Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension

onlinear Differ. Equ. Appl. 07 4:50 c 07 Springer International Publishing AG DOI 0.007/s00030-07-0473-7 onlinear Differential Equations and Applications odea Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension Zhisu Liu, Marco Squassina and Jianjun Zhang Abstract. We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential V. Under suitable assumptions on V, using the monotonicity trick and the profile decomposition, we prove the existence of ground states. In particular, the nonlinearity does not satisfy the Ambrosetti Rabinowitz type condition or monotonicity assumptions. Mathematics Subject Classification. 35Q55, 35Q5, 53C35. Keywords. Fractional Kirchhoff type problems, Ground state solutions, Profile decomposition. Contents. Introduction and results.. Overview.. Main results.3. Main difficulties. Variational setting 3. The perturbed functional 4. Upper estimate of c λ and limit problems 4.. An energy estimate 4.. The limit problem 5. Behaviour of Palais Smale sequences 5.. Splitting lemmas 5.. Profile decomposition Z. Liu is supported by the SFC 667. M. Squassina is member of the Gruppo azionale per l Analisi Matematica, la Probabilità e le loro Applicazioni GAMPA of the Istituto azionale di Alta Matematica IdAM. J. J. Zhang was partially supported by the Science Foundation of Chongqing Jiaotong University 5JDKJC-B033. 03456789.: V,-vol

50 Page of 3 Z. Liu, M. Squassina and J. Zhang odea 6. Proof of the main results 6.. ontrivial critical points of I λ 6.. Completion of the proof Acknowledgements References. Introduction and results.. Overview In this paper we are concerned with the existence of positive ground state solutions to the following nonlinear fractional Kirchhoff equation a + b Δ u dx Δ u + V xu = fu in, R K u H, u > 0 in, where a, b are positive constants, 0, and >. The operator Δ is the fractional Laplacian defined as F ξ F u, where F denotes the Fourier transform on. When a =andb = 0, then K reduces to the following fractional Schrödinger equation Δ u + V xu = fu in,. which has been proposed by Laskin [0] 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 nonlocal problems, Caffarelli and Silvestre [8] expressed Δ as a Dirichlet eumann 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 [6, 34] and to the references therein. Investigated first in [, 3] via variational methods, there has been a lot of interest in the study of the existence and multiplicity of solutions for. when V and f satisfy general conditions. We cite [,33,36] with no attempts to provide a complete list of references. If =, then problem K formally reduces to the well-known Kirchhoff equation a + b u dx Δu + V xu = fu in,. related to the stationary analogue of the Kirchhoff Schrödinger type equation u t a + b u dx Δu = ft, x, u, Ω where Ω is a bounded domain in, 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 [9] to describe the transversal oscillations of a stretched string. Besides, we also point

odea Ground states for fractional critical Kirchhoff equations Page 3 of 3 50 out that such nonlocal problems appear in other fields like biological systems, where u describes a process depending on the average of itself. We refer readers to Chipot and Lovat [0], Alves and Corrêa []. However, the solvability of the Kirchhoff type equations has been well studied in a general dimension by various authors only after Lions [3] introduced an abstract framework to such problems. For more recent results concerning Kirchhoff-type equations we refer e.g. to [4,7,6,8]. In [], by using a monotonicity trick and a global compactness lemma, LiandYeprovedthatforfu = u p u and p 3, /, problem. has a positive ground state. Subsequently, Liu and Guo [5] extended the above result to p, /. Fiscella and Valdinoci [4], proposed the following stationary Kirchhoff variational equation with critical growth M Δ u dx Δ u = λfx, u+ u u in Ω, R.3 u =0 in \Ω, which models nonlocal aspects of the tension arising from measurements of the fractional length of the string. They obtained the existence of non-negative solutions when M and f are continuous functions satisfying suitable assumptions. Autuori et al. [3] considered the existence and the asymptotic behavior of non-negative solutions of.3. Pucci and Saldi [30] established multiplicity of nontrivial solutions. Via a three critical points theorem, yamoradi [7] studied the subcritical case of.3 and obtained three solutions. See also [9,5,9,3,39] for related results. To the best of our knowledge, there are few papers in the literature on fractional Kirchhoff equations in. Recently, Ambrosio and Isernia [] considered the fractional Kirchhoff problem a + b Δ u dx Δ u = fu in,.4 where f is an odd subcritical nonlinearity satisfying the well known Berestycki and Lions [6] assumptions. By minimax arguments, the authors establish a multiplicity result in the radial space Hrad R when the parameter b is sufficiently small. Asin[], Teng [37] also searched for ground state solutions for the fractional Schrödinger Poisson system in R 3 with critical growth { Δ u + V xu + φu = μ u q u + u u in R 3, Δ t φ = u in R 3. We point out that, in [,37] the corresponding limit problems play an important role. In order to get the existence of ground state solutions of the limit problems, the authors used a constrained minimization on a manifold M obtained by combining the ehari and Pohožaev manifolds... Main results Motivated by the works above, in this paper we aim to study the existence of positive ground state solutions to the fractional Kirchhoff equation with the Berestycki Lions type conditions of critical type, firstly introduced in [40].

50 Page 4 of 3 Z. Liu, M. Squassina and J. Zhang odea... Assumptions on V. On the external potential we assume the following: V V C, R and, setting W x := max{x Vx, 0}, we assume Δ u dx W < as R L R, S := inf u D, /, u 0 dx := ; V there exists V R such that u V x lim V y =V, for all x ; y V 3 the operator a Δ + V x :H H satisfies inf u H u 0 a Δ u + V xu dx > 0. u dx... Assumptions on f. We assume that ft = 0 for all t 0and f f C R + ft, R and lim t 0 t =0; ft f lim t =; t f 3 there are D>0 and <q< such that ft t + Dt q for any t 0. ow we state our first result. Theorem.. Assume V V 3, f f 3 and = with, or =3with 3 4,. i If q,, there is D > 0 such that, for D D, K admits a positive ground state solution. 4 ii If q,, for any D>0, K admits a positive ground state solution. Remark.. It is worth pointing out that we have to restrict, when =or 3 4, if = 3 in the process of proving the Mountain-Pass geometry for the corresponding energy functional. Moreover, in order to get a positive ground state solution, we construct a perturbed functional whose nontrivial critical points can be proved in such restrictions on. However, when is small, the argument in proving the two statements above does not work any more. For the details, see Lemmas 3.3 and 6.. We point out that without any symmetry assumption on V, the ground state solution obtained above maybe is not radially symmetric. In the following, we impose a monotonicity assumption of V and show that K admits a radially symmetric solution. Assume now that V is radially symmetric and increasing, that is for all x, y : x y V x V y. V 4

