Self-generated interior blow-up solutions in fractional elliptic equation with absorption
aa r X i v : . [ m a t h . A P ] N ov Self-generated interior blow-up solutions in fractionalelliptic equation with absorption
Huyuan Chen, Patricio FelmerDepartamento de Ingenier´ıa Matem´atica and Centro de ModelamientoMatem´atico UMR2071 CNRS-UChile, Universidad de ChileCasilla 170 Correo 3, Santiago, Chile. ([email protected], [email protected]) andAlexander QuaasDepartamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıaCasilla: V-110, Avda. Espa˜na 1680, Valpara´ıso, Chile ([email protected])
Abstract
In this paper we study positive solutions to problem involving thefractional Laplacian ( − ∆) α u ( x ) + | u | p − u ( x ) = 0 , x ∈ Ω \ C ,u ( x ) = 0 , x ∈ Ω c , lim x ∈ Ω \C , x →C u ( x ) = + ∞ , (0.1)where p > C domain in R N , C ⊂
Ω is a compact C manifold with N − − ∆) α with α ∈ (0 ,
1) is the fractionalLaplacian.We consider the existence of positive solutions for problem (0.1).Moreover, we further analyze uniqueness, asymptotic behaviour andnonexistence.
Key words : Fractional Laplacian, Existence, Uniqueness, Asymptoticbehavior, Blow-up solution.1
Introduction
In 1957, a fundamental contribution due to Keller in [11] and Osserman in[19] is the study of boundary blow-up solutions for the non-linear ellipticequation ( − ∆ u + h ( u ) = 0 in Ω , lim x ∈ Ω ,x → ∂ Ω u ( x ) = + ∞ . (1.1)They proved the existence of solutions to (1.1) when h : R → [0 , + ∞ ) isa locally Lipschitz continuous function which is nondecreasing and satisfiesthe so called Keller-Osserman condition. From then on, the result of Kellerand Osserman has been extended by numerous mathematicians in variousways, weakening the assumptions on the domain, generalizing the differen-tial operator and the nonlinear term for equations and systems. The case of h ( u ) = u p + with p = N +2 N − is studied by Loewner and Nirenberg [15], wherein particular uniqueness and asymptotic behavior were obtained. After that,Bandle and Marcus [2] obtained uniqueness and asymptotic for more gen-eral non-linearties h . Later, Le Gall in [9] established a uniqueness result ofproblem (1.1) in the domain whose boundary is non-smooth when h ( u ) = u .Marcus and V´eron [16, 18] extended the uniqueness of blow-up solution for(1.1) in general domains whose boundary is locally represented as a graph ofa continuous function when h ( u ) = u p + for p >
1. Under this special assump-tion on h , Kim [12] studied the existence and uniqueness of boundary blow-upsolution to (1.1) in bounded domains Ω satisfying ∂ Ω = ∂ ¯Ω. For anotherinteresting contributions to boundary blow-up solutions see for example Kon-dratev, Nikishkin [13], Lazer, McKenna [14], Arrieta and Rodr´ıguez-Bernal[1], Chuaqui, Cort´azar, Elgueta and J. Garc´ıa-Meli´an [4], del Pino and Lete-lier [5], D´ıaz and Letelier [6], Du and Huang [7], Garc´ıa-Meli´an [10], V´eron[20], and the reference therein.In a recent work, Felmer and Quaas [8] considered a version of Keller andOsserman problem for a class of non-local operator. Being more precise, theyconsidered as a particular case the fractional elliptic problem ( − ∆) α u ( x ) + | u | p − u ( x ) = f ( x ) , x ∈ Ω ,u ( x ) = g ( x ) , x ∈ ¯Ω c , lim x ∈ Ω , x → ∂ Ω u ( x ) = + ∞ , (1.2)where p > f and g are appropriate functions and Ω is a bounded domainwith C boundary. The operator ( − ∆) α is the fractional Laplacian which isdefined as ( − ∆) α u ( x ) = − Z R N δ ( u, x, y ) | y | N +2 α dy, x ∈ Ω , (1.3)2ith α ∈ (0 ,
1) and δ ( u, x, y ) = u ( x + y ) + u ( x − y ) − u ( x ).In [8] the authors proved the existence of a solution to (1.2) provided that g explodes at the boundary and satisfies other technical conditions. In casethe function g blows up with an explosion rate as d ( x ) β , with β ∈ [ − αp − , d ( x ) = dist ( x, ∂ Ω), it is shown that the solution satisfies0 < lim inf x ∈ Ω ,x → ∂ Ω u ( x ) d ( x ) − β ≤ lim sup x ∈ Ω ,x → ∂ Ω u ( x ) d ( x ) αp − < + ∞ . Here the explosion is driven by the external value g and the external source f has a secondary role, not intervening in the explosive character of thesolution.More recently, Chen, Felmer and Quaas [3] extended the results in [8]studying existence, uniqueness and non-existence of boundary blow-up so-lutions when the function g vanishes and the explosion on the boundary isdriven by the external source f , with weak or strong explosion rate. More-over, the results are extended even to the case where the boundary blow-upsolutions in driven internally, when the external source and value, f and g ,vanish. Existence, uniqueness, asymptotic behavior and non-existence resultsfor blow-up solutions of (1.2) are considered in [3]. In the analysis developedin [3], a key role is played by the function C : ( − , → R , that governs thebehavior of the solution near the boundary. The function C is defined as C ( τ ) = Z + ∞ χ (0 , ( t ) | − t | τ + (1 + t ) τ − t α dt (1.4)and it possess exactly one zero in ( − ,
