A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian
aa r X i v : . [ m a t h . A P ] N ov A Liouville type theorem for Lane-Emden systems involving thefractional Laplacian
Alexander Quaas and Aliang XiaDepartamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıaCasilla: V-110, Avda. Espa˜na 1680, Valpara´ıso, Chile. ([email protected] and [email protected])
Abstract
We establish a Liouville type theorem for the fractional Lane-Emden system: (cid:26) ( − ∆) α u = v q in R N , ( − ∆) α v = u p in R N , where α ∈ (0 , N > α and p, q are positive real numbers andin an appropriate new range. To prove our result we will use thelocal realization of fractional Laplacian, which can be constructed asDirichlet-to-Neumann operator of a degenerate elliptic equation in thespirit of Caffarelli and Silvestre [5]. Our proof is based on a mono-tonicity argument for suitable transformed functions and the methodof moving planes in an infinity half cylinder based on some maximumprinciples which obtained by some barrier functions and a couplingargument using fractional Sobolev trace inequality. During the last years there has been a renewed and increasing interest inthe study of linear and nonlinear integral operators, including the fractionalLaplacian, motivated by many applications and by important advances on thetheory of nonlinear partial differential equations. This because these typesof systems appear as limiting equations of many phenomena, such as pat-tern formation, population evolution, chemical reaction, etc. Some of theseequations are named by Lotka-Volterra, Bose-Einstein, Schr¨odinger system,Gierer-Meinhardt. The solutions in most of case represent concentrationsin the process and thus naturally positive solutions of the systems are ofparticular interest.Most of the results in this field are obtained assuming that the diffusionis governed by the Laplacian or other more general local elliptic operators.1he mathematical literature in the case of elliptic systems when the diffusionis governed by L´evy stable process and the elliptic operator turns to be thefractional order of Laplacian is very recent. See [10], [19], [21], [29], [30], [31]and [32]. Notice that the fractional Laplacian appears in different diffusionmodels, see for example [1], [3], [4], [15], [20], [27] and the references therein.As far as we know, there aren’t existence results for systems withoutvariation structure. When the variational structure breaks, the methods de-veloped to prove existence results for local elliptic systems are obtained byPerron’s Method or topological arguments; for example, the Leray-Schauderdegree or Krasnoselskii’s index theory, where the many assumptions in usingthese theories are the priori bound for solutions. These bounds are obtainedin many settings by the classical scaling or blow-up argument due to Gidasand Spruck [14] in the scalar case and [11] for systems case, see also ref-erences therein. Liouville type Theorems are crucial to get a contradictionfor the limiting system or equation. Roughly speaking, better Liouville typeTheorems give more general existence results. Observe that there are stillsome problems even in the case ( α = 1) which is known as Lane-Emdenconjecture, see [26], [23] and blow.The aim of this paper is establish a new Liouville type theorem for thefollowing Lane-Emden system involving the fractional Laplacian: (cid:26) ( − ∆) α u = v q in R N , ( − ∆) α v = u p in R N . (1.1)where α ∈ (0 ,
1) and
N > α .The fractional Laplacian ( − ∆) α can be defined, for example, by theFourier transform. Namely, for a function u in the Schwartz class S , wehave \ ( − ∆) α u ( ξ ) = | ξ | α ˆ u ( ξ ) . Furthermore, consider the space L α ( R N ) := (cid:26) u (cid:12)(cid:12)(cid:12) u : R N → R sucht that Z R N | u ( y ) | | y | N +2 α dy < ∞ (cid:27) . endowed with the norm k u k L α ( R N ) := Z R N | u ( y ) | | y | N +2 α dy < ∞ . If u ∈ L α ( R N ) (see [22]), then ( − ∆) α u can be defined as a distribution, thatis, for any ϕ ∈ S , Z R N ϕ ( − ∆) α udx = Z R N u ( − ∆) α ϕdx.
2n additional, for some σ >
0, suppose that u ∈ L α ( R N ) ∩ C α + σ ( R N ) if0 < α < / u ∈ L α ( R N ) ∩ C , α + σ − ( R N ) if α ≥ /
2, then we have( − ∆) α u ( x ) = C N,α
P.V. Z R N u ( x ) − u ( y ) | x − y | N +2 α dy for x ∈ R N , where P.V. denotes the principal value of the integral and C N,α is a normal-ization constant.When α = 1, the Lane-Emden system for Laplace operator (cid:26) − ∆ u = v q in R N , − ∆ v = u p in R N , (1.2)has been extensively studied in the literature, see [2, 12, 18, 25, 26]. It hasbeen conjectured that the Sobolev’s hyperbola (cid:26) p > , q > p + 1 + 1 q + 1 = 1 − N (cid:27) , is the dividing curve between existence and nonexistence for (1.2). For theradial case, this conjecture was completely solved by [18, 25]. In fact, if thepair ( p, q ) lies below Sobolev’s hyperbola, that is,1 p + 1 + 1 q + 1 > − N , (1.3)then there are no radial positive solutions to system (1.2), see [18] (for p > , q >
