Maximum principles, Liouville theorem and symmetry results for the fractional g-Laplacian
aa r X i v : . [ m a t h . A P ] F e b MAXIMUM PRINCIPLES, LIOUVILLE THEOREM ANDSYMMETRY RESULTS FOR THE FRACTIONAL g − LAPLACIAN
SANDRA MOLINA, ARIEL SALORT AND HERN ´AN VIVAS
Abstract.
We study different maximum principles for non-local non-linearoperators with non-standard growth that arise naturally in the context offractional Orlicz-Sobolev spaces and whose most notable representative is thefractional g − Laplacian:( − ∆ g ) s u ( x ) := p.v. Z R n g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19) dy | x − y | n + s , being g the derivative of a Young function.We further derive qualitative properties of solutions such as a Liouvilletype theorem and symmetry results and present several possible extensionsand some interesting open questions. These are the first results of this typeproved in this setting. Contents
1. Introduction and main results 12. Preliminaries 63. Proof of the maximum principle on domains, maximum principle onhyperplanes and Liouville theorem 134. Proof of Theorem 1.4 and Proposition 1.5 175. Symmetry results 196. Extensions and applications 22References 231.
Introduction and main results
The aim of this manuscript is to study qualitative properties of the so-calledfractional g − Laplacian; for s ∈ (0 ,
1) the fractional g − Laplacian is defined by( − ∆ g ) s u ( x ) := p.v. Z R n g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19) dy | x − y | n + s where p.v. stands for the principal value and g = G ′ is the derivative of a Youngfunction G (for this and other definitions see Section 2). Mathematics Subject Classification.
Key words and phrases.
Fractional g − Laplacian; maximum principles; qualitative properties.
This operator was introduced in [14] and has received an increasing attentionin the last years since it allows to model non-local problems obeying a non-powerbehavior. See for instance [1, 2, 3, 4, 15, 12, 21, 20] and the references therein.We will be particularly interested in different versions of maximum principles for the fractional g − Laplacian, from where many qualitative properties of solutionswill be obtained.The literature on maximum principles and the consequential qualitative proper-ties of solutions (such as symmetry, for instance) is nowadays huge, and differenttechniques were developed in order to overcome technical difficulties arisen by theparticular nature of the operators under study. For instance, the square power case(i.e. G ( t ) = t ) corresponds with the fractional Laplacian , and several tools suchas representation formulas for solutions, the Fourier transform or the Caffarelli-Silvestre extension method have shown to be useful, and a series of successful re-sults have been obtained (see [9, 10, 19] and the references therein). However,when G ( t ) = t p (the well-known fractional p − Laplacian ) these effective techinquesno longer work due to the nonlinearity of the operator and new techniques and ideasneed to be developed. Several results regarding maximum principles and qualitativeproperties of solutions were proved in [7, 8, 22, 23, 25, 24], just to mention somerecent works. Furthermore, the method of moving planes introduced in the cele-brated paper by Gidas, Ni and Nirenberg [16], or the sliding method developed byBerestycki and Nirenberg [5, 6] provide a flexible alternative to approach symmetryand related issues, and have been adapted to the nonlocal setting in the upper citedpapers, among others.In this manuscript we have as main goal to introduce several formulations ofthe maximum principle for the fractional g − Laplacian, from where we will deducesome interesting qualitative results such as a Liouville type theorem, or symmetryof solutions in a ball.The novelty of our results is that they can be applied to non-local operatorsadmitting behaviors more general than powers such as G ( t ) = t p log(1 + t ), p ≥ G ( t ) = t p + t q , where p, q ≥
2. Tothe best of the authors’ knowledge, these are the first results of this kind availablein the literature for non-standard growth models.We further highlight that the possible lack of homogeneity of G will be one of themain obstacles to overcome, and leads us to develop specific tools for this setting.Throughout the paper, solutions of equations involving the fractional g − Laplacianwill be assumed to be of class C , ( R n ) ∩ L g , being L g the tail space defined as L g := (cid:26) u ∈ L ( R n ) : Z R n g (cid:18) | u ( x ) | | x | s (cid:19) dx | x | n + s < ∞ (cid:27) . That regularity ensures the operator to be well-posed, see Lemma 2.4.Our first result is a rather standard maximum principle for the fractional g − Laplacian:
AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 3
Theorem 1.1 (Maximum principle on domains) . Let Ω be a bounded domain in R n . Assume that u ∈ C , loc ( R n ) ∩ L g and satisfies ( ( − ∆ g ) s u ( x ) ≥ if x ∈ Ω u ( x ) ≥ if x ∈ R n \ Ω . Then u ≥ in Ω . Moreover, if u ( x ) = 0 at some point x ∈ Ω , then u ≡ in R n . In our next result we prove that if u is a bounded g − harmonic function, then itis symmetric about any given hyper-plane in R n and hence it must be constant: Theorem 1.2 (Liouville) . Let u ∈ C , loc ( R n ) ∩ L g satisfying ( − ∆ g ) s u = 0 in R n . If u is bounded, then u is constant in R n . The idea in order to obtain symmetry of u with respect to a given hyper-plane H is to consider the function w ( x ) = u (˜ x ) − u ( x ), where ˜ x denotes for the reflectionof x with respect to H . If we can prove that w ( x ) ≤ H , then interchanging theroles of x and ˜ x , we could deduce that w ( x ) ≡ H , and therefore u ( x ) wouldbe symmetric with respect to the plane H . Since the fractional g − Laplacian isinvariant under rotations and translations this gives that u must be constant.The aforementioned strategy is reached by means of the following maximumprinciple for antisymmetric functions on hyperplanes: Theorem 1.3 (Maximum principle on hyperplanes) . Let H be a hyperplane in R n , Σ the half space at one side of the plane and ˜ x be the reflection of x across H . Let u ∈ C , loc ( R n ) ∩ L g and define e u ( x ) := u (˜ x ) , and w ( x ) := e u ( x ) − u ( x ) . Assume w is bounded in Σ . If for any x ∈ Σ such that e u ( x ) > u ( x ) we have (1.1) ( − ∆ g ) s e u ( x ) − ( − ∆ g ) s u ( x ) ≤ then (1.2) w ( x ) ≤ in Σ . Theorems like 1.2 are often generalized to allow some growth at infinity on thefunction u ; indeed, the classical Liouville theorem for harmonic functions statesthat(1.3) ∆ u = 0 in R n and u = O ( | x | k ) as | x | → ∞ imply that u is a polynomial of order at most k . Even if the techniques displayedhere do not seem to be adaptable to get a result under assumptions similar to (1.3),the problem is interesting and worth pointing out.We will also be interested in studying nonlinear equations of the form( − ∆ g ) s u = f ( u )under suitable assumptions on the nonlinearity f . The classical method in thisscenario is the method of moving planes; before stating the results we introducesome notation (which is fairly standard): let λ ∈ R and T λ := { x ∈ R n : x n = λ for λ ∈ R } SANDRA MOLINA, ARIEL SALORT AND HERN´AN VIVAS be the hyperplane at height λ ; letΣ λ := { x ∈ R n : x n < λ } be the upper half-space. For each x ∈ Σ λ let x λ := ( x , x , · · · , λ − x n )be its reflection about the plane T λ . Finally, we will denote w λ ( x ) := u ( x λ ) − u ( x )(notice that w