0) and we call it τ ( α ). In what followswe explain with more details the results in the case of vanishing externalsource and values, that is f = 0 in Ω and g = 0 in ¯Ω c , which is the case wewill consider in this paper. In Theorem 1.1 in [3], we proved that whenever1 + 2 α < p < − ατ ( α ) , then problem (1.2) admits a unique positive solution u such that0 < lim inf x ∈ Ω ,x → ∂ Ω u ( x ) d ( x ) αp − ≤ lim sup x ∈ Ω ,x → ∂ Ω u ( x ) d ( x ) αp − < + ∞ . On the other hand, we proved that when p ≥ , then problem (1.2) does notadmit any solution u such that0 < lim inf x ∈ Ω ,x → ∂ Ω u ( x ) d ( x ) − τ ≤ lim sup x ∈ Ω ,x → ∂ Ω u ( x ) d ( x ) − τ < + ∞ , (1.5)3or any τ ∈ ( − , \ { τ ( α ) , − αp − } . We observe that the non-existence resultdoes not include the case when u has an asymptotic behavior of the form d ( x ) τ ( α ) , where τ ( α ) is precisely where C vanishes. We have a a specialexistence result in this case, precisely ifmax { − ατ ( α ) + τ ( α ) + 1 τ ( α ) , } < p < − ατ ( α ) , then, for any t >
0, problem (1.2) admits a positive solution u such thatlim x ∈ Ω ,x → ∂ Ω u ( x ) d ( x ) − τ ( α ) = t. Motivated by these results and in view of the non-local character of thefractional Laplacian we are interested in another class of blow-up solutions.We want to study solutions that vanish at the boundary of the domain Ωbut that explodes at the interior of the domain, near a prescribed embeddedmanifold. From now on, we assume that Ω is an open bounded domain in R N with C boundary, and that there is a C , ( N − C without boundary, embedded in Ω, such that, it separates Ω \ C in exactlytwo connected components. We denote by Ω the inner component and by Ω the external component, that is ¯Ω ∩ ∂ Ω = ∅ and ¯Ω ∩ ∂ Ω = ∂ Ω . Throughoutthe paper we will consider the distance function D : Ω \ C → R + , D ( x ) = dist( x, C ) . (1.6)Let us consider the equations, for i = 1 , ( − ∆) α u ( x ) + | u | p − u ( x ) = 0 , x ∈ Ω i ,u ( x ) = 0 , x ∈ ¯Ω ci , lim x ∈ Ω i , x → ∂ Ω i u ( x ) = + ∞ , (1.7)which have solutions u and u , for i = 1 , α = 1, these two solutionscertainly do not interact among each other, but when α ∈ (0 , ( − ∆) α u ( x ) + | u | p − u ( x ) = 0 , x ∈ Ω \ C ,u ( x ) = 0 , x ∈ Ω c , lim x ∈ Ω \C , x →C u ( x ) = + ∞ , (1.8)4here p >
1, Ω and
C ⊂
Ω are as described above. The explosion of thesolution near C is governed by a function c : ( − , → R , defined as c ( τ ) = Z + ∞ | − t | τ + (1 + t ) τ − t α dt. (1.9)This function plays the role of the function C used in [3], but it has certaindifferences. In Section § α ∈ (0 ,
1) suchthat α ∈ [ α ,
1) the function c is always positive in ( − , α ∈ (0 , α )then there exists exists a unique τ ( α ) ∈ ( − ,
0) such that c ( τ ( α )) = 0 and c ( τ ) > − , τ ( α )) and c ( τ ) < τ ( α ) , τ ( α ) > τ ( α ) if α ∈ (0 , α ).Now we are ready to state our main theorems on the existence unique-ness and asymptotic behavior of interior blow-up solutions to equation (1.8).These theorems deal separately the case α ∈ (0 , α ) and α ∈ [ α , Theorem 1.1
Assume that α ∈ (0 , α ) and the assumptions on Ω and C .Then we have: ( i ) If α < p < − ατ ( α ) , (1.10) then problem (1.8) admits a unique positive solution u satisfying < lim inf x ∈ Ω \C ,x →C u ( x ) D ( x ) αp − ≤ lim sup x ∈ Ω \C ,x →C u ( x ) D ( x ) αp − < + ∞ . (1.11)( ii ) If max { − ατ ( α ) + τ ( α ) + 1 τ ( α ) , } < p < − ατ ( α ) . (1.12) Then, for any t > , there is a positive solution u of problem (1.8) satisfying lim x ∈ Ω \C ,x →C u ( x ) D ( x ) − τ ( α ) = t. (1.13)( iii ) If one of the following three conditions holds a) 1 < p ≤ α and τ ∈ ( − , \ { τ ( α ) } , b) 1 + 2 α < p < − ατ ( α ) and τ ∈ ( − , \ { τ ( α ) , − αp − } or c) p ≥ − ατ ( α ) and τ ∈ ( − , ,then problem (1.8) does not admit any solution u satisfying < lim inf x ∈ Ω \C ,x →C u ( x ) D ( x ) − τ ≤ lim sup x ∈ Ω \C ,x →C u ( x ) D ( x ) − τ < + ∞ . (1.14)5e observe that this theorem is similar to Theorem 1.1 in [3], where therole of τ ( α ) is played here by τ ( α ). A quite different situation occurs when α ∈ [ α ,
1) and the function c never vanishes in ( − , Theorem 1.2
