the following p-Kirchho problem involving critical Hardy-Sobolev nonlinearity where p * s (α) = (N−α)p N−ps is the critical Hardy-Sobolev exponent with α ∈ [ , ps), f and g are subcritical nonlinear terms, and V ∈ C(R N , R) with inf x∈R N V(x) ≥ V > . Moreover, the existence of in nitely many solutions for problem (1.1) is investigated, assuming inf x∈R N V(x) = and the subcritical nonlinear term f satis es superlinear growth condition. In [37], Byeon and Wang rst studied the asymptotic behavior of positive solutions to Schrodinger equations under the condition inf x∈R N V(x) = , which is called critical frequency. In [38], Cao and Noussair extended the results of Byeon and Wang, and studied multi-bump standing waves for nonlinear Schrödinger equations. In this paper, we follow the ideas of [39][40][41]. Although the ideas were used before for other problems, the adaptation of the procedure to our problem is not trivial because of the appearance of degenerate Kirchho function and the nonlocal nature of the fractional p-Laplacian. For this, we need more delicate estimates and computations.
To show our main results, we rst give some notations. For λ > , let W λ be the closure of C ∞ (R N ) with respect to the norm Then (W λ , · λ ) is a uniformly convex Banach space, see [26] for the details. Moreover, under the condition (V), for each λ > the embedding W λ → W s,p (R N ) is continuous. Indeed, for each u ∈ W λ , we have where S > is de ned as follows From this it follows that the embedding W λ → W s,p (R N ) is continuous. Next we give the de nition of solutions for problem (1.1).
De nition 1.1. We say that u ∈ W λ is a (weak) solution of equation ( for any φ ∈ W λ . Now we are in a position to state the rst result of our paper as follows: Then for any λ > , there exists λ * > such that problem (1.1) has a nontrivial solution u λ for any λ ∈ ( , λ * ) which satis es is λm > such that problem (1.1) admits at least m pairs of solutions u λ,i (i = , , · · · , m) which satisfy (1.3) whenever < λ ≤ λm.
The proof of Theorem 1.1 is mainly based on the application of the concentration compactness lemma in fractional Sobolev spaces developed by Xiang et al. in [34]. We show that the energy functional I λ associated to problem (1.1) satis es (PS)c condition for c > small and λ > small. To get the multiplicity of solutions for problem (1.1), we nd a special nite dimensional subspaces by which we construct su ciently small minimax levels. It is worth to point out that the authors in [42] just concerned with the case that M(t) = a + bt θ− with a, b > , which just focused on the non-degenerate Kirchho problems, that is M( ) > . Moreover, for the nonlinear term f , our assumption (f ) is more general than (h ) and (h ) in [42]. Finally, we consider the critical case θ = p * s /p. To this aim, we assume the subcritical term f satis es following assumptions.
(f ) f : R N × R → R is a Carathéodory function and there exists q ∈ (p, p * s ) such that for any ε > there exists Cε > and |f (x, t)| ≤ pε|t| p− + qCε|t| q− for a.e. x ∈ R N and all t ∈ R; (f ) There exists q ∈ (p, p * s ) such that F(x, t) ≥ a |t| q for a.e. x ∈ R N and all t ∈ R. For the critical case θ = p * s /p, the method used in Theorem 1.1 seems to be invalid. For this, we will use Krasnoselskii's genus theory to prove Theorem 1.2, see also [43] about the application of the same method to the multiplicity of solutions for a class of fractional Choquard-Kirchho equations. Furthermore, as usual for elliptic problems involving critical nonlinearities, we must pay attention to the lack of compactness. To overcome this di culty, we x parameter λ larger than a suitable threshold. We would like to point out that the authors in [34] just obtained the existence of two weak solutions for a variant of problem (1.1) by using Ekeland's variational principle and the mountain pass theorem. To our best knowledge, this is the rst time to consider the existence of in nitely many solutions for the critical case θ = p * s /p in the study of general Kirchho problems.
The rest of our paper is organized as follows. In Section 2, we give the proof of Theorems 1.1. In Section 3, we consider the critical case θ = p * s /p and obtain the proof of the Theorem 1.2.

Proof of Theorem 1.1
In this section, we prove the main result of this paper. In the following, we shortly denote the norm of L ν (R N ) by | · |ν.
Obviously, the energy functional I λ : W λ → R associated with problem (1.1) It is easy to verify that as argued in [26], I λ ∈ C (W λ , R) and its critical points are solutions of (1.1). Under our assumptions, we can show that functional has mountain pass geometry.
For any u ∈ W λ , by (M), Hölder's inequality and the fractional Sobolev inequality, one has Note that by the fractional Sobolev embedding theorem (see [1]), there exists C > such that |u| θp ≤ C u λ and |u|q ≤ C u λ .
