Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction

In this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity ε2αa+ε4α-3b∫R3|(-Δ)α2u|2dx(-Δ)αu+V(x)u=H(u-β)f(u)inR3,u∈Hα(R3),u>0inR3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \left( \varepsilon ^{2\alpha }a+\varepsilon ^{4\alpha -3}b\int _{{\mathbb {R}}^3}|(-\Delta )^{\frac{\alpha }{2}} u|^2{{\mathrm{d}}}x\right) (-\Delta )^\alpha {u}+V(x)u = H(u-\beta )f(u) &{} \quad \text{ in }\,\,{\mathbb {R}}^3, \\ u\in H^\alpha ({\mathbb {R}}^3),\quad u>0 &{} \quad \text{ in }\,\, {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$\end{document}where ε,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ,\beta >0$$\end{document} are small parameters, α∈(34,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (\frac{3}{4},1)$$\end{document} and a, b are positive constants, (-Δ)α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{\alpha }$$\end{document} is the fractional Laplacian operator, H is the Heaviside function, V is a positive continuous potential, and f is a superlinear continuous function with subcritical growth. By using minimax theorems together with the non-smooth theory, we obtain existence and concentration properties of positive solutions to this non-local system.


Introduction and results
This paper is devoted to the qualitative analysis of solutions for the fractional Kirchhoff equation in R 3 . We are concerned with the existence and multiplicity of solutions, as well as with concentration properties of solutions for small values of two positive parameters. A feature of this paper is that the reaction has lack of regularity, which allows to consider larger classes of nonlinearities. The main result is described in the final part of this section.

