Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians
aa r X i v : . [ m a t h . A P ] S e p SUPER POLY-HARMONIC PROPERTIES, LIOUVILLE THEOREMS ANDCLASSIFICATION OF NONNEGATIVE SOLUTIONS TO EQUATIONSINVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS
DAOMIN CAO, WEI DAI, GUOLIN QIN
Abstract.
In this paper, we are concerned with equations (1.1) involving higher-order frac-tional Laplacians. By introducing a new approach, we prove the super poly-harmonic prop-erties for nonnegative solutions to (1.1) (Theorem 1.1). Our theorem seems to be the firstresult on this problem. As a consequence, we derive many important applications of the superpoly-harmonic properties. For instance, we establish Liouville theorems, integral represen-tation formula and classification results for nonnegative solutions to fractional higher-orderequations (1.1) with general nonlinearities f ( x, u, Du, · · · ) including conformally invariantand odd order cases. In particular, our results completely improve the classification resultsfor third order equations in Dai and Qin [21] by removing the assumptions on integrability.We also derive a characterization for α -harmonic functions via averages in the appendix. Keywords:
Super poly-harmonic properties; Higher-order fractional Laplacians; Conformally in-variant equations; Nonnegative classical solutions; Classification of solutions; Liouville theorems.
Primary: 35R11; Secondary: 35C15, 35B53, 35B06. Introduction
Background and setting of the problem.
In this paper, we mainly consider nonneg-ative classical solutions to the following equations involving higher-order fractional Laplacians(1.1) ( ( − ∆) m + α u ( x ) = f ( x, u, Du, · · · ) , x ∈ R n ,u ∈ C m +[ α ] , { α } + ǫloc ∩ L α ( R n ) , u ( x ) ≥ , x ∈ R n , where n ≥
2, 1 ≤ m < + ∞ is an integer, 0 < α < ǫ > α ] denotesthe integer part of α , { α } := α − [ α ], the higher-order fractional Laplacians ( − ∆) m + α :=( − ∆) m ( − ∆) α and nonlinearity f ( x, u, Du, · · · ) ≥ x , u and derivatives of u ) which is continuous with respect to x ∈ R n .For any u ∈ C [ α ] , { α } + ǫloc ( R n ) ∩ L α ( R n ), the nonlocal operator ( − ∆) α (0 < α <
2) is definedby (see [6, 12, 21, 22, 37, 41])(1.2) ( − ∆) α u ( x ) = C n,α P.V. Z R n u ( x ) − u ( y ) | x − y | n + α dy := C n lim ε → Z | y − x |≥ ε u ( x ) − u ( y ) | x − y | n + α dy, where the function space(1.3) L α ( R n ) := n u : R n → R (cid:12)(cid:12) Z R n | u ( x ) | | x | n + α dx < ∞ o . D. Cao was supported by NNSF of China (No. 11771469) and Chinese Academy of Sciences (No. QYZDJ-SSW-SYS021). W. Dai was supported by the NNSF of China (No. 11971049), the Fundamental ResearchFunds for the Central Universities and the State Scholarship Fund of China (No. 201806025011).
The fractional Laplacians ( − ∆) α can also be defined equivalently (see [13]) by Caffarelli andSilvestre’s extension method (see [15]) for u ∈ C [ α ] , { α } + ǫloc ( R n ) ∩L α ( R n ). Throughout this paper,we define ( − ∆) m + α u := ( − ∆) m ( − ∆) α u for u ∈ C m +[ α ] , { α } + ǫloc ( R n ) ∩ L α ( R n ), where ( − ∆) α u is defined by definition (1.2). Due to the nonlocal feature of ( − ∆) α , we need to assume u ∈ C m +[ α ] , { α } + ǫloc ( R n ) with arbitrarily small ǫ > u ∈ C m +[ α ] , { α } is not enough)to guarantee that ( − ∆) α u ∈ C m ( R n ) (see [13, 37]), and hence u is a classical solution toequation (1.1) in the sense that ( − ∆) m + α u is pointwise well-defined and continuous in thewhole R n .For 0 < γ < + ∞ , PDEs of the form(1.4) ( − ∆) γ u ( x ) = f ( x, u, · · · )have numerous important applications in conformal geometry and Sobolev inequalities, whichalso model many phenomena in mathematical physics, astrophysics, probability and finance(see [3, 12, 13, 15, 16, 17, 31, 39, 40] and the references therein). We say that equation (1.4)is in critical order if γ = n , is in sub-critical order if 0 < γ < n and is in super-critical orderif n < γ < + ∞ .1.2. Super poly-harmonic properties of nonnegative solutions.
First, we will investi-gate the super poly-harmonic properties of nonnegative solutions to (1.1). It is well known thatthe super poly-harmonic properties of nonnegative solutions play a crucial role in establishingthe integral representation formulae, Liouville type theorems and classification of solutions tohigher order PDEs in R n or R n + (see [2, 3, 4, 5, 11, 18, 19, 20, 21, 23, 25, 26, 31, 33, 39] andthe references therein).For integer higher-order equations (i.e., α = 0 in (1.1)), the super poly-harmonic propertiesof nonnegative solutions usually can be derived via the “spherical average, re-centers anditeration” arguments in conjunction with careful ODE analysis (we refer to [5, 31, 33, 39], seealso [3, 4, 11, 20, 23, 25] and the references therein). However, for the fractional higher-orderequation (1.1) , so far there is no result on the super poly-harmonic properties. The reason forthis is that ( − ∆) α is nonlocal and ( − ∆) α f ( r ) can not be calculated or expanded accurately(0 < α < f ( r ) is a radially symmetric function), thus the strategy for integer higher-order equations does not work any more for equation (1.1) involving higher-order fractionalLaplacians. To overcome these difficulties we need to implement new ideas and arguments.In this paper, by taking full advantage of the Poisson representation formulae for ( − ∆) α and developing some new integral estimates on the average R + ∞ R R α r ( r − R ) α ¯ u ( r ) dr and iterationtechniques, we will introduce a new approach to overcome these difficulties and establish thesuper poly-harmonic properties of nonnegative classical solutions to (1.1) (see Section 2). Ourtheorem seems to be the first result on this problem. Theorem 1.1.
Assume n ≥ , m ≥ , < α < and f ≥ is continuous w.r.t. x ∈ R n .Suppose that u is a nonnegative classical solution to (1.1) . Then, we have, for every i =0 , , · · · , m − , (1.5) ( − ∆) i + α u ( x ) ≥ , ∀ x ∈ R n . Now suppose the nonlinearity f ( x, u, Du, · · · ) ≤ x , u and derivatives of u ) which is continuous with respect to x ∈ R n , by usingthe ideas in proving the super poly-harmonic properties in Theorem 1.1, we can derive thefollowing sub poly-harmonic properties of nonnegative classical solutions to (1.1). DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 3
Theorem 1.2.
Assume n ≥ , m ≥ , < α < and f ≤ is continuous w.r.t. x ∈ R n .Suppose that u is a nonnegative classical solution to (1.1) . Then, we have, for every i =0 , , · · · , m − , (1.6) ( − ∆) i + α u ( x ) ≤ , ∀ x ∈ R n . Liouville theorems, integral representation formula and classification of non-negative solutions.
In this subsection, by applying the super poly-harmonic properties inTheorem 1.1, we will derive some important results on equations involving higher-order frac-tional Laplacians. (i) Liouville theorem for fractional poly-harmonic functions in R n . Assume u ≥ R n , that is,(1.7) ( − ∆) m + α u ( x ) = 0 , ∀ x ∈ R n , where n ≥
2, 1 ≤ m < + ∞ is an integer and 0 < α < − ∆) i + α u ≡ R n for every i =0 , · · · , m −
1. In particular, one has ( − ∆) α u ≡ R n , and hence from the Liouville theoremfor fractional Laplacians ( − ∆) α with 0 < α < u ≡ C in R n forsome nonnegative constant C ≥
0. Therefore, we have the following Liouville theorem forfractional poly-harmonic functions in R n . Theorem 1.3.
Assume n ≥ , m ≥ and < α < . Suppose u is a nonnegative fractionalpoly-harmonic functions in R n satisfying (1.7) , then u ≡ C ≥ in R n .(ii) Subcritical order cases m + α < n . Equation (1.1) is closely related to the following integral equation(1.8) u ( x ) = Z R n R m + α,n | x − y | n − m − α f ( y, u ( y ) , · · · ) dy, where the Riesz potential’s constants R γ,n := Γ( n − γ ) π n γ Γ( γ ) for 0 < γ < n (see [38]).From the super poly-harmonic properties of nonnegative solutions in Theorem 1.1, by usingthe methods in [6, 41], we can deduce the following equivalence between PDEs (1.1) and IEs(1.8). Theorem 1.4.