λ is anti-symmetric), and(1.4) λ := sup { λ ≤ w µ ≥ µ for any µ ≤ λ } . This notation will be used throughout the paper.The first step for the moving planes technique is to provide for a starting point tomove the plane: for λ sufficiently negative, it must be showed that w λ ( x ) ≥ λ .This can be ensured by using the following maximum principle for anti-symmetricfunctions in bounded domains: Theorem 1.4 (Maximum principle on bounded domains in hyperplanes) . Let T λ , Σ λ , w λ be defined as above, u ∈ C , loc ( R n ) ∩ L g .If (cid:26) ( − ∆ g ) s u ( x λ ) − ( − ∆ g ) s u ( x ) ≥ in Ω w λ ≥ in Σ λ \ Ω then w λ ( x ) ≥ in Σ λ . Moreover, if w λ = 0 at some point in Ω , then w λ ≡ in R n .Moreover, the result holds true for unbounded domains if we further assume that w λ ( x ) ≥ as | x | → ∞ . The second step consists in proving that λ = 0 which, applying the result to w λ and − w λ , implies that u is symmetric about the plane { x n = 0 } . This canbe proved by means of a contradiction argument: by assuming that λ < λ j ց λ , and x j ∈ Σ λ j such that w λ j ( x j ) = min Σ λj w λ j ≤ . We will show that such a sequence contradicts the following boundary estimate:
Proposition 1.5 (Boundary estimate) . Let λ be given by (1.4) and assume it isfinite and that w λ > in Σ λ . Suppose that there exists a sequence λ j ց λ and x j ∈ Σ λ j such that w λ j ( x j ) = min Σ λj w λ j ≤ and lim j →∞ x j = ¯ x ∈ T λ . Let δ j = dist( x j , T λ j ) . Then lim sup j →∞ δ j (cid:16) ( − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) (cid:17) < . AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 5
With the aid of the previous results, we can establish the symmetry of positivesolutions under natural assumptions on the right hand side f ; this is the contentof the following two theorems, concerning bounded domains and the whole space,respectively. Theorem 1.6 (Symmetry for solutions in a ball) . Let B be the unit ball in R n and u ∈ C , loc ( B ) ∩ C ( B ) be a positive function in B satisfying ( ( − ∆ g ) s u = f ( u ) in Bu = 0 in R n \ B, (1.5) where f is a Lipschitz function with f ′ nondecreasing and satisfying the followinggrowth condition: (1.6) g ′ ( t ) ≤ Cf ′ ( t ) for < t < and some C > . Then u is radially symmetric and monotone nondecreasing around the origin. Theorem 1.7 (Symmetry for solutions in whole space: decreasing RHS) . Let u ∈ C , loc ( R n ) ∩ L g satisfy (1.7) ( − ∆ g ) s u = f ( u ) and u > in R n . Assume (1.8) f ′ ( t ) ≤ for t ≤ , (1.9) lim | x |→∞ u ( x ) = 0 . Then u is radially symmetric around some point in R n . We leave as an open question to find which are the (best) conditions on f andon the decay of u at infinity in order to ensure symmetry of positive solutions u ∈ C , ( R n ) ∩ L g of( − ∆ g ) s u = f ( u ) and u > R n in the case in which f is an increasing function.Further interesting research directions would be to address qualitative propertiesof solutions unbounded domains, for instance( − ∆ g ) s u = f ( u ) in { x n > } and u = 0 on { x n = 0 } , or more general unbounded domains such as those given by the epigraph of aLipschitz function. Organization of the paper.
This article is organized as follows. Section 2 isdevoted to introduce the notion of Young function and the proof of some usefulinequalities, and several properties that the fractional g − Laplacian fulfills. In sec-tion 3 we prove the maximum principles in domains and hyperplanes as well as theLiouville theorem, namely, Theorems 1.1, 1.3 and 1.2. Section 4 contains the proofsof the maximum principle on bounded domains in hyperplanes, i.e., Theorem 1.4,and the boundary estimate stated in Proposition 1.5. In Section 5 we deliver theproof of our symmetry results, namely Theorems 1.6 and 1.7. Finally, in section 6we introduce some applications and extensions of our results.
SANDRA MOLINA, ARIEL SALORT AND HERN´AN VIVAS Preliminaries
In this section we give some preliminary definitions and technical results thatwill be used throughout the paper. We recall the notion of Young function andpresent some simple technical inequalities that will be helpful. Then, we define thefractional g − Laplacian and prove some important properties of it, both useful forthe rest of the paper and of independent interest.2.1.
Young functions.
An application G : [0 , ∞ ) −→ [0 , ∞ ) is said to be a Youngfunction if it admits the integral representation G ( t ) = Z t g ( τ ) dτ, where the right-continuous function g defined on [0 , ∞ ) has the following properties: g (0) = 0 , g ( t ) > t > ,g is nondecreasing on (0 , ∞ )lim t →∞ g ( t ) = ∞ . From these properties it is easy to see that a Young function G is continuous,nonnegative, strictly increasing and convex on [0 , ∞ ). Further, we recall that wemay extend g to the whole R in an odd fashion: for t < g ( t ) = − g ( − t ).We will consider the class of Young functions such that g = G ′ is an absolutelycontinuous function that satisfies the condition(2.1) 1 < p − − ≤ tg ′ ( t ) g ( t ) ≤ p + − < ∞ , t > . This condition was first considered in the seminal work of G. Lieberman [18] and isthe analogous to the ellipticity condition in the linear theory as it will be apparentlater on; it essentially says that (2.1) means that g ( t ) is “trapped between powers”.Moreover, integrating (2.1) we have that G verifies(2.2) 2 < p − ≤ tg ( t ) G ( t ) ≤ p + < ∞ , t > . In [17, Theorem 4.1] it is shown that the upper bound in (2.1) (or in (2.2)) isequivalent to the so-called ∆ condition (or doubling condition) , namely(∆ ) g (2 t ) ≤ p + − g ( t ) , G (2 t ) ≤ p + G ( t ) t ≥ . It is easy to verify that this condition implies the existence of constants C , C > a, b ≥ g ( a + b ) ≤ C ( g ( a ) + g ( b )) , G ( a + b ) ≤ C ( G ( a ) + G ( b )) . Further, the inequalities(2.4) min { α p − − , α p + − } g ( t ) ≤ g ( αt ) ≤ max { α p − − , α p + − } g ( t )and(2.5) min { α p − , α p + } p + G ( t ) ≤ G ( αt ) ≤ p + max { α p − , α p + } G ( t ) AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 7 hold for any, α, t ≥ g is that its derivative g ′ (that exists a.e.) isnondecreasing; we point out that this is analogous of dealing with the degeneratecase p ≥ p − Laplacian.2.2.
Some useful inequalities.
We include here some technical inequalities thatwill be used throughout the paper; the proofs are simple but included for the sakeof completeness.
Lemma 2.1.
Let G be a Young function such that g = G ′ satisfies (2.1) . Thenthere exists C = C ( p + , p − ) > such that g ( b ) − g ( a ) ≥ Cg ( b − a ) . for all b ≥ a .Proof. We split the proof in several cases.
Case 1: b ≥ a ≥ a ≥ b , then, for some ξ ∈ ( a, b ) and using (2.1), g ( b ) − g ( a ) = g ′ ( ξ )( b − a ) ≥ g ′ (cid:18) b (cid:19) ( b − a ) ≥ g ′ (cid:18) b − a (cid:19) ( b − a ) ≥ p − − g (cid:18) b − a (cid:19) ≥ − p + ( p − − g ( b − a )where we have used (2.4) for the last inequality.If 0 < a < b , then we use (2.4) again and the fact that g is nondecreasing: g ( b ) − g ( a ) ≥ g ( b ) − g (cid:18) b (cid:19) ≥ (cid:16) − − p + (cid:17) g ( b ) ≥ (cid:16) − − p + (cid:17) g ( b − a ) . Case 2: a ≤ b ≤ | a | ≥ | b | ≥ g ( b ) − g ( a ) = g ( | a | ) − g ( | b | ) ≥ Cg ( | a | − | b | ) = Cg ( b − a ) . Case 3: a ≤ ≤ b .Here a and b have different signs, then since g is odd g ( b ) − g ( a ) = g ( | b | ) + g ( | a | ) ≥ Cg ( | b | + | a | ) = Cg ( b − a ) , where we have used (2.3) for g . The proof is now completed. (cid:3) SANDRA MOLINA, ARIEL SALORT AND HERN´AN VIVAS
Lemma 2.2.