odea Ground states for fractional critical Kirchhoff equations Page 5 of 3 50 Theorem.3. Under the assumptions of Theorem. and V 4, K admits a radially symmetric positive solution at the global unrestricted to radial paths mountain pass energy level. As a main tool to prove Theorem. we shall give the profile decomposition of the Palais Smale sequences by which we can derive some compactness and get a positive ground states for K. The main tool for the proof of Theorem.3 is a symmetric version of the monotonicity trick [35]. We recall that Zhang and Zou [4] studied the critical case for Berestycki Lions theorem of the Schrödinger equation Δu+V xu = fu. They obtained positive ground state solutions when V satisfies similar assumptions as V V 3, f satisfies f f 3 and f 4 f 4 t C + t, for t>0 and some C>0. We should mention that in the present paper, f 4 is removed..3. Main difficulties We mention the difficulties and the idea in proving Theorem.. Firstly, without the Ambrosetti Rabinowitz condition, it is difficult to get the boundedness of Palais Smale sequences. In order to overcome this difficulty, inspired by [], we will use the monotonicity trick developed by Jeanjean [8], introduce a family of functionals I λ and obtain a bounded PS cλ sequence for I λ for almost all λ in an interval J, where c λ is given in Sect. 3. Secondly, by the presence of the Kirchhoff term, one obstacle arises in getting the compactness of I λ, even in the subcritical case. Precisely, this does not hold in general: for any ϕ C0, Δ un dx Δ un Δ ϕdx R Δ u dx Δ u Δ ϕdx, where {u n } n is a PS-sequence of I λ satisfying u n uin H. Then, even in the subcritical case, it is not clear that weak limits are critical points of I λ.in[], for.4 the compactness was recovered by restricting I λ to the radial space Hrad R, which is compactly embedded L s for all s,. For the related works in the bounded domains, see e.g. [4,30,39]. In the present paper, we do not impose any symmetry and just consider K in H. So the arguments mentioned above cannot be applied. Inspired by [], in place of I λ, we consider a family of related functionals J λ,whose corresponding problem is a non Kirchhoff equation. Thirdly, the critical exponent makes the problem rather tough. The PS- condition does not hold in general and to overcome this difficulty, we show that the mountain pass level c λ is strictly less than some critical level c λ. For Δu + V xu = λfu with critical growth, if S is the best constant of D, L, one can show that [7] c λ = S λ.

50 Page 6 of 3 Z. Liu, M. Squassina and J. Zhang odea For a + b u dx Δu + V xu = λfu inr 3 involving critical growth R 3 [, 4] c λ = ab 4λ S3 + [b S 4 +4λaS] 3 4λ + b3 S 6 4λ. However, for fractional Kirchhoff equations, to give the exact value of c λ is complicated, since one cannot solve precisely a fractional order algebra equation. A careful analysis is needed at this stage. With an estimate of c λ, inspired by [4], we establish a profile decomposition of the Palais Smale sequence {u n } n Lemma 5.4 related to J λ. Thanks to this result, for almost every λ [/, ] we obtain a nontrivial critical point u λ of I λ at the level c λ.finally, choosing a sequence λ n [/, ] with λ n, thanks to the Pohožaev identity we obtain a bounded PS c -sequence of the original functional I. Using the idea above again, we obtain a nontrivial solution of problem K. Throughout this paper, C will denote a generic positive constant. The paper is organized as follows. In Sect., the variational setting and some preliminary lemmas are presented. In Sect. 3, we consider a perturbation of the original problem K. Then using the monotonicity trick developed by Jeanjean, we obtain the bounded PS cλ -sequence {u n } n for almost all λ. In Sect. 4, an upper estimate of the mountain pass value is obtained and the limit problem is discussed. In Sect. 5, we give the profile decomposition of {u n } n. In Sect. 6, Theorem. and.3 are finally proved.. Variational setting In this section we outline the variational framework for K and recall some preliminary lemmas. For any 0,, the fractional Sobolev space H R 3 is defined by { } ux uy H := It is known that u L : R ux uy x y + dxdy =Cn, x y + L Δ u dx, where cos ζ Cn, = R ζ + dζ. We endow the space H with the norm / u H := u dx + Δ u dx..

odea Ground states for fractional critical Kirchhoff equations Page 7 of 3 50 H is also the completion of C0 with H and it is continuously embedded into L q forq [, ]. The homogeneous space D, is { } D, := u L ux uy : L, x y + and it is also the completion of C0 with respect to the norm / u D, := Δ u dx. Lemma.. orm equivalence Assume V V 3. Then there exist ε 0 > 0 and ω>0 with a ε Δ u + V xu dx ω u dx, for every u H and all ε 0,ε 0. Proof. By contradiction, for ε 0 = ω =/n there exist ε n 0and{u n } n H with a εn Δ un + V xu n dx u n dx. n Then, up to a standard nomalization, we may assume that u n H = and a εn Δ un + V xu n dx R n. In view of V 3, we get u n 0, which implies from V and the above inequality that {u n } n goes to zero in D,. Therefore u n 0inH, which contradicts the normalization. Let H := { } u H : V xu dx < be the Hilbert space equipped with the inner product u, v H := a Δ u Δ vdx+ V xuv dx, R and the corresponding induced norm / u := a Δ u dx + V xu dx. From Lemma., it easily follows that the above norm is equivalent to H. A function u His a weak solution to problem K if, for every ϕ H,we