1) and [25] (for p > , q > p, q ) lies above Sobolev’s hyperbola (seealso [25]).The conjecture for more general case, i.e., without the assumption ofradial symmetry, has not been completely answered yet. Partial results fornonexistence are known. Define γ = 2( q + 1) pq − , γ = 2( p + 1) pq − pq > . (1.4)There are no positive classical supersolutions to (1.2) if pq ≤ pq > { γ , γ } ≥ N − , (1.5)see [18, 24, 26]. Moreover, we know that the condition (1.4) is optimal forsupersolution. For positive solution, Felmer and Figueiredo [12] proved thatif 0 < p, q ≤ N + 2 N − , ( p, q ) = (cid:18) N + 2 N − , N + 2 N − (cid:19) , (1.6)3hen problem (1.2) has no classical positive solutions. Notice that for N ≥ N = 3 ,
4, see [26] and [23]. In the high dimensions, apart from (1.5),the conjecture is only known to be true in some subregion of subcritial range:min { γ , γ } ≥ N −
22 and ( γ , γ ) = (cid:18) N − , N − (cid:19) (1.7)by Busca and Man´asevich [2]. Note that the condition (1.6) in particularcontains where both exponents are subcritical, that is, the region consideredin [12].The aim of the present paper is to show that the result of Busca andMan´asevich [2] can be extended to system (1.1). We prove the followingresult. Theorem 1.1
Let p, q > and pq > and set β = 2 α ( q + 1) pq − , β = 2 α ( p + 1) pq − . (1.8) Suppose β , β ∈ (cid:20) N − α , N − α (cid:19) and ( β , β ) = (cid:18) N − α , N − α (cid:19) . (1.9) Then, for some σ > , there exists no positive L α ( R N ) ∩ C α + σ ( R N ) if <α < / or in L α ( R N ) ∩ C , α + σ − ( R N ) if α ≥ / type solution to system(1.1). Remark 1.1
1) Observe that region (1.9) in particular contains where bothexponents are subcritical, that is NN − α < p, q ≤ N + 2 αN − α , with ( p, q ) = (cid:18) N + 2 αN − α , N + 2 αN − α (cid:19) . (1.10) By Theorem 3 in [19] and Theorem 3 in [34], we know there are no positive u, v ∈ L α ( R N ) ∩ C α + σ ( R N ) if < α < / or in L α ( R N ) ∩ C , α + σ − ( R N ) if α ≥ / type solution to system (1.1) if (1.10) holds. Hence, Theorem 1.1is valid for a large region of ( p, q ) in comparison with Theorem 3 in [19].2) If we take q = 1 we can obtained Liouville type resutlts for bi-fractionalequations. emark 1.2 We note that region (1.9) does not include the point ( β , β ) = (cid:18) N − α , N − α (cid:19) . Indeed, if β = β = N − α , then p = q = N + 2 αN − α , and problem ( − ∆) α u = u N +2 αN − α . (1.11) has nontrivial nonnegative solutions called fractional bubble, see Chen-Li-Ou[7, 8], Jin-Li-Xiong [16] and also Y.Y. Li [17]. In [5], Caffarelli and Silvestre introduced a local realization of the frac-tional Laplacian ( − ∆) α in R N through the Dirichlet-Neumann map of anappropriate degenerate elliptic operator in upper half space R N +1+ . Moreprecisely, consider an extension of u to the upper half space R N +1+ so that U ( x,
0) = u ( x ) and∆ x U + 1 − αy U y + U yy = 0 for X = ( x, y ) ∈ R N +1+ . Let P α ( x, y ) denote the corresponding Poisson kernel P α ( x, y ) = c N,α y α ( | x | + y ) ( N +2 α ) / for x ∈ R N and y > . where c N,α is a normalization constant ( for an explicit value of c N,α see [6]).If u ∈ L α ( R N ), we can define U ( x, y ) = P α ( · , y ) ∗ u = c N,α Z R N y α ( | x − ξ | + y ) ( N +2 α ) / u ( ξ ) dξ. Moreover, for some σ >
0, suppose that u ∈ L α ( R N ) ∩ C α + σ ( R N ) if 0 <α < / u ∈ L α ( R N ) ∩ C , α + σ − ( R N ) if α ≥ /
2, then U ∈ C ( R N +1+ ) ∩ C ( R N +1+ ), y − α ∂ y U ∈ C ( R N +1+ ) and div ( y − α ∇ U ) = 0 in R N +1+ ,U = u on ∂ R N +1+ , − lim y → + y − α U y = κ α ( − ∆) α u on ∂ R N +1+ , κ α = Γ(1 − α )2 α − Γ( α )with Γ being the Gamma function, see Theorem 1.3 in [9] and also [5, 6, 12,24].Using the local formulation established by Caffarelli and Silvestre [5],the above theorem will follow as a corollary of the following Liouville typeresult for a degenerated systems with a coupling with a nonlinear Neumanncondition in the upper half space R N +1+ . Theorem 1.2