For any a, b ∈ R and g an absolutely continuous function such that g ′ is nondecreasing, it holds that | g ( a + b ) − g ( a ) | ≤ | b | g ′ ( | a | + | b | ) . Proof.
A straightforward computation gives (recall that g ′ is nondecreasing) | g ( a + b ) − g ( a ) | = (cid:12)(cid:12)(cid:12)(cid:12) b Z g ′ ( a + tb ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ | b | Z g ′ ( | a | + t | b | ) dt ≤ | b | g ′ ( | a | + | b | ) , from where the lemma follows. (cid:3) Lemma 2.3.
Let G be a Young function such that G ′ = g satisfies (2.1) . Thereexists a constant < C = C ( p − , p + ) ≤ such that, if we write g ( b ) − g ( a ) = g ′ ( ξ )( b − a ) , then (2.6) | ξ | ≥ C max {| b | , | a |} . Proof.
Without loss of generality we may assume that | b | > | a | . Case 1: | a | ≥ | b | .If a and b have the same sign, then ξ is between a and b and (2.6) holds with C = 1. If a and b are of opposite signs, since g is odd, g ( a ) and g ( b ) are of oppositesigns. It follows that2 g ′ ( | ξ | ) | b | ≥ | g ′ ( ξ ) || b − a | = | g ( b ) − g ( a ) | ≥ | g ( b ) | . Then, by using (2.1) and the fact that | g ( b ) | = | g ( | b | ) | ,2 g ′ ( | ξ | ) ≥ | g ( b ) || b | ≥ g ′ ( | b | ) p + − g ′ is nondecreasing. Case 2: | a | ≤ | b | .In this case (2.1) implies that g ( | b | ) ≥ g (2 | a | ) ≥ p − − g ( | a | )from where2 g ′ ( | ξ | ) | b | ≥ | g ( b ) − g ( a ) | ≥ | g ( | b | ) | − | g ( | a | ) | ≥ (1 − − ( p − − ) g ( | b | )and (2.6) follows. (cid:3) AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 9
The fractional g − Laplacian.
Recall the fractional g − Laplacian defined in[14] by:(2.7) ( − ∆ g ) s u ( x ) := p.v. Z R n g ( D s u ( x, y )) dy | x − y | n + s where the notation D s u ( x, y ) := u ( x ) − u ( y ) | x − y | s is (and will be) used.In this section we include some elementary properties of the fractional g − Laplacian.( − ∆ g ) s is an operator “of order 2 s ”, so to ensure that ( − ∆ g ) s is well defined at x we need u ∈ C s + δ at x for some δ >
0; for the purposes of this paper it will beenough to assume that u ∈ C , at x in the sense that there exists C ≥ | u ( x ) − u ( y ) − ∇ u ( x ) · ( y − x ) | ≤ C | y − x | for | y − x | small enough.This is a stronger regularity assumption than C s + δ but serves for the sake ofclarity.On the other hand, because of the nonlocal nature of ( − ∆ g ) s we need to controlthe behavior of u at infinity; we will denote(2.9) L g := (cid:26) u ∈ L ( R n ) : Z R n g (cid:18) | u ( x ) | | x | s (cid:19) dx | x | n + s < ∞ (cid:27) . Notice that the inclusion(2.10) L g ⊂ L g ′ holds; indeed, if u ∈ L g we can split Z R n g ′ (cid:18) | u ( x ) | | x | s (cid:19) dx | x | n + s = Z { x : | u ( x ) | | x | s ≤ } + Z { x : | u ( x ) | | x | s > } ! g ′ (cid:18) | u ( x ) | | x | s (cid:19) dx | x | n + s . Since g ′ is nondecreasing the first term is bounded by g ′ (1) Z R n dx | x | n + s < ∞ whereas the second term is bounded (using the (2.1) and the fact that | u ( x ) | | x | s > p + − Z R n g (cid:18) | u ( x ) | | x | s (cid:19) dx | x | n + s < ∞ . The next lemma shows that (2.8) and (2.9) are enough for (2.7) to be well defined.
Lemma 2.4.
Let u ∈ C , ∩ L g at x ∈ R n , then ( − ∆ g ) s u ( x ) is well defined.Proof. Let 0 < ε < Z R n \ B ε ( x ) g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19) dy | x − y | n + s = I + I with I := Z B ( x ) \ B ε ( x ) g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19) dy | x − y | n + s and I := Z R n \ B ( x ) g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19) dy | x − y | n + s . On one hand we have u ( x ) − u ( y ) = ∇ u ( x ) · ( x − y ) + O ( | x − y | )as y → x so Lemma 2.2 gives (recall | x − y | < (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) ∇ u ( x ) · ( x − y ) + O ( | x − y | ) | x − y | s (cid:19) − g (cid:18) ∇ u ( x ) · ( x − y ) | x − y | s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x − y | − s g ′ (cid:0) C | x − y | − s (cid:1) Next notice that g (cid:16) ∇ u ( x ) · ( x − y ) | x − y | s (cid:17) is odd so its integral over B ( x ) \ B ε ( x ) vanishes,whence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( x ) \ B ε ( x ) g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19) dy | x − y | n + s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( x ) \ B ε ( x ) h g (cid:18) ∇ u ( x ) · ( x − y ) + O ( | x − y | ) | x − y | s (cid:19) − g (cid:18) ∇ u ( x ) · ( x − y ) | x − y | s (cid:19) dy | x − y | n + s i(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B ( x ) \ B ε ( x ) C | x − y | − s g ′ (cid:0) C | x − y | − s (cid:1) dy | x − y | n + s ≤ Cg ′ ( C ) Z B ( x ) \ B ε ( x ) dy | x − y | n − − s ) so I converges as ε → + .On the other hand, by (2.3) Z R n \ B ( x ) g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19) dy | x − y | n + s ≤ C Z R n \ B ( x ) g (cid:18) | u ( x ) || x − y | s (cid:19) dy | x − y | n + s + Z R n \ B ( x ) g (cid:18) | u ( y ) || x − y | s (cid:19) dy | x − y | n + s ! and noticing that for y outside B ( x ) the polynomials | x − y | s and 1 + | y | s and | x − y | n + s and 1 + | y | n + s are comparable we get | I | ≤ C g ( | u ( x ) | ) Z R n \ B ( x ) dy | x − y | n + s + Z R n \ B ( x ) g (cid:18) | u ( y ) | | y | s (cid:19) dy | y | n + s ! . The first term is obviously integrable and so is the second owing to (2.9). Theresult follows. (cid:3)
The following simple result shows that ( − ∆ g ) s is rotation invariant: Lemma 2.5.
Let u ∈ C , ∩ L g at x ∈ R n and let Q ∈ R n × n be an orthogonalmatrix. Define u Q ( x ) := u ( Qx ) . Then ( − ∆ g ) s u Q ( x ) = ( − ∆ g ) s u ( Qx ) . AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 11
Proof.
The proof is an immediate change of variables:( − ∆ g ) s u Q ( x ) = lim ε → Z R n \ B ε ( x ) g (cid:18) u ( Qx ) − u ( Qy ) | x − y | s (cid:19) dy | x − y | n + s = lim ε → Z R n \ B ε ( x ) g (cid:18) u ( Qx ) − u ( Qy ) | Qx − Qy | s (cid:19) dy | Qx − Qy | n + s = lim ε → Z R n \ B ε ( Qx ) g (cid:18) u ( Qx ) − u ( z ) | Qx − z | s (cid:19) dz | Qx − z | n + s = ( − ∆ g ) s u ( Qx ) . (cid:3) We will further need the following result concerning the fractional g − Laplacianof a cut-off function:
Lemma 2.6.