50 Page 8 of 3 Z. Liu, M. Squassina and J. Zhang odea have a + b Δ u dx Δ / u Δ / ϕdx + V xuϕdx = fuϕdx. We stress that, under assumptions V V 3 andf f 3, if u is a weak solution to the above problem, then u is globally bounded and Hölder regular allowing the pointwise reppresentation of Δ u by the results of [3]. In particular u>0 a.e. wherever u 0. In what follows, we recall a fractional varsion of Lions lemma whose proof can be seen in [3]. Lemma.. Lions lemma Assume that {u n } n is bounded in H and lim sup u n dx =0, n y B ry for some r>0. Thenu n 0 in L s for all s,. The energy functional associated with K, I : H R, is defined as Iu = u + b Δ u dx F udx, u H, 4 with F u = u 0 ftdt. Obviously I C H and its critical points are weak solutions to K. 3. The perturbed functional Since we do not impose the well-known Ambrosetti Rabinowitz condition, the boundedness of the Palais Smale sequence becomes complicated. To overcome this difficulty, we adopt a monotonicity trick due to Jeanjean [8]. Theorem 3.. Monotonicity trick [8] Let E, be a real Banach space with its dual space E and J R + an interval. Consider the family of C functionals on E I λ = Au λbu, λ J, with B nonnegative and either Au + or Bu + as u, satisfying I λ 0 = 0. We set Γ λ := {γ C[0, ],E γ0 = 0, I λ γ < 0}, for all λ J. If for every λ J, Γ λ is nonempty and c λ = inf max I λγs > 0, γ Γ λ s [0,] then for almost any λ J, I λ admits a bounded Palais Smale sequence {u n } n E, namely sup n u n <, I λ u n c λ and I λ u n 0 in E. Moreover λ c λ is left continuous.

odea Ground states for fractional critical Kirchhoff equations Page 9 of 3 50 Set J := [, ], E := H and Au := u + b Δ u dx, Bu := F udx. 4 We consider the family of functionals I λ : H R defined by I λ u =Au λbu, that is I λ u = u + b Δ u dx λ F udx. 4 It is easy to see that Bu 0 for all u Hand Au + as u. In the following H denotes a closed half-space of containing the origin, 0 H. We denote by H the set of closed half-spaces of containing the origin. We shall equip H with a topology ensuring that H n H as n if there is a sequence of isometries i n : such that H n = i n H and i n converges to the identity. Given x, the reflected point σ H x will also be denoted by x H. The polarization of a nonnegative function u : R + with respect to H is defined as { u H max{ux,uσ H x}, for x H, x := min{ux,uσ H x}, for x \H. Given u, the Schwarz symmetrization u of u is the unique function such that u and u are equimeasurable and u x =h x, where h :0, R + is a continuous and decreasing function. We set H + := {u H: u 0}. ow we state a symmetric version of Theorem 3.. Lemma 3.. Symmetric monotonicity trick [35] Under the assumptions of Theorem 3. for E = H, assume that I λ u I λ u for any λ J and u H and I λ u H I λ u, for any λ J, u H + and H H. Then, for any p [, ], I λ has a bounded Palais Smale sequence {u n } n H with u n u n p 0. Lemma 3.3. Uniform Mountain-Pass geometry Assume that f f 3 and V V 3 hold. Furthermore let =with, or =3with 3 4,. Then we have: Γ λ, for every λ J; there exist r, η > 0 independent of λ, such that u = r implies I λ u η. In particular c λ η. Proof. For every ϕ H + \{0}, taking into account of f 3, we have I λ ϕ ϕ + b Δ ϕ dx D ϕ q dx 4 R q R ϕ dx. Under the assumptions on and, it follows that > 4. Then there exists t 0 > 0 sufficiently large, independent of λ J, such that I λ t 0 ϕ < 0. Setting

50 Page 0 of 3 Z. Liu, M. Squassina and J. Zhang odea w := t 0 ϕ H,wehaveI λ w < 0 and we can define the corresponding Γ λ. Then, setting γt :=tw, wehaveγ Γ λ for every λ J. ii f -f imply that, for any ε>0, there exists C ε > 0 such that F s ε s + C ε s, for all s R. Then there exist σ,σ > 0 such that I λ ϕ σ ϕ σ ϕ, for every ϕ H. Hence there exist r, η > 0, independent of λ, such that for u = r, I λ u η and I λ ϕ > 0assoonas ϕ r with ϕ 0. ow fix λ J and γ Γ λ. Since γ0 = 0 and I λ γ < 0, certainly γ >r. By continuity, we conclude that there exists t γ 0, such that γt γ = r. Therefore, for every λ J, we conclude c λ inf γ Γλ I λ γt γ η>0. Lemma 3.4. I λ decreases upon polarization Assume V 4 holds. Then for any λ J, for all u H + and H H there holds I λ u H I λ u. Proof. It is knownsee[5, Theorem ] that R u H x u H y R ux uy x y + dxdy x y + dxdy, for all u H+. Furthermore, we have see [38] F u H dx = F udx, for all u H +, and, by the monotonicity assumptions on V, V xu H dx V xu dx, for all u H +, which concludes the proof by the definition of I λ. Assume V V 3 andf f 3. As a consequence we now get the following result. Corollary 3.5. Bounded Palais Smale with sign For almost every λ J, there is a bounded sequence {u n } n H + such that I λ u n c λ, I λ u n 0. Furthermore, u n u n 0ifV 4 is assumed. Proof. For a.a. λ J, a bounded PS-sequence {u n } n Hfor I λ is provided by combining Theorem 3. with Lemmas 3.3 and 3.4. Furthermore, if V 4 holds, using Lemma 3. in place of Theorem 3., we also get u n u n 0. ext we show that we can assume that u n is nonnegative. Indeed, we know that I λ u n,u n = μ n,u n with μ n 0inH as n, with u n = min{u n, 0}, namely fs =0fors 0 a + b Δ un dx Δ u n u n dx+ V x u n dx = μ n,u n. As it is readily checked, for all x, y,wehave u n x u n yu n x u n y u n x u n y,