Assume m + α < n , m ≥ , < α < and f ≥ is continuous w.r.t. x ∈ R n .Suppose that u is a nonnegative classical solution to (1.1) , then u is also a nonnegative solutionto integral equation (1.8) , and vice versa.Remark . Based on Theorem 1.1, the proof of Theorem 1.4 is entirely similar to [6, 41] (seealso [20, 22]), so we omit the details here.
Remark . One can observe that, Theorem 1.1 and Theorem 1.4 hold for PDEs (1.1) andIEs (1.8) if we take the nonlinearities f = | x | a u p ( a ≥ p > f = | x | a e nu ( a ≥
0) or f = | x | a u p (1 + |∇ u | ) κ ( a ≥ p > κ >
0) and so on · · · . If we consider positive solution u >
0, then Theorem 1.1 and Theorem 1.4 are also valid for PDEs (1.1) and IEs (1.8) with f = | x | a u − q ( a ≥ q > f = u n +2 m + αn − m − α , which is geometrically interesting. DAOMIN CAO, WEI DAI, GUOLIN QIN
The quantitative and qualitative properties of solutions to fractional order or higher orderconformally invariant equations of the form(1.9) ( − ∆) γ u = u n + γn − γ with 0 < γ < n have been extensively studied (see [7, 12, 14, 16, 20, 21, 27, 31, 32, 39, 40] and the referencestherein). The classification results for conformally invariant equations (1.9) have importantapplications in many problems from conformal geometry (i.e., prescribing scalar curvatureproblems, variational problems involving Paneitz operators on compact Riemannian manifolds,see [7, 8, 9, 10, 14, 16, 21, 31, 32, 33, 34, 39, 40]). In [14], by developing the method of movingplanes in integral forms, Chen, Li and Ou classified all positive L nn − γ loc solutions to the equivalentintegral equation of PDE (1.9) for general γ ∈ (0 , n ). As a consequence, they obtained theclassification for positive weak solutions to PDE (1.9). As to the classification theorems forpositive classical solutions to PDE (1.9), all known results are focused on the cases that0 < γ <
2, or 2 ≤ γ < n is an even integer (see Caffarelli, Gidas and Spruck [7], Chen, Li andLi [12], Chen, Li and Ou [14], Gidas, Ni and Nirenberg [27], Lin [31], Wei and Xu [39]).One should observe that, when γ ∈ (2 , n ) is an odd integer, or more general, when γ =2 m + α < n with m ≥ < α <
2, classification for positive classical solutions to (1.9) isstill open. In the particular case γ = 3, by applying the harmonic asymptotic expansions for( − ∆) ¯ u (¯ u is the Kelvin transform of u ) and the method of moving planes to the third-orderequation (1.9) directly, Dai and Qin [21] derived the classification of nonnegative classicalsolutions to (1.9) under additional weak integrability assumption R R n u n +3 n − | x | n − dx < ∞ .In this paper, by the classification of positive L nn − m − α loc solutions to integral equation (1.8) in[14] (Theorem 1 in [14]) and the equivalence between PDE (1.1) and integral equation (1.8) inTheorem 1.4, we can classify all positive classical solutions to (1.1) in the conformally invariantcases f = u n +2 m + αn − m − α without any assumptions on integrability or decay of u .Our classification result for (1.1) in the conformally invariant cases is as follows. Theorem 1.7.
Assume m + α < n , m ≥ , < α < and f = u n +2 m + αn − m − α . Suppose that u isa nonnegative classical solution of (1.1) , then either u ≡ or u is of the following form u ( x ) = µ n − m − α Q (cid:0) µ ( x − x ) (cid:1) for some µ > and x ∈ R n , where Q ( x ) := R m + α,n I (cid:0) n − m − α (cid:1) ! n − m − α m + α ) (cid:16)
11 + | x | (cid:17) n − m − α with I ( s ) := π n Γ (cid:0) n − s (cid:1) Γ( n − s ) for < s < n .Remark . Theorem 1.7 follows directly from Theorem 1 in [14] and Theorem 1.4, so weomit the details here. The exact constants in the expression of Q ( x ) are given by formula (37)in Lemma 4.1 in [17]. Remark . Combining Theorem 1.7 with the classification theorems in [7, 12, 14, 27, 31, 39]gives us the complete classification results for conformally invariant equations (1.9) in all thecases 0 < γ < n . If we take α = 1, then Theorem 1.7 gives the classification results for allthe odd order conformally invariant equations (1.1). In particular, Theorem 1.7 completely DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 5 improves the classification results for third order conformally invariant equations in [21] byremoving the integrability assumption R R n u n +3 n − | x | n − dx < ∞ .Next, we take f = | x | a u p ( a ≥ p >
0) and study the Liouville property of nonnegativesolutions in the subcritical cases.For PDEs (1.1) and IEs (1.8), we say the Hardy-H´enon type nonlinearities f = | x | a u p issubcritical if 0 < p < p c ( a ) := n +2 m + α +2 an − m − α , critical if p = p c ( a ) and super-critical if p > p c ( a ).There are also lots of literature on Liouville type theorems for fractional order or higher orderHardy-H´enon type equations in the subcritical cases, and we refer to [4, 5, 6, 7, 12, 18, 19,21, 22, 24, 25, 26, 31, 35, 39, 41] and the references therein. It should be noted that, all theknown results focused on the cases m = 0 or α = 0, hence Liouville type theorems for generalfractional higher-order cases m ≥ < α < m = α = 1 and f = u p with 1 ≤ p < n +3 n − , Dai and Qin [21] derived Liouville type theoremfor nonnegative classical solutions to (1.1) under additional weak integrability assumption R R n u p | x | n − dx < ∞ .In this paper, by applying the method of scaling spheres developed recently by Dai and Qin[22] (see also [23, 24, 26]), we will establish Liouville type theorem for nonnegative solutionsto IEs (1.8). Our Liouville type result for IEs (1.8) is as follows. Theorem 1.10.
Assume m + α < n , m ≥ , < α < and f = | x | a u p with a ≥ and < p < p c ( a ) . Suppose u ∈ C ( R n ) is a nonnegative solution to IEs (1.8) , then u ≡ in R n .Remark . It is clear from the proof of Theorem 1.10 that (see (4.41) in Section 3), theLiouville type results in Theorem 1.10 are also valid for f = | x i | a u p ( i = 1 , , · · · , n ) with a ≥ < p < p c ( a ). Theorem 1.10 can also be available for more general nonlinearities f ( x, u )satisfying appropriate assumptions, we leave the details to readers (we refer to [22, 23, 24, 26]).From the equivalence between PDEs (1.1) and IEs (1.8) in Theorem 1.4 and Theorem 1.10,we derive the following Liouville type result for nonnegative classical solutions to PDEs (1.1)immediately. Corollary 1.12.
Assume m + α < n , m ≥ , < α < and f ( x, u ) = | x | a u p with a ≥ and < p < p c ( a ) . Suppose u is a nonnegative classical solution to PDEs (1.1) , then u ≡ in R n .Remark . If we take α = 1, then Corollary 1.12 gives Liouville type results for all theodd order equations (1.1) with f = | x | a u p in subcritical cases 0 < p < p c ( a ). In particular,Corollary 1.12 completely improves the Liouville theorem for third order equations (1.1) with f = u p (1 ≤ p < n +3 n − ) in [21] by removing the integrability assumption R R n u p | x | n − dx < ∞ andextending 1 ≤ p < n +3 n − to the full subcritical range 0 < p < n +3 n − . (iii) Critical and super-critical order cases: n ≤ m + α < + ∞ . As an immediate consequence of the super poly-harmonic properties in Theorem 1.1, byarguments developed by Chen, Dai and Qin [3], we can establish Liouville type theorem fornonnegative solutions to (1.1) with general nonlinearities f in both critical and super-criticalorder cases. For the particular case α = 0, Liouville type theorems for integer higher-orderH´enon-Hardy type equations in R n or R n + have been derived by Chen, Dai and Qin [3] and Daiand Qin [23] in both critical and super-critical order cases. Our result will extend the resultsin [3] to general fractional higher-order cases 0 < α < f ( x, u, · · · ).For the critical and super-critical order cases we have the following result. DAOMIN CAO, WEI DAI, GUOLIN QIN
Theorem 1.14.