Let p, q > , pq > and (1.9) holds. Then there exists nopositive C ( R N +1+ ) ∩ C ( R N +1+ ) and y − α ∂ y ( · ) ∈ C ( R N +1+ ) type solution of div ( y − α ∇ U ) = 0 in R N +1+ , − lim y → + y − α U y = V q on ∂ R N +1+ ,div ( y − α ∇ V ) = 0 in R N +1+ , − lim y → + y − α V y = U p on ∂ R N +1+ . (1.12)Our proof follows the idea in [2]. Roughly speaking, as in [2], we firsttransform the elliptic equation (1.12) in upper half space R N +1+ to an appro-priate equation in upper half infinite cylinder R × S N + (see (2.3) and (2.5)),where S N + is the upper half unit sphere. Then, we study the nonexistenceresult via a symmetry and monotonicity result (i.e., Lemma 3.2) which ob-tained by the method of moving planes. However, there are some difficultiesappear compared our article with [2] that the operator in (1.12) is a degen-erated operator and the nonlinearity is at the boundary. In particular, toprove the maximum principle for ”narrow” domains which permits us to getthe moving planes started. For this purpose we follow some similar argu-ments as in [13] which are for the single equation and a coupling argumentestablishd by a fractional Sobolev trace inequality (see Lemma 3.1). We alsoneed prove two Hopf’s lemmas where barrier functions need to be construct,see Lemmas 2.1 and 2.2 in section 2.We end the introduction by mention that we can use Theorem 1.1 toobtain a priori estimate and existence result for positive solutions of nonlinearelliptic systems involving the fractional Laplacian.The paper is organized as follows. In section 2, we do a transformationas in [2] to problem (1.12) and present some preliminary results, the Hopf’slemmas and the strong maximum principle. A monotonicity and symmetryresult is shown by the method of moving planes in section 3 and we provethe nonexistence result (Theorem 1.2) at the end of section 3.6 Preliminaries
This section is devote to introduce some preliminary results, the Hopf’s lem-mas and the strong maximum principle. We start this section by transforming(1.12) as in [2] by using polar coordinates and Emden-Fowler variables. Wetake standard polar coordinates in R N +1+ : X = ( x, y ) = rθ , where r = | X | and θ = X/ | X | . Denote θ = ( θ , θ , · · · , θ N , θ N +1 ) and let θ N +1 = y/ | X | denote the component of θ in the y direction and S N + = { X ∈ R N +1+ : r =1 , θ N +1 > } denote the upper unit half space.For a given function w of X ∈ R N +1+ , we write, using the same symbol w without risk of confusion, w ( X ) = w ( r, θ ) . Thus we have the following formula∆ w = 1 r ∆ θ w + Nr ∂w∂r + ∂ w∂r , (2.1)where ∆ θ denotes the Laplace-Beltrami operator on S N .Set (cid:26) U ( t, θ ) = r β U ( r, θ ) V ( t, θ ) = r β V ( r, θ ) (2.2)for β , β to be fixed, where t = log r . Using the formula (2.1), an easycalculation verifies that θ α − N +1 div ( θ − αN +1 ∇ θ U ) + U tt − δ U t − ν U = 0 in R × S N + , − lim θ N +1 → + θ − αN +1 ∂ θ N +1 U = r β +2 α − qβ V q on R × ∂S N + ,θ α − N +1 div ( θ − αN +1 ∇ θ V ) + V tt − δ V t − ν V = 0 in R × S N + , − lim θ N +1 → + θ − αN +1 ∂ θ N +1 V = r β +2 α − pβ U p on R × ∂S N + , (2.3)where (cid:26) δ = 2 β − ( N − α ) , ν = β (( N − α ) − β ) ,δ = 2 β − ( N − α ) , ν = β (( N − α ) − β ) . (2.4)For ease of the notation, we define the operators L α U := θ α − N +1 div ( θ − αN +1 ∇ U ) = ∆ θ U + 1 − αθ N +1 ∂U∂θ N +1 L α V := θ α − N +1 div ( θ − αN +1 ∇ V ) = ∆ θ V + 1 − αθ N +1 ∂U∂θ N +1 .
7f we define β and β as in (1.8), then we write (2.3) as L α U + U tt − δ U t − ν U = 0 in R × S N + , − lim θ N +1 → + θ − αN +1 ∂ θ N +1 U = V q on R × ∂S N + ,L α V + V tt − δ V t − ν V = 0 in R × S N + , − lim θ N +1 → + θ − αN +1 ∂ θ N +1 V = U p on R × ∂S N + . (2.5)Here we have used the facts β + 2 α − qβ = 0 and β + 2 α − pβ = 0 by (1.8).Moreover, with these notations in (2.4), the assumptions (1.9) is equivalentto δ , δ ≥ , ( δ , δ ) = (0 , ,ν , ν > ,p, q > , pq > . (2.6)In order to prove Theorem 1.2, we will use the method of moving planes.The key tools for use the method of moving planes is the Hopf’s lemma andthe strong maximum principle. The remain of the section is devote to provethese results related the operators we studied. We first show the followingweak maximum principle. Proposition 2.1
Let Ω be an bounded domain in R × S N + and w ∈ C (Ω) ∩ C ( ¯Ω) . Suppose L α w + w tt + a ( t, θ ) w t ≤ in Ω , where | a ( t, θ ) | ≤ a =constant in Ω . Then the nonnegative minimum of w in ¯Ω is achieved on ∂ Ω , that is, inf Ω w = inf ∂ Ω w. Proof.
It is clear that if L α w + w tt + a ( t, θ ) w t < w cannot achieve an interior nonnegativeminimum in ¯Ω. Indeed, if ( t , θ ) ∈ Ω, then w tt ( t , θ ) ≥ , ∆ θ w ( t , θ ) ≥ , w t ( t , θ ) = 0 , and ∇ θ w ( t , θ ) = 0 . This implies L α w ( t , θ ) + w tt ( t , θ ) + a ( t , θ ) w t ( t , θ ) ≥ γ >
0, gives,( e γt ) tt + a ( e γt ) t = e γt ( γ + aγ ) ≥ e γt ( γ − a γ ) . So we can choose γ large enough such that ( e γt ) tt + a ( e γt ) t >
0. Hence, forany ε > L α ( w − εe γt ) + ( w − εe γt ) tt + a ( w − εe γt ) t <
08n Ω so that inf Ω ( w − εe γt ) = inf ∂ Ω ( w − εe γt ) . Letting ε →