odea Ground states for fractional critical Kirchhoff equations Page of 3 50 which yields that Cn, R Δ u n u n dx 3 u n x u n yu n x u n y = R x y + dxdy R u n x u n y x y + dxdy =Cn, u n D,. Thus u n = o n, which also yields that {u + n } n is bounded. We can now prove that I λ u + n c λ and I λ u+ n 0. Of course u n = u + n + o n and Δ un dx = Δ u + n dx + o n. otice that from f f, we get F u n dx F u + n dx C u n + u n u n C u n + C u n = o n. This shows that I λ u + n c λ. We claim that I λ u+ n 0. Setting w n := I λ u n I λ u+ n, it is enough to prove that w n 0inH. For any ϕ Hwith ϕ H, we have w n,ϕ = a + b Δ un dx Δ / u n Δ / ϕdx R a + b Δ u + n dx Δ / u + n Δ / ϕdx R + V xu n ϕdx λ fu n fu + n ϕdx R = a + b Δ u + n dx Δ / u n Δ / ϕdx R + V xu n ϕdx λ fu n ϕdx + ξ n,ϕ, for some ξ n 0inH. Then, by f -f, w n,ϕ C u n H + ξ n H, proving the claim. Observe now that by the triangular inequality and the contractility property of the Schwarz symmetrization in L p -spaces i.e. w z p w z p for all w, z L p with w, z 0, we get u + n u + n u n u n u n +u + n u n u n + u + n u n u n + u + n u n = u n C u n H = o n. Since u n u n 0, we have u + n u + n 0asn. This ends the proof.

50 Page of 3 Z. Liu, M. Squassina and J. Zhang odea 4. Upper estimate of c λ and limit problems In this section, we give an upper estimate of the mountain pass value c λ. Moreover, the corresponding limit problem is discussed. 4.. An energy estimate ext we provide a crucial energy estimate for c λ. Lemma 4.. Energy estimate Suppose that f f 3 and V V 3 hold. For any λ [, ], assume that ] 4 q, 4 or q, with D large enough. Then we have c λ <c λ, c λ := as T + bs 4 T 4 λ T, where T = T λ > 0 continuously depends on λ. Proof. Let η C0 R 3 be a cut-off function with support in B 0 such that η onb 0 and η [0, ] on B 0. It is well known that S is achieved by T x :=κ μ + x x 0 for arbitrary κ R, μ>0andx 0. Then, taking x 0 = 0, we can define T x/s v ε x :=ηxu ε x, u ε x =ε u x/ε, u / x :=. T Then Δ u ε = u ε u ε and Δ u ε = u ε = S.Asin[33], we have A ε := Δ vε x dx = S + Oε. 4. On the other hand, for any q [,, we obtain v ε q dx ε q κ q T q S S / εs / r dr, 0 μ + r q where S is the unit sphere in. Observe that, as ε 0, εs / r c 0,, if q>, dr = Olog 0 μ + r q ε, if q =, = Oε q, if q<. Then q Oε, if q> C ε := v ε q, dx Olog R q ε ε, if q =, 4. Oε q, if q<.

odea Ground states for fractional critical Kirchhoff equations Page 3 of 3 50 Since <, Similar as above, v ε dx Oε. v ε dx ε κ T S S / Oε. εs / 0 r μ + r dr So that we have B ε := v ε dx = Oε. 4.3 As can be seen in [33], it holds D ε := v ε dx = S + Oε. Step. For any ε>0 small there exists t 0 > 0 such that I λ γ ε t 0 < 0, where γ ε t :=v ε /t. Indeed, by V andf 3, for any t>0, I λ γ ε t a Δ γε t dx + b Δ γε t dx R 4 + V [ γ ε t γε t dx λ R + D γ εt q ] dx q = aa ε t + ba ε V B ε 4 t 4 + λd ε λdc ε t. 4.4 q oting that <<4, wehave0< 4 <. Then by 4.3, V B ε λd ε λs, as ε 0. So it follows from 4. that for any ε>0 small enough, I λ γ ε t as t +. Then there exists t 0 > 0 such that I λ γ ε t 0 < 0. Step. otice that, as t 0 +,wehave [ Δ γε t + γ ε t ] dx = t A ε + t B ε 0 uniformly for ε>0 small. We set γ ε 0 = 0. Then γ ε t 0 Γ λ, where Γ λ is as in Theorem 3. and c λ sup I λ γ ε t. t 0 Recalling that c λ > 0, by 4.4, there exists t ε > 0 such that sup I λ γ ε t = I λ γ ε t ε. t 0

50 Page 4 of 3 Z. Liu, M. Squassina and J. Zhang odea By 4., 4. and 4.4, we get I λ γ ε t 0 + as t 0 + and I λ γ ε t as t + uniformly for ε>0 small. Then there exist t,t > 0 independent of ε>0 such that t t ε t.let J ε t := aa ε t + ba ε 4 t 4 λd ε t, then V B ε c λ sup J ε t+ λdc ε t ε t 0 q By formula 4., for any q,, we have Then by 4.3, we conclude that q C ε Oε. c λ sup J ε t+oε q ODε. t 0 oting that >0and q/ > 0, we have sup t 0 J ε t c λ / uniformly for ε>0 small. As above, there are t 3,t 4 > 0 independent of ε>0 such that sup t 0 J ε t =sup t [t3,t 4] J ε t. By 4., where c λ sup K t 0 Observe that for t>0, S t + O ε O Dε q Kt = as t + bs 4 t 4 λ t., 4.5 K t = t Kt, where Kt :=as + bs t λt, and K t =t bs λt 4. Since 4 >, there is a unique T>0 such that Kt > 0ift 0,Tand Kt < 0ift>T. Hence, T is the unique maximum point of K. Then by 4.5, c λ KT +Oε O Dε q. 4.6 If q>4/, then 0 < q/ <, which implies by 4.6 that for any fixed D>0, c λ <KT forε>0 small. If <q 4/, for ε>0 small and D ε q/, then also in this case c λ <KT, which completes the proof.

odea Ground states for fractional critical Kirchhoff equations Page 5 of 3 50 4.. The limit problem ote that V x V as x. For any λ [/, ], we consider the problem a + b Δ u dx Δ u + V u = λfu in, R u H, u > 0 in, whose energy functional is defined by Iλ u := a Δ u + V u dx + b Δ u dx R 4 R λ F udx. We will use of the following Pohožaev type identity, whose proof is similar as in []. Lemma 4.. Pohožaev identity Let u be a critical point of Iλ in H for λ [, ]. ThenP λu =0, P λ u := a Δ u dx + b Δ u dx R + V u dx λ F udx. 4.7 otice that P λ u = d dt I λ u /t t=. Lemma 4.3. For λ [, ], ifw λ H\{0} solves P λ w λ =0, then there exists γ λ C[0, ], H such that γ λ 0 = 0, Iλ γ λ < 0, w λ γ λ [0, ], 0 γ λ 0, ] and max t [0,] I λ γ λ t = Iλ w λ. Proof. ote that Iλ w λ /t = t a Δ wλ dx + bt 4 Δ wλ dx R 4 R + t V w λdx t λ F w λ dx =0, which, by 4.7, yields lim t I λ w λ /t < 0. Then there is t 0 > 0 such that Iλ w λ /t 0 < 0. Let γ λ t = w λ /tt 0 for 0 <t andγ λ 0 = 0. Then γ λ C[0, ], H, w λ γ λ [0, ] and max t [0,] Iλ γ λt = Iλ w λast = t 0 is the unique maximum point of t Iλ γ λt by Lemma 4..