Assume that α ∈ [ α , and the assumptions on Ω and C .Then we have: ( i ) If p > α , then problem (1.8) admits a unique positive solution u satisfying (1.11). ( ii ) If p > , then problem (1.8) does not admit any solution u satisfying(1.14) for any τ ∈ ( − , \ {− αp − } . Comparing Theorem 1.1 with Theorem 1.2 we see that the range of exis-tence for the absorption term is quite larger for the second one, no constraintfrom above. The main difference with the results in [3], Theorem 1.1, withvanishing f and g occurs when α is large and the function c does not vanish,allowing thus for existence for all p large. This difference comes from the factthat the fractional Laplacian is a non-local operator so that in the interiorblow-up, in each of the domains Ω and Ω there is a non-zero external value,the solutions itself acting on the other side of C .The proof of our theorems is obtained through the use of super and sub-solutions as in [3]. The main difficulty here is to find the appropriate superand sub-solutions to apply the iteration technique to fractional elliptic prob-lem (1.8). Here we make use of some precise estimates based on the function c and the distance function D near C .This article is organized as follows. In section §
2, we introduce some pre-liminaries and we prove the main estimates of the behavior of the fractionalLaplacian when applied to suitable powers of the function D . In section § § In this section, we recall some basic results from [3] and obtain some use-ful estimate, which will be used in constructing super and sub-solutions ofproblem (1.8). The first result states as:
Theorem 2.1
Assume that p > and there are super-solution ¯ U and sub-solution U of problem (1.8) such that ¯ U ≥ U in Ω \ C , lim inf x ∈ Ω \C ,x →C U ( x ) = + ∞ , ¯ U = U = 0 in Ω c . hen problem (1.8) admits at least one positive solution u such that U ≤ u ≤ ¯ U in Ω \ C . Proof.
The procedure is similar to the proof of Theorem 2.6 in [3], here wegive the main differences.Let us define Ω n := { x ∈ Ω | D ( x ) > /n } then we solve ( − ∆) α u n ( x ) + | u n | p − u n ( x ) = 0 , x ∈ Ω n ,u n ( x ) = U , x ∈ Ω cn . (2.1)To find these solutions of (2.1) we observe that for fix n the method ofsection 3 of [8] applies even if the domain is not connected since the estimateof Lemma 3.2 holds with δ < / n (see also Proposition 3.2 part ii) in [3]),form here sub and super-solution can be construct for the Dirichlet problemand then existence holds for (2.1) by an iteration technique (see also section2 of [3] for that procedure). Then as in Theorem 2.6 in [3] we have U ≤ u n ≤ u n +1 ≤ ¯ U in Ω . By monotonicity of u n , we can define u ( x ) := lim n → + ∞ u n ( x ) , x ∈ Ω and u ( x ) := 0 , x ∈ Ω c . Which, by a stability property, is a solution of problem (1.8) with the desiredproperties. (cid:3)
In order to prove our existence result, it is crucial to have available superand sub-solutions to problem (1.8). To this end, we start describing theproperties of c ( τ ) defined in (1.9), which is a C function in ( − , Proposition 2.1
There exists a unique α ∈ (0 , such that ( i ) For α ∈ [ α , , we have c ( τ ) > , for all τ ∈ ( − , ii ) For any α ∈ (0 , α ) , there exists unique τ ( α ) ∈ ( − , satisfying c ( τ ) > , if τ ∈ ( − , τ ( α )) , = 0 , if τ = τ ( α ) ,< , if τ ∈ ( τ ( α ) ,
0) (2.2) and lim α → α − τ ( α ) = 0 and lim α → + τ ( α ) = − . (2.3) Moreover, τ ( α ) > τ ( α ) , for all α ∈ (0 , α ) , where τ ( α ) ∈ ( − , is theunique zero of C ( τ ) , defined in (1.4). roof. From (1.9), differentiating twice we find that c ′′ ( τ ) = Z + ∞ | − t | τ (log | − t | ) + (1 + t ) τ (log(1 + t )) t α dt > , (2.4)so that c is strictly convex in ( − , c (0) = 0 and lim τ →− + c ( τ ) = ∞ . (2.5)Thus, if c ′ (0) ≤ c ( τ ) > τ ∈ ( − ,
0) and if c ′ (0) >
0, thenthere exists τ ( α ) ∈ ( − ,
0) such that c ( τ ) > τ ∈ ( − , τ ( α )), c ( τ ) < τ ∈ ( τ ( α ) ,
0) and c ( τ ( α )) = 0. In order to complete our proof, we haveto analyze the sign of c ′ (0), which depends on α and to make this dependenceexplicit, we write c ′ (0) = T ( α ). We compute T ( α ) from (1.9), differentiatingand evaluating in τ = 0 T ( α ) = Z + ∞ log | − t | t α dt. (2.6)We have to prove that T possesses a unique zero in the interval (0 , α → − T ( α ) = −∞ and lim α → + T ( α ) = + ∞ . (2.7)The first limit follows from the fact that log(1 − s ) ≤ − s, for all s ∈ [0 , / α → − Z log(1 − t ) t α dt ≤ − lim α → − Z t − α dt = −∞ and the fact that exists a constant t such that Z + ∞ log | − t | t α dt ≤ t , for all α ∈ (1 / , . The second limit in (2.7) follows fromlim α → + Z + ∞ log | − t | t α dt ≥ log 3 lim α → + Z + ∞ t − − α dt = + ∞ and the fact that there exists a constant t such that Z log | − t | t α dt ≤ t , for all α ∈ (0 , / .