Overview
In the last decade, the investigation of nonlinear problems involving fractional and non-local operators has achieved an immense popularity. This is due to the fundamental role of such problems in the analysis of several complex phenomena such as phase transition, game theory, image processing, population dynamics, minimal surfaces and anomalous diffusion, as they are the typical outcome of stochastically stabilization of Lévy processes; see, for instance, the monograph [35] for more details. Moreover, such equations and the associated fractional operators allow us to develop a generalization of quantum mechanics and also to describe the motion of a chain or an array of particles that are connected by elastic springs as well as unusual diffusion processes in turbulent fluid motions and material transports in fractured media; for more details, see [13,14] and the references therein.
The purpose of this paper is to study the existence and concentration of positive solutions for the following fractional Kirchhoff-type equation: where α ∈ (0, 1) and a, b are positive constants, ε, β > 0 are positive parameters, H is the Heaviside function given by The operator (−Δ) α is the fractional Laplacian defined as F −1 (|ξ| 2α F (u)), where F denotes the Fourier transform on R 3 . The potential V : R 3 → R is a continuous function satisfying the following conditions introduced by Rabinowitz in [40]: and f : R → R is a continuous function fulfilling the following hypotheses: (f 1 ) f (t) = 0 for all t < 0 and f (t) = o(t 3 ) as t → 0 + . (f 2 ) There exists 4 < p < 2 * α − 1 such that t 3 is increasing in (0, ∞). (f 4 ) f (t) ≥ γt σ for all t > 0 with some γ > 0 and σ ∈ (3, p − 1). Obviously, it follows from the conditions of (f 1 )-(f 3 ) that where F (t) = t 0 f (s)ds. When a = 1, b = 0, (K) reduces to the following fractional Schrödinger equation which has been proposed by Laskin [26] in fractional quantum mechanics as a result of extending the Feynman integrals from the Brownian like to the Lévy like quantum mechanical paths. For such a class of fractional and non-local problems, Caffarelli and Silvestre [14] expressed (−Δ) α as a Dirichlet-Neumann map for a certain local elliptic boundary value problem on the half-space. This method is a valid tool to deal with equations involving fractional operators to get regularity and handle variational methods. We refer the readers to [22,43] and to the references therein. Investigated first in [20] via variational methods, there has been a lot of interest in the study of the existence and multiplicity of solutions for (1.2) when V and f satisfy general conditions. We cite [17,42] with no attempts to provide a complete list of references. If α = ε = 1 and R 3 is replaced by bounded domain Ω, then problem (K) formally reduces to the well-known Kirchhoff equation related to the stationary analogue of the Kirchhoff-Schrödinger-type equation where u denotes the displacement, f is the external force, b is the initial tension, and a is related to the intrinsic properties of the string. Equations of this type were first proposed by Kirchhoff [25] in the onedimensional case, without forcing term and with Dirichlet boundary conditions, in order to describe the transversal free vibrations of a clamped string in which the dependence of the tension on the deformation cannot be neglected. This is a quasilinear partial differential equation; namely, the nonlinear part of the equation contains as many derivatives as the linear differential operator. The Kirchhoff equation is an extension of the classical d'Alembert wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. Besides, we also point out that such non-local problems appear in other fields like biological systems, where u describes a process depending on the average of itself; see Alves et al. [1]. The solvability of the Kirchhoff-type equations has been well studied in a general dimension by various authors only after J.-L. Lions [28] introduced an abstract framework to such problems. For more recent results concerning Kirchhoff-type equations in bounded or unbounded domain, we refer, e.g., to [11,23,27,29,30,33,34,37,45,48] and their references.
In the non-local fractional framework, Fiscella and Valdinoci in [21], proposed the following stationary Kirchhoff variational equation with critical growth which models non-local aspects of the tension arising from measurements of the fractional length of the string. They in [21] obtained the existence of non-negative solutions when M and f are continuous functions satisfying suitable assumptions. After that, some existence and multiplicity results to problem (1.4) were obtained in [9,10,38,39,46] and their references. Recently, several authors have also been paid attention to the existence and multiplicity of solutions for fractional Kirchhoff equations in R N via variational and topological methods. Precisely, Ambrosio and Isernia [7] considered the fractional where f is an odd subcritical nonlinearity satisfying the well known Berestycki-Lions assumptions. By minimax arguments, the authors establish a multiplicity result in the radial space H α rad (R 3 ) when the parameter b is sufficiently small. Liu et al. [31] used the monotonicity trick and the profile decomposition to prove the existence of ground states to a fractional Kirchhoff equation with critical nonlinearity in low dimension. In [8], the authors employed penalization method and Lusternik-Schnirelmann category theory to study the existence and multiplicity of solutions for a fractional Schrödinger-Kirchhoff equation with subcritical nonlinearities. see also [6,24] and their references.
Though there have been many works on the existence and concentration of solutions for Kirchhoff-type problem involving continuous nonlinearities, to the best of our knowledge, it seems that no result has been done for the discontinuous case. In the present paper, we will study a class of fractional Kirchhoff-type problem involving discontinuous nonlinearities. We emphasize that since many obstacle problems and free boundary problems may be reduced to partial differential equations with discontinuous nonlinearities [15,16], the existence, multiplicity and concentration of solutions for the elliptic problem with discontinuous nonlinearities have been studied in recent years, see [1][2][3][4][5]41,47] and their references.

Main results and strategy
Motivated by the works above, in this paper we aim to study the existence and concentration of positive solutions to the fractional Kirchhoff equation with a discontinuous nonlinearity. In order to study (K), we use the change of variable x → εx and we will look for solutions to Then, there exist ε * , β * > 0 such that (K ε ) has a positive solution u ε,β for ε ∈ (0, ε * ) and β ∈ (0, β * ). Moreover, there exists a maximum point x ε,β ∈ R 3 of u ε,β such that Since we deal with the fractional Kirchhoff-type equation with a discontinuous nonlinearity, some estimates are totally different from those used in the mentioned paper. The minimax method and the non-smooth theory are our main approach in present paper, which are motivated by [3,29,44]. The main obstacles are as follows.
Firstly, observe that the energy functional is only locally Lipschitz continuous due to the effect of the Heaviside function, so that we are not able to use variational methods for C 1 -functionals. For this item, we have to use the variational framework for non-differentiable functionals which will be introduced in Sect. 2. Secondly, for the case with continuous nonlinearity, one can establish one equivalent relationship between the mountain pass level and infimum of energy functional on Nehari manifold, and then use the Fatou lemma to prove the existence of positive ground state solutions for the corresponding limit equation. However, the method of Nehari manifold does not work for locally Lipschitz continuous functionals, and so some new technique needs to be developed to obtain fine estimates to the mountain pass levels. Finally, with the presence of the Kirchhoff term, the main obstacle arises in getting the compactness of the corresponding locally Lipschitz continuous energy functional. Precisely, this does not hold in general: where {u n } n∈N is a (PS)-sequence of the energy functional satisfying u n u in H α (R 3 ). Then, it is not clear that weak limits are critical points of energy functional, which is totally different from those in [2,3,23]. For this reason, it is necessary to give one specific profile decomposition of the Kirchhoff term ( R 3 |(−Δ) α 2 u n | 2 dx) 2 which enables us to establish refined energy estimate to the mountain pass level. Finally, the above information together with the mountain pass geometry behaviors of energy functional helps us to obtain the compactness (see Lemma 3.6).
Throughout this paper, C will denote a generic positive constant. We denote by | · | r the L r -norm and use o(1) to denote any quantity which trends to zero when n → ∞. For any ρ > 0 and z ∈ R 3 , stands for the weak convergence in space E and its dual space E * .
The paper is organized as follows. In Sect. 2, the variational setting and some preliminary lemmas are presented. In Sect. 3, we study existence of positive solutions to (K ε ) with ε = 1. Section 4 is devoted by the existence and concentration of positive solutions to (K ε ).

