EXISTENCE OF HIGH-ENERGY SOLUTIONS FOR SUPERCRITICAL FRACTIONAL SCHRÖDINGER EQUATIONS IN R N

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1 Electronic Journal of Differential Equations, Vol 06 (06, No 3, pp 7 ISSN: URL: or EXISTENCE OF HIGH-ENERGY SOLUTIONS FOR SUPERCRITICAL FRACTIONAL SCHRÖDINGER EQUATIONS IN LU GAN, WEIMING LIU Abstract In this article, we study supercritical fractional Schrödinger equations Applying the finite-dimensional reduction method and the penalization method, we obtain the high-energy solutions for this equation Introduction and statement of main results This article is devoted to the study of the problem ( s u = V (xu (s+ɛ, u > 0, x, u 0 as x +, ( where (s = N Ns, N > s, 0 < s <, ɛ > 0, V is a positive continuous potential Here, the fractional Laplacian of a function f : R is expressed by the formula ( s f(x f(y f(x = CN,s p v dy R x y N+s N ( f(x f(y = C N,s lim δ 0 x y N+s dy, \B δ (x where C N,s is some normalization constant The operator ( s can be seen as the infinitesimal generators of Lévy stable diffusion processes (see [] The Lévy processes occur widely in physics, biology, chemistry and finance (see [, 3] The stable Lévy processes that give rise to equations with fractional Laplacians have recently attracted much research interest, and there are a lot of results in the literature on the existence of such solutions In [5], Barrios et al studied the existence and multiplicity of solutions to the following critical problem with convex-concave nonlinearities ( s u = λu q + u (s, u > 0, x Ω, u = 0, x \ Ω (3 As we now, the fractional power of the Laplacian can also be defined by using spectral decomposition The same problem considered in [5] but for this spectral fractional Laplacian has been treated in [6] As in [6] the purpose of this paper 00 Mathematics Subject Classification 35J0, 35B09, 35J60 Key words and phrases Fractional Schrödinger equation; supercritical; finite-dimensional reduction method c 06 Texas State University Submitted October 3, 06 Published December 0, 06

2 L GAN, W LIU EJDE-06/3 is to study the existence of wea solutions for (3 In [7], in order to construct solutions to the problem of the form ( s u = ɛhu q + + u p +, Dipierro et al used the Lyapunov-Schmidt reduction, that taes advantage of the variational structure of the problem For related results, we refer the reader to [7, 4, 9, 3, 4, 7, 9, 33, 34, 35] Let us come bac to equation ( Recently, many results on the existence of solutions for problem ( when ɛ = 0 have been obtained Liu [5] obtained infinitely many concentration solutions for ( under certain conditions Assume V = + τk and K has at least two critical points satisfying some local conditions, Chen and Zheng [3] proved the existence of two-pea solutions when the positive number τ is small enough When V, the existence of finite-energy sign-changing solutions to ( has been established by Garrido and Musso [0] In particular, DelaTorre et al [6] constructed a class of Delaunay-type solutions for ( When s =, problem ( reduces formally, to the classical Schrödinger equation u = V (xu +ɛ, u > 0, u 0 as x + (4 The study of problem (4 has attracted considerable attention in recent years, and there are several results in the literature on the existence of solutions When V is a perturbation of the constant, Ambrosetti et al [] and Cao et al [9] proved the existence of two or many positive solutions Li [5] proved that (4 has infinitely many positive solutions if V is periodic, while similar result was obtained in [38] if V has a sequence of strict local maximum points tending to infinity Wei and Yan [36] obtained solutions with large number of bumps near infinity for (4 with V being radial Meanwhile, they proved that the energy of these solutions can be arbitrarily large For related results, we refer the readers to [0,,, ] and the references therein The aim of this article is to show the existence of high-energy solutions for the fractional Schrödinger equation with slightly supercritical exponent We assume that the positive continuous potential V satisfies the following conditions: (A There exist constants q [0, s and C > 0, such that V (x C(+ x q for all x ; (A For some µ, r > 0, V C,µ (B r (y 0, and V (y 0 > 0, where y 0 is a strict local minimum point of V Now, we recall the basic theory on fractional Laplacian operator For s (0,, the nonlocal operator ( s in is defined on the Schwartz class through the Fourier transform ( s f(ξ = ξ s f(ξ, or via the Riesz potential And is the Fourier transform When f has some sufficiently regularity, the fractional Laplacian of a function f : R is expressed as ( That integral maes sense directly when s < / and f C 0,γ ( with γ > s, or if f C,γ ( with + γ > s It is well nown that ( s on with 0 < s < is a nonlocal operator In the remarable wor by Caffarelli and Silvestre [], this nonlocal operator was expressed as a generalized Dirichlet-to- Neumann map for a certain elliptic boundary value problem with a local differential operator defined on the upper half-space + + := {(x, y : x, y > 0} That is, for a function f : R, we consider the extension u : [0, + R