Assume n ≥ , m ≥ , < α < , n ≤ m + α < + ∞ , f ≥ is continuousw.r.t. x ∈ R n and f > at some point in R n if u > in the whole R n . Suppose that u is anonnegative classical solution to (1.1) , then u ≡ in R n .Remark . If we take α = 1, then Theorem 1.14 gives Liouville type results for all thecritical and super-critical order equations (1.1) involving odd order Laplacians. One shouldobserve that, if f ≥ x ∈ R n and f ≥ C | x | a u p for some a ∈ R , p > C > x = 0 in R n , then f satisfies the assumptions in Theorem 1.14 andhence Theorem 1.14 is valid for equations (1.1) with such kind of nonlinearities f ( x, u, · · · ). Remark . If we consider positive solution u >
0, suppose f ≥ x ∈ R n and f ≥ C | x | a u − q for some a ∈ R , q > C > x = 0 in R n , then Theorem1.14 implies nonexistence of positive solutions and thus extend Theorems 1.2 and 1.3 in [36]to general fractional higher-order cases 0 < α < f ( x, u, · · · ).This paper is organized as follows. In Section 2, we will carry out our proof of Theorem1.1. In Section 3, we will prove Theorem 1.2. Section 4 and 5 are devoted to proving Theo-rem 1.10 and 1.14 respectively. In the Appendix, we establish an important characterizationfor α -harmonic functions via the averages R + ∞ R R α r ( r − R ) α u ( r ) dr and deduce some importantproperties for α -harmonic functions.Throughout this paper, we will use C to denote a general positive constant that may dependon u and the quantities appearing in the subscript, and whose value may differ from line toline. 2. Proof of Theorem 1.1
In this section, we will carry out our proof of the super poly-harmonic properties for non-negative solutions to (1.1) (i.e., Theorem 1.1) via contradiction arguments.Let v i := ( − ∆) i + α u for i = 0 , , · · · , m −
1, then it follows from equation (1.1) that(2.1) ( − ∆) α u = v in R n , − ∆ v = v in R n , · · · · · ·− ∆ v m − = f ≥ R n . Suppose that Theorem 1.1 does not hold, then there must exist a largest integer 0 ≤ k ≤ m − x ∈ R n such that(2.2) v k ( x ) = ( − ∆) k + α u ( x ) < . Let(2.3) ¯ g ( r ) = ¯ g (cid:0) | x − x | (cid:1) := 1 | ∂B r ( x ) | Z ∂B r ( x ) g ( x ) dσ be the spherical average of a function g with respect to the center x .First, we will show that 0 ≤ k ≤ m − k is an odd integer.From (2.1) and the well-known property ∆ u = ∆¯ u , we get(2.4) v k ( r ) ≤ v k (0) := − c < , ∀ r > . It follows immediately that(2.5) v k − ( r ) ≥ v k − (0) + c n r , ∀ r > , DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 7 and(2.6) v k − ( r ) ≤ v k − (0) − r n v k − (0) − c n ( n + 2) r , ∀ r > . Repeating the above argument, we get(2.7) v ( r ) ≥ v (0) + c r + c r + · · · + c k r k , ∀ r > , where c k >
0. From (2.7), we infer that there exists a r large enough, such that(2.8) v ( r ) ≥ c k r k , ∀ r > r . From the first equation in (2.1), we conclude that, for arbitrary
R > u ( x ) = Z B R ( x ) G αR ( x, y ) v ( y ) dy + Z | y − x | >R P αR ( x, y ) u ( y ) dy, ∀ x ∈ B R ( x ) , where the Green’s function for ( − ∆) α with 0 < α < B R ( x ) is given by(2.10) G αR ( x, y ) := C n,α | x − y | n − α Z tRsR b α − (1 + b ) n db if x, y ∈ B R ( x )with s R = | x − y | R , t R = (cid:16) − | x − x | R (cid:17) (cid:16) − | y − x | R (cid:17) , and G αR ( x, y ) = 0 if x or y ∈ R n \ B R ( x )(see [28]), and the Poisson kernel P αR ( x, y ) for ( − ∆) α in B R ( x ) is defined by P αR ( x, y ) := 0for | y − x | < R and(2.11) P αR ( x, y ) := Γ( n ) π n +1 sin πα (cid:18) R − | x − x | | y − x | − R (cid:19) α | x − y | n for | y − x | > R (see [13]). Therefore, we have+ ∞ > u ( x ) = Z B R ( x ) C n,α | y − x | n − α Z R | y − x | − b α − (1 + b ) n db ! v ( y ) dy (2.12) + C ′ n,α Z | y − x | >R R α ( | y − x | − R ) α · u ( y ) | y − x | n dy = C n,α Z R r α − Z R r − b α − (1 + b ) n db ! v ( r ) dr + C ′ n,α Z + ∞ R R α r ( r − R ) α u ( r ) dr. Observe that, if 0 < r ≤ R , then 3 ≤ R r − < + ∞ , and hence(2.13) Z b α − (1 + b ) n db ≤ Z R r − b α − (1 + b ) n db ≤ Z + ∞ b α − (1 + b ) n db. As a consequence of (2.7), (2.8), (2.12) and (2.13), we deduce that u ( x ) ≥ C n,α Z R r r α − v ( r ) dr − e C n,α Z r r α − | v ( r ) | dr (2.14) ≥ C Z R r r k + α − dr − e C ≥ CR k + α − e C DAOMIN CAO, WEI DAI, GUOLIN QIN for any
R > r . By letting R → + ∞ in (2.14), we get immediately a contradiction. Therefore, k must be even.Next, we will show that k = 0. Suppose on contrary that 2 ≤ k ≤ m − v ( r ) ≤ v (0) − c r − c r − · · · − c k r k , ∀ r > , where c k >
0. Thus there exists a r > v ( r ) ≤ − c k r k , ∀ r > r . Observe that, if R < r < R , then 0 < R r − <
3, and hence(2.17) Z R r − b α − (1 + b ) n db ≥ Z R r − b α − n db ≥ C n,α (cid:18) R r − (cid:19) α . It follows from (2.12), (2.13), (2.15), (2.16) and (2.17) that, for any
R > r , Z + ∞ R R α u ( r ) r ( r − R ) α dr ≥ − C n,α Z R r α − Z R r − b α − (1 + b ) n db ! v ( r ) dr (2.18) ≥ C Z R r r k + α − dr − e C Z r r α − | v ( r ) | dr + C Z R R r k + α − (cid:18) R r − (cid:19) α dr ≥ CR k + α − e C. Thus there exists a r > r large enough such that(2.19) Z + ∞ R R α u ( r ) r ( r − R ) α dr ≥ CR k + α , ∀ R > r . Since u ∈ L α ( R n ), we have(2.20) Z | x − x | > u ( x ) | x − x | n + α dx = C Z + ∞ u ( r ) r α dr < + ∞ , and hence, for any δ > Z + ∞ R α + δ Z + ∞ R R α u ( r ) r ( r − R ) α drdR = Z + ∞ u ( r ) r Z r R δ ( r − R ) α dRdr (2.21) ≤ C Z + ∞ u ( r ) r α Z r R δ dRdr + C Z + ∞ u ( r ) r δ Z r r r α ( r − R ) α dRdr ≤ C δ Z + ∞ u ( r ) r α dr + C Z + ∞ u ( r ) r α dr < + ∞ , which is a contradiction with (2.19) and thus k = 0.Since k = 0, we deduce that(2.22) v ( r ) ≤ v (0) := − c < , ∀ r > . DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 9
Thus (2.12), (2.13), (2.17) and (2.22) yield that, for any