Variational setting
In this section, we outline the variational framework for (K) and recall some preliminary lemmas. First, we fix the notations and we recall some useful preliminary results on fractional Sobolev spaces, see [35].

Concentration of solutions for fractional Kirchhoff
Page 5 of 23 211 For any α ∈ (0, 1), the fractional Sobolev space H α (R 3 ) is defined by It is known that and it is also the completion of C ∞ 0 (R 3 ) with respect to the norm be the Hilbert space equipped with the inner product and the corresponding induced norm We now recall some definitions and basic results on the critical point theory of locally Lipschitz continuous functionals as developed by Chang [16], Clarke [18]. The directional derivative of I at u in the direction of v ∈ E is defined by So I 0 (u; ·) is continuous, convex and its subdifferential at z ∈ E is given by where ·, · is the duality pairing between E * and E. The generalized gradient of I at u is the set Since I 0 (u; 0) = 0, ∂I(u) is the subdifferential of I 0 (u; 0). A few definitions and properties will be recalled below. ∂I(u) ⊂ E * is convex, non-empty and weak * -compact, and A critical point of I is an element u 0 ∈ E such that 0 ∈ ∂I(u 0 ) and a critical value of I is a real number c such that I(u 0 ) = c for some critical point u 0 ∈ E.
By a solution for (K ε ), we understand as a function u ∈ W 2α, p+1 . Now, we recall the following mountain pass theorem which was established in Radulescu [41]. (i) there are r > 0 and ρ > 0 such that We set

Existence of positive solutions to (K ε ) with ε = 1
The energy functional associated with (K ε ) with ε = 1, I : E → R is defined as . We now verify the functional I β satisfies the mountain pass geometry.
Lemma 3.1. The following properties hold: So from Sobolev's imbedding inequality, we infer that Hence, there exist r, ρ > 0, independent of β, such that for u = r, with e > 0 and Θ := mes{x| e(x) > β} > 0, then by (f 4 ) one has which implies that Combining Lemma 3.1 with Theorem 2.1, there exists a sequence {u n } ⊂ E satisfying where c β is the mountain pass level of the functional I β . In what follows, we show that sequence {u n } is bounded in E.

Proof. Set
It then follows from (3.4) that which finishes the proof of the lemma. Recalling Proof. Since u n u in E, one has u n → u in L p+1 loc (R 3 ). Hence, for any ϕ ∈ C ∞ 0 (R 3 ), the following holds