3 EJDE-06/3 FRACTIONAL SCHRÖDINGER EQUATIONS 3 that satisfies the equations Equation (6 can also be written as u(x, 0 = f(x, (5 x u + s u y + u yy = 0 (6 y div(y s u = 0, (7 which is clearly the Euler-Lagrange equation for the functional J(u = u y s dx dy From (5-(7 it follows that y>0 C( s f = lim y 0 + ys u y = s lim y 0 + u(x, y u(x, 0 y s In the rest of this article, the homogeneous fractional Sobolev space is given by { } D s ( = u L N Ns ( : ξ s û(ξ < + with the norm ( /, u = ξ s û(ξ which is induced by the inner product ( / u, v s = ξ s û(ξ v(ξ The so-called Gagliardo semi-norm of u is defined as ( RN u(x u(y / [u] H s ( := x y N+s dx dy It can be proved [3, Proposition 34 and 36] that ( /= [u] Hs ( = C ξ s û(ξ s C ( u L ( for a suitable positive constant C depending only on s and N We consider the equation ( s u = u (s, u > 0 on (8 It has been proved in [8, 6] that the following function, for y and λ > 0, ( λ Ns U y,λ = C 0 + λ x y, x, where C 0 = C 0 (N, s > 0, solves (8 on For any positive integer m, y = (y, y,, y m R mn, λ = (λ, λ,, λ m, and λ > 0, =,,, m, we define { E y,λ = w D s ( : w, U y,λ λ s = w, U y,λ y s = 0, } i =,, m, i =,, N Our main result in this paper can be stated as follows

4 4 L GAN, W LIU EJDE-06/3 Theorem Suppose that N > s, 0 < s < If V satisfies (A and (A, then for any positive integer m, there exists an ε 0 > 0, such that for each ε (0, ε 0 ], the problem ( has a solution of the form u ε = m = U y ε,λ ε + ω ε, where ω ε E y,λ, y = (y ɛ, y ɛ,, y ɛm and as ε 0, y ε y 0, λ ε +, =,, m, ω ε 0, The proof of our results is inspired by the methods of [8, 37], we will combine a penalization argument and Lyapunov-Schmidt reduction scheme which are similar to [8, 37] to prove our main result This article is organized as follows In section, we introduce the penalization problem, give some preliminary estimates and carry out the finite dimensional reduction In section 3, we give the proof of Theorem Some technical estimates are left in the appendix Throughout this paper, we simply write f to mean the Lebesgue integral of f(x in The ordinary inner product between two vectors a, b will be denoted by a b, and C, C, c i denote generic constants, which may vary inside a chain of inequalities We use O(t, o(t to mean O(t C t, o(t t 0 as t 0; o( denotes quantities that tend to 0 as x Preliminaries and finite dimensional reduction In this section, we give some preliminary results, which are crucial in the proof of the main theorem and the finite-dimensional reduction Problem ( is the Euler-Lagrange equation of functional I (u = u, u s V (xu (s+ɛ (s + ɛ As we now, under the conditions (A and (A, the functional I (u will not be well defined and differentiable in D s ( Inspired by the idea introduced by Yan [37] and Deng et al [8], we modify the nonlinearity as in [37] To this end, we need to fix some notation Choose R > 0 large enough Define f(x, u = χ BR (0(xf (u + χ B C R (0(xf (u, where χ BR (0 denotes the characteristic function of B R (0, and u (s+ɛ, if 0 u ɛ N, f (u = a ɛ u (s + b ɛ, if u ɛ N, f (u, if u < 0, where a ɛ = ( ɛ + ɛ Nɛ, (s ɛ b ɛ = (s ɛn( (s+ɛ, and > 0 is a constant to be determined in Proposition 3 Moreover, f (x, u = x f N+s+ɛ(Ns ( x Ns u, where the nonnegative C function f satisfies: 0 if u, f (u = u (s+ɛ if u [0, ], f (u if u < 0