R > Z + ∞ R R α u ( r ) r ( r − R ) α dr ≥ C Z R r α − dr + C Z R R r α − (cid:18) R r − (cid:19) α dr (2.23) ≥ CR α + C Z R R R α − ( R − r ) α dr ≥ CR α . Since u ∈ L α ( R n ), we have(2.24) Z | x − x | >N u ( x ) | x − x | n + α dx = C Z + ∞ N u ( r ) r α dr = o N (1)as N → + ∞ , and hence(2.25) Z + ∞ R R α u ( r ) r ( r − R ) α dr ≤ CR α Z + ∞ R u ( r ) r α dr = o R (1) R α as R → + ∞ . We can choose R > o R (1) < C for any R > R with the same constant C as in the RHS of (2.23). Consequently, it follows from (2.23) and(2.25) that(2.26) Z RR R α u ( r ) r ( r − R ) α dr > CR α , ∀ R > R . By (2.20), we arrive at Z + ∞ R α Z RR R α u ( r ) r ( r − R ) α drdR = Z + ∞ u ( r ) r Z r r R ( r − R ) α dRdr (2.27) ≤ C Z + ∞ u ( r ) r α Z r r r − R ) α dRdr ≤ C Z + ∞ u ( r ) r α dr < + ∞ , which is a contradiction with (2.26). Therefore, the super poly-harmonic properties in Theo-rem 1.1 holds and hence Theorem 1.1 is proved.3. Proof of Theorem 1.2
In this section, we show sub poly-harmonic properties for nonnegative classical solutions toequations (1.1) with f ≤
0, i.e. Theorem 1.2.Suppose that u is a nonnegative classical solution to (1.1). Let u ( x ) := ( − ∆) α u ( x ) and u i ( x ) := ( − ∆) i − u ( x ) for i = 2 , · · · , m . Then, from equations (1.1), we have(3.1) ( − ∆) α u ( x ) = u ( x ) in R n , − ∆ u ( x ) = u ( x ) in R n , · · · · · ·− ∆ u m ( x ) = f ≤ R n . Our aim is to prove that u i ≤ R n for every i = 1 , · · · , m .First, we will prove u m ≤ x ∈ R n such that u m ( x ) >
0. By taking spherical average w.r.t. center x to all equations except the first equation in (3.1), we have(3.2) ( − ∆) α u ( x ) = u ( x ) in R n , − ∆ u ( r ) = u ( r ) , ∀ r ≥ , · · · · · ·− ∆ u m ( r ) = f ( r ) ≤ , ∀ r ≥ . From last equation of (3.2), one has(3.3) − r n − (cid:16) r n − u m ′ ( r ) (cid:17) ′ ≤ , ∀ r ≥ . Integrating both sides of (3.3) twice gives(3.4) u m ( r ) ≥ u m (0) = u m ( x ) =: c > , for any r ≥
0. Then from the last but one equation of (3.2) we derive(3.5) − r n − (cid:16) r n − u m − ′ ( r ) (cid:17) ′ ≥ c , ∀ r ≥ . Again, by integrating both sides of (3.5) twice, we arrive at(3.6) u m − ( r ) ≤ u m − (0) − c r , ∀ r ≥ , where c = c n >
0. Continuing this way, we finally obtain that(3.7) ( − m − u ( r ) ≥ a m − r m − + · · · + a , ∀ r ≥ , where a m − >
0. Hence, we have(3.8) ( − m − u ( r ) ≥ Cr m − , for any r > R with R sufficiently large. One should observe that it is very difficult totake spherical average to the first equation in (3.2), since the fractional Laplacian ( − ∆) α isa nonlocal operator. Instead of taking spherical average, we will apply the Green-Poissonrepresentation formulae for ( − ∆) α to the first equation of (3.2) to overcome this difficulty.From the first equation in (3.2), we conclude that, for arbitrary R > u ( x ) = Z B R ( x ) G αR ( x, y ) u ( y ) dy + Z | y − x | >R P αR ( x, y ) u ( y ) dy, ∀ x ∈ B R ( x ) , where G αR is the Green’s function for ( − ∆) α with 0 < α < B R ( x ) and P αR ( x, y ) is thePoisson kernel for ( − ∆) α in B R ( x ). Taking x = x in (3.9) gives u ( x ) = Z B R ( x ) C n,α | y − x | n − α Z R | y − x | − b α − (1 + b ) n db ! u ( y ) dy (3.10) + C ′ n,α Z | y − x | >R R α ( | y − x | − R ) α u ( y ) | y − x | n dy = Z R e C n,α r − α Z R r − b α − (1 + b ) n db ! u ( r ) dr + C n,α Z + ∞ R R α ¯ u ( r ) r ( r − R ) α dr. DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 11
One can easily observe that for 0 < r ≤ R , R r − ≥ R R r − b α − (1+ b ) n db ≥ R b α − (1+ b ) n db =: C >
0. For R < r < R , one has 0 < R r − <
3, thus R R r − b α − (1+ b ) n db > R R r − b α − n db =: C (cid:16) R − r r (cid:17) α . Thus, we conclude that(3.11) Z R r − b α − (1 + b ) n db ≥ C χ 0. Therefore, we only need to consider the case that m iseven. In such cases, (3.12) implies(3.13) Z + ∞ R R α ¯ u ( r ) r ( r − R ) α dr ≥ CR m − α for R sufficiently large. Since u ∈ L α ( R n ), we have(3.14) Z | x − x | > u ( x ) | x − x | n + α dx = C Z + ∞ ¯ u ( r ) r α dr < + ∞ . Then, by (3.14) and the fact that 2 m − ≥ 0, for R sufficiently large, we have Z RR R α ¯ u ( r ) r ( r − R ) α dr ≥ CR m − α − Z + ∞ R R α ¯ u ( r ) r ( r − R ) α dr (3.15) ≥ CR m − α − C ′ R α Z + ∞ R ¯ u ( r ) r α dr ≥ CR m − α − o R (1) R α ≥ CR m − α . On the one hand, by (3.14), we have Z + ∞ R α Z RR R α ¯ u ( r ) r ( r − R ) α drdR = Z + ∞ ¯ ur Z r r R ( r − R ) α dRdr (3.16) ≤ C Z + ∞ ¯ u ( r ) r α dr < + ∞ . On the other hand, by (3.15), we derive(3.17) Z + ∞ R α Z RR R α ¯ u ( r ) r ( r − R ) α drdR ≥ Z + ∞ N R dR = + ∞ , where N is sufficiently large such that (3.15) holds for any R > N . Combining (3.16) with(3.17), we get a contradiction. Hence, we must have u m ≤ R n . One should observe that,in the proof of u m ≤ 0, we have mainly used the property − ∆ u m ≤ 0. Therefore, through asimilar argument as above, one can prove that u m − ≤ 0. Continuing this way, we obtain that u i = ( − ∆) i − α u ≤ i = 1 , , · · · , m . This completes the proof of Theorem 1.2.4. Proof of Theorem 1.10 In this section, we will prove Theorem 1.10 by way of contradiction and the method of scalingspheres developed by Dai and Qin [22] (see also [23, 24, 26]). For more related literature onthe method of moving planes (spheres), we refer to [2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 17, 18, 19,21, 25, 27, 30, 31, 32, 34, 39, 40] and the references therein.Now suppose, on the contrary, that u ≥ u is notidentically zero, then there exists a ponit ¯ x ∈ R n such that u (¯ x ) > 0. It follows from (1.8)immediately that(4.1) u ( x ) > , ∀ x ∈ R n , i.e., u is actually a positive solution in R n . Moreover, there exists a constant C > 0, such thatthe solution u satisfies the following lower bound:(4.2) u ( x ) ≥ C | x | n − m − α for | x | ≥ . Indeed, since u > u ( x ) ≥ C n,m,α Z | y |≤ | y | a | x − y | n − m − α u p ( y ) dy (4.3) ≥ C | x | n − m − α Z | y |≤ | y | a u p ( y ) dy =: C | x | n − m − α for all | x | ≥ u , which contradict with the integral equations (1.8) for 0 < p < n +2 m + α +2 an − m − α . Theorem 4.1. Assume m ≥ , < α < , m + α < n , ≤ a < + ∞ , < p < n +2 m + α +2 an − m − α .Suppose u ∈ C ( R n ) is a positive solution to (1.8) , then it satisfies the following lower bound DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 13 estimates: for | x | ≥ , (4.4) u ( x ) ≥ C κ | x | κ ∀ κ < m + α + a − p , if < p < u ( x ) ≥ C κ | x | κ ∀ κ < + ∞ , if ≤ p < n + 2 m + α + 2 an − m − α . Proof. Given any λ > 0, we first define the Kelvin transform of a function u : R n → R centered at 0 by(4.6) u λ ( x ) = (cid:18) λ | x | (cid:19) n − m − α u (cid:18) λ x | x | (cid:19) for arbitrary x ∈ R n \ { } . It’s obvious that the Kelvin transform u λ may have singularity at 0and lim | x |→∞ | x | n − m − α u λ ( x ) = λ n − m − α u (0) > 0. By (4.6), one can infer from the regularityassumptions on u that u λ ∈ C ( R n \ { } ).Next, we will carry out the process of scaling spheres with respect to the origin 0 ∈ R n .To this end, let λ > ω λ ( x ) := u λ ( x ) − u ( x )for any x ∈ B λ (0) \ { } . We will first show that, for λ > ω λ ( x ) ≥ , ∀ x ∈ B λ (0) \ { } . Then, we start dilating the sphere S λ from a place near the origin 0 outward as long as (4.8)holds, until its limiting position λ = + ∞ and derive lower bound estimates on u . Therefore,the scaling sphere process can be divided into two steps. Step 1. Start dilating the sphere from near λ = 0 . Define(4.9) B − λ := { x ∈ B λ (0) \ { } | ω λ ( x ) < } . We will show that, for λ > B − λ = ∅ . Since u ∈ C ( R n ) is a positive solution to integral equations (1.8), through direct calculations,we get(4.11) u ( x ) = C Z B λ (0) | y | a | x − y | n − m − α u p ( y ) dy + C Z B λ (0) | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α (cid:18) λ | y | (cid:19) τ u pλ ( y ) dy for any x ∈ R n , where τ := n + 2 m + α + 2 a − p ( n − m − α ) > 0. Direct calculations deducethat u λ satisfies the following integral equation(4.12) u λ ( x ) = C Z R n | y | a | x − y | n − m − α (cid:18) λ | y | (cid:19) τ u pλ ( y ) dy for any x ∈ R n \ { } , and hence, it follows immediately that(4.13) u λ ( x ) = C Z B λ (0) | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α u p ( y ) dy + C Z B λ (0) | y | a | x − y | n − m − α (cid:18) λ | y | (cid:19) τ u pλ ( y ) dy. From the integral equations (4.11) and (4.13), one can derive that, for any x ∈ B − λ ,0 > ω λ ( x ) = u λ ( x ) − u ( x )(4.14) = C Z B λ (0) | y | a | x − y | n − m − α − | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α ! (cid:18)(cid:18) λ | y | (cid:19) τ u pλ ( y ) − u p ( y ) (cid:19) dy> C Z B − λ | y | a | x − y | n − m − α − | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α ! max (cid:8) u p − ( y ) , u p − λ ( y ) (cid:9) ω λ ( y ) dy ≥ C Z B − λ | y | a | x − y | n − m − α max (cid:8) u p − ( y ) , u p − λ ( y ) (cid:9) ω λ ( y ) dy. By Hardy-Littlewood-Sobolev inequality and (4.14), we have, for any nn − m − α < q < ∞ , k ω λ k L q ( B − λ ) ≤ C (cid:13)(cid:13) | x | a max (cid:8) u p − , u p − λ (cid:9) ω λ (cid:13)(cid:13) L nqn +(2 m + α ) q ( B − λ ) (4.15) ≤ C (cid:13)(cid:13) | x | a max (cid:8) u p − , u p − λ (cid:9)(cid:13)(cid:13) L n m + α ( B − λ ) k ω λ k L q ( B − λ ) . Since (4.3) implies that(4.16) inf x ∈ B λ (0) \{ } u λ ( x ) ≥ C for any λ ≤ 1, there exists a ǫ > C (cid:13)(cid:13) | x | a max (cid:8) u p − , u p − λ (cid:9)(cid:13)(cid:13) L n m + α ( B − λ ) ≤ < λ ≤ ǫ . Thus (4.15) implies(4.18) k ω λ k L q ( B − λ ) = 0 , ∀ < λ ≤ ǫ , which means B − λ = ∅ . Consequently for all 0 < λ ≤ ǫ ,(4.19) ω λ ( x ) ≥ , ∀ x ∈ B λ (0) \ { } , which completes Step 1. Step 2. Dilate the sphere S λ outward until λ = + ∞ to derive lower bound estimates on u . Step 1 provides us a start point to dilate the sphere S λ from place near λ = 0. Now we dilatethe sphere S λ outward as long as (4.8) holds. Let(4.20) λ := sup { λ > | ω µ ≥ in B µ (0) \ { } , ∀ < µ ≤ λ } ∈ (0 , + ∞ ] , and hence, one has(4.21) ω λ ( x ) ≥ , ∀ x ∈ B λ (0) \ { } . In what follows, we will prove λ = + ∞ by contradiction arguments.Suppose on contrary that 0 < λ < + ∞ . In order to get a contradiction, we will first showthat(4.22) ω λ ( x ) > , ∀ x ∈ B λ (0) \ { } . Then, we will obtain a contradiction with (4.20) via showing that the sphere S λ can be dilatedoutward a little bit further. More precisely, there exists a ε > ω λ ≥ B λ (0) \ { } for all λ ∈ [ λ , λ + ε ]. DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 15 Now we start to prove (4.22). Indeed, if we suppose that(4.23) ω λ ( x ) ≡ , ∀ x ∈ B λ (0) \ { } , then by the second equality in (4.14) and (4.23), we arrive at0 = ω λ ( x ) = u λ ( x ) − u ( x )(4.24) = C Z B λ (0) | y | a | x − y | n − m − α − | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α ! (cid:18)(cid:18) λ | y | (cid:19) τ − (cid:19) u p ( y ) dy > x ∈ B λ (0) \ { } , which is absurd. Thus there exists a point x ∈ B λ (0) \ { } suchthat ω λ ( x ) > 0, which implies that by continuity, there exists a small δ > c > B δ ( x ) ⊂ B λ (0) \ { } and ω λ ( x ) ≥ c > , ∀ x ∈ B δ ( x ) . From (4.25) and the integral equations (4.11) and (4.13), one can derive that, for any x ∈ B λ (0) \ { } , ω λ ( x ) = u λ ( x ) − u ( x )(4.26)= C Z B λ (0) | y | a | x − y | n − m − α − | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α ! (cid:18)(cid:18) λ | y | (cid:19) τ u pλ ( y ) − u p ( y ) (cid:19) dy> C Z B λ (0) | y | a | x − y | n − m − α − | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α ! (cid:0) u pλ ( y ) − u p ( y ) (cid:1) dy ≥ C Z B λ (0) | y | a | x − y | n − m − α − | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α ! min (cid:8) u p − ( y ) , u p − λ ( y ) (cid:9) ω λ ( y ) dy ≥ C Z B δ ( x ) | y | a | x − y | n − m − α − | y | a (cid:12)(cid:12)(cid:12) | y | λ x − λ | y | y (cid:12)(cid:12)(cid:12) n − m − α ! min (cid:8) u p − ( y ) , u p − λ ( y ) (cid:9) ω λ ( y ) dy > , and thus we arrive at (4.22). Furthermore, (4.26) also implies that there exists a 0 < η < λ small enough such that, for any x ∈ B η (0) \ { } ,(4.27) ω λ ( x ) > c + C Z B δ ( x ) c a c c p − c dy =: e c > . Next, we will show that the sphere S λ can be dilated outward a little bit further and henceobtain a contradiction with the definition (4.20) of λ .To this end, we fixed 0 < r < λ small enough, such that(4.28) C (cid:13)(cid:13) | x | a max (cid:8) u p − , u p − λ (cid:9)(cid:13)(cid:13) L n m + α ( A λ r , r ) ≤ λ ∈ [ λ , λ + r ], where the constant C is the same as in (4.15) and the narrow region(4.29) A λ + r , r := { x ∈ B λ + r (0) | | x | > λ − r } . By (4.14), one can easily verify that inequality as (4.15) (with the same constant C ) also holdsfor any λ ∈ [ λ , λ + r ], that is, for any nn − m − α < q < ∞ ,(4.30) k ω λ k L q ( B − λ ) ≤ C (cid:13)(cid:13) | x | a max (cid:8) u p − , u