Let ϕ ∈ C ∞ c ( B ) be a radially symmetric function decreasing with | x | , then ( − ∆ g ) s ϕ ( x ) is well defined and (2.11) | ( − ∆ g ) s ϕ ( x ) | ≤ C for some C depending of n , s and k ϕ k C ( B ) . Furthermore, ( − ∆ g ) s ϕ ( x ) is also aradial function.Proof. The bound in (2.11) follows simply by repeating the steps of the proof ofLemma 2.4. Let us show that ( − ∆ g ) s ϕ ( x ) is radial; making the change of variablesto spherical coordinates and denoting x = rx ′ , y = ρy ′ with | x ′ | = | y ′ | = 1 we cancompute( − ∆ g ) s ϕ ( x ) = p.v. Z R n g (cid:18) ϕ ( x ) − ϕ ( y ) | x − y | s (cid:19) dy | x − y | n + s = p.v. Z ∞ Z ∂B g (cid:18) ϕ ( r ) − ϕ ( ρ ) | rx ′ − ρy ′ | s (cid:19) ρ n − dy ′ dρ | rx ′ − ρy ′ | n + s = p.v. Z ∞ Z ∂B g (cid:18) ϕ ( r ) − ϕ ( ρ ) r s | x ′ − ρr y ′ | s (cid:19) ρ n − dy ′ dρr n + s | x ′ − ρr y ′ | n + s ( ρ = rτ ) = r − n − s p.v. Z ∞ Z ∂B ( rτ ) n − g (cid:18) ϕ ( r ) − ϕ ( rτ ) r s | x ′ − τ y ′ | s (cid:19) rdy ′ dτ | x ′ − τ y ′ | n + s = r − s p.v. Z ∞ τ n − (cid:18)Z ∂B g (cid:18) ϕ ( r ) − ϕ ( rτ ) r s | x ′ − τ y ′ | s (cid:19) dy ′ | x ′ − τ y ′ | n + s (cid:19) dτ = r − s p.v. Z ∞ τ n − h ( τ ) dτ. It is not obvious a priori that h is a function of τ alone (and not of x ′ ), but if we let z ′ ∈ ∂B and Q be a orthogonal matrix such that z ′ = Qx ′ by changing variables y ′ = Qw ′ we have Z ∂B g (cid:18) ϕ ( r ) − ϕ ( rτ ) r s | z ′ − τ y ′ | s (cid:19) dy ′ | z ′ − τ y ′ | n + s = Z ∂B g (cid:18) ϕ ( r ) − ϕ ( rτ ) r s | Qx ′ − τ Qw ′ | s (cid:19) dw ′ | Qx ′ − τ Qw ′ | n + s = Z ∂B g (cid:18) ϕ ( r ) − ϕ ( rτ ) r s | x ′ − τ w ′ | s (cid:19) dw ′ | x ′ − τ w ′ | n + s . Therefore h is indeed independent of x ′ and ( − ∆ g ) s ϕ ( x ) depends only on r asdesired. (cid:3) We end this section with a technical lemma that gives control of ( − ∆ g ) s u if weperturb it by a smooth function. Lemma 2.7.
Let u ∈ C , ∩ L g at x and ψ ∈ C ∞ ( R n ) , then for all δ > thereexists C δ > such that | ( − ∆ g ) s ( u + εψ )( x ) − ( − ∆ g ) s u ( x ) | ≤ C δ ε + ω ( δ ) with ω a continuous function of δ satisfying ω (0) = 0 .Proof. Denote v ε ( x ) := u ( x ) + εψ ( x ), then( − ∆ g ) s v ε ( x ) − ( − ∆ g ) s u ( x ) == p.v. Z B cδ ( x ) + Z B δ ( x ) ! (cid:20) g (cid:18) v ε ( x ) − v ε ( y ) | x − y | s (cid:19) − g (cid:18) u ( x ) − u ( y ) | x − y | s (cid:19)(cid:21) dy | x − y | n + s := I + I . Let us bound the first integral. Recall g ( b ) − g ( a ) = ( b − a ) Z g ′ ( a + t ( b − a )) dt = ( b − a ) g ′ ( a + t ( b − a ))for some t ∈ (0 , a = u ( x ) − u ( y ) | x − y | s and b = v ε ( x ) − v ε ( y ) | x − y | s to get g ( D s v ε ) − g ( D s u ) = ε ψ ( x ) − ψ ( y ) | x − y | s g ′ (cid:18) u ( x ) − u ( y ) + t ε ( ψ ( x ) − ψ ( y )) | x − y | s (cid:19) . Then, | I | ≤ Z B cδ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ε ( ψ ( x ) − ψ ( y )) | x − y | s g ′ (cid:18) u ( x ) − u ( y ) + t ε ( ψ ( x ) − ψ ( y )) | x − y | s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dy | x − y | n + s ≤ ε k ψ k ∞ Z B cδ ( x ) g ′ (cid:18) | u ( x ) | + | u ( y ) | + 2 k ψ k ∞ | x − y | s (cid:19) dy | x − y | n +2 s ≤ εC k ψ k ∞ Z B cδ ( x ) g ′ (cid:18) | u ( x ) | + | u ( y ) | + 2 k ψ k ∞ | y | s (cid:19) dy | y | n +2 s , where we have also used that | x − y | s ∼ | y | s and | x − y | n +2 s ∼ | y | n +2 s if y ∈ B cδ ( x ) (the constant C depends of course on δ ). Proceeding as in the proof of(2.10) and denoting E := (cid:26) y ∈ R n : | u ( x ) | + | u ( y ) | + 2 k ψ k ∞ | y | s ≤ (cid:27) we further obtain | I | ≤ εC k ψ k ∞ Z B cδ ( x ) ∩ E g ′ (1) dy | y | n +2 s + Z B cδ ( x ) ∩ E c g (cid:18) | u ( x ) | + | u ( y ) | + 2 k ψ k ∞ | y | s (cid:19) dy | y | n +2 s ! AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 13 but Z B cδ ( x ) ∩ E c g (cid:18) | u ( x ) | + | u ( y ) | + 2 k ψ k ∞ | y | s (cid:19) dy | y | n +2 s ≤ C Z B cδ ( x ) ∩ E c g (cid:18) | u ( y ) | | y | s (cid:19) dy | y | n +2 s + g ( | u ( x ) | + 2 k ψ k ∞ ) Z B cδ ( x ) ∩ E c dy | y | n +2 s ! . Given that the second integral is finite and so is the first (since u ∈ L g ) we obtain | I | ≤ C δ ε. Now we turn to I : (2.8) allows us to write v ε ( x ) − v ε ( y ) = ∇ v ε ( x ) · ( x − y ) + O ( | x − y | ) . Also, ∇ v ε ( x ) · ( x − y ) is anti-symmetric for y ∈ B δ ( x ) and g is an odd function wehave p.v. Z B δ ( x ) g (cid:18) ∇ v ε ( x ) · ( x − y ) | x − y | s (cid:19) dy | x − y | n + s = 0 . Then, using Lemma 2.2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p.v. Z B δ ( x ) g ( D s v ε ) dy | x − y | n + s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p.v. Z B δ ( x ) g ( D s v ε ) dy | x − y | n + s − p.v. Z B δ ( x ) g (cid:18) ∇ v ε ( x ) · ( x − y ) | x − y | s (cid:19) dy | x − y | n + s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p.v. Z B δ ( x ) O ( | x − y | − s ) g ′ (cid:0) |∇ v ε ( x ) || x − y | − s + O ( | x − y | − s ) (cid:1) dy | x − y | n + s ≤ p.v. C δ Z B δ ( x ) g ′ (cid:0) |∇ v ε ( x ) | δ − s + δ − s ) (cid:1) dy | x − y | n + s − ≤ p.v. C δ g ′ ( C δ ) Z B δ ( x ) dy | x − y | n + s − < ∞ . The bound for g ( D s u ) is analogous and we get | I | ≤ p.v. C δ g ′ ( C δ ) Z B δ ( x ) dy | x − y | n + s − , and the conclusion of the lemma follows. (cid:3) Remark . It may be worth pointing out that the constant C δ depends on otherquantities besides from δ ; it depends on n, s, k ψ k ∞ and on u itself. However, for thepurposes of our application in the proof of Theorem 1.3, the important property isthat it does not depend on ε .3. Proof of the maximum principle on domains, maximum principle onhyperplanes and Liouville theorem
This section is devoted to the proof of Theorems 1.1, 1.3 and 1.2. The first oneis rather simple owing to the nonlocal nature of the operator:
Proof of Theorem 1.1.