8n the other hand we claim that T ′ ( α ) = − Z + ∞ log | − t | t α log tdt < , α ∈ (0 , . (2.8)In fact, since log | − t | log t is negative only for t ∈ (1 , √ Z + ∞ log | − t | t α log tdt > Z √ − log(1 − t ) t α log tdt + Z √ log( t −
1) log tdt ≥ Z √ − − t t α log tdt + Z √ log( t −
1) log tdt = − Z √ − t − α log tdt + Z √ − log(1 + t ) log tdt ≥ − Z √ − t − α log tdt + Z √ − t log tdt > . Then, (2.7) and (2.8) the existence of the desired α ∈ (0 ,
1) with the requiredproperties follows, completing ( i ) and (2.2) in ( ii ).To continue with the proof of our proposition, we study the first limit in(2.3). We assume that there exist a sequence α n ∈ (0 , α ) and ˜ τ ∈ ( − , n → + ∞ α n = α and lim n → + ∞ τ ( α n ) = ˜ τ and so c (˜ τ ) = 0. Moreover c (0) = 0 and c ′ (0) = T ( α ) = 0, contradictingthe strict convexity of c given by (2.4). Next we prove the second limit in(2.3). We proceed by contradiction, assuming that there exist a sequence { α n } ⊂ (0 ,
1) and ¯ τ ∈ ( − ,
0) such thatlim n → + ∞ α n = 0 and τ ( α n ) ≥ ¯ τ > − , for all n ∈ N . Then there exist C , C >
0, depending on ¯ τ , such that Z | | − t | τ ( α n ) + (1 + t ) τ ( α n ) − t α n | dt ≤ C andlim n →∞ Z + ∞ | − t | τ ( α n ) + (1 + t ) τ ( α n ) − t α n dt ≤ − C lim n →∞ Z + ∞ t α n dt = −∞ . Then c ( τ ( α n )) → −∞ as n → + ∞ , which is impossible since c ( τ ( α n )) = 0 .
9e finally prove the last statement of the proposition. Since τ ( α ) ∈ ( − ,
0) is such that C ( τ ( α )) = 0 and we have, by definition, that c ( τ ) = C ( τ ) + Z + ∞ ( t − τ t α dt, we find that c ( τ ( α )) >
0, which together with (2.2), implies that τ ( α ) ∈ ( − , τ ( α )) . (cid:3) Next we prove the main proposition in this section, which is on the basisof the construction of super and sub-solutions. By hypothesis on the domainΩ and the manifold C , there exists δ > d ( · ), to ∂ Ω, and D ( · ), to C , are of class C in B δ and A δ , respectively, and dist ( A δ , B δ ) >
0, where A δ = { x ∈ Ω | D ( x ) < δ } and B δ = { x ∈ Ω | d ( x ) <δ } . Now we define the basic function V τ as follows V τ ( x ) := D ( x ) τ , x ∈ A δ \ C ,d ( x ) , x ∈ B δ ,l ( x ) , x ∈ Ω \ ( A δ ∪ B δ ) , , x ∈ Ω c , (2.9)where τ is a parameter in ( − ,
0) and the function l is positive such that V τ is of class C in R N \ C . Proposition 2.2
Let α and τ ( α ) be as in Proposition 2.1. ( i ) If ( α, τ ) ∈ [ α , × ( − , or ( α, τ ) ∈ (0 , α ) × ( − , τ ( α )) , then thereexist δ ∈ (0 , δ ] and C > such that C D ( x ) τ − α ≤ − ( − ∆) α V τ ( x ) ≤ CD ( x ) τ − α , x ∈ A δ \ C . ( ii ) If ( α, τ ) ∈ (0 , α ) × ( τ ( α ) , , then there exist δ ∈ (0 , δ ] and C > suchthat C D ( x ) τ − α ≤ ( − ∆) α V τ ( x ) ≤ CD ( x ) τ − α , x ∈ A δ \ C . ( iii ) If ( α, τ ) ∈ (0 , α ) × { τ ( α ) } , then there exist δ ∈ (0 , δ ] and C > suchthat | ( − ∆) α V τ ( x ) | ≤ CD ( x ) min { τ, τ − α +1 } , x ∈ A δ \ C . This proposition and its proof has many similarities with Proposition 3.2in [3], but it has also important differences so we give a complete proof of it.10 roof.