0, we see that inf Ω w = inf ∂ Ω w. as asserted in the proposition. (cid:3) Next we suppose more generally that L α w + w tt + a ( t, θ ) w t − b ( t, θ ) w ≤ , where | a ( t, θ ) | ≤ a and b is a nonnegative function in Ω. By consideringthe sunset Ω − ⊂ Ω in which w <
0. We can observe that if L α w + w tt + a ( t, θ ) w t − b ( t, θ ) w ≤ L α w + w tt + a ( t, θ ) w t ≤ b ( t, θ ) w ≤ − and hence the minimum of w on Ω − must be achieved on ∂ Ω − and hencealso on ∂ Ω. Thus, writing w − = min { w, } we obtain: Theorem 2.1
Let Ω be an bounded domain in R × S N + and w ∈ C (Ω) ∩ C ( ¯Ω) .Suppose L α w + w tt + a ( t, θ ) w t − b ( t, θ ) w ≤ in Ω , where | a ( t, θ ) | ≤ a and b is a nonnegative function in Ω . Then inf Ω w ≥ inf ∂ Ω w − . Next, we prove two Hopf’s Lemmas. Let D is an bounded domain of S N + and ¯ t ∈ R , we define Ω δ = {| t − ¯ t | ≤ δ } × D ⊂ R × S N + for some δ >
0. The first Hopf’s lemma is
Lemma 2.1
Suppose that ( t , θ ) ∈ ∂ Ω δ and w ∈ C (Ω δ ) ∩ C (Ω δ ∪ ( t , θ )) be a solution of L α w + w tt + a ( t, θ ) w t − b ( t, θ ) w ≤ in Ω δ , where a and b are bounded functions and b is nonnegative. Assume in addi-tion that w ( t, θ ) > for every ( t, θ ) ∈ Ω δ and t = t . Moreover, w ( t , θ ) = 0 if ( t , θ ) ∈ Ω δ . Then lim t → t w ( t, θ ) − w ( t , θ ) t − t < . roof. For 0 < ρ < δ , we define an auxiliary function φ as φ ( t ) = e − β | t − ¯ t | − e − βδ , where | t − ¯ t | > ρ and β is a positive constant to be determined later. Wenotice that 0 < φ ( t ) < φ tt + aφ t − bφ = e − β | t − ¯ t | (cid:0) β | t − ¯ t | + 2 β ( a | t − ¯ t | − − b (cid:1) . Hence we can chose β large enough such that φ tt + aφ t − bφ ≥ δ \ Ω δ/ . Since w > t, θ ) ∈ Ω δ and t = t and φ ( t ) = 0, there is a ε > w − εφ ≥ ∂ Ω δ ∪ ∂ Ω δ/ . Moreover, we have L α ( w − εφ ) + ( w − εφ ) tt + c ( w − εφ ) t − c ( w − εφ ) ≤ δ \ Ω δ/ . Hence, by the weak maximum principle (see Theorem 2.1)implies that w − εφ ≥ δ \ Ω δ/ . Taking the outer normal derivative at ( t , θ ), we obtainlim t → t w ( t, θ ) − w ( t , θ ) t − t ≤ ε lim t → t φ ( t ) − φ ( t ) t − t = − εβδe − βδ < , as we required. (cid:3) Remark 2.1
If in addition w ∈ C (Ω δ ∪ ( t , θ )) , then we have ∂w∂t ( t , θ ) < . Next we established the second Hopf’s lemma on the boundary R × ∂S N + .We denote as before that θ = (˜ θ, θ N +1 ) ∈ S N + with θ N +1 >
0. For (˜ θ , ∈ ∂S N + , we define Γ R = { ˜ θ ∈ S N − | d S N − (˜ θ, ˜ θ ) ≤ R } , as a neighbourhood of (˜ θ ,
0) on ∂S N + , where d S N − denotes the distance in S N − . Then, for some cosntant τ >
0, we let C R,τ ( t , ˜ θ ,
0) := ( t − τ, t + τ ) × Γ R ⊂ R × ∂S N + . For notational simplicity we denote C R,τ = C R,τ ( t , ˜ θ ,
0) in what follows.10 emma 2.2
Assume ( t , ˜ θ , ∈ R × ∂S N + and consider the subset C R,τ × (0 , ̺ ) of R × S N + for τ > and < ̺ < . Let w ∈ C ( C R,τ × (0 , ̺ )) ∩ C ( C R,τ × (0 , ̺ )) satisfies L α w + w tt + a ( t, θ ) w t − b ( t, θ ) w ≤ in C R,τ × (0 , ̺ ) ,w > in C R,τ × (0 , ̺ ) ,w ( t , ˜ θ ,
0) = 0 , where a and b are bounded nonnegative functions.Then, − lim sup θ N +1 → + θ − αN +1 w ( t , ˜ θ , θ N +1 ) θ N +1 < . Proof.
Here we follow the argument in [6]. Consider the function φ = φ ( t ) on C R,τ and satisfies the following ODE − φ tt − aφ t + bφ = 0 in ( t − τ / , t + τ / ,φ > t − τ / , t + τ / ,φ ( t − τ /
2) = 0 , where a = inf C R,τ × (0 ,̺ ) a ≥ b = sup C R,τ × (0 ,̺ ) b ≥
0. Since b ≥
0, then wecan write φ ( t ) = C e (cid:16) − a + √ a +4 b (cid:17) t + C e (cid:16) − a − √ a +4 b (cid:17) t with C ≥ ≥ C . This implies φ t ≥ t − τ / , t + τ / k φ k L ∞ ≤ C .Hence, we have L α φ + φ tt + aφ t − bφ = ( a − a ) φ t + ( b − b ) φ ≥ C R/ ,τ/ × (0 , ̺ ) ,φ ≥ C R/ ,τ/ × (0 , ̺ ) ,φ = 0 , on ∂ C R/ ,τ/ × [0 , ̺ ) , Therefore, for ε > L α ( w − εφ ) + ( w − εφ ) tt + a ( w − εφ ) t − b ( w − εφ ) ≤ C R/ ,τ/ × (0 , ̺ )and w − εφ = w ≥ ∂ C R/ ,τ/ × [0 , ̺ ). Moreover, taking ε > w ≥ εφ on C R/ ,τ/ × ( { θ N +1 = ̺/ } ∪ { θ N +1 = 0 } ) since w is continuous and positive on C R/ ,τ/ × (0 , ̺/ L α ( w − εφ ) + ( w − εφ ) tt + a ( w − εφ ) t − b ( w − εφ ) ≤ C R/ ,τ/ × (0 , ̺/
2) with w − εφ ≥ ∂ ( C R/ ,τ/ × (0 , ̺/ w − εφ ≥ C R/ ,τ/ × (0 , ̺/ . This implies that w ≥ εφ ≥ εθ αN +1 φ in C R/ ,τ/ × (0 , ̺/ . Consequently, this leads tolim sup θ N +1 → + − θ − αN +1 w ( t , ˜ θ , θ N +1 ) θ N +1 ≤ ε lim sup θ N +1 → + − θ − αN +1 θ αN +1 φ ( t ) θ N +1 = − εφ ( t ) < , as claimed in the proposition. (cid:3) Remark 2.2
If in addition θ − αN +1 w θ N +1 ∈ C ( C R,τ × (0 , ̺ )) , we have that ∂ ν α w ( t , ˜ θ ,
0) = − lim θ N +1 → + θ − αN +1 w ( t , ˜ θ , θ N +1 ) θ N +1 < . Finally, by the above two Hopf’s lemmas and a similar argument as Corol-lary 4.12 in [6], we obtain the following version strong maximum principle.