5 EJDE-06/3 FRACTIONAL SCHRÖDINGER EQUATIONS 5 Let F (x, u = u f(x, τdτ, then we have the following Lemma 0 Lemma Assume that V (x satisfies (A Then V (xf (x, u is well defined on D s ( Proof Using (A and the definition of f(x, u, we obtain V (xf (x, udx R N V (xf (x, u dx + V (xf (x, u dx C C \B R (0 \B R (0 \B R (0 + C u C C \B R (0 B R (0 ( + x q( u f(x, τdτ 0 ( + x q x N+s+ε(Ns ( u (s dx ( + x q ( x Ns u x N+ε(Ns dx + C R + x N Nq+(Nsε dx + C u (s dx B R (0 0 0 f ( x Ns τdτ u (s dx < + Consequently, the result follows from the above estimate Now we consider the penalization problem ( s u = V (xf(x, u, x, u D s ( dx f (τdτ dx + C u (s dx ( The functional associated with problem ( is given by I(u = u, u s V (xf (x, u, u D s ( ( It follows from Lemma that I C is well defined in E y,λ and hence its critical points are solutions of problem ( Denote U(y, λ = and set = = m J(y, λ, w = I + w, (y, λ, w M y,λ, (3 where M y,λ = {(y, λ, w : w E y,λ, (y, λ D y,m, w δ}, (4 { D y,m = y = (y, y,, y m R mn, λ = (λ, λ,, λ m, y B δ (y 0, λ [ɛ, ɛ ], =,, m, ɛ L, j },

6 6 L GAN, W LIU EJDE-06/3 where small > 0 and large > 0 are constants to be determined in Proposition 3, L > 0 large enough, and ( λj ɛ = + λ + λ j λ y j y sn λ λ j Lemma There exists ε 0 > 0, such that, for ε (0, ε 0 ], L > 0 large enough, δ > 0 small enough, (y, λ, w M y,λ is a critical point of J if and only if u = m = U y,λ + w is a critical point of I in D s ( The proof of Lemma is standard, since can be complete it with the same arguments as those in [30, 3], we omit it Without loss of generality, we assume that y 0 = 0 and V (0 = Expanding J(y, λ, w, we obtain J(y, λ, w = J(y, λ, 0 + l y,λ (w + L y,λw, w s + R y,λ (w, where m (s+ε l y,λ (w = V (x w + Uy,λ, w s, = m L y,λ w, w s = w, w s ( (s+ε (s + ε V (x w, = R y,λ,w = V (xf (x, U y,λ + w + V (xf (x, U y,λ R N m (s+ε + V (x w = + (s + ε m (s+ε V (x w Now, we state a lemma which is very important for our precise estimate on the functional energy and can be found in [3] Lemma 3 For < q 3 and a > b, a + b q a q b q q a q b q b q a C b q a For q > 3, a + b q a q b q q a q b q b q a C( a q b + b q a Next, we show the invertibility of L y,λ Lemma 4 There exist constants ε 0 > 0 and C > 0 such that for (y, λ D y,m, = L y,λ w C w, w E y,λ Proof We proceed by contradiction Assume that there exist ε n 0, δ n 0, λ n, (y n, λ n = (y n,, y n m, λ n,, λ n m D y,m and w n E y,λ, such that L y n,λ nw n, ϕ s = o n ( w n ϕ, ϕ E y,λ (5 Without loss of generality, we assume that w n = Let w n, (x = (λ n sn w n ((λ n x + y n, =,, m