Suppose that the conclusion is false. Then, since u is con-tinuous in Ω, there exists ¯ x ∈ Ω such that u (¯ x ) = min Ω u < . By Lemma 2.4 we can evaluate point-wisely the operator, then the last claimtogether with the fact the u ≥ R n \ Ω give that( − ∆ g ) s u (¯ x ) = p.v. Z Ω + Z R n \ Ω ! g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19) dy | ¯ x − y | n + s < Z R n \ Ω g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19) dy | ¯ x − y | n + s ≤ . This contradicts that ( − ∆ g ) s u (¯ x ) ≥ u ( x ) ≥ u (¯ x ) = 0 at some ¯ x ∈ Ω, then0 ≤ ( − ∆ g ) s u (¯ x ) = p.v. Z R n g (cid:18) − u ( y ) | ¯ x − y | s (cid:19) dy | ¯ x − y | n + s ≤ u is non-negative, we concludethat u ( x ) ≡ R n and the proof concludes. (cid:3) Next we give the
Proof of Theorem 1.3.
By the rotation and translation invariance of ( − ∆ g ) s wemay assume that H = { x ∈ R n : x = 0 } and Σ = { x ∈ R n : x < } . By contradiction, let us suppose (1.2) is false and let A := sup Σ w ( x ) > . Then, for γ ∈ (0 ,
1) to be chosen later there exists ¯ x ∈ Σ such that w (¯ x ) ≥ γA. Let η ∈ C ∞ c ( B ) be a radially symmetric, decreasing function satisfying0 ≤ η ≤ , η (0) = 1 . Recall that, due to Lemma 2.6, | ( − ∆ g ) s η | ≤ C and ( − ∆ g ) s η is a radial function. Let us further set ψ ( x ) := η ( x + ¯ x ) and e ψ ( x ) := η ( x − ¯ x )and notice that e ψ ( x ) − ψ ( x ) is antisymmetric with respect to H, e ψ ( x ) = 0 in R n \ B (¯ x ) ,ψ ( x ) = 0 in R n \ B ( − ¯ x ) (in particular in Σ) . AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 15
The idea is to construct an appropriate perturbation of w and use (1.1) to get acontradiction. We start by choosing ε > w (¯ x ) + ε e ψ (¯ x ) − εψ (¯ x ) = w (¯ x ) + ε ≥ A and notice that by construction w (¯ x ) + ε e ψ (¯ x ) − εψ (¯ x ) ≥ w ( x ) + ε e ψ ( x ) − εψ ( x ) ∀ x ∈ Σ \ B (¯ x )and thereforemax x ∈ Σ ( w ( x ) + ε e ψ ( x ) − εψ ( x )) = w (¯ x ) + ε e ψ (¯ x ) − εψ (¯ x ) for some ¯ x ∈ B (¯ x ) . We will estimate(3.1) ( − ∆ g ) s ( e u + ε e ψ )(¯ x ) − ( − ∆ g ) s ( u + εψ )(¯ x )by above and below to reach a contradiction.We start computing( − ∆ g ) s ( e u + ε e ψ )(¯ x ) − ( − ∆ g ) s ( u + εψ )(¯ x ) == Z R n " g e u (¯ x ) + ε e ψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x − y | s ! − g (cid:18) u (¯ x ) + εψ (¯ x ) − u ( y ) − εψ ( y ) | ¯ x − y | s (cid:19) dy | ¯ x − y | n + s . Splitting R n as Σ ∪ Σ c , and performing a change of variables the expression abovereads as Z Σ " g e u (¯ x ) + ε e ψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x − y | s ! − g (cid:18) u (¯ x ) + εψ (¯ x ) − u ( y ) − εψ ( y ) | ¯ x − y | s (cid:19) dy | ¯ x − y | n + s + Z Σ " g e u (¯ x ) + ε e ψ (¯ x ) − u ( y ) − εψ ( y ) | ¯ x + y | s ! − g u (¯ x ) + εψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x + y | s ! dy | ¯ x + y | n + s where we have used the definition of e u and ˜ ψ . We can further rewrite this as Z Σ (cid:18) | ¯ x − y | n + s − | ¯ x + y | n + s (cid:19) " g e u (¯ x ) + ε e ψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x − y | s ! − g (cid:18) u (¯ x ) + εψ (¯ x ) − u ( y ) − εψ ( y ) | ¯ x − y | s (cid:19)(cid:21) dy + Z Σ " g e u (¯ x ) + ε e ψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x − y | s ! − g u (¯ x ) + εψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x + y | s ! + g e u (¯ x ) + ε e ψ (¯ x ) − u ( y ) − εψ ( y ) | ¯ x + y | s ! − g (cid:18) u (¯ x ) + εψ (¯ x ) − u ( y ) − εψ ( y ) | ¯ x − y | s (cid:19) dy | ¯ x + y | n + s := I + I + I . We bound each term separately. Notice that(3.2) 1 | ¯ x − y | n + s ≥ | ¯ x + y | n + s for y ∈ Σ and observe that0 ≤ ( w (¯ x ) + ε e ψ (¯ x ) − εψ (¯ x )) − ( w ( y ) + ε e ψ ( y ) − εψ ( y ))= ( e u (¯ x ) + ε e ψ (¯ x ) − e u ( y ) − ε e ψ ( y )) − ( u (¯ x ) + εψ (¯ x ) − u ( y ) − εψ ( y ))so the monotonicity of g implies that I ≥ I , first observe that( e u (¯ x ) + ε e ψ (¯ x ) − e u ( y ) − ε e ψ ( y )) − ( u (¯ x ) + εψ (¯ x ) − e u ( y ) − ε e ψ ( y ))= w (¯ x ) + ε e ψ (¯ x ) − εψ (¯ x ) ≥ A, therefore, by (3.2) and Lemma 2.1 we get g e u (¯ x ) + ε e ψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x − y | s ! − g u (¯ x ) + εψ (¯ x ) − e u ( y ) − ε e ψ ( y ) | ¯ x + y | s ! ≥ Cg (cid:18) A | ¯ x − y | s (cid:19) . and therefore I ≥ C Z Σ g (cid:18) A | ¯ x − y | s (cid:19) dy | ¯ x + y | n + s = C g ( A ) . Finally, notice that the bound for I is similar to the one for I and we get(3.3) (3.1) ≥ C g ( A ) . To get an upper bound, we use Lemma 2.7 and (1.1)(3.1) = ( − ∆ g ) s ( e u + ε e ψ )(¯ x ) − ( − ∆ g ) s e u (¯ x ) − (( − ∆ g ) s ( u + εψ )(¯ x ) − ( − ∆ g ) s u (¯ x )) + ( − ∆ g ) s e u (¯ x ) − ( − ∆ g ) s u (¯ x ) ≤ C δ ε + ω ( δ )) + ( − ∆ g ) s e u (¯ x ) − ( − ∆ g ) s u (¯ x ) ≤ C δ ε + ω ( δ )) . The last inequality, together with (3.3) gives C g ( A ) ≤ C δ ε + ω ( δ )which, taking δ such that ω ( δ ) ≤ C g ( A )2 gives C g ( A )2 C δ ≤ ε, but since ε can be chosen as small as needed we have reached a contradiction. (cid:3) As mentioned in the Introduction, the previous theorem is sufficient to give the
Proof of Theorem 1.2.