Theorem 2.2
Let Ω ⊂ R × S N + be an open bounded set with a part of bound-ary Γ ⊂ R × ∂S N + . Assume w ∈ C (Ω) ∩ C ( ¯Ω) and θ − αN +1 w θ N +1 ∈ C ( ¯Ω) satisfies L α w + w tt + a ( t, θ ) w t − b ( t, θ ) w ≤ in Ω , − lim θ N +1 → + θ − αN +1 w θ N +1 ≥ on Γ ,w ≥ , w on Ω , where a and b are bounded functions and b is nonnegative. Then w > in Ω ∪ Γ . We prove our main result in this section via the method of moving planes.For which we give some preliminary notations, we defineΣ µ = { ( t, θ ) : t ∈ ( −∞ , µ ) , θ ∈ S N + } ,T µ = { ( t, θ ) : t = µ, θ ∈ S N + } , µ ( t, θ ) = U (2 µ − t, θ ) − U ( t, θ ) ,z µ ( t, θ ) = V (2 µ − t, θ ) − V ( t, θ ) . A direct calculation shows the comparison functions w µ and z µ satisfy L α w µ + w µtt + δ w µt − ν w µ = − δ U t in Σ µ , − lim θ N +1 → + θ − αN +1 ∂ θ N +1 w µ = c µ z µ on ∂ L Σ µ ,L α z µ + z µtt + δ z µt − ν z µ = − δ V t in Σ µ , − lim θ N +1 → + θ − αN +1 ∂ θ N +1 z µ = d µ w µ on ∂ L Σ µ , (3.1)where ∂ L Σ µ := ( R × ∂S N + ) ∩ Σ µ , c µ ( t, θ ) = ( ( V (2 µ − t,θ ) ) q − ( V ( t,θ ) ) q V (2 µ − t,θ ) − V ( t,θ ) if V (2 µ − t, θ ) = V ( t, θ ) , V (2 µ − t, θ ) = V ( t, θ ) , (3.2)and d µ ( t, θ ) = ( ( U (2 µ − t,θ ) ) p − ( U ( t,θ ) ) p U (2 µ − t,θ ) − U ( t,θ ) if U (2 µ − t, θ ) = U ( t, θ ) , U (2 µ − t, θ ) = U ( t, θ ) . (3.3)From (3.2) and (3.3), we have that c µ ≥ , d µ ≥ R × ∂S N + . (3.4)Moreover, the definitions of w µ and z µ imply w µ ≡ z µ ≡ T µ . (3.5)Next, we show that U and V decay monotonically near −∞ . In fact, bydifferentiating (2.2), we find that U t = r β ( β U + rU r ) and V t = r β ( β V + rV r ) . (3.6)So take into account β , β > r = e t , U >
V >
0, we can obtain t for which U t > V t > t , (3.7)and 0 < U ( t, θ ) , V ( t, θ ) < ε in Σ t , (3.8)where 0 < ε << t such that (3.7) and (3.8) holds. The following maximumprinciple for system (3.1) near −∞ is needed, which permits us to get themoving planes method started. 13 emma 3.1 (1) For all µ ∈ ( −∞ , t ] one has w µ ≥ and z µ ≥ in Σ µ .(2) Suppose that for µ ∈ ( t , + ∞ ) , we have w µ ≥ and z µ ≥ on T t . Then w µ ≥ and z µ ≥ in Σ t . Proof.
Observe that in both cases (1) and (2) we have w µ ≥ z µ ≥ T t ∧ µ by (3.5), where t ∧ µ = min { t , µ } . Therefore, we treat bothcases at the same time by a contradiction argument, assuming thatmin (cid:26) inf Σ t ∧ µ w µ , inf Σ t ∧ µ z µ (cid:27) < . (3.9)Up to an symmetry in the argument, there are two cases to rule out.Case I: inf Σ t ∧ µ w µ < Σ t ∧ µ z µ ≥ . We consider the function W ( x ) = ( max {− w µ ( x ) , } , x ∈ Σ t ∧ µ , , x ∈ (cid:0) Σ t ∧ µ (cid:1) c . (3.10)Therefore, using the equation (3.1), we have0 ≤ Z Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt = − Z Σ t ∧ µ θ − αN +1 ∇ w µ ∇ W · e δ t dθdt = Z Σ t ∧ µ div ( θ − αN +1 ∇ w µ ) W e δ t dθdt − Z ∂ L Σ t ∧ µ θ − αN +1 ∂w µ ∂ν W e δ t dθdt = − Z Σ t ∧ µ θ − αN +1 ( w µtt + δ w µt ) W e δ t dθdt + Z Σ t ∧ µ θ − αN +1 ν w µ W e δ t dθdt − Z Σ t ∧ µ θ − αN +1 δ U t W e δ t dθdt − Z ∂ L Σ t ∧ µ c µ z µ W e δ t dθdt ≤ − Z Σ t ∧ µ θ − αN +1 ( w µtt + δ w µt ) W e δ t dθdt (3.11)since (3.7), δ ≥ ν > z µ ≥ ∂ L Σ t ∧ µ by the continuity of z µ .Since β >
0, the definitions of U (see (2.2)) and w µ implylim inf t →−∞ inf θ ∈ S N + w µ ( t, θ ) ≥ . This implies that W ≡ t → −∞ and θ ∈ S N + . Therefore, we can estimatethat Z Σ t ∧ µ θ − αN +1 ( w µtt + δ w µt ) W e δ t dθdt = Z Σ t ∧ µ θ − αN +1 (cid:0) e δ t w µt (cid:1) t W dθdt − Z Σ t ∧ µ θ − αN +1 e δ t w µt W t dθdt = Z Σ t ∧ µ θ − αN +1 e δ t | W t | dθdt ≥ . (3.12)Together (3.11) and (3.12), we have Z Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt ≡ . This is impossible unless W ≡ t ∧ µ and therefore w µ ≥ t ∧ µ . Thiscontradicts with inf Σ t ∧ µ w µ < Σ t ∧ µ w µ < Σ t ∧ µ z µ < . Since w µ ≥ T t ∧ µ and lim inf t →−∞ inf θ ∈ S N + w µ ( t, θ ) ≥