p − λ (cid:9)(cid:13)(cid:13) L n m + α ( B − λ ) k ω λ k L q ( B − λ ) . From (4.22) and (4.27), we can infer that(4.31) m := inf x ∈ B λ − r (0) \{ } ω λ ( x ) > . Since u is uniformly continuous on arbitrary compact set K ⊂ R n (say, K = B λ (0)), wecan deduce from (4.31) that, there exists a 0 < ε < r sufficiently small, such that, for any λ ∈ [ λ , λ + ε ],(4.32) ω λ ( x ) ≥ m > , ∀ x ∈ B λ − r (0) \ { } . In order to prove (4.32), one should observe that (4.31) is equivalent to(4.33) | x | n − m − α u ( x ) − λ n − m − α u ( x λ ) ≥ m λ n − m − α , ∀ | x | ≥ λ λ − r . Since u is uniformly continuous on B λ (0), we infer from (4.33) that there exists a 0 < ε < r sufficiently small, such that, for any λ ∈ [ λ , λ + ε ],(4.34) | x | n − m − α u ( x ) − λ n − m − α u ( x λ ) ≥ m λ n − m − α , ∀ | x | ≥ λ λ − r , which is equivalent to (4.32), hence we have proved (4.32).By (4.32), we know that for any λ ∈ [ λ , λ + ε ],(4.35) B − λ ⊂ A λ + r , r , and hence, estimates (4.28) and (4.30) yields(4.36) k ω λ k L q ( B − λ ) = 0 . Therefore, for any λ ∈ [ λ , λ + ε ], we deduce from (4.36) that, B − λ = ∅ , that is,(4.37) ω λ ( x ) ≥ , ∀ x ∈ B λ (0) \ { } , which contradicts with the definition (4.20) of λ . Thus we must have λ = + ∞ , that is,(4.38) u ( x ) ≥ (cid:18) λ | x | (cid:19) n − m − α u (cid:18) λ x | x | (cid:19) , ∀ | x | ≥ λ, ∀ < λ < + ∞ . For arbitrary | x | ≥ 1, let λ := p | x | , then (4.38) yields that(4.39) u ( x ) ≥ | x | n − m − α u (cid:18) x | x | (cid:19) , and hence, we arrive at the following lower bound estimate:(4.40) u ( x ) ≥ (cid:18) min x ∈ S u ( x ) (cid:19) | x | n − m − α := C | x | n − m − α , ∀ | x | ≥ . The lower bound estimate (4.40) can be improved remarkably for 0 < p < n +2 m + α +2 an − m − α usingthe “Bootstrap” iteration technique and the integral equations (1.8). DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 17 In fact, let µ := n − m − α , we infer from the integral equations (1.8) and (4.40) that, for | x | ≥ u ( x ) ≥ C Z | x |≤| y |≤ | x | | x − y | n − m − α | y | pµ − a dy (4.41) ≥ C | x | n − m − α Z | x |≤| y |≤ | x | | y | pµ − a dy ≥ C | x | n − m − α Z | x | | x | r n − − pµ + a dr ≥ C | x | pµ − ( a +2 m + α ) . Let µ := pµ − ( a + 2 m + α ). Due to 0 < p < n +2 m + α +2 an − m − α , our important observation is(4.42) µ := pµ − ( a + 2 m + α ) < µ . Thus we have obtained a better lower bound estimate than (4.40) after one iteration, that is,(4.43) u ( x ) ≥ C | x | µ , ∀ | x | ≥ . For k = 0 , , , · · · , define(4.44) µ k +1 := pµ k − ( a + 2 m + α ) . Since 0 < p < n +2 m + α +2 an − m − α , it is easy to see that the sequence { µ k } is monotone decreasing withrespect to k . Repeating the above iteration process involving the integral equation (1.8), wehave the following lower bound estimates for every k = 0 , , , · · · ,(4.45) u ( x ) ≥ C k | x | µ k , ∀ | x | ≥ . Now Theorem 4.1 follows easily from the obvious properties that as k → + ∞ ,(4.46) µ k → − a + 2 m + α − p if 0 < p < µ k → −∞ if 1 ≤ p < n + 2 m + α + 2 an − m − α . This finishes our proof of Theorem 4.1. (cid:3) We have proved that a nontrivial nonnegative solution u to integral equations (1.8) is ac-tually a positive solution. For 0 < p < n +2 m + α +2 an − m − α , the lower bound estimates in Theorem 4.1contradicts with the following integrability(4.47) C Z R n u p ( x ) | x | n − m − α − a dx = u (0) < + ∞ indicated by integral equations (1.8). Therefore, u ≡ R n , that is, the unique nonnegativesolution to IEs (1.8) is u ≡ R n . The proof of Theorem 1.10 is therefore completed. Proof of Theorem 1.14 In this section, using Theorem 1.1 and the arguments from Chen, Dai and Qin [3], we willprove the Liouville properties in Theorem 1.14 in both critical order cases m + α = n andsuper-critical order cases m + α > n .We will prove Theorem 1.14 by using contradiction arguments. Suppose on the contrarythat u ≥ u is not identically zero, then there exists a point ¯ x ∈ R n such that u (¯ x ) > 0. By Theorem 1.1, we can deduce from ( − ∆) α u ≥ u ≥ u (¯ x ) > u ( x ) > , ∀ x ∈ R n . Suppose not, then there exists a point e x ∈ R n such that u ( e x ) = 0, and hence we have(5.2) ( − ∆) α u ( e x ) = C n,α P.V. Z R n − u ( y ) | e x − y | n + α dy < , which is absurd. Moreover, by maximum principle and induction, we can also infer furtherfrom ( − ∆) i + α u ≥ i = 0 , · · · , m − u > 0, the assumptions on f and equation (1.1) that(5.3) ( − ∆) i + α u ( x ) > , ∀ i = 0 , · · · , m − , ∀ x ∈ R n . Since m + α ≥ n , it follows immediately that either m = n − with n odd or m ≥ ⌈ n ⌉ , where ⌈ x ⌉ denotes the least integer not less than x .In the following, we will try to obtain contradictions by discussing the two different cases m = n − with n odd and m ≥ ⌈ n ⌉ separately. Case i): m = n − and n is odd. Since m + α ≥ n , we have 1 ≤ α < 2. Now we will firstshow that ( − ∆) m − α u satisfies the following integral equation(5.4) ( − ∆) m − α u ( x ) = Z R n R ,n | x − y | n − f ( y, u ( y ) , · · · ) dy, ∀ x ∈ R n , where the Riesz potential’s constants R α,n := Γ( n − α ) π n α Γ( α ) for 0 < α < n .To this end, for arbitrary R > 0, let f ( u )( x ) := f ( x, u ( x ) , · · · ) and(5.5) v R ( x ) := Z B R (0) G R ( x, y ) f ( u )( y ) dy, where the Green’s function for − ∆ on B R (0) is given by(5.6) G R ( x, y ) = R ,n (cid:20) | x − y | n − − (cid:0) | x | · (cid:12)(cid:12) Rx | x | − yR (cid:12)(cid:12)(cid:1) n − (cid:21) , if x, y ∈ B R (0) , and G R ( x, y ) = 0 if x or y ∈ R n \ B R (0). Then, we can derive that v R ∈ C ( R n ) and satisfies(5.7) ( − ∆ v R ( x ) = f ( x, u ( x ) , · · · ) , x ∈ B R (0) ,v R ( x ) = 0 , x ∈ R n \ B R (0) . Let w R ( x ) := ( − ∆) m − α u ( x ) − v R ( x ). By Theorem 1.1, (1.1) and (5.7), we have w R ∈ C ( R n )and satisfies(5.8) ( − ∆ w R ( x ) = 0 , x ∈ B R (0) ,w R ( x ) > , x ∈ R n \ B R (0) . DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 19 By maximum principle, we deduce that for any R > w R ( x ) = ( − ∆) m − α u ( x ) − v R ( x ) > , ∀ x ∈ R n . Now, for each fixed x ∈ R n , letting R → ∞ in (5.9), we have(5.10) ( − ∆) m − α u ( x ) ≥ Z R n R ,n | x − y | n − f ( u )( y ) dy =: v ( x ) > . Take x = 0 in (5.10), we get(5.11) Z R n f ( y, u ( y ) , · · · ) | y | n − dy < + ∞ . One can easily observe that v ∈ C ( R n ) is a solution of(5.12) − ∆ v ( x ) = f ( x, u ( x ) , · · · ) , x ∈ R n . Define w ( x ) := ( − ∆) m − α u ( x ) − v ( x ). Then, by (1.1), (5.10) and (5.12), we have w ∈ C ( R n ) and satisfies(5.13) ( − ∆ w ( x ) = 0 , x ∈ R n ,w ( x ) ≥ , x ∈ R n . From Liouville theorem for harmonic functions, we can deduce that(5.14) w ( x ) = ( − ∆) m − α u ( x ) − v ( x ) ≡ C ≥ . Therefore, we have(5.15) ( − ∆) m − α u ( x ) = Z R n R ,n | x − y | n − f ( y, u ( y ) , · · · ) dy + C =: f ( u )( x ) > C ≥ . Next, for arbitrary R > 0, let(5.16) v R ( x ) := Z B R (0) G R ( x, y ) f ( u )( y ) dy. Then, we can get(5.17) ( − ∆ v R ( x ) = f ( u )( x ) , x ∈ B R (0) ,v R ( x ) = 0 , x ∈ R n \ B R (0) . Let w R ( x ) := ( − ∆) m − α u ( x ) − v R ( x ). By Theorem 1.1, (5.15) and (5.17), we have(5.18) ( − ∆ w R ( x ) = 0 , x ∈ B R (0) ,w R ( x ) > , x ∈ R n \ B R (0) . By maximum principle, we deduce that for any R > w R ( x ) = ( − ∆) m − α u ( x ) − v R ( x ) > , ∀ x ∈ R n . Now, for each fixed x ∈ R n , letting R → ∞ in (5.19), we have(5.20) ( − ∆) m − α u ( x ) ≥ Z R n R ,n | x − y | n − f ( u )( y ) dy =: v ( x ) > . Take x = 0 in (5.20), we get(5.21) Z R n C | y | n − dy ≤ Z R n f ( u )( y ) | y | n − dy < + ∞ , it follows easily that C = 0, and hence we have proved (5.4), that is,(5.22) ( − ∆) m − α u ( x ) = f ( u )( x ) = Z R n R ,n | x − y | n − f ( y, u ( y ) , · · · ) dy. One can easily observe that v is a solution of(5.23) − ∆ v ( x ) = f ( u )( x ) , x ∈ R n . Define w ( x ) := ( − ∆) m − α u ( x ) − v ( x ), then it satisfies(5.24) ( − ∆ w ( x ) = 0 , x ∈ R n ,w ( x ) ≥ , x ∈ R n . From Liouville theorem for harmonic functions, we can deduce that(5.25) w ( x ) = ( − ∆) m − α u ( x ) − v ( x ) ≡ C ≥ . Therefore, we have proved that(5.26) ( − ∆) m − α u ( x ) = Z R n R ,n | x − y | n − f ( u )( y ) dy + C =: f ( u )( x ) > C ≥ . By the same methods as above, we can prove that C = 0, and hence(5.27) ( − ∆) m − α u ( x ) = f ( u )( x ) = Z R n R ,n | x − y | n − f ( u )( y ) dy. Repeating the above argument, defining(5.28) f k +1 ( u )( x ) := Z R n R ,n | x − y | n − f k ( u )( y ) dy for k = 1 , , · · · , m , then by Theorem 1.1 and induction, we have(5.29) ( − ∆) m − k + α u ( x ) = f k +1 ( u )( x ) = Z R n R ,n | x − y | n − f k ( u )( y ) dy for k = 1 , , · · · , m − 1, and(5.30) ( − ∆) α u ( x ) = Z R n R ,n | x − y | n − f m ( u )( y ) dy + C m = f m +1 ( u )( x ) + C m > C m ≥ . For arbitrary R > 0, let(5.31) v Rm +1 ( x ) := Z B R (0) G αR ( x, y ) ( f m +1 ( u )( y ) + C m ) dy, where the Green’s function for ( − ∆) α with 0 < α < B R (0) is given by(5.32) G αR ( x, y ) := C n,α | x − y | n − α Z tRsR b α − (1 + b ) n db if x, y ∈ B R (0)with s R = | x − y | R , t R = (cid:16) − | x | R (cid:17) (cid:16) − | y | R (cid:17) , and G αR ( x, y ) = 0 if x or y ∈ R n \ B R (0) (see [28]).Then, we can get(5.33) ( ( − ∆) α v Rm +1 ( x ) = f m +1 ( u )( x ) + C m , x ∈ B R (0) ,v Rm +1 ( x ) = 0 , x ∈ R n \ B R (0) . DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 21 Let w Rm +1 ( x ) := u ( x ) − v Rm +1 ( x ). By Theorem 1.1, (5.30) and (5.33), we have(5.34) ( ( − ∆) α w Rm +1 ( x ) = 0 , x ∈ B R (0) ,w Rm +1 ( x ) > , x ∈ R n \ B R (0) . Now we need the following maximum principle for fractional Laplacians ( − ∆) α , which canbeen found in [12, 37]. Lemma 5.1. Let Ω be a bounded domain in R n and < α < . Assume that u ∈ L α ∩ C , loc (Ω) and is l.s.c. on Ω . If ( − ∆) α u ≥ in Ω and u ≥ in R n \ Ω , then u ≥ in R n . Moreover, if u = 0 at some point in Ω , then u = 0 a.e. in R n . These conclusions also hold for unboundeddomain Ω if we assume further that lim inf | x |→∞ u ( x ) ≥ . By Lemma 5.1, we can deduce immediately from (5.34) that for any R > w Rm +1 ( x ) = u ( x ) − v Rm +1 ( x ) > , ∀ x ∈ R n . Now, for each fixed x ∈ R n , letting R → ∞ in (5.35), we have(5.36) u ( x ) ≥ Z R n R α,n | x − y | n − α ( f m +1 ( u )( y ) + C m ) dy > . Take x = 0 in (5.36), we get(5.37) Z R n C m | y | n − α dy ≤ Z R n f m +1 ( u )( y ) + C m | y | n − α dy < + ∞ , it follows easily that C m = 0, and hence we have(5.38) ( − ∆) α u ( x ) = f m +1 ( u )( x ) = Z R n R ,n | x − y | n − f m ( u )( y ) dy, and(5.39) u ( x ) ≥ Z R n R α,n | x − y | n − α f m +1 ( u )( y ) dy. In particular, it follows from (5.29), (5.38) and (5.39) that+ ∞ > ( − ∆) m − k + α u (0) = Z R n R ,n | y | n − f k ( u )( y ) dy (5.40) ≥ Z R n R ,n | y k | n − Z R n R ,n | y k − y k − | n − · · · Z R n R ,n | y − y | n − f ( y , u ( y ) , · · · ) dy dy · · · dy k for k = 1 , , · · · , m , and+ ∞ > u (0) ≥ Z R n R α,n | y | n − α f m +1 ( u )( y ) dy (5.41) ≥ Z R n R α,n | y m +1 | n − α (cid:18)Z R n R ,n | y m +1 − y m | n − · · · Z R n R ,n | y − y | n − f ( y , · · · ) dy · · · dy m (cid:19) dy m +1 . From the properties of Riesz potential, for any α , α ∈ (0 , n ) such that α + α ∈ (0 , n ), onehas(see [38])(5.42) Z R n R α ,n | x − y | n − α · R α ,n | y − z | n − α dy = R α + α ,n | x − z | n − ( α + α ) . By applying (5.42) and direct calculations, we obtain that Z R n R ,n | y m +1 − y m | n − · · · Z R n R ,n | y − y | n − · R ,n | y − y | n − dy · · · dy m (5.43) = R m,n | y m +1 − y | n − m . Now, note that m = n − and n is odd, we can deduce from (5.41), (5.43) and Fubini’stheorem that+ ∞ > u (0) ≥ Z R n R α,n | y m +1 | n − α (cid:18)Z R n R n − α,n | y m +1 − y | f ( y , u ( y ) , · · · ) dy (cid:19) dy m +1 (5.44) = 1(2 π ) n Z R n | y | n − α (cid:18)Z R n | y − z | f ( z, u ( z ) , · · · ) dz (cid:19) dy. We will get a contradiction from (5.44). Indeed, if we assume that u is not identically zero,then by (5.1), u > R n . By the assumptions on f , we have, there exists a point x ∈ R n such that f > x . Hence by the integrability (5.11), we have(5.45) 0 < C := Z R n f ( z, u ( z ) , · · · ) | z | n − dz < + ∞ . For any given | y | ≥ 3, if | z | ≥ (cid:0) ln | y | (cid:1) − n − , then one has immediately(5.46) | y − z | ≤ | y | + | z | ≤ (cid:16) | y | (cid:0) ln | y | (cid:1) n − + 1 (cid:17) | z | ≤ | y | (cid:0) ln | y | (cid:1) n − | z | . Thus it