Then choosing ε ∈ ( , m /( θpC θp )), we have Proof. By (M) and F(x, t) ≥ for a.e. x ∈ R N and all t ∈ R, we have Note that all norms in a nite dimensional space are equivalent. Hence there exists C E > such that |u| p * s ≥ C E u λ for all u ∈ E. Then, It follows from p * s > θp that I λ (u) → −∞ as u ∈ E, u λ → ∞.  (2.4) in the measure sense, where δx j is the Dirac measure concentrated x j . Moreover, where S > is the best constant of the embedding D s,p (R N ) → L p * s (R N ). Next we prove that J = ∅. Otherwise, suppose that J ≠ ∅, then for xed j ∈ J and ε > , choose φ ε,j ∈ C ∞ (R N ) such that and |∇φ ε,j | ≤ /ε. Evidently, φ ε,j un ∈ D s.p (R N ). Hence it follows from I λ (un), φ ε,j un → that By using Hölder's inequality and Lemma 2.
From the above inequality, together with (2.5), it follows that Hence, (2.11) On the other hand, by (2.3) and (2.4), we obtain which is a contradiction. Hence the desired conclusion holds.
Letting R > , we de ne thanks to the assumption θ > . It is easy to see that which is absurd. Hence, we have ν∞ = . In view of J = ∅ and (2.18), we have (2.24) Now we show that un → u in W λ . To this aim, we rst assume that d := inf n≥ un λ > . Since Here we used the following fact: (2.25) Now we show that (2.25) is true. By Theorem 2.1 of [26], we know that the embedding W λ → L ν (R N ) is compact for any ν ∈ [p, p * s ). Thus, up to a subsequence, we have un → u in L ν (R N ) for any ν ∈ [p, p * s ). According to (f ) and (f ), for any ε > we have Thus, we obtain Hence, we conclude from (2.24) that This, together with d := inf n≥ un > , implies that (2.26) Let us now recall the well-known inequalities: for all a, b ∈ R N . Similar to the proof of Lemma 6 in [26], it is easy to deduce from (2.27) that un → u strongly in W λ as n → ∞.
In the end, we consider the case inf n un λ = . Then either 0 is an accumulation point of the sequence {un}n and so there exists a subsequence of {un}n strongly converging to u = , or 0 is an isolated point of the sequence {un}n and so there exists a subsequence, still denoted by {un}n, such that infn un > . In the rst case we are done, while in the latter case we can proceed as above.
Since the functional I λ satis es the (PS)c condition for small c > , we will nd a special nite dimensional subspaces by which we construct su ciently small minimax levels. By (V), we know that V(x ) = min x∈R N V(x) = . Without loss of generality, we assume from now on that x = . By means of (M) and (f ), we have Then I λ (u) ≤ J λ (u) for all u ∈ W λ . Hence it su ces to construct small minimax levels for J λ . For any δ > , one can choose ϕ δ ∈ C ∞ (R N ) with ϕ δ q = and supp ϕ δ ⊂ Br δ ( ) such that [ϕ δ ] p s,p < δ. Let Then supp e λ ⊂ B λ θp * s N(p * s −θp) r δ ( ). Thus, for t ≥ , we have It follows from λ ∈ ( , ) and θ > that Observe that V( ) = and V ∈ C(R N ), then there exists Λ δ > such that for all |x| ≤ r δ and < λ ≤ Λ δ . It follows from [ϕ δ ] p s,p < δ that Proof. Let δ > small enough such that Taking Λ = Λ δ and choosing t λ > such that t λ e λ λ > ρ λ and I λ (te λ ) < for all t ≥ t λ . The result follows by letting e λ = t λ e λ .
Then for each u = m i= c i e i λ ∈ E m λ,δ , we have Hence, and as above  [45]) Let G = R N and ∂Ω be the boundary of an open, symmetric, and bounded subset Ω ⊂ R N with ∈ Ω. Then γ(∂Ω) = N.
Denote by S N− the surface of the unit sphere in R N . Then we can deduce from Lemma 3.1 that γ(S N− ) = N.
We shall use the following theorem to obtain the existence of in nitely many solutions for (1.1).
Theorem 3.1. (see [46]) Let T ∈ C (G, R) be an even functional satisfying the (PS) condition. Furthermore, (1) T is bounded from below and even; (2) there is a compact set E ∈ Γ such that γ(E) = k and sup u∈E T(u) < T( ). Then T has at least k pairs of distinct critical points and their corresponding critical values are less than T( ).

Lemma 3.2.
Assume that s ∈ ( , ), ≤ p < N/s, θ = p * s /p, p < q < p * s and f satis es (f ). Then functional I λ satis es the (PS)c conditions in W λ for all λ > p S −p * s /p /m .
as n → ∞. Similar to the discussion as in Section 2, we have  Letting n → ∞, we have p m (η p + u p λ ) (θ− )p η p ≤ S −p * s /p λ − η p * s .