0, there existsa point (¯ t, ¯ θ ) ∈ Σ t ∧ µ ∪ ∂ L Σ t ∧ µ such that the negative infimum of w µ isachieved, that is, w µ (¯ t, ¯ θ ) = inf Σ t ∧ µ w µ < . If (¯ t, ¯ θ ) ∈ Σ t ∧ µ , then w µtt (¯ t, ¯ θ ) ≥ , ∆ θ w µ (¯ t, ¯ θ ) ≥ , w µt (¯ t, ¯ θ ) = 0 , and ∇ w µ (¯ t, ¯ θ ) = 0 . Thus, from (2.6), (3.7) and the first equation of (3.1), we have that0 < − ν w µ (¯ t, ¯ θ ) ≤ − δ U t (¯ t, ¯ θ ) ≤ , since ν > t, ¯ θ ) ∈ ∂ L Σ t ∧ µ , which impliesthat ∂ θ N +1 w µ (¯ t, ¯ θ ) ≥
0. Therefore, c µ (¯ t, ¯ θ ) z µ (¯ t, ¯ θ ) ≤ z µ (¯ t, ¯ θ ) ≤ c µ ≥
0. By the continuity of z µ , we know z µ ( t, θ ) ≤ ∂ L Σ t ∧ µ . Then0 < V (2 µ − t, θ ) ≤ V ( t, θ ) < ε on ∂ L Σ t ∧ µ . By the mean value principle, we have c µ ( t, θ ) ≤ qε q − on ∂ L Σ t ∧ µ . Similarly,we have d µ ( t, θ ) ≤ pε p − on ∂ L Σ t ∧ µ .We define function Z ( x ) = ( max {− z µ ( x ) , } , x ∈ Σ t ∧ µ , , x ∈ (cid:0) Σ t ∧ µ (cid:1) c . (3.13)15ithout of loss generality, we suppose δ ≤ δ . By a similar estimate inCase I, we have Z Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt ≤ Z Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt ≤ − Z ∂ L Σ t ∧ µ d µ w µ Ze δ t dθdt = Z ∂ L Σ t ∧ µ d µ W Ze δ t dθdt = Z ∂ L Σ t ∧ µ d µ W Ze ( δ − δ ) t e δ t dθdt ≤ pε p − e ( δ − δ ) t Z ∂ L Σ t ∧ µ W Ze δ t dθdt. By the H¨older and fractional Sobolev trace inequalities as in [28] (see also[33] for the Sobolev trace inequality in all R n ), we know R ∂ L Σ t ∧ µ W Ze δ t d ˜ θdt ≤ (cid:16)R ∂ L Σ t ∧ µ | e δ t/ Z | d ˜ θdt (cid:17) / (cid:16)R ∂ L Σ t ∧ µ | e δ t/ W | d ˜ θdt (cid:17) / ≤ C N,α (cid:16)R Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt (cid:17) / (cid:16)R Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt (cid:17) / , where θ = (˜ θ, θ N +1 ) ∈ S N + and C N,α is a positive constant depending only on N and α . Hence, Z Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt ! / ≤ pε p − C N,α e ( δ − δ ) t Z Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt ! / . (3.14)Moreover, as the argument in Case I, we have Z Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt ≤ − Z ∂ L Σ t ∧ µ c µ z µ W e δ t d ˜ θdt = Z ∂ L Σ t ∧ µ c µ ZW e δ t d ˜ θdt. Similarly, by the H¨older and fractional Sobolev trace inequalities, we have R Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt ≤ qε q − (cid:16)R ∂ L Σ t ∧ µ | e δ t/ Z | d ˜ θdt (cid:17) / (cid:16)R ∂ L Σ t ∧ µ | e δ t/ W | d ˜ θdt (cid:17) / ≤ qε q − C N,α (cid:16)R Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt (cid:17) / (cid:16)R Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt (cid:17) / . Z Σ t ∧ µ θ − αN +1 |∇ W | e δ t ! / dθdt ≤ qε q − C N,α Z Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt ! / . (3.15)Then, combining (3.14) and (3.15), we have (cid:16)R Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt (cid:17) / ≤ pqC N,α e ( δ − δ ) t ε p + q − (cid:16)R Σ t ∧ µ θ − αN +1 |∇ W | e δ t dθdt (cid:17) / and (cid:16)R Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt (cid:17) / ≤ pqC N,α e ( δ − δ ) t ε p + q − (cid:16)R Σ t ∧ µ θ − αN +1 |∇ Z | e δ t dθdt (cid:17) / . These are impossible since ε << p + q > pq > W ≡ Z ≡ t ∧ µ and thus w µ ≥ z µ ≥ t ∧ µ , which contradictwith our assumption. We complete the proof of Lemma 3.1. (cid:3) By Case (1) of Lemma 3.1, we have w µ ≥ z µ ≥ µ for all µ ∈ ( −∞ , t ]. This enables us to define a maximal value of µ up to whichthe positivity of these functions holds. This is the purpose of the followinglemma. Lemma 3.2
We have either (1) : there exists ¯ µ ∈ R such that w ¯ µ ≡ and z ¯ µ ≡ in Σ ¯ µ , or (2) : for any µ ∈ R one has w µ > and z µ > in Σ µ .Moreover, in the latter case one has U t > and V t > in R × S N + . (3.16) Proof.