Let us see that u is symmetric with respect to any hyper-plane, as a consequence of Theorem 1.3. Indeed, if H is any hyperplane and Σ isthe semi-space on one side of H , we can define w as in Theorem 1.3 and notice thatsince u is bounded so is w . Further, since u is g − harmonic in R n ,( − ∆ g ) s e u ( x ) − ( − ∆ g ) s u ( x ) = 0 , in particular if e u ( x ) > u ( x ) for some x ∈ Σ.By Theorem 1.3 w ( x ) ≤ w ( x ) ≥ w ( x ) ≡ u ( x ) is symmetric with respect to H . Since H can be AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 17 chosen arbitrarily, u is radially symmetric about any point, giving that u ( x ) ≡ C and concluding the proof. (cid:3) Proof of Theorem 1.4 and Proposition 1.5
This section is dedicated to the proofs of Theorem 1.4 and Proposition 1.5,starting with the former:
Proof of Theorems 1.4.
Assume that the thesis of the theorem fails to hold, that isfor some ¯ x ∈ Ω w λ (¯ x ) = min Σ λ w = min Ω w < . Let us compute, splitting R n as Σ λ ∪ Σ cλ ,( − ∆ g ) s u (¯ x λ ) − ( − ∆ g ) s u (¯ x ) = Z R n (cid:20) g (cid:18) u (¯ x λ ) − u ( y λ ) | ¯ x − y | s (cid:19) − g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19)(cid:21) dy | ¯ x − y | n + s = Z Σ λ (cid:20) g (cid:18) u (¯ x λ ) − u ( y λ ) | ¯ x − y | s (cid:19) − g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19)(cid:21) dy | ¯ x − y | n + s + Z Σ λ (cid:20) g (cid:18) u (¯ x λ ) − u ( y ) | ¯ x − y λ | s (cid:19) − g (cid:18) u (¯ x ) − u ( y λ ) | ¯ x − y λ | s (cid:19)(cid:21) dy | ¯ x − y λ | n + s = I + I with I := Z Σ λ (cid:20) g (cid:18) u (¯ x λ ) − u ( y λ ) | ¯ x − y | s (cid:19) − g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19)(cid:21) (cid:18) | ¯ x − y | n + s − | ¯ x − y λ | n + s (cid:19) dy and I := Z Σ λ (cid:20) g (cid:18) u (¯ x λ ) − u ( y ) | ¯ x − y λ | s (cid:19) − g (cid:18) u (¯ x ) − u ( y λ ) | ¯ x − y λ | s (cid:19) + g (cid:18) u (¯ x λ ) − u ( y λ ) | ¯ x − y | s (cid:19) − g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19)(cid:21) dy | ¯ x − y λ | n + s . Now, 1 | ¯ x − y | n + s − | ¯ x − y λ | n + s > w λ (¯ x ) − w λ ( y ) = u (¯ x λ ) − u ( y λ ) − ( u (¯ x ) − u ( y )) ≤ , we get g (cid:18) u (¯ x λ ) − u ( y λ ) | ¯ x − y | s (cid:19) − g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19) ≤ I ≤ . For the other term we have I = w λ (¯ x ) Z Σ λ g ′ ( ξ ( y )) + g ′ ( ζ ( y )) | ¯ x − y λ | n +2 s dy with ξ ( y ) between u (¯ x λ ) − u ( y ) and u (¯ x ) − u ( y )and ζ ( y ) between u (¯ x λ ) − u ( y λ ) and u (¯ x ) − u ( y λ ) . Since by the contradiction assumption w λ (¯ x ) < G is convex (which implies g ′ >
0) we have(4.2) I < . Putting together (4.1) and (4.2) we get(4.3) ( − ∆ g ) s u (¯ x λ ) − ( − ∆ g ) s u (¯ x ) < w λ ( x ) = 0 at some x ∈ Ω, then x is a minimum of w in Ω.Therefore, by using the hypothesis and splitting the integrals as before,0 ≤ ( − ∆ g ) s u (¯ x λ ) − ( − ∆ g ) s u (¯ x ) = I + I with I = 0, and then I ≥
0. This implies that g (cid:18) u (¯ x λ ) − u ( y λ ) | ¯ x − y | s (cid:19) − g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19) ≥ . Observe that the monotonicity of g implies that (cid:18) g (cid:18) u (¯ x λ ) − u ( y λ ) | ¯ x − y | s (cid:19) − g (cid:18) u (¯ x ) − u ( y ) | ¯ x − y | s (cid:19)(cid:19) ( u (¯ x λ ) − u (¯ x )) − ( u ( y ) − u ( y λ )) | x − y | s ≥ u (¯ x λ ) − u (¯ x )) − ( u ( y ) − u ( y λ )) = w λ ( x ) − w λ ( y ) = − w λ ( y ) ≥ , giving that w λ ( y ) = 0 in Σ λ , and from the antisymmetry of w λ , in R n .Finally, when Ω is unbounded, if we further assume that w ( x ) ≥ | x | → ∞ , ifit assumed that w ( x ) ≥ λ does not hold, a similar reasoning can be performedto reach a contradiction. (cid:3) Recall that, to prove Theorem 1.6 we need Proposition 1.5 so that is the nextproof we address.
Proof of Proposition 1.5.
Proceeding as in the previous proof we compute1 δ j (cid:16) ( − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) (cid:17) == 1 δ j Z Σ λj " g u ( x λ j j ) − u ( y λ j ) | x j − y | s ! − g (cid:18) u ( x j ) − u ( y ) | x j − y | s (cid:19) | x j − y | n + s − | x j − y λ j | n + s (cid:19) dy + w λ j ( x j ) δ j Z Σ λj g ′ ( ξ ( y )) + g ′ ( ζ ( y )) | x j − y λ j | n +2 s dy. Recall that(4.4) w λ j ( x j ) δ j Z Σ λj g ′ ( ξ ( y )) + g ′ ( ζ ( y )) | x j − y λ j | n +2 s dy ≤ AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 19 since w λ j ( x j ) ≤ g ′ > j →∞ u ( x λ j j ) − u ( y λ j ) − u ( x j ) − u ( y ) = lim j →∞ w λ j ( x j ) − w λ j ( y ) = w λ (¯ x ) − w λ ( y ) < j →∞ g u ( x λ j j ) − u ( x j ) | ¯ x − y | s ! − g (cid:18) u ( x j ) − u ( y ) | ¯ x − y | s (cid:19) < . Also, 1 | x j − y | n + s − | x j − y λ j | n + s = − ( n + s )2 | η ( y ) | n + s +2 (cid:0) | x j − y | − | x j − y λ j | (cid:1) and so(4.6) lim j →∞ δ j (cid:18) | x j − y | n + s − | x j − y λ j | n + s (cid:19) > . Gathering (4.5)-(4.6) with (4.4) and taking lim sup giveslim sup j →∞ δ j (cid:16) ( − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) (cid:17) < (cid:3) Symmetry results
Next we give present the proofs of our symmetry results, Theorems 1.6 and 1.7:
Proof of Theorem 1.6.