By compactness of C , we just need to prove that the correspondinginequality holds in a neighborhood of any point ¯ x ∈ C and, without loss ofgenerality, we may assume ¯ x = 0. For a given 0 < η ≤ δ , we define Q η = ( − η, η ) × B η ⊂ R × R N − , where B η denotes the ball centered at the origin and with radius η in R N − .We observe that Q η ⊂ Ω . Let ϕ : R N − → R be a C function such that( z , z ′ ) ∈ C ∩ Q δ if and only if z = ϕ ( z ′ ). We further assume that e isnormal to C at ¯ x and then there exists C > | ϕ ( z ′ ) | ≤ C | z ′ | for | z ′ | ≤ δ . Thus, choosing η > | ϕ ( z ′ ) | < η for | z ′ | ≤ η . In the proof of our inequalities, we will consider ageneric point along the normal x = ( x , ∈ A η/ , with 0 < | x | < η/
4. Weobserve that | x − ¯ x | = D ( x ) = | x | . By definition we have − ( − ∆) α V τ ( x ) = 12 Z Q η δ ( V τ , x, y ) | y | N +2 α dy + 12 Z R N \ Q η δ ( V τ , x, y ) | y | N +2 α dy. (2.10)It is not difficult to see that the second integral is bounded by Cx τ , for anappropriate constant C >
0, so that we only need to study the first integral,that from now on we denote by E ( x ).Our first goal is to obtain positive constants c , c so that lower boundfor E ( x ) E ( x ) ≥ c c ( τ ) | x | τ − α − c | x | min { τ, τ − α +1 } (2.11)holds, for all | x | ≤ η/
4. For this purpose we assume that 0 < η ≤ δ/
2, thenfor all y = ( y , y ′ ) ∈ Q η we have that x ± y ∈ Q δ , so that D ( x ± y ) ≤ | x ± y − ϕ ( ± y ′ ) | , for all y ∈ Q η . From here and the fact that τ ∈ ( − , E ( x ) = Z Q η δ ( V τ , x, y ) | y | N +2 α dy ≥ Z Q η I ( y ) | y | N +2 α dy + Z Q η J ( y ) + J ( − y ) | y | N +2 α dy, (2.12)where the functions I and J are defined, for y ∈ Q η , as I ( y ) = | x − y | τ + | x + y | τ − x τ (2.13)and J ( y ) = | x + y − ϕ ( y ′ ) | τ − | x + y | τ . (2.14)In what follows we assume x > x < Z Q η I ( y ) | y | N +2 α dy = x τ − α Z Q ηx | − z | τ + | z | τ − | z | N +2 α dz.
11n one hand we have that, for a constant c , we have Z R N | − z | τ + | z | τ − | z | N +2 α dz = 2 c ( τ ) Z R N − | z ′ | + 1) N +2 α dz ′ = c c ( τ ) , and, on the other hand, for constants C and C we have | Z ηx − ηx Z | z ′ |≥ ηx | − z | τ + | z | τ − | z | N +2 α dz |≤ Z ηx − ηx ( | − z | τ + | z | τ + 2) dz Z | z ′ |≥ ηx dz ′ | z ′ | N +2 α ≤ C x α and | Z | z |≥ ηx Z R N − | − z | τ + | z | τ − | z | N +2 α dz |≤ Z + ∞ ηx | − z | τ + | z | τ + 2 z α dz Z R N − | z ′ | ) N +2 α dz ′ ≤ C x α . Consequently, for an appropriate constant c | Z Q η I ( y ) | y | N +2 α dy − c c ( τ ) x τ − α | ≤ c x τ . (2.15)Next we estimate the second term of the right hand side in (2.12). Since Z Q η J ( − y ) | y | N +2 α dy = Z Q η J ( y ) | y | N +2 α dy, we only need to estimate Z Q η J ( y ) | y | N +2 α dy = Z B η Z η − η | x + y − ϕ ( y ′ ) | τ − | x + y | τ ( y + | y ′ | ) N +2 α dy dy ′ . (2.16)We notice that | x + y − ϕ ( y ′ ) | ≥ | x + y | if and only if ϕ ( y ′ )( x + y − ϕ ( y ′ )2 ) ≤ . From here and (2.16), we have Z Q η J ( y ) | y | N +2 α dy ≥ Z B η Z − x + ϕ +( y ′ )2 − η | x + y − ϕ + ( y ′ ) | τ − | x + y | τ ( y + | y ′ | ) N +2 α dy dy ′ + Z B η Z η − x + ϕ − ( y ′ )2 | x + y − ϕ − ( y ′ ) | τ − | x + y | τ ( y + | y ′ | ) N +2 α dy dy ′ = E ( x ) + E ( x ) , ϕ + ( y ′ ) = max { ϕ ( y ′ ) , } and