DefineΛ = sup { µ ∈ R : ∀ λ ∈ ( −∞ , µ ) , w λ ≥ z λ ≥ λ } . (3.17)By Lemma 3.1, it is clear that Λ > −∞ . Next, we prove that either Λ < + ∞ ,in which case (1) holds with ¯ µ = Λ, or Λ = + ∞ in which case we have case(2) together with (3.16). 17f Λ = + ∞ , then case (2) is trivially satisfied. Moreover, in this case(3.16) is a consequence of the following argument. Since (3.5), we have ∂w µ ∂t ≤ ∂z µ ∂t ≤ µ . By the definitions of w µ and z µ , we know ∂w µ ∂t = − U t and ∂z µ ∂t − V t on T µ . (3.18)Therefore, U t ≥ V t ≥ T µ for all µ ∈ R and thus throughout R × S N + . Then, by (3.1), we have (cid:26) L α w µ + w µtt + δ w µt − ν w µ ≤ µ ,L α z µ + z µtt + δ z µt − ν z µ ≤ µ . (3.19)Applying the Hopf lemma to each equation in (3.19) yields ∂w µ ∂t < ∂z µ ∂t < T µ , and thus we have (3.16) thanks to (3.18).Suppose that Λ < + ∞ . We prove case (1) by contradiction and assumethat w Λ z Λ Λ . For all −∞ < µ ≤ Λ, by the strongermaximum principle, we know that w µ > z µ > µ . The abovearguments imply that U t > V t > T µ for −∞ < µ < Λ. Hence, U t > V t > Λ .Therefore, by (3.1) we have that, for −∞ < µ < Λ, (cid:26) L α w µ + w µtt + δ w µt − ν w µ = − δ U t ≤ µ ,L α z µ + z µtt + δ z µt − ν z µ = − δ V t ≤ µ . (3.20)Now, evaluating (3.20) at µ = Λ by continuity, we obtain L α w Λ + w Λ tt + δ w Λ t − ν w Λ ≤ Λ , (3.21)and L α z Λ + z Λ tt + δ z Λ t − ν z Λ ≤ Λ . (3.22)An application of the strong maximum principle (see Theorem 2.2) to (3.21)and (3.22) implies that either w Λ > w Λ ≡ Λ on the one hand z Λ > z Λ ≡ Λ on the other hand.18t is easy to check that (3.1) and Theorem 2.2 rules out the cases w Λ > z Λ ≡ Λ as well as w Λ ≡ z Λ > Λ . Here we only showthe case w Λ > z Λ ≡ Λ is impossible. In fact, since z Λ ≡ Λ ,by the continuous up to the boundary, we have z Λ = 0 and θ − αN +1 ∂ θ N +1 z Λ = 0on ∂ L Σ Λ . If w Λ > Λ , then applying Theorem 2.2 to (3.1) we know w Λ > ∂ L Σ Λ and thus d Λ >
0. So by (3.1) we have0 = − lim θ N +1 → + θ − αN +1 ∂ θ N +1 z Λ = d Λ ( t, θ ) w Λ > ∂ L Σ Λ . This is a contradiction.Hence in the following we may assume that both w Λ and z Λ are strictlypositive in Σ Λ . By the Hopf’s lemma (see Lemma 2.1) we have ∂w Λ ∂t < ∂z Λ ∂t < T Λ , (3.24)since w Λ = 0 and z Λ = 0 on T Λ .Next, we claim that there exists ε > w µ ≥ z µ ≥ µ for all µ ∈ (Λ , Λ + ε ). This will result in a contradiction with the definitionof Λ; hence the lemma will be proved. This is done in the following way.We split the domain into three disjoint subsets:Σ µ = Σ t ∪ (Σ Λ − δ \ Σ t ) ∪ (Σ µ \ Σ Λ − δ ) , for some small δ > δ >
0, we know w Λ > z Λ > Λ − δ \ Σ t . Therefore, a straightforwardcontinuity argument implies that there exists ε = ε ( δ ) > µ ∈ [Λ , Λ + ε ), min ( inf Σ Λ − δ \ Σ t w µ , inf Σ Λ − δ \ Σ t z µ ) > . We carry on the analysis by examining the first part of the domain. Bythe above consideration, for all µ ∈ [Λ , Λ + ε ), we have w µ ≥ z µ ≥ T t ⊂ (Σ Λ − δ \ Σ t ). An application of case (2) of Lemma 3.1, we have w µ > z µ > t .Finally, we do the analysis on the third part of the domain, namely Σ µ \ Σ Λ − δ . A simple continuity argument shows that (3.24) remains valid if Λ isreplaced by any µ in a small right neighborhood of Λ, that is, there exists ε > µ ∈ [Λ , Λ + ε ), ∂w µ ∂t < ∂z µ ∂t < T µ .
19y elliptic estimates give locally uniform C bounds for w µ and z µ in ( t, θ )as well as in µ . Henceinf µ − ε 1) and θ ∈ S N + . Similarly for z µ . This means that we canchoose ε ′ ∈ (0 , ε ) such that ∂w µ /∂t < ∂z µ /∂t < { ( t, θ ) : µ − ε ′ < t < µ, θ ∈ S N + } for all µ ∈ [Λ , Λ+ ε ′ / w µ = 0 and z µ = 0on T µ , we know w µ > z µ > { ( t, θ ) : µ − ε ′ < t < µ, θ ∈ S N + } for all λ ∈ [Λ , Λ + ε ′ / δ = ε ′ / ε = min { ε ( δ ) , ε } , summing up the above re-sults, we have w µ > z µ > µ for all µ ∈ (Λ , Λ+ ε ). This contradictsto the definition of Λ (see (3.17)). (cid:3) Proof of Theorem 1.2. Suppose that ( U, V ) is positive solution to(1.12). Then the comparison functions w µ and z µ satisfy the alternative inLemma 3.1. Next, we prove both cases (1) and (2) in Lemma 3.2 cannothappen.First, we show the case (1) in Lemma 3.2 provides a contradiction. Inorder to get the contradiction, we translate the origin to ¯ µ , that is, defineˆ U ( t, θ ) = U ( t + ¯ µ, θ ) and ˆ V ( t, θ ) = V ( t + ¯ µ, θ ). Since w ¯ µ ≡ z ¯ µ ≡ ¯ µ , then those two functions are even in the variable t , i.e.,ˆ U ( − t, θ ) = ˆ U ( t, θ ) and ˆ V ( − t, θ ) = ˆ V ( t, θ ) , (3.25)This implies U t and V t are odd functions. On the other hand, by the firstand third equations of (3.1) and ( δ , δ ) = (0 , 0) (see 2.6), we know U t or V t are even in variable t . So we can conclude that U or V must be constant inthe whole domain R × S N + . This contradicts the regularity of U or V at theorigin, see (2.2).We complete the proof of Theorem 1.2 by showing that case (2) in Lemma3.2 is also impossible. Observe that (1.12) is translation invariant in x direc-tion for X = ( x, y ) ∈ R N × R + . Hence we can change the initial function,i.e., we define U x ( x, y ) = U ( x − x , y ), V x ( x, y ) = V ( x − x , y ) (and