50 Page 6 of 3 Z. Liu, M. Squassina and J. Zhang odea 5. Behaviour of Palais Smale sequences By Corollary 3.5, for almost every λ [/, ], there exists a bounded Palais Smale sequence {u n } n H for I λ at the level c λ. Then there exists a subsequence of {u n } n, still denoted by {u n } n, such that u n u 0 in H and u n u 0 a.e. in as n.let vn := u n u 0. Then vn 0inH and vn 0 a.e. otice that, since u n 0, the dominated convergence theorem implies that vn 0inL q for any q. 5.. Splitting lemmas Let us set gt :=ft t +, Gt := t 0 gsds. In order to get the profile decomposition of {u n } n, we state the following splitting lemmas. Lemma 5.. Splitting lemma I We have gu n gu 0 gvnϕdx o n ϕ, 5. where o n 0 as n, uniformly for any ϕ C0. Proof. For each n, there exists θ n 0, such that gu n gvn g vn + θ n u 0 u 0. 5. In view of f -f 3, for any ε>0, there exists D >0 such that gt ε t, for t D/. 5.3 Let Ω n D :={x : u n x D} and for r>0, B r := {x : x < r}, Br c := \B r 0. Since u 0 H,wehave BR c { u 0x D/} 0as R. Then for ε given as above, there exist R>0and Ω R with Ω R Λ ε such that u 0 x < D/ forx BR c \Ω R, where Λ ε > 0 will be chosen later small enough. Then, by Hölder s inequality, 5. and 5.3, we have gu n gvn ϕ dx BR c \ΩR gu n gvn ϕ dx BR c \ΩR Ωn D + gu n gvn ϕ dx B c R \ΩR Ωc n D εc u n + vn ϕ + max g t t D B c R u 0xdx / ϕ. 5.4

odea Ground states for fractional critical Kirchhoff equations Page 7 of 3 50 It follows from f andf that, for ε>0 given, there exists C ε = C ε f > 0 such that gu n gvn ϕ dx Ω R ε u n + vn ϕ dx + C ε εc Ω R u n + vn ϕ + C ε Ω R un + vn ϕ dx Ω R un + v n ϕ. 5.5 By 5.4 and 5.5, by choosing Λ ε such that C ε Λ / ε ε, there exists C>0 with gu n g vn ϕ dx Cε ϕ. 5.6 B c R Moreover, gu 0 ϕ dx C B c R C B c R u 0 ϕ dx + u 0 ϕ dx BR c / u 0 dx / ϕ + C u 0 dx ϕ. B c R 5.7 It follows from 5.6 and 5.7 that, for ε>0 above, we choose R>0above large enough such that gu n gu 0 gvnϕdx Cε ϕ, 5.8 B c R where C is independent of n, ε and ϕ C0. On the other hand, gu n gu 0 ϕ dx B R gu n gu 0 B R Observe that / dx g t 0 + g / t lim = lim t + t B c R / BR / t t / =0. ϕ Then gu n gu 0 / 0inL B R. Hence, we deduce /. gu n gu 0 ϕ dx o n ϕ. 5.9 B R Similarly, we also obtain that gvn ϕdx o n ϕ, 5.0 B R for any ϕ C0. It follows from 5.8, 5.9 and 5.0 that 5. holds.

50 Page 8 of 3 Z. Liu, M. Squassina and J. Zhang odea Lemma 5.. Splitting lemma II We have u n u n u 0 u 0 vn vn ϕdx o n ϕ, where o n 0 as n, uniformly for any ϕ C0. Proof. For any ε>0, there exists R = Rε > 0 such that u n u n u 0 u 0 vn v n ϕdx \B R0 u n u n vn v n ϕdx \B R0 + u 0 5. u 0 ϕdx \B R0 C u n + vn u 0 ϕ dx \B R0 + u 0 ϕ dx Cε ϕ. \B R0 On the other hand, for every r>0, we have u n u n u 0 u vn vn ϕdx B R0 u n u n u 0 u 0 vn v B R0 { vn r} n ϕdx + u n u n u 0 u 0 vn vn ϕdx =: I + I. B R0 { v n r} ow, there exists r = rr such that r B R 0 / ε. Therefore, we have I C u n + u 0 + vn vnϕ dx B R0 { v n r} Cr B R 0 / ϕ Cε ϕ. 5. For such r, R fixed above, u n converges to u in measure in B R 0, i.e. B R 0 { v n r} 0forn. Therefore, for n large, I C u n + vn u 0 ϕ dx B R0 { vn r} + u 0 ϕ dx Cε ϕ. 5.3 B R0 { vn r} Then 5., 5. and 5.3 yield the assertion. Lemma 5.3. Splitting lemma III We have fu n u n dx = fvnv ndx + fu 0 u 0 dx + o n,

odea Ground states for fractional critical Kirchhoff equations Page 9 of 3 50 where o n 0 as n. Furthermore F u n dx = F vndx + F u 0 dx + o n. Proof. Since ft =gt+t for t 0, by the standard Brezis Lieb lemma, it suffices to prove gu n u n dx = gvnv ndx + gu 0 u 0 dx + o n, where o n 0asn.Fixedε>0, there exists C ε > 0 such that gt εt + C ε t, t 0. 5.4 Then there exists R = Rε > 0 large enough such that gu 0 vndx gu 0 vn dx R B + gu 0 vn dx R BR c ε u 0 + C ε u 0 vn dx + ε vn + vn and B R Cε + C ε o n. g v n u0 dx g R B vn u0 dx + g vn u0 dx R BR c ε vn + C ε vn u 0 dx B R + ε vn + C ε vn u 0 dx B c R Cε + C ε o n. 5.5 5.6 It follows from 5.5, 5.6 and Lemma 5. that gu n u n gu 0 u 0 gvnv ndx gu n gu 0 gvnu R R n dx + gvnu 0 dx + gu 0 vn dx o n u n + Cε + C ε o n. Letting n and ε 0 + completes the proof of the first assertion. The second assertion follows from the standard Brezis Lieb lemma and Gu n dx = Gvndx + Gu 0 dx + o n, whose proof is left to the reader.