ϕ − ( y ′ ) = min { ϕ ( y ′ ) , } . We only estimate E ( x ) ( E ( x ) is similar). Using integration by parts, we obtain E ( x )= Z B η Z ϕ +( y ′ )2 x − η | y − ϕ + ( y ′ ) | τ − | y | τ (( y − x ) + | y ′ | ) N +2 α dy dy ′ = Z B η Z x − η ( ϕ + ( y ′ ) − y ) τ − ( − y ) τ (( y − x ) + | y ′ | ) N +2 α dy dy ′ + Z B η Z ϕ +( y ′ )2 ( ϕ + ( y ′ ) − y ) τ − y τ (( y − x ) + | y ′ | ) N +2 α dy dy ′ = 1 τ + 1 Z B η [ − ϕ + ( y ′ ) τ +1 ( x + | y ′ | ) N +2 α + ( η − x + ϕ + ( y ′ )) τ +1 − ( η − x ) τ +1 ( η + | y ′ | ) N +2 α ] dy ′ − N + 2 ατ + 1 Z B η Z x − η ( ϕ + ( y ′ ) − y ) τ +1 − ( − y ) τ +1 (( y − x ) + | y ′ | ) N +2 α +1 ( y − x ) dy dy ′ + 1 τ + 1 Z B η [ − − τ ϕ + ( y ′ ) τ +1 (( ϕ + ( y ′ )2 − x ) + | y ′ | ) N +2 α + ϕ + ( y ′ ) τ +1 ( x + | y ′ | ) N +2 α ] dy ′ + N + 2 ατ + 1 Z B η Z ϕ +( y ′ )2 ( ϕ + ( y ′ ) − y ) τ +1 + y τ +11 (( y − x ) + | y ′ | ) N +2 α +1 ( y − x ) dy dy ′ ≥ − − τ τ + 1 Z B η ϕ + ( y ′ ) τ +1 (( ϕ + ( y ′ )2 − x ) + | y ′ | ) N +2 α dy ′ + N + 2 ατ + 1 Z B η Z min { ϕ +( y ′ )2 ,x } ( ϕ + ( y ′ ) − y ) τ +1 + y τ +11 (( y − x ) + | y ′ | ) N +2 α +1 ( y − x ) dy dy ′ = A ( x ) + A ( x ) . (2.17)In order to estimate A ( x ), we split B η in O = { y ′ ∈ B η : | ϕ + ( y ′ )2 − x | ≥ x } and B η \ O . On one hand we have Z O | y ′ | τ +2 (( ϕ + ( y ′ )2 − x ) + | y ′ | ) N +2 α dy ′ ≤ x τ − α +11 Z B η/x | z ′ | τ +2 (1 / | z ′ | ) N +2 α dz ′ ≤ C ( x τ − α +11 + x τ ) . On the other hand, for y ′ ∈ B η \ O we have that | y ′ | ≥ c √ x , for someconstant c , and then Z B η \ O | y ′ | τ +2 (( ϕ + ( y ′ )2 − x ) + | y ′ | ) N +2 α dy ′ ≤ Z B η \ B c √ x | y ′ | τ +2 − N − α dy ′ C ( x τ − α + + 1) . Thus, for some
C > A ( x ) ≥ − Cx min { τ, τ − α +1 } . (2.18)Next we estimate A ( x ): A ( x ) ≥ N + 2 α ) τ + 1 Z B η Z x ϕ + ( y ′ ) τ +1 ( y − x )(( y − x ) + | y ′ | ) N +2 α +1 dy dy ′ ≥ C Z B η Z x | y ′ | τ +2 ( y − x )(( y − x ) + | y ′ | ) N +2 α +1 dy dy ′ ≥ Cx τ − α +11 Z B η/x Z | z ′ | τ +2 ( z − z − + | z ′ | ) N +2 α +1 dz dz ′ ≥ − C x min { τ, τ − α +1 } , for some C, C >
0. From here, (2.17) and (2.18) we obtain, for some
C > E ( x ) ≥ − Cx min { τ, τ − α +1 } . Using the similar estimate for E ( x ), we obtain Z Q η J ( y ) + J ( − y ) | y | N +2 α dy ≥ − Cx min { τ, τ − α +1 } . (2.19)Thus, from (2.12), (2.15), (2.19) and noticing that these inequalities alsohold with x < E ( x ) we gave in (2.11). Our second goal is to get an upper bound for E ( x )and for this, we first recall Lemma 3.1 in [3] to obtain D ( x ± y ) τ ≤ ( x ± y − ϕ ( y ′ )) τ (1+ C | y ′ | ) , for all | x | ≤ η/ , y = ( y , y ′ ) ∈ Q η . From here we see that E ( x ) ≤ Z Q η I ( y ) | y | N +2 α dy + Z Q η J ( y ) + J ( − y ) | y | N +2 α dy + C Z Q η I ( y ) + J ( y ) + J ( − y ) | y | N +2 α | y ′ | dy. (2.20)We denote by E ( x ) the third integral above. The first integral was studiedin (2.15), so we study the second integral and that we only need to consider14he term J ( y ), since the other is completely analogous. We see that | x + y − ϕ ( y ′ ) | ≤ | x + y | if and only if ϕ ( y ′ )( x + y − ϕ ( y ′ )2 ) ≥ . As before, we will consider only the case x >