7 EJDE-06/3 FRACTIONAL SCHRÖDINGER EQUATIONS 7 Assume that From w n, w, =,, m, as n, w n, w, strongly in L loc(, =,, m, as n Uy n,λ n, w n λ n s Uy n =,λ n y n, w n = 0, s i =,, m, i =,, N, we obtain U0, U0, x=0, w n, =, w n, = 0, λ λ= s x i s for =,, m, i =,, N So w satisfies U0,, w U0, x=0 =, w = 0, (6 λ λ= s x i s for =,, m, i =,, N Define Ẽ y,λ = {ϕ D s ( U0, U0, x=0,, ϕ =, ϕ = 0, i =,, N} λ λ= s x i s Note that o( ϕ = w n, ϕ s V (xf (x, U yn,λ nw nϕ (7 Let ϕ C0 ( Ẽy,λ and tae ϕ n (x := (λ n sn ϕ[(x y nλn ] Letting n, we obtain w, ϕ s ( (s (s 0, wϕ = 0 It is easy to prove that w, ϕ s ( (s U U (s 0, w ϕ = 0, ϕ Ẽy,λ (8 x=0 Thus, (8 is true for U But (8 is true for ϕ = c 0, 0 λ λ= + n i= c i U0, x i any ϕ E y,λ, and hence w = c 0 U0, λ + n i= c i U0, x=0 x λ= i It follows from (6 that c i = 0 (i = 0,,, N and w = 0 Therefore, letting ε n, δ n > 0 small enough, λ n big enough, o( = w n, w n s V (xf (x, U y n,λ nw n R N (9 C (s 0, wn = R ( + o( This contradicts (5 U Proposition 5 For ε > 0 sufficiently small and (y, λ D y,m, there exists a C -map w(y, λ : D y,m M y,λ such that w(y, λ satisfies J(w w =, ϕ 0 for all ϕ M y,λ Moreover, m ( DV (y w C + λ λ + V (0V (y λ N+s +ε ln λ + ε +τ (0 =

8 8 L GAN, W LIU EJDE-06/3 Proof To find a critical point for J(w, we only need to solve l y,λ + L y,λ w, w + R (w = 0 ( From Lemma 4, we now that L y,λ is invertible Therefore, ( can be rewritten as w = A(w =: L y,λ l y,λ L y,λ R (w Set N = { w E y,λ : w + ε δ ln λ + = DV (y λ δ ε ( +τ(δ }, + λ (δ where δ > 0 is small enough As in [37], R(w is the higher order term satisfying + R i (w = O( w +θi, i = 0,,, λ N+s (δ V (0 V (y where θ > 0 is some constant Hence, Lemma 6 below implies A(w C l y,λ + C w +θ m ( DV (y C + λ λ + = m + C + = DV (y λ δ ε ( +τ(δ + +θ λ (δ λ N+s + V (0 V (y + ε ln λ + λ N+s (δ ε +τ V (0 V (y + ε δ ln λ ( = + Meanwhile, DV (y λ δ + ε ( +τ(δ λ (δ + λ N+s (δ V (0 V (y + ε δ ln λ A(w A(w = L y,λ R (w L y,λ R (w C R (w R (w C R (εw + ( εw w w C( w θ + w θ w w w w, where ε (0, Thus, A maps N to N and A is a contraction map By the contraction mapping theorem, we see that there is a unique w such that ( holds, and from ( that (0 holds