50 Page 0 of 3 Z. Liu, M. Squassina and J. Zhang odea 5.. Profile decomposition In the following, we give the profile decomposition of {u n } n, which plays a crucial role in getting the compactness. Since c λ > 0, for some B >0wehave Δ un dx B, as n. ow, for any u H,let J λ u := a + b B Δ u dx + V x u dx λ F udx R and Jλ u := a + b B Δ u dx + V u dx λ F udx, R which are respectively the corresponding functional of the following problems a + b B Δ u + V xu = fu, a + b B Δ u + V u = fu, u H. Here we point out that in contrast with the original problem K, the problems above are both non Kirchhoff. ow we take advantage of this to get the profile decomposition of {u n } n. Lemma 5.4. Profile decomposition Let {u n } n Hbe the sequence mentioned above and assume that conditions V -V 3, f -f 3 hold and <4. Then J λ u 0=0with u 0 0, and there exist a number k {0}, nontrivial positive critical points w,...,w k H of Jλ which decay polynomially at infinity as w j x x + = O, such that i yn +, j yn j yn + i if i j, i, j k, n +, ii c λ + b B 4 4 = J λ u 0 + k Jλ wj, j= iii u n u 0 k w j yn j 0, j= iv B = Δ u 0 + k Δ w j. j= Moreover, we agree that in the case k =0the above holds without w j.in addition, if V 4 holds, then k =0and u 0 H rad R. Proof. Observe that, from I λ u n =c λ + o n and I λ u n 0inH,we obtain J λ u n =c λ + b B 4 4 + o n, J λu n 0 in H.

odea Ground states for fractional critical Kirchhoff equations Page of 3 50 Then, it is standard to get J λ u 0ϕ = 0 for all ϕ H. From Lemma 5.3, we get F vndx = F u n dx F u 0 dx + o n, R fvnv ndx = fu n u n dx fu 0 u 0 dx + o n. It follows that J λ u n =J λ vn+j λ u 0 +o n, 5.7 J λv nv n = J λu n u n J λu 0 u 0 + o n = o n. 5.8 On the other hand, by a slight variant of [, Proposition 4.], u 0 satisfies the Pohǒzaev identity a + b Δ u0 dx + V x xu R 0dx + V xu 0dx λ F u 0 dx =0. Then by V and<4, wehave J λ u 0 =a + b B Δ u0 dx V x xu R 0dx R a + b B Δ u0 dx R W u 0 a + b B Δ u0 dx a Δ u0 dx R = b B Δ u0 dx > 0, which implies that J λ u 0 b B Δ u0 dx. 4 5.9 We claim that one of the following conclusions holds for vn: v vn 0inH, or v there exist r > 0, σ>0 and a sequence {yn} n such that lim inf n vn dx σ>0. B r yn 5.0 Indeed, suppose that v does not occur. Then for any r>0, we have lim vn dx =0. n y B ry Therefore, it follows from Lemma. that vn 0inL s fors,. It follows from 5.4 that for any ε>0, there exists C ε > 0 such that gvnv n dx ε vn + vn dx + C ε vn q dx.

50 Page of 3 Z. Liu, M. Squassina and J. Zhang odea So from vn 0inL q and the arbitrariness of ε, we can easily obtain that fvnv ndx = vn + dx + on. Furthermore, from J λ v nvn = o n in 5.8, we have vn + b B Δ v n dx = λ vn + + o n. 5. In view of conditions V -V 3, we can check that V > 0. And so we can also get V x vn dx = V + x vn dx + o n, which, together with the definition of S and 5., implies that as vn 4 dx + bs vn dx λ vn dx + on. 5. Let l 0 be such that v R n dx l.ifl>0, then it follows from 5. that K l l = as l + bs l 4 λl 0, where K has been defined in Lemma 4.. This also implies that l T T is the unique maximum point of K. On the other hand, by 5.7 and 5.9, we have c λ + b B 4 4 a + b = B R + J λ u 0 +o n a + b B 4 + b B 4 4 + o n, Δ v n + V x v n λ v n Δ v n + V x v n λ + v n dx + which, together with 5. and the definition of S, implies that c λ a Δ v n dx + R 4 b Δ v n dx + o n + o n. as v n dx + 4 dx 4 bs v n dx

odea Ground states for fractional critical Kirchhoff equations Page 3 of 3 50 Thus, combining v R n dx l and l T, K T =0,wehave c λ as l + 4 bsl 4 as T + 4 bst 4 = as T + 4 bs T 4 λ T = c λ, contradicting c λ <c λ. Hence, l = 0. It follows from 5. that v n 0, that is, u n u 0 in H. Then Lemma 5.4 hold with k = 0 if v does not occur. In particular, if we assume V 4 holds, then by Corollary 3.5, u n u n 0. Obviously, { u n } n H rad R is bounded and u n u n q 0for q,. Since { u n } n has a strongly convergent subsequence in L q for q,, without loss of generality, we assume that u n u 0 in L q for q, andu 0 = u 0. As a consequence, v does not hold and as above, u n u 0 in H. In the following, otherwise, suppose that v holds, that is 5.0 holds. Consider v n +y n. The boundedness of {v n} n and 5.0 imply that v n + y n w 0 in H. Thus, it follows from v n 0 in H that {y n} n is unbounded and, up to a subsequence, y n +. Let us prove that J λ w = 0. It suffices to show that J λ v n + y nϕ 0 for any ϕ C 0. Combining Lemmas 5. and 5., we obtain J λu n ϕ J λu 0 ϕ J λv nϕ o n ϕ, ϕ C 0, which implies that J λ v nϕ o n ϕ, for all ϕ C0, as n. otice that J λv nϕ y n Cn, = a + b B v nx vnyϕx yn ϕy yn R x y + dxdy + V xvnxϕx yndx λ gvnxϕx yndx R λ vnx + ϕx yndx = o n ϕ yn = o n ϕ. Thus, as n, it follows that Cn, a + b B v nx + yn v n y + ynϕx ϕy R x y + dxdy + V x + ynv nx + ynϕxdx λ gvnx + ynϕxdx R vnx + yn + ϕxdx = o n ϕ. 5.3

50 Page 4 of 3 Z. Liu, M. Squassina and J. Zhang odea Since y n and ϕ C 0, we obtain V x + yn V vnx + ynϕxdx 0. 5.4 Thus, combining 5.3 and 5.4, we have for any ϕ C0, Jλ vn + ynϕ Cn, = a + b v nx + yn v n y + ynϕx ϕy R x y + dxdy + V vnx + ynϕxdx λ R gvnx + ynϕxdx λ vnx + yn + ϕxdx = o n. Then, Jλ w =0,w > 0andw x x + = O as x. Finally, let us set vnx =vnx w x yn, 5.5 then vn 0inH. Since V x V as x and vn 0 strongly in L loc R, we have V x V vn dx = o n. It follows that V x vn dx = V x vn dx + V x + yn w x dx V x + ynv nx + ynw xdx R = V u n dx V u 0 dx R V w dx + o n = V x u n dx R V x u 0 dx V w dx + o n, 5.6 and it is easy to see that vn =o n also Δ vn = Δ u n Δ u 0 Δ w + o n, 5.7 vn + = u n u 0 w + o n, Gvndx = Gu n dx Gu 0 dx Gw dx. 5.8 It is readily checked that we also have gvnϕdx = gu n ϕdx gu 0 ϕdx R 5.9 gw ynϕdx + o n ϕ,

odea Ground states for fractional critical Kirchhoff equations Page 5 of 3 50 for any ϕ C0. Combining 5.6, 5.7, 5.8 and 5.9, we deduce that J λ vn=j λ u n J λ u 0 Jλ w +o n, J λv nϕ = J λu n ϕ J λu 0 ϕ Jλ w ynϕ + o n ϕ = o n ϕ, 3 Jλ vn=j λ vn Jλ w +o n for any ϕ C0. Thus {vn} n is a Palais Smale sequence and we get J λ vn=c λ + b B 4 4 J λu 0 Jλ w +o n <c λ + b B 4 4. Remark that one of v and v holds for vn.ifv n 0inH, then Lemma 5.4 holds with k =. Otherwise, {vn} is non-vanishing, that is, v holds for vn. Similarly, we repeat the arguments. By iterating this procedure we obtain sequences of points {yn} R j such that yn +, j yn j yn + i if i j as n + and vn j = vn j w j x yn j like 5.5 with j such that vn j 0inH, Jλ w j = 0. Using the properties of the weak convergence, we have k k a u n u 0 w j yn j = u n u 0 w j yn j + o n, j= bj λ u n =J λ u 0 + k j= j= J λ w j +J λ v k+ n +o n. 5.30 ote that there is ρ>0 such that w ρ for every nontrivial critical point w of Jλ and {u n} n is bounded in H. By5.30a, the iteration stops at some k. Thatis,vn k+ 0inH. We stress that the polynomial decay of the limiting profiles w j can be justified as in [3, Theorem 3.4 and Theorem.5]. The proof is now complete. 6. Proof of the main results In order to obtain the existence of ground state solutions of problem K, our strategy is that we firstly obtain the existence nontrivial solutions of the perturbed problem, then as λ goes to, we get a nontrivial solution of the original problem. Finally, thanks to the profile decomposition of the PS- sequence, we obtain the existence of ground state solutions of problem K. 6.. ontrivial critical points of I λ Lemma 6.. Assume that V -V 3 and f -f 3 hold. For almost every λ [/, ], there exists u λ H\{0} such that I λ u λ =c λ and I λ u λ=0.in addition, if V 4 holds, then u λ H rad R.

50 Page 6 of 3 Z. Liu, M. Squassina and J. Zhang odea Proof. For almost all λ [/, ], there is a bounded sequence {u n } n H such that I λ u n c λ, I λ u n 0. From Lemma 5.4, up to a subsequence, there exist u 0 Hand B >0 such that u n u 0 in H, Δ un dx B, as n and J λ u 0 = 0. Furthermore, there exist k {0}, nontrivial critical points w,...,w k of Jλ and k sequences of points {yj n}, j k, such that k u n u 0 w j yn j 0, c λ + b B 4 k 4 = J λu 0 + Jλ w j 6. and j= B = Δ u0 + j= k Δ w j. 6. ow we claim that if u 0 0, then by <4, J λ u 0 > b B Δ u0 dx. 6.3 4 Indeed, since J λ u 0 = 0, similar as in [], we get P λ u 0 := a + b B Δ u0 dx + V xu R 0dx + V x,xu 0dx λ F u 0 dx =0. By hypothesis V wehave J λ u 0 = a + b B Δ u0 dx V x xu R 0dx > b B Δ u0 dx, which implies that 6.3 holds. For each nontrivial critical point w j, j =,..., k ofjλ, a + b B Δ w j dx + V w j dx R R λ F w j dx = Pλ w j =0. Then it follows from 6. that a Δ w j b dx + Δ w j dx R + V w j dx λ F w j dx 0. j=

odea Ground states for fractional critical Kirchhoff equations Page 7 of 3 50 Then there exists t j 0, ] such that at j Δ w j dx + bt 4 j + t j Δ w j dx 6.4 V w j dx t j λ F w j dx =0. That is, w j /t j satisfies the identity P λ u = 0 and it follows from Lemma 4.3 that there exists γ λ C[0, ], H such that γ λ 0 = 0, Iλ γ λ < 0, w j γ λ [0, ] and Iλ w j /t j = max t [0,] I λ γ λ t. By hypothesis V, we have max t [0,] Iλ γ λt max t [0,] I λ γ λ t, which, by the definition of c λ, implies that Iλ wj t j c λ. In particular, if V x V, then Iλ w j /t j >c λ. 6.5 So by 6.4 wehave Δ w j dx Jλ w j =Jλ w j P λ w j =a + b B a Δ w j x dx R t j + 4 b Δ w j x dx R t j + b B Δ w j dx 4 R = Iλ w j t j P λw j + b B Δ w j dx t j 4 R = Iλ w j + b B Δ w j dx t j 4 6.6 and then we conclude that Jλ w j c λ + b B Δ w j dx, 4 where the inequality is strict if V x V. Then by formulas 6.-6.3, c λ + b B 4 4 = J λu 0 + k Jλ w j kc λ + b B 4 j= 4, 6.7 with strict inequality if V x V or u 0 0. If k = 0, we are done. If condition V 4 holds, then k =0andu 0 is radial. Then it follows that I λ u 0 =J λ u 0 b B 4 /4=c λ and I λ u 0= J λ u 0=0. If k =andv x V or u 0 0, then 6.7 yields a contradiction

50 Page 8 of 3 Z. Liu, M. Squassina and J. Zhang odea If k =andv x V and u 0 = 0, then B = Δ w and it follows from 6. that c λ = Jλ w b B 4 4 = J λ w b 4 Δ w 4 = I λ w, I λw =0, as desired. Hence, in any case, the assertion follows. 6.. Completion of the proof Choosing a sequence {λ n } n [, ] satisfying λ n, we find a sequence of nontrivial critical points {u λn } n still denoted by {u n } n ofi λn and I λn u n =c λn. In particular, if V 4 holds, then {u n } n Hrad R. ow we show that {u n } is bounded in H. Remark that u n satisfies the Pohožaev identity as follows a Δ un dx + b Δ un dx R + V xu ndx + V x xu R ndx λ F u n dx =0. It follows that I λn u n = a Δ un dx + b Δ un dx R 4 V x xu ndx. Since c λ is continuous on λ, I λ n u n =c λn + o n <c λ n. It follows from V that there is a positive number κ 0, a such that W κs. Hence, a κ Δ un dx I λn u n, which implies that a Δ u n dx is bounded from above. By V 3, f - f andi λ n u n u n = 0, there is ν>0such that for any ε>0, there exists C ε > 0 with ν u ndx a Δ un dx + V xu ndx ε u ndx +C ε u n dx, which yields that {u n } n is bounded in L. Then {u n } n is bounded in H. By Theorem 3., lim Iu n = lim I λn u n +λ n F u n dx = lim c λ n n n n = c and for any ϕ C0, lim n I u n ϕ = lim I λ n n u n ϕ +λ n fu n ϕdx =0.

odea Ground states for fractional critical Kirchhoff equations Page 9 of 3 50 That is, {u n } n is a bounded Palais Smale sequence for I at level c. Then by Lemma 6., there is a nontrivial critical point u 0 Hradial, if V 4 holds for I and Iu 0 =c. Set ν =inf{iu :u H\{0}, I u =0}. Of course 0 <ν Iu 0 =c <. By the definition of ν, there is {u n } n H with Iu n ν and I u n = 0. We deduce that {u n } n is bounded in H. Up to a sequence, for some B >0, Δ un dx B. Let us set Ju :=J u andj u :=J u, for any u H. From Lemma 5.4 there exists u 0 Hsuch that u n u 0 in H and J u 0 = 0. Furthermore, there exist k {0}, nontrivial critical points w,...,w k of J and k sequences of points {yn} j n, j k, such that k u n u 0 w j yn j 4 b B k 0, ν + j= 4 = Ju 0+ J w j 6.8 j= and k B = Δ u0 + Δ w j. If k = 0, we are done. If k, assume by contradiction that u 0 0. Then, as in Lemma 6., Ju 0 > b B Δ u0 dx, 6.9 4 for each j there is t j 0, ] such that I w j /t j c, which is strict if V x V,and J w j c + b B Δ w j dx, 4 where the inequality is strict if V x V. Then by formulas 6.8-6.9 and ν c,weget c + b B 4 4 ν + b B 4 4 = Ju 0+ j= k j= J w j >kc + b B 4 4, a contradiction. Hence u 0 =0andk =, in which case a contradiction follows as in the proof of Lemma 6.. The proof is complete. Acknowledgements The authors would like to express their sincere gratitude to the anonymous referee for his/her valuable suggestions and comments.

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50 Page 3 of 3 Z. Liu, M. Squassina and J. Zhang odea [33] Servadei, R., Valdinoci, E.: The Brezis irenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67 0 05 [34] Silvestre, L.: Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. J. Math. 55, 55 74 006 [35] Squassina, M.: On the Struwe Jeanjean Toland monotonicity trick. Proc. R. Soc. Edinb. Sect. A 4, 55 69 0 [36] Tao, F., Wu, X.: Existence and multiplicity of positive solutions for fractional Schrödinger equations with critical growth. onlinear Anal. RWA 35, 58 74 07 [37] Teng, K.: Existence of ground state solutions for the nonlinear fractional Schrödinger Poisson system with critical Sobolev exponent. J. Differ. Equ. 6, 306 306 06 [38] Van Schaftingen, J.: Symmetrization and minimax principles. Commun. Contemp. Math. 7, 463 48 005 [39] Xiang, M., Zhang, B., Guo, X.: Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem. onlinear Anal. 0, 99 33 05 [40] Zhang, J.J., Zou, W.M.: A Berestycki Lions theorem revisted. Commun. Contemp. Math. 4, 4 0 [4] Zhang, J., Zou, W.M.: The critical case for a Berestycki Lions theorem. Sci. China Math. 4, 54 554 04 Zhisu Liu School of Mathematics and Physics University of South China Hengyang 400 Hunan People s Republic of China e-mail: liuzhisu83@sina.com Marco Squassina Dipartimento di Matematica e Fisica Universita Cattolica del Sacro Cuore Via Musei 4 5 Brescia Italy e-mail: marco.squassina@unicatt.it Jianjun Zhang College of Mathematics and Statistics Chongqing Jiaotong University Chongqing 400074 People s Republic of China e-mail: zhangjianjun09@tsinghua.org.cn Received: 6 December 06. Accepted: 5 July 07.