0, since the other one isanalogous. From (2.16) we have Z Q η J ( y ) | y | N +2 α dy ≤ Z B η Z − x + ϕ − ( y ′ )2 − η | x + y − ϕ − ( y ′ ) | τ − | x + y | τ ( y + | y ′ | ) N +2 α dy dy ′ + Z B η Z η − x + ϕ +( y ′ )2 | x + y − ϕ + ( y ′ ) | τ − | x + y | τ ( y + | y ′ | ) N +2 α dy dy ′ = F ( x ) + F ( x ) . Next we estimate F ( x ) ( F ( x ) is similar), using integration by parts F ( x )= Z B η Z ϕ − ( y ′ )2 x − η | y − ϕ − ( y ′ ) | τ − | y | τ (( y − x ) + | y ′ | ) N +2 α dy dy ′ = Z B η Z ϕ − ( y ′ ) x − η ( ϕ − ( y ′ ) − y ) τ − ( − y ) τ (( y − x ) + | y ′ | ) N +2 α dy dy ′ + Z B η Z ϕ − ( y ′ )2 ϕ − ( y ′ ) ( y − ϕ − ( y ′ )) τ − ( − y ) τ (( y − x ) + | y ′ | ) N +2 α dy dy ′ = 1 τ + 1 Z B η [ ( − ϕ − ( y ′ )) τ +1 (( x − ϕ − ( y ′ )) + | y ′ | ) N +2 α + ( η − x + ϕ − ( y ′ )) τ +1 − ( η − x ) τ +1 ( η + | y ′ | ) N +2 α ] dy ′ − N + 2 ατ + 1 Z B η Z ϕ − ( y ′ ) x − η ( ϕ − ( y ′ ) − y ) τ +1 − ( − y ) τ +1 (( y − x ) + | y ′ | ) N +2 α +1 ( y − x ) dy dy ′ + 1 τ + 1 Z B η [ 2 − τ ( − ϕ − ( y ′ )) τ +1 (( ϕ − ( y ′ )2 − x ) + | y ′ | ) N +2 α + − ( − ϕ − ( y ′ )) τ +1 (( x − ϕ − ( y ′ )) + | y ′ | ) N +2 α ] dy ′ + N + 2 ατ + 1 Z B η Z ϕ − ( y ′ )2 ϕ − ( y ′ ) ( y − ϕ − ( y ′ )) τ +1 + ( − y ) τ +1 (( y − x ) + | y ′ | ) N +2 α +1 ( y − x ) dy dy ′ ≤ τ + 1 Z B η − τ ( − ϕ − ( y ′ )) τ +1 (( ϕ − ( y ′ )2 − x ) + | y ′ | ) N +2 α dy ′ = B ( x ) . Since ( ϕ − ( y ′ )2 − x ) ≥ x , we have B ( x ) ≤ − τ τ + 1 Z B η ( − ϕ − ( y ′ )) τ +1 ( x + | y ′ | ) N +2 α dy ′ C Z B η | y ′ | τ +2 ( x + | y ′ | ) N +2 α dy ′ ≤ Cx min { τ, τ − α +1 } , for some C > x . Thus we have obtained that F ( x ) ≤ Cx min { τ, τ − α +1 } . (2.21)Similarly, we can get an analogous estimate for F ( x ) and these two esti-mates imply Z Q η J ( y ) + J ( − y ) | y | N +2 α dy ≤ Cx min { τ, τ − α +1 } . (2.22)Finally we obtain Z Q η I ( y ) | y | N +2 α | y ′ | dy = x τ − α +21 Z Q ηx | − z | τ + | z | τ − | z | N +2 α | z ′ | dz ≤ Cx min { τ,τ − α +2 } and, in a similar way, Z Q η J ( y ) | y ′ | | y | N +2 α dy ≤ Cx min { τ, τ − α +1 } . From the last two inequalities we obtain E ( x ) ≤ Cx min { τ, τ − α +1 } . (2.23)Then, taking into account (2.20), (2.15), (2.22), (2.23) and considering alsothe case x <
0, we obtain E ( x ) ≤ c c ( τ ) | x | τ − α + c | x | min { τ, τ − α +1 } . (2.24)From inequalities (2.11), (2.24) and Proposition 2.1 the result follows. (cid:3) This section is devoted to use Proposition 2.2 to prove the existence of solu-tion of problem (1.8). To this purpose, our main goal is to construct appro-priate sub-solution and super-solution of problem (1.8) under the hypothesesof Theorem 1.1 ( i ), ( ii ) and Theorem 1.2 ( i ).We begin with a simple lemma that reduces the problem to find themonly in A δ \ C . 16 emma 3.1 Let U and W be classical ordered super and sub-solution of(1.8) in the sub-domain A δ \ C . Then there exists λ large such that U λ = U + λ ¯ V and W λ = W − λ ¯ V , are ordered super and sub-solution of (1.8),where ¯ V is the solution of ( ( − ∆) α ¯ V ( x ) = 1 , x ∈ Ω , ¯ V ( x ) = 0 , x ∈ Ω c . (3.1) Remark 3.1
Here
U, W : IR N → R are classical ordered of super and sub-solution of (1.8) in the sub-domain A δ \ C if U satisfies ( − ∆) α U + | U | p − U ≥ in A δ \ C and W satisfies the reverse inequality. Moreover, they satisfy U ≥ W in Ω \ C , lim inf x ∈ Ω \C ,x →C W ( x ) = + ∞ , U = W = 0 in Ω c . Proof.