follows from (5.45) and (5.46) that, there exists a R ≥ | y | ≥ R , we have Z R n f ( z, u ( z ) , · · · ) | y − z | dz ≥ | y | ln | y | Z | z |≥ (cid:0) ln | y | (cid:1) − n − f ( z, u ( z ) , · · · ) | z | n − dz (5.47) ≥ | y | ln | y | Z R n f ( z, u ( z ) , · · · ) | z | n − dz ≥ C | y | ln | y | . As a consequence, we can finally deduce from (5.44), (5.47) and 1 ≤ α < ∞ > u (0) ≥ C π ) n Z | y |≥ R | y | n − α +1 ln | y | dy = + ∞ , which is a contradiction. Therefore u ≡ R n . This proves Theorem 1.14 in Case i): m = n − and n is odd. Case ii): m ≥ ⌈ n ⌉ . Let(5.49) f k +1 ( u )( x ) := Z R n R ,n | x − y | n − f k ( u )( y ) dy for k = 1 , , · · · , ⌈ n ⌉ , by a quite similar way as in the proof for Case i), we can infer fromTheorem 1.1 and induction that(5.50) ( − ∆) m − k + α u ( x ) = f k +1 ( u )( x ) = Z R n R ,n | x − y | n − f k ( u )( y ) dy DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 23 for k = 1 , , · · · , ⌈ n ⌉ − 1, and(5.51) ( − ∆) m −⌈ n ⌉ + α u ( x ) ≥ f ⌈ n ⌉ +1 ( u )( x ) = Z R n R ,n | x − y | n − f ⌈ n ⌉ ( u )( y ) dy. In particular, it follows from (5.50) and (5.51) that+ ∞ > ( − ∆) m − k + α u (0) = Z R n R ,n | y | n − f k ( u )( y ) dy (5.52) ≥ Z R n R ,n | y k | n − Z R n R ,n | y k − y k − | n − · · · Z R n R ,n | y − y | n − f ( y , u ( y ) , · · · ) dy dy · · · dy k for k = 1 , , · · · , ⌈ n ⌉ − 1, and+ ∞ > ( − ∆) m −⌈ n ⌉ + α u (0) ≥ Z R n R ,n | y | n − f ⌈ n ⌉ ( u )( y ) dy ≥ Z R n R ,n | y ⌈ n ⌉ | n − (5.53) × (cid:18)Z R n R ,n | y ⌈ n ⌉ − y ⌈ n ⌉− | n − · · · Z R n R ,n | y − y | n − f ( y , · · · ) dy · · · dy ⌈ n ⌉− (cid:19) dy ⌈ n ⌉ . By applying the formula (5.42) and direct calculations, we obtain that Z R n R ,n | y ⌈ n ⌉ − y ⌈ n ⌉− | n − · · · Z R n R ,n | y − y | n − · R ,n | y − y | n − dy · · · dy ⌈ n ⌉− (5.54) = R ⌈ n ⌉− ,n | y ⌈ n ⌉ − y | n − ⌈ n ⌉ +2 . Now, we can deduce from (5.53), (5.54) and Fubini’s theorem that+ ∞ > ( − ∆) m −⌈ n ⌉ + α u (0)(5.55) ≥ Z R n R ,n | y ⌈ n ⌉ | n − (cid:18)Z R n R ⌈ n ⌉− ,n | y ⌈ n ⌉ − y | n − ⌈ n ⌉ +2 f ( y , u ( y ) , · · · ) dy (cid:19) dy ⌈ n ⌉ = C n Z R n | y | n − (cid:18)Z R n | y − z | n − ⌈ n ⌉ +2 f ( z, u ( z ) , · · · ) dz (cid:19) dy. We will get a contradiction from (5.55). To do this, let τ ( n ) := n − ⌈ n ⌉ + 2 ∈ { , } , thenit follows from (5.45) and (5.46) that, there exists a R ≥ | y | ≥ R , Z R n | y − z | τ ( n ) f ( z, u ( z ) , · · · ) dz (5.56) ≥ τ ( n ) | y | τ ( n ) ln | y | Z | z |≥ (cid:0) ln | y | (cid:1) − n − | z | n − f ( z, u ( z ) , · · · ) dz ≥ τ ( n )+1 | y | τ ( n ) ln | y | Z R n f ( z, u ( z ) , · · · ) | z | n − dz ≥ C τ ( n )+1 | y | τ ( n ) ln | y | . Therefore, we can finally deduce from (5.55) and (5.56) that(5.57) + ∞ > ( − ∆) m −⌈ n ⌉ + α u (0) ≥ C C n τ ( n )+1 Z | y |≥ R | y | n − τ ( n ) ln | y | dy = + ∞ , which is a contradiction again. Therefore, u ≡ R n in Case ii): m ≥ ⌈ n ⌉ .This concludes our proof of Theorem 1.14. Appendix: A characterization for α -harmonic functions via averages One can observe from the proof of Theorem 1.1 and 1.2 that, the average R + ∞ R R α r ( r − R ) α u ( r ) dr plays an basic and important role for the nonlocal fractional Laplacians ( − ∆) α (0 < α < u ( r ) for the Laplacian − ∆. In this appendix, we will characterize the α -harmonic functions by using the averages R + ∞ R R α r ( r − R ) α u ( r ) dr and deduce some importantproperties for α -harmonic functions.Let Ω ⊆ R n be a (bounded or unbounded) domain. We have the following characterizationfor α -harmonic functions in Ω. Theorem 6.1. Assume < α < . Let u ∈ C [ α ] , { α } + ǫloc (Ω) ∩ L α ( R n ) (with ǫ > arbitrarilysmall) satisfy ( − ∆) α u ≥ ( ≤ ) in Ω , then for any ball B = B R ( y ) ⊂⊂ Ω , we have (6.1) u ( y ) ≥ ( ≤ ) C n,α Z + ∞ R R α r ( r − R ) α u ( r ) dr, where C n,α := Γ( n ) π n sin πα and u ( r ) denotes the spherical average of u w.r.t. y . Furthermore, ( − ∆) α u = 0 in Ω if and only if (6.2) u ( y ) = C n,α Z + ∞ R R α r ( r − R ) α u ( r ) dr for any ball B = B R ( y ) ⊂⊂ Ω .Proof. Suppose ( − ∆) α u ≥ B = B R ( y ) ⊂⊂ Ω, u ( y ) = Z B R ( y ) G αR ( y, z )( − ∆) α u ( z ) dz + Z | z − y | >R P αR ( y, z ) u ( z ) dz (6.3) = Z B R ( y ) e C n,α | z − y | n − α Z R | z − y | − b α − (1 + b ) n db ! ( − ∆) α u ( z ) dz + C n,α Z | z − y | >R R α ( | z − y | − R ) α · u ( z ) | z − y | n dz ≥ C n,α Z + ∞ R R α r ( r − R ) α u ( r ) dr. If ( − ∆) α u ≤ u satisfies the average property (6.2) for any ball B = B R ( y ) ⊂⊂ Ω, wewill show that u is α -harmonic in Ω. To this end, for any ball B = B R ( y ) ⊂⊂ Ω, let us define(6.4) h ( x ) := C n,α Z | z − y | >R (cid:18) R − | x − y | | z − y | − R (cid:19) α · u ( z ) | x − z | n dz, ∀ x ∈ B R ( y ) , and h ( x ) := u ( x ) if | x − y | ≥ R . Then it follows that ( − ∆) α h = 0 in B R ( y ) and hence satisfiesthe average property (6.2) for any ball B ⊂⊂ B R ( y ). Define w := u − h , then w ( x ) = 0 if | x − y | ≥ R and w satisfies the average property (6.2) for any ball B ⊂⊂ B R ( y ). Our aim isto show that w = 0 in B R ( y ). DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 25 Indeed, suppose there exists a point ¯ x ∈ B R ( y ) such that w (¯ x ) = M := max x ∈ B R ( y ) w ( x ) > B ¯ R (¯ x ) ⊂⊂ B R ( y ),(6.5) 0 = w (¯ x ) − M = C n,α Z + ∞ ¯ R ¯ R α r ( r − ¯ R ) α w − M ( r ) dr < , where w − M ( r ) denotes the spherical average of w − M w.r.t. ¯ x . This is absurd, thus w ≤ B R ( y ). Similarly, we can show that m := min x ∈ B R ( y ) w ( x ) = 0. 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Math., (2005), 485-503.[41] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian , arXiv: 1401.7402. DES INVOLVING HIGHER-ORDER FRACTIONAL LAPLACIANS 27 Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, and Uni-versity of Chinese Academy of Sciences, Beijing 100049, P. R. China E-mail address : [email protected] School of Mathematics and Systems Science, Beihang University (BUAA), Beijing 100083,P. R. China, and LAGA, Universit´e Paris 13 (UMR 7539), Paris, France E-mail address : [email protected] Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, and Uni-versity of Chinese Academy of Sciences, Beijing 100049, P. R. China E-mail address ::