Recalling Lemma 2.2 we get
The proof is now complete.
This implies that u n → 0 in E. Furthermore, I β (u n ) → 0 as n → ∞. This contradicts c β > 0. The proof is complete.
Let c ∞ be the mountain pass level associated with the functional Proof. Suppose on the contrary that u 0 = 0. The Sobolev's embedding together with Lemma 4.2 yields {y n } is unbounded. That is, up to subsequence, |y n | → +∞. Let us set v n (x) := u n (x + y n ). It is easy to (3.12) It follows from (3.10) that (3.14) Using the Hölder inequality, the boundedness of {u n } in E and |∇ϕ R | ≤ C R , we have where we have used polar coordinate transformation in the third inequality. Taking into account (3.11)-(3.15), by Fatou's lemma we infer that Since v = 0, there exists t ∈ (0, 1) such that tv ∈ N (see [23]), where N is the Nehari manifold associated with I ∞ given by As a consequence, using (f 3 ) and Fatou's lemma, we have Recalling the definitions of f H and f H , we deduce from (f 1 ) that there exists β 1 > 0 such that f (t) ≤ V 0 t 3 for t ∈ (0, β 1 ) and then by (3.5) and c β < c ∞ , we have for large n for any fixed β ∈ (0, β 1 ). Putting (3.19) into (3.18), by c ∞ = inf u∈N I(u) (see [23]) and (3.18), one has which is a contradiction. Therefore, u 0 ≥ 0 and u 0 = 0.
The next result establishes the existence of mountain pass solutions to (K ε ) with ε = 1; that is, there exists u 0 ∈ E satisfying where c β is the mountain pass level associated with I β .
It is easy to see from (4.5) that u 0 is a non-trivial weak solution of (K 1 ). Moreover, by the Hölder inequality and the fractional Gagliardo-Nirenberg-Sobolev inequality, we have As a consequence, letting v = u n in (3.20) and v = u 0 in (3.22), one has We conclude that u n → u 0 in E as n → ∞. Using the chain rule for locally Lipschitz continuous function, there is w ∈ ∂I β (tu 0 ) such that h (t) = w, u 0 . That is to say, there exists ρ 0 ∈ ∂B(tu 0 ) satisfying It follows from 0 ∈ ∂J β (u 0 ), |Λ| = 0 and (3.22) that (a + bB) Based on the above facts, one has which implies by |Λ| = 0 and the fact that ρ 0 ∈ [f H (tu 0 (x)), f H (tu 0 (x))] a.e in R 3 that (3.27) Similarly, we can also obtain Define H : (0, +∞) → R by Obviously, H(1) < 0 due to u 0 2 D α,2 < B. It follows from (f 1 )-(f 3 ) and (3.27) that there existst ∈ (0, 1) such that H(t) = 0 and h (t) < 0 for all t ∈ (t, +∞) ∩ I c . Based on (3.25) and the above facts, h has a global maximum in t = t * ∈ (0,t]. Thus, conclusion (a) holds and then the claim is true. Recalling the definition of c β , we obtain c β ≤ I β (t * u 0 ). (3.28) Observe by |Λ| = 0 and the definition of H that Since t * is a global maximum point of h(t), there exist sequence {t n } ⊂ (t * , +∞) ∩ I c with t n → t * and ρ n ∈ ∂B(t n u 0 ) such that which implies by (3.28), (3.29), the definition of I β and t n ∈ (0, 1) that which is a contradiction. Here, ω n has been defined in Lemma 3.2. We conclude that Case 2 does not occur. The proof is complete.

Existence and concentration of positive solutions to (K ε )
Let us consider the following Hilbert space The energy functional associated with (K ε ) is given by Using the same argument as Lemma 3.1, we can prove that I ε,β has the corresponding mountain pass geometry. The mountain pass level of I ε,β is denoted by c ε,β , where is defined by

Existence
Define There exists a positive function v ∈ H α (R 3 ) (see [31]) such that I V0 (v) = 0 and I V0 (v) = c V0 , where c V0 is the mountain pass level. Define the corresponding manifold of I V0 by as R → +∞.