9 EJDE-06/3 FRACTIONAL SCHRÖDINGER EQUATIONS 9 Lemma 6 m ( DV (y l y,λ C + λ λ + = = λ N+s Proof First, we now that m U y,λ, w s =, w = s Next, we estimate V (0 V (y + ε ln λ + = ε +τ RN U N+s Ns y,λ w (3 V (xf(x, U y,λ w m (s+ε = V (x w = m = V (xu R (s+ε N y,λ w = +O( V (xu R (s+ε N y j,λ j U (s+ε y,λ w, if < (s + ε, m = V (xu R (s+ε N y,λ w ( ε w, if (s + ε >, ( m = V (xu R (s+ε = N y,λ w ε +τ w, if < (s + ε, m = V (xu R (s+ε N y,λ w ( ε w, if (s + ε > We also have V (xu (s+ε y,λ w = V (xu (s y,λ w (ε ln λ w R N ( = V (y U (s DV (y y,λ w + R λ N λ + ε ln λ w Combining above estimates, we obtain RN l y,λ w = U N+s RN Ns y,λ w V (y U N+s Ns y,λ w = = m ( DV (y + λ λ = = = + ε ln λ + RN V (0 V (y U N+s Ns y,λ w m ( DV (y + λ λ = m ( DV (y = O + λ λ + = w + ε +τ + ε ln λ + λ N+s ε +τ w ε +τ w V (0 V (y + ε ln λ (4

10 0 L GAN, W LIU EJDE-06/3 3 Proof of main result Let w(y, λ be the map obtained in Proposition 5 Define Ĩ(y, λ := I(y, λ, w(y, λ, (y, λ D y,m Let (y ε, λ ε D y,m be any point for which Ĩ(y ε, λ ε = sup{ĩ(y, λ : (y, λ D y,m} (3 The next Proposition shows that for small ε > 0, (y ε, λ ε is an interior point of D y,m, and hence a critical point of Ĩ Proposition 3 Let (y ε, λ ε satisfy (3 Then as ε 0, y ε 0, =,, m, λ ε [ɛ, ɛ ], =,, m, for some positive constant <, ɛ 0, j Proof It follows from Lemma 4, Lemma 6 and Proposition 5 that I(y, λ, w(y, λ = I(y, λ, 0 ( l y,λ w y,λ + w y,λ ( m = V (y U (s (s + ε = R (s + ε N [ ( RN εv (y ln λ Ns U (s (s + ε m = = V (x ( m ε +τ +O = ( m j=+ DV (y λ m ( ε ln λ + ε ln λ λ = V (y λ x U (s ln U] = U λ +µ = ( V (y +τ U yj,λ j (s+ε +O ( m +O ( m Denote z ε = εe, λ ε = ε, =,, m Some unit vectors e,, e m with e e j ( j, for Then z j ε z ε = ε e j e 0, λ ε, as ε 0 Then = I(y ε, λ ε, w ε (y ε, λ ε I(z ε, λ ε, w ε (z ε, λ ε ( = m (s + ε U (s mcε ln ε (ε (s

11 EJDE-06/3 FRACTIONAL SCHRÖDINGER EQUATIONS Thus ( m (s + ε (s + ε m = (s + ε = = [ ( RN εv (yɛ ln(λ ɛ Ns U (s = V (xu y ɛ,λ ɛ ( m ε +τ +O = λ ɛ V (yɛ U (s = ( m j=+ U y j ɛ,λ j ɛ DV (y ɛ (λ ɛ +O m ( ε ln λ ɛ + ε ln λ ɛ ( m (s + ε Moreover, m = This and (3 imply U V (xu y ɛ,λ ɛ = U (s+ε +O ( m = V (y ɛ (λ ɛ (s ln U] (λ ɛ +µ m ( V (yɛ +τ = (s mcε ln ε (ε ( m j=+ U y j ɛ,λ j ɛ x U (s (3 (s+ε C ɛ (33 0 V (yɛ j Cɛ ln (ɛ, (34 ɛ ɛ Cɛ ln (ɛ, j, (35 ɛ εv (yɛ ln(λ ɛ Ns U (s + V (y ɛ R (λ N ɛ x U (s m DV (yɛ ( (λ ɛ (36 (λ ɛ +µ = ( (s + ɛmcɛ ln ɛ (ɛ, which implies λ ε + for =,, m, and ɛ 0 for j, as ɛ 0,, j =,,, m If λ ε = ɛ for some, then from (36, we obtain ɛ V (y ɛ x U ( m DV (yɛ ɛ Cɛ ln R ɛ N This is a contradiction if > 0 small enough If λ ε = ɛ for some, then from (36, we obtain N s ε ln ɛ V (y ɛ (s ( (s + ɛmcɛ ln ɛ (ɛ, U which is impossible if we choose > max{ = ( (s+mc (Ns R U (s, result λ ε [ɛ, ɛ ] ( =,, m follows from the above estimate } Consequently, the