corre-sponding U x , V x ) for any x ∈ R N . Repeat the whole discussion for thesenew functions, only two cases can be arise. First, there exists an origin x such that case (1) holds. As we have done, it is impossible. Another case is,for any origin x ∈ R N , (2) holds for the transformed functions U x and V x .So by (3.6), we have in particular β U ( X ) + ∇ U ( X ) · ( x − x , y ) ≥ X = ( x, y ) ∈ R N × R + and x ∈ R N . Hence we can obtain ∇ U ( X ) · ( x − x | ( x − x , y ) | , y | ( x − x , y ) | ) ≥ − β U ( X ) | ( x − x , y ) | (3.26)for all X ∈ ( R N \ { x } ) × R + . Now we let e = ( e , e , · · · , e N +1 ) ∈ S N + with e N +1 > x = x − σ ( e , · · · , e N ) and y = σe N +1 for σ > ∇ U ( X ) · e ≥ − β U ( X ) σ . Letting σ → + ∞ yields ∇ U ( X ) · e ≥ . (3.27)Since (3.27) holds for any e ∈ S N + and X ∈ R N +1+ , then we deduce that ∇ U ( X ) ≡ . This is impossible by the second equation of (1.12) since the solution we aredealing with is nontrivial. (cid:3) A. Q. was partially supported by Fondecyt Grant No. 1151180 ProgramaBasal, CMM. U. de Chile and Millennium Nucleus Center for Analysis ofPDE NC130017 and A. Xia was supported by the ”Programa de Iniciaci´ona la Investigaci´on Cient´ıfica” (PIIC) UTFSM 2014. References [1] D. Applebaum, L´evy processes: From probability to Finance and quan-tum groups, Not. Am. Math. Soc. 51 (2004), 1336-1347.[2] J. Busca and R. Man´asevich, A Liouville-type theorem for Lane-Emdensystems. Indiana Univ. Math. J. 51 (2002), no. 1, 37-51.[3] J. Bertoin, L´evy Processes, Cambridge Tracts in Mathematics, 121 Cam-bridge University Press, Cambridge, 1996.[4] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered me-dia: Statistical mechanics, models and physical applications, Physicsreports 195 (1990). 215] L. Caffarelli and L. Silvestre, An extension problem related to the frac-tional Laplacian. Comm. Partial Differential Equations 32 (2007), no.7-9, 1245-1260.[6] X. Cabr´e and Y. Sire, Nonlinear equations for fractional Laplacians, I:Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst.H. Poincar´e Anal. Non Lin´eaire 31 (2014), no. 1, 23-53.[7] W.X. Chen, C.M. Li and B. Ou, Classification of solutions for an integralequation. Comm. Pure Appl. Math. 59 (2006), no. 3, 330-343.[8] W.X. Chen, C.M. Li and B. Ou, Qualitative properties of solutions foran integral equation. Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 347-354.[9] J. D´avila, L. Dupaigne and J.C. Wei, On the fractional Lane-Emdenequation. arXiv:1404.3694v1 [math.AP].[10] S. Dipierroa and A. Pinamontib, A geometric inequality and a symme-try result for elliptic systems involving the fractional Laplacian, J. ofDifferential Equa. 255, (2013), 85-119.[11] D.G. de Figueiredo and B. Sirakov, Liouville type theorems, monotonic-ity results and a priori bounds for positive solutions of elliptic systems.Math. Ann. 333 (2005), no. 2, 231-260.[12] P. Felmer and D.G. de Figueiredo, A Liouville-type theorem for systems.Ann. Sc. Norm. Sup. Pisa XXI (1994), 387-397.[13] M. Fall and T. Weth, Nonexistence results for a class of fractional ellipticboundary value problems. J. Funct. Anal. 263 (2012), no. 8, 2205-2227.[14] B. Gidas and J. Spruck, Global and local behavior of positive solutionsof nonlinear elliptic equations. Comm. Pure Appl. Math. 34, (1981),525-598.[15] Humphries et al., Environmental context explains Levy and Browianmovement patterns of marine predators, Nature 465, June 2010, 1066-1069.[16] T.L. Jin, Y.Y. Li and J.G. Xiong, On a fractional Nirenberg problem,part I: blow up analysis and compactness of solutions. J. Eur. Math.Soc. (JEMS) 16 (2014), no. 6, 1111-1171.2217] Y.Y. Li, Remark on some conformally invariant integral equations: themethod of moving spheres. J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2,153-180.[18] E. Mitidieri, Non-existence of Positive Solutions of Semilinear EllipticSystems in R N , Differential Integral Equations 9 (1996), 465-479.[19] L. Ma and D. Chen, A Liouville type theorem for an integral system.Commun. Pure Appl. Anal. 5 (2006), no. 4, 855-859.[20] E. di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to thefractional Sobolev spaces, Bull. Sci. Math. (2012), no. 5, 521–573.[21] A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equa-tions and systems involving fractional Laplacian in the half space, Calc.of Variations, 52 (2015) 641-659.[22] L. Silvestre, Regularity of the obstacle problem for a fractional power ofthe Laplace operator. Comm. Pure Appl. Math. 60 (2007), no. 1, 67-112.[23] P. Souplet, The proof of the Lane-Emden conjecture in four space di-mensions. Adv. Math. 221 (2009), no. 5, 1409-1427.[24] M.A.S. Souto, A priori estimates and existence of positive solutions ofnon-linear cooperative elliptic systems, Diff. Int. Eq. 8 (1995) 1245-1258.[25] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emdensystems, Atti Sem. Mat. Fis. Univ. Modena 46, suppl. (1998), 369-380.[26] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emdensystems, Diff. Int. Eq. 9 (1996), 635-653.[27] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul.11