As mentioned, the proof follows the scheme of the movingplanes method. Let us set Ω λ := Σ λ ∩ B. The first step is to show that for λ > − − w λ ≥ λ . Let us assume for the sake of contradiction that this is not the case.Then w λ (¯ x ) = min Ω λ w λ < λ will be suitably chosen later).On one hand, (1.5) gives( − ∆ g ) s u (¯ x λ ) − ( − ∆ g ) s u (¯ x ) = f ( u (¯ x λ )) − f ( u (¯ x )) = f ′ ( ξ ) w λ (¯ x )for some ξ that lies between u (¯ x λ ) and u (¯ x ). Since f ′ is nondecreasing and byhypothesis w λ (¯ x ) < − ∆ g ) s u (¯ x λ ) − ( − ∆ g ) s u (¯ x ) ≥ f ′ ( u (¯ x )) w λ (¯ x ) . On the other hand, we can proceed as in the proof of Theorem 1.4 to get( − ∆ g ) s u (¯ x λ ) − ( − ∆ g ) s u (¯ x ) ≤ w λ (¯ x ) Z Σ λ g ′ ( ξ ( y )) + g ′ ( ζ ( y )) | ¯ x − y λ | n +2 s dy =: w λ (¯ x ) I with u (¯ x λ ) − u ( y λ ) < ξ ( y ) < u (¯ x ) − u ( y λ ) u (¯ x λ ) − u ( y ) < ζ ( y ) < u (¯ x ) − u ( y ) . We will use Lemma 2.3 to bound I by below: I ≥ Z Σ λ g ′ ( C max {| u (¯ x λ ) − u ( y λ ) | , | u (¯ x ) − u ( y λ ) |} ) + g ′ ( C max {| u (¯ x λ ) − u ( y ) | , | u (¯ x ) − u ( y ) |} ) | ¯ x − y λ | n +2 s dy ≥ Z Σ λ \ Ω λ g ′ ( C max {| u (¯ x λ ) | , | u (¯ x ) |} ) | ¯ x − y λ | n +2 s dy ≥ Z Σ λ \ Ω λ g ′ ( C | u (¯ x ) | ) | ¯ x − y λ | n +2 s dy ≥ Z Ω λ +1 \ Ω λ g ′ ( C | u (¯ x ) | ) | ¯ x − y λ | n +2 s dy ≥ g ′ ( C | u (¯ x ) | ) | λ | s where we have used that u ( y ) = 0 in Σ λ \ Ω λ and w λ (¯ x ) = u (¯ x λ ) − u (¯ x ) < ≤ (cid:18) g ′ ( C u (¯ x )) | λ + 1 | s − f ′ ( u (¯ x )) (cid:19) w λ (¯ x )but thanks to (1.6) (and the fact that C ≤
1) we know that f ′ ( u (¯ x )) ≥ f ′ ( C u (¯ x )) ≥ C g ′ ( C u (¯ x ))so that0 ≤ (cid:18) g ′ ( C u (¯ x )) | λ + 1 | s − f ′ ( u (¯ x )) (cid:19) w λ (¯ x ) ≤ (cid:18) C | λ + 1 | s − (cid:19) g ′ ( C u (¯ x )) w λ (¯ x )and we can choose λ sufficiently close to − (cid:18) C | λ + 1 | s − (cid:19) > w λ ≥ λ for some λ > −
1; next we want to show that λ = 0.Assume the contrary; then the Maximum Principle 1.1 implies that(5.2) w λ > λ . By definition of supremum, the exists a sequence λ j such that λ j ≤ λ j − , λ j ≤ , lim j →∞ λ j = λ and w λ j ( x j ) = min Ω λj w λ j < x j ∈ Ω λ j . We may assume further (up to taking a subsequence if needed)that lim j →∞ x j = x and w λ ( x ) ≤ x ∈ T λ . AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 21
Further, setting δ j := dist( x j , T λ j ) = | x j − z j | for some z j ∈ T λ j the equationgives 1 δ j (cid:16) ( − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) (cid:17) = f ′ ( ξ j ) w λ j ( x j ) δ j = f ′ ( ξ j ) w λ j ( x j ) | x j − z j | . Notice that w λ j ≡ T λ j solim j →∞ w λ j ( x j ) | z j − x j | = lim j →∞ w λ j ( x j ) − w λ j ( z j ) | z j − x j | = lim j →∞ ∇ w λ j ( z j ) · ( z j − x j ) + o ( | z j − x j | ) | z j − x j | = 0which, together with the previous line implieslim j →∞ δ j (cid:16) ( − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) (cid:17) = 0(notice that ξ j is a bounded sequence and hence so is f ′ ( ξ j )).But Proposition 1.5 giveslim sup j →∞ δ j (cid:16) ( − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) (cid:17) < , a contradiction.Therefore, λ = 0; since we can choose the opposite direction to reflect we get thesymmetry of u about the x n direction. Finally, the rotation invariance of ( − ∆ g ) s implies that we can repeat the argument in any direction, so u is symmetric aboutthe origin. (cid:3) Proof of Theorem 1.7.
We split the proof in the two steps in order to apply themoving planes method.
Step 1.
Let us see that for λ sufficiently negative, w λ ( x ) ≥ x ∈ Σ λ .Let us assume the opposite and obtain a contradiction. Due to the decay condi-tion (1.9) on u , there exists ¯ x ∈ Σ λ such that w λ (¯ x ) = min Σ λ w λ < − ∆ g ) s u ( x λ ) − ( − ∆ g ) s u ( x ) = f ( u ( x λ )) − f ( u ( x )) = f ′ ( ξ ) w λ ( x )where ξ lies between u ( x λ ) and u ( x ). In particular, we have u (¯ x λ ) ≤ ξ (¯ x ) ≤ u (¯ x ) . Because of the decay assumption on u , for λ sufficiently negative, u (¯ x ) is small, andthen ξ (¯ x ) is small, giving that f ′ ( ξ (¯ x )) ≤ − ∆ g ) s u ( x λ ) − ( − ∆ g ) s u ( x ) = f ( u ( x λ )) − f ( u ( x )) ≥ . However, as seen in (4.3), under these conditions we have that( − ∆ g ) s u ( x λ ) − ( − ∆ g ) s u ( x ) = f ( u ( x λ )) − f ( u ( x )) < , which is a contradiction. Therefore w λ ( x ) ≥ x ∈ Σ λ for λ sufficientlynegative. Step 2 . Define λ = sup { λ : w µ ( x ) ≥ , x ∈ Σ µ , µ ≤ λ } .Let us see that u is symmetric about the limiting plane T λ , or(5.4) w λ ( x ) ≡ , x ∈ Σ λ . The proof of this fact runs similarly as the second step of the proof of Theorem1.6: suppose that (5.4) does not hold, then by Theorem 1.4 w λ ( x ) > , ∀ x ∈ Σ λ . Observe that, by definition of λ , there is a sequence λ j ց λ and x j ∈ Σ λ j suchthat(5.5) w λ j ( x j ) = min Σ λj w λ j < , and ∇ w λ j ( x j ) = 0 . From condition (1.8) we can guarantee that, up to a subsequence, { x j } k ∈ N con-verges to some ¯ x . In fact, using (1.9), if | x j | is sufficiently large we have that u ( x j )is small and hence ξ λ j ( x j ) is small, which implies that f ′ ( ξ λ j ( x j )) ≤ − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) = f ( u ( x λ j j )) − f ( u ( x j ))= f ′ ( ξ λ j ( x j )) w λ j ( x j ) ≥ , which contradicts that fact that x j is a negative minimum of w λ j since by the anal-ysis derived in (4.3) we should have ( − ∆ g ) s u ( x λ j j ) − ( − ∆ g ) s u ( x j ) <
0. Therefore, { x j } j ∈ N must be bounded.Finally, from (5.5), w λ (¯ x ) ≤ x ∈ T λ , and ∇ w λ (¯ x ) = 0. Then, asin Theorem 1.6 we deduce thatlim j →∞ w λ j ( x j ) δ j = 0 , δ j := dist( x j , Σ λ j )which, in light of (5.3), contradicts Proposition 1.5 and gives the result. (cid:3) Extensions and applications
In this section we present some extensions, applications and further discussionsof our results that we consider to be of interest. We start by pointing out thatwhen the Young function G is given by a power, i.e. G ( t ) = t p p , p ≥ p − Laplacian, thus our results can be considera generalization or extension of these to the nonhomogeneous scenario. Further-more, as mentioned in the Introduction our setting allows for more general growthconditions such as G ( t ) = t p log(1 + t ).Another special type of Young function that falls into the category studied hereis G ( t ) = t p + t q for q > p ≥
2. This structure is closely related to a special type of problems referredto as double phase variational problems where the aim is to study minimizers of thefunctional(6.1) I ( u ) = Z Ω ( |∇ u | p + a ( x ) |∇ u | q ) dx ;these have attracted much interest in the PDE community since the seminal workof Colombo and Mingione [11]. Our work is a step in the direction of obtainingqualitative properties of solution of the fractional analog of (6.1). AXIMUM PRINCIPLES, LIOUVILLE THEOREM AND SYMMETRY RESULTS 23
In a different direction, it is worth to mention that all the results stated in thismanuscript hold true for a more general operators of the form L g,s ( u ) := p.v. Z R n g (cid:18) u ( x ) − u ( y ) k ( | x − y | ) (cid:19) dyk ( | x − y | ) | x − y | n where, for fixed constants c , c > k : R n × R n × R → R is such that c t s ≤ k ( t ) ≤ c t s for any t ≥ local case, that is operators of the formdiv (cid:18) g ( |∇ u | ) ∇ u |∇ u | (cid:19) . In that regard, a rather intriguing question is whether such local results could berecovered as a limit as s → + . Acknowledgements.