12 L GAN, W LIU EJDE-06/3 Proof of Theorem By Proposition 3, we can chec that (3 is achieved by (y ε, λ ε which is an interior point of D y,m for small ɛ, It follows from Proposition 5 that (y ε, λ ε is a critical point of J Using Lemma, we can obtain u = m = U y,λ + w is a critical point of I Note that ( ( s w = V (xf x, On the other hand, ( V (xf x, + w = = = + w = U (s y,λ U (s y,λ C w (s + Cɛ (N+s Since w ɛ 0 as ɛ 0, by adapting the same approach explored in [4] and [5], we deduce that w ɛ C w ɛ L (s (B (0 + Cɛ (N+s, x B (0, which implies u ɛ ɛ N for all x < R Let ū ɛ (x = x (Ns u ɛ (x/ x which satisfies ( x (Nsɛ x V ( s x f (ū ū ɛ = g(xū ɛ := ɛ, if x ( R, x (N+s x V x f ( x Ns ū ɛ, if R x R We can choose γ > 0 small enough, and it also follows from the same approach explored in [4] and [5] that ū ɛ L (B γ(0 C ū ɛ L (s (B γ(0 0, as ɛ 0, which implies x Ns u ɛ 0 as ɛ 0 From above estimates, we can obtain that u solves indeed the original problem ( 4 Appendix In this section, we prove some estimates needed in the proof of our main results Lemma 4 For any (y, λ D y,m, we have I(U y,λ ( m = (s + ε m = (s + ε λ (+µ = V (y U (s m ( ε ln λ (s + ε V (y λ x U ((ε + ε λ j = +O + ε ln λ +O λ = ( εv (y ln λ Ns U (s εv (y = m ( V (y +τ = ( ε +τ j, m = V (x ( m j=+ U (s ln U U yj,λ j (s+ε (s

13 EJDE-06/3 FRACTIONAL SCHRÖDINGER EQUATIONS 3 Proof Firstly, for any (y, λ D y,m, we see that I(U y,λ = U y,λ, U y,λ s V (xf (x, U y,λ = U y,λ, U y,λ s V (xu y,λ (s + ε = y,λ, s + = U U yj,λ j, s j< V (xu (s+ε y,λ (s + ε = y,λ, s + = U U (s y j,λ j j< V (xu (s+ε y,λ (s + ε = (s+ε Next, we estimate V (xu (s+ε y,λ Using Lemma 3, we have V (xu (s+ε y,λ ( (s+ε = V (x U y,λ + = m = V (xu (s+ε (s+ε y,λ + V (x = + ( (s + ε V (xu (s+ε y,λ = m + ( (s+ε ( (s + ε V (xu y,λ +O ε +τ j, By repeated applications of Lemma 3, we deduce V (xu (s+ε y,λ = V (xu (s+ε y,λ + ( (s + ε V (xu (s+ε y,λ = = ( + ( (s + ε V (xu y,λ ( (s+ε We also deduce (s+ε = ε +τ j, V (xu y,λ = V (xu (s y,λ + ε V (xu (s y,λ ln (ε ln λ R N ( = V (xu (s y,λ + ε V (y U (s ε ln λ y,λ ln R λ N (4 (4