Concentration
Assume that u ε,β is the solution of equation (K ε ) obtained above. Then, there exists ρ ε,β ∈ L p+1 p (R 3 ) such that u ε,β solves a.e. in R 3 . Take ε n , β n → 0 arbitrarily, we denote by u n = u εn,βn and ρ n = ρ εn,βn . It is important to compare the minimax levels c V0 and c εn,βn in our arguments.
Proof. Suppose on the contrary that the conclusion does not hold. Using the similar arguments as in Lemma 3.2, we have that {u n } is bounded in H α (R 3 ). It then follows from Lemma 2.4 that u n → 0 in L s (R 3 ) for s ∈ (2, 2 * α ). By virtue of (f 1 ) and (f 2 ), for any ε > 0, there exists C ε > 0 such that f (u n ) ≤ ε|u n | + C ε |u n | p . So from the definition of F H and the fact that ρ n ∈ [f H (u n ), f H (u n )] a.e. in R 3 , we deduce that R 3 F H (u n )dx → 0 and R 3 ρ n u n dx → 0 as n → ∞. It follows that which contradicts Lemma 4.1. The proof is complete.
a.e., in R 3 . Then, v n solves the following equation Proof. For each v n , choosing t n > 0 such thatv n := t n v n ∈ N V0 , we deduce from 0 ∈ ∂I εn,ηn (u n ) and Lemma 3.6 that We can show that {t n } is bounded. Indeed, since u n 2 εn is bounded uniformly for n, we can easy obtain I V0 (v n ) → −∞ when t n → +∞. This is not possible since I V0 (v n ) ≥ c V0 for all n ∈ N. Hence, {t n } is bounded. Up to subsequence, we assume that By uniqueness, we deducē v = tv. Using Ekeland's variational principle in [19], we can prove that {v n } ⊂ N V0 is a Palais-Smale sequence of I V0 (see Lemma 5.1 in [32]), that is, (4.8) Although I V0 is of C 1 class, we can still use the similar arguments as in Lemma 3.6 to obtainv n →v in H α (R 3 ) andv ∈ N V0 , and so v n → v in H α (R 3 ). Let us show that y n := ε nỹn is bounded. If not, then |y n | → ∞. It follows fromv n ∈ N V0 , the Fatou Lemma and Lemma 3.6 that which is a contradiction. So {y n } is bounded. Up to subsequence, y n → y * . Moreover, since f (t) = 0 for all t ≤ 0, we have as β → 0 Hence, based on the facts thatρ n (x) ∈ [f H (v n (x)), f H (v n (x))] a.e. in R 3 and v n → v in H α (R 3 ), using the Lebesgue dominated convergence theorem, we deduce that for any Therefore, the limit v of sequence {v n } solves the equation Define the functional If V (y * ) > V 0 , then we can get a contradiction by similar arguments as above. So V (y * ) = V 0 and I y * (v) = c V0 . That is to say, v is a positive ground state solutions of (4.9). The proof is complete.
We use the Moser iteration method [36] to prove L ∞ -estimate for problem (4.7).
Proof. Since |Λ * | = 0 with Λ * := {x ∈ R 3 | v n (x) = β}, our argument is similar to that in Lemma 3.2 of [6]. So we omit the details of the proof. Now, we claim that there exists c > 0 such that v n ∞ ≥ c > 0. Otherwise, v n ∞ → 0 as n → ∞. By (f 1 )-(f 3 ) and the definition of f H , for n large enough, we have which is a contradiction. Hence, from Lemma 4.4 there exist c, C > 0 independent of n such that c ≤ v n ∞ ≤ C. (4.10) Observe from (4.7) that v n solves and there exits C > 0 independent of n such that χ n ∞ ≤ C. Based on the above, v n can be expressed as v n (x) := (K * χ n )(x) = where K is the Bessel kernel and satisfies the following properties (see [20]): (1) K is positive, radially symmetric and smooth in R 3 \ {0}, (2) there is C > 0 such that K(x) ≤ C |x| 3+2α for any x ∈ R 3 \ {0}, (3) K ∈ L r (R 3 ) for any r ∈ [1, 3 3−2α ). Arguing as in Theorem 3.4 in [20], we have that v n (x) → 0 as |x| → ∞ uniformly in n ∈ N.
For c > 0 given in (4.10), we can find R > 0 such that v n (x) < c for all |x| ≥ R and uniformly for n ∈ N. Let x n denote the maximum point of v n , then |x n | ≤ R. Moreover, we have z n = x n +ỹ n where z n is one maximum point of v n . It is easy to check from Lemma 4.3 that ε n z n = ε n x n + ε nỹn → y * ∈ Θ.
By the continuity of V , we obtain lim n→∞ V (ε n z n ) = V (y * ) = V 0 .
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