This work was partially supported by Consejo Nacional deInvestigaciones Cient´ıficas y T´ecnicas. (CONICET).
References [1] Angela Alberico, Andrea Cianchi, Luboˇs Pick, and Lenka Slav´ıkov´a,
Fractional Orlicz-Sobolevembeddings , Journal de Math´ematiques Pures et Appliqu´ees (2020). 2[2] ,
On the limit as s → + of fractional Orlicz-Sobolev spaces , J. Fourier Anal. Appl. (2020), no. 6, Paper No. 80, 19. MR 4165063 2[3] Sabri Bahrouni and Hichem Ounaies, Embedding theorems in the fractional Orlicz-Sobolevspace and applications to non-local problems , Discrete Contin. Dyn. Syst. (2020), no. 5,2917–2944. MR 4097484 2[4] Sabri Bahrouni and Ariel Salort, Neumann and Robin type boundary conditions in fractionalorlicz-sobolev spaces , ESAIM: COCV, forthcoming article. 2[5] Henri Berestycki and Louis Nirenberg,
Monotonicity, symmetry and antisymmetry of solu-tions of semilinear elliptic equations , J. Geom. Phys. (1988), no. 2, 237–275. MR 10294292[6] , On the method of moving planes and the sliding method , Bol. Soc. Brasil. Mat. (N.S.) (1991), no. 1, 1–37. MR 1159383 2[7] Wenxiong Chen and Congming Li, Maximum principles for the fractional p -Laplacian andsymmetry of solutions , Adv. Math. (2018), 735–758. MR 3836677 2[8] Wenxiong Chen and Leyun Wu, A maximum principle on unbounded domains and a liouvilletheorem for fractional p-harmonic functions , 2019. 2[9] Wenxiong Chen and Jiuyi Zhu,
Indefinite fractional elliptic problem and Liouville theorems ,J. Differential Equations (2016), no. 5, 4758–4785. MR 3437604 2[10] Tingzhi Cheng, Genggeng Huang, and Congming Li,
The maximum principles for fractionalLaplacian equations and their applications , Commun. Contemp. Math. (2017), no. 6,1750018, 12. MR 3691506 2[11] Maria Colombo and Giuseppe Mingione, Regularity for double phase variational problems ,Archive for Rational Mechanics and Analysis (2015), no. 2, 443–496. 22[12] Pablo De N´apoli, Juli´an Fern´andez Bonder, and Ariel Salort,
A P´olya-Szeg¨o principle forgeneral fractional Orlicz-Sobolev spaces , Complex Variables and Elliptic Equations (2020),no. 0, 1–23. 2[13] Juli´an Fern´andez Bonder, Mayte P´erez-Llanos, and Ariel Salort, A H¨older Infinity Laplacianobtained as limit of Orlicz Fractional Laplacians , arXiv preprint arXiv:1807.01669 (2018). 7[14] Juli´an Fern´andez Bonder and Ariel Salort,
Fractional order Orlicz-Sobolev spaces , J. Funct.Anal. (2019), no. 2, 333–367. MR 3952156 2, 9 [15] Juli´an Fern´andez Bonder, Ariel Salort, and Hern´an Vivas,
Interior and up to the boundaryregularity for the fractional g -laplacian: the convex case , arXiv preprint arXiv: 2008.05543(2020). 2[16] Basilis Gidas, Wei-Ming Ni, and Louis Nirenberg, Symmetry and related properties via themaximum principle , Communications in Mathematical Physics (1979), no. 3, 209–243. 2[17] M. A. Krasnosel’ski˘ı and Ja. B. Rutickii, Convex functions and Orlicz spaces , Translated fromthe first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. MR 01267226[18] Gary M. Lieberman,
The natural generalization of the natural conditions of Ladyzhenskayaand Uraltseva for elliptic equations , Comm. Partial Differential Equations (1991), no. 2-3,311–361. MR 1104103 6[19] Roberta Musina and Alexander I. Nazarov, Strong maximum principles for fractional Lapla-cians , Proc. Roy. Soc. Edinburgh Sect. A (2019), no. 5, 1223–1240. MR 4010521 2[20] Ariel Salort,
Eigenvalues and minimizers for a non-standard growth non-local operator , J.Differential Equations (2020), no. 9, 5413–5439. MR 4066053 2[21] Ariel Salort and Hern´an Vivas,
Fractional eigenvalues in orlicz spaces with no ∆ condition ,arXiv preprint arXiv:2005.01847 (2020). 2[22] Guotao Wang, Xueyan Ren, Zhanbing Bai, and Wenwen Hou, Radial symmetry of standingwaves for nonlinear fractional Hardy-Schr¨odinger equation , Appl. Math. Lett. (2019),131–137. MR 3948870 2[23] Leyun Wu and Wenxiong Chen, The sliding methods for the fractional p -Laplacian , Adv.Math. (2020), 106933, 26. MR 4038146 2[24] Lihong Zhang, Bashir Ahmad, Guotao Wang, and Xueyan Ren, Radial symmetry of solutionfor fractional p -Laplacian system , Nonlinear Anal. (2020), 111801, 16. MR 4066751 2[25] Lihong Zhang and Wenwen Hou, Standing waves of nonlinear fractional p -LaplacianSchr¨odinger equation involving logarithmic nonlinearity , Appl. Math. Lett. (2020),106149, 6. MR 4037705 2(SM) Centro Marplatense de Investigaciones matem´aticas, CIC-UNMdP, Mar delPlata, Argentina
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