14 4 L GAN, W LIU EJDE-06/3 (ε ln λ ( RN = V (xu (s y,λ + εv (y ln λ Ns U (s U (s ln U R ( N ε ln λ +O(ε ln λ (43 λ From (4 and (43, we have V (xu (s+ε y,λ [ ( RN = V (xu (s y,λ + εv (y ln λ Ns U (s = ( ε ln λ ] +O(ε ln λ +( (s + ε V (xu λ j< + ( (s + ε m = V (x ( m j=+ U (s ln U (s+ε y j,λ j U yj,λ j (s+ε +O ( ε +τ j, We also have V (xu (s+ε y j,λ j = V (xu (s y j,λ j (ε j, ε R N = V (y j U (s y j,λ j (ε j, ε ( ε j, R λ N i and V (xu (s y,λ = V (y U (s y,λ + DV (y, x y U (s y,λ R N + D V (y (y x, (y x U (s y,λ (λ (+µ = V (y (s + V (y x U (s (λ (+µ U λ Combining above estimates, we obtain I(U y,λ = y,λ, s + = U U (s y j,λ j j< [ ( RN V (xu (s y (s + ε,λ + εv (y ln λ Ns = ( U ε ln (s λ ] ln U +O +O(ε ln λ R λ N V (xu (s+ε y j,λ j j< m m (s+ε ( V (x U yj,λ j +O ε +τ j, = j=+ U (s

15 EJDE-06/3 FRACTIONAL SCHRÖDINGER EQUATIONS 5 = ( m (s + ε = V (y U (s V (y (s + ε λ x U (s ( m = = m ( ε ln λ ( + ε ln λ +O (ε + ε λ λ j = ( (s + ε εv (y ε +τ j, = m V (x = ( εv (y ln λ Ns U ( m j=+ U (s m (s ln U +O ( V (y +τ = λ (+µ U yj,λ j (s+ε Acnowledgements The authors than the referee and the editor for their helpful discussions and suggestions This research was supported by the NSFC No 6039 References [] D Applebaum; Lévy processes: From probability to finance and quantum groups, Not Am Math Soc, 5 (004, [] A Ambrosetti, J Garcia Azorero, I Peral; Perturbation of u = u N+ N, the scalar curvature problem in, and related topics, J Funct Anal, 65 (999, 7-49 [3] D Applebaum; Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics vol 6, Cambridge University Press, (009 [4] C O Alves, O H Miyagai; Existence and concentration of solution for a class of fractional elliptic equation in via penalization method, Calc Var Partial Differ Equ, 55 (06, -9 [5] B Barrios, E Colorado, R Servadei, F Soria; A critical fractional equation with concaveconvex power nonlinearities, Ann Inst H Poincaré Anal Non Linéaire 3 (05, no 4, [6] B Barrios, E Colorado, A de Pablo, U Sánchez; On some critical problems for the fractional Laplacian operator, J Differ Equ, 5 (0, [7] L Caffarelli, J M Roquejoffre, Y Sire; Variational problems with free boundaries for the fractional Laplacian, J Eur Math Soc, (00, 5-79 [8] W Chen, C Li, B Ou; Classification of solutions for an integral equation, Comm Pure Appl Math, 59 (006, [9] D Cao, E Noussair, S Yan; On the scalar curvature equation u = ( + ɛku N+ N in, Calc Var Partial Differ Equ, 5 (00, [0] D Cao, S Peng; Concentration of solutions for the Yamabe problem on half-spaces, Proc Roy Soc Edinburgh Sect A, 43 (03, [] D Cao, S Peng, S Yan; On the Webster scalar curvature problem on the CR sphere with a cylindricaltype symmetry, J Geom Anal, 3 ( [] L Caffarelli, L Silvestre; An extension problem related to the fractional Laplacian, Comm Partial Differential Equations, 3(007, [3] G Chen, Y Zheng; Pea solutions for the fractional Nirenberg problem, Nonlinear Anal, (05, 00-4

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