Notice that by the maximum principle ¯ V is nonnegative in Ω, there-fore U λ ≥ U and W λ ≤ W , so they are still ordered. In addition U λ satisfies( − ∆) α U λ + | U λ | p − U λ ≥ ( − ∆) α U + | U | p − U + λ > , in Ω \ C . This inequality holds because of our assumption in A δ \ C and the fact that( − ∆) α U + | U | p − U is continuous in Ω \ A δ and by taking λ large enough.By the same type of arguments we find that W λ is a sub-solution. (cid:3) Proof of existence results in Theorem 1.1 ( i ) and Theorem 1.2 ( i ) . We define U µ ( x ) = µV τ ( x ) and W µ ( x ) = µV τ ( x ) , x ∈ R N \ C , (3.2)where V τ is defined in (2.9) with τ = − αq − U µ is Super-solution. By hypotheses of Theorem 1.1 ( i ) and Theo-rem 1.2 ( i ), we notice that τ ∈ ( − , , for α ∈ [ α , ,τ ∈ ( − , τ ( α )) , for α ∈ (0 , α )and τ p = τ − α , then we use Proposition 2.2 part ( i ) to obtain that thereexist δ ∈ (0 , δ ] and C > − ∆) α U µ ( x ) + U pµ ( x ) ≥ − CµD ( x ) τ − α + µ p D ( x ) τp , x ∈ A δ \ C . µ > µ ≥ µ , we have( − ∆) α U µ ( x ) + U pµ ( x ) ≥ , x ∈ A δ \ C . W µ is Sub-solution. We use Proposition 2.2 part ( i ) to obtain thatthere exist δ ∈ (0 , δ ] and C > x ∈ A δ \ C , we have( − ∆) α W µ ( x ) + | W µ | p − W µ ( x ) ≤ − µC D ( x ) τ − α + µ p D ( x ) τp ≤ ( − µC + µ p ) D ( x ) τ − α . Then there exists µ ∈ (0 ,
1) such that for all µ ∈ (0 , µ ), it has( − ∆) α W µ ( x ) + | W µ | p − W µ ( x ) ≤ , x ∈ A δ \ C . To conclude the proof we use Lemma 3.1 and Proposition 2.2. (cid:3)
Proof of Theorem 1.1 ( ii ) . For any given t >
0, we denote U ( x ) = tV τ ( α ) ( x ) , x ∈ R N \ C ,W µ ( x ) = tV τ ( α ) ( x ) − µV ¯ τ ( x ) , x ∈ R N \ C where ¯ τ = min { τ ( α ) p + 2 α, τ ( α ) } <
0. By (1.12), we have¯ τ ∈ ( τ ( α ) , , ¯ τ − α < min { τ ( α ) , τ ( α ) − α + 1 } and ¯ τ − α < τ ( α )p . (3.3) U is Super-solution. We use Proposition 2.2 ( iii ) to obtain that forany x ∈ A δ \ C ,( − ∆) α U ( x ) + U p ( x ) ≥ − CtD ( x ) min { τ ( α ) , τ ( α ) − α +1 } + t p D ( x ) τ ( α ) p , together with τ ( α ) p < min { τ ( α ) , τ ( α ) − α + 1 } , then there exists δ ∈ (0 , δ ] such that ( − ∆) α U ( x ) + U p ( x ) ≥ , x ∈ A δ \ C . W µ is Sub-solution. We use Proposition 2.2 ( ii ) and ( iii ) to obtain thatfor x ∈ A δ \ C ,( − ∆) α W µ ( x ) + | W µ | p − W µ ( x ) ≤ CtD ( x ) min { τ ( α ) , τ ( α ) − α +1 } − µC D ( x ) ¯ τ − α + t p D ( x ) τ ( α ) p . Then there exists δ ∈ (0 , δ ] such that for any µ ≥
1, we have( − ∆) α W µ ( x ) + | W µ | p − W µ ( x ) ≤ , x ∈ A δ \ C . To conclude the proof we use Lemma 3.1 and Proposition 2.2. (cid:3) Uniqueness and nonexistence
We prove the uniqueness statement by contradiction. Assume that u and v are solutions of problem (1.8) satisfying (1.11). Then there exist C ≥ δ ∈ (0 , δ ) such that1 C ≤ v ( x ) D ( x ) − τ , u ( x ) D ( x ) − τ ≤ C , ∀ x ∈ A ¯ δ \ C , (4.4)where τ = − αp − . We denote A = { x ∈ Ω \ C | u ( x ) > v ( x ) } . (4.5)Then A is open and A ⊂
Ω. Then the uniqueness in Theorem 1.2 ( i ) andTheorem 1.1 ( i ) is a consequence of the following result: Proposition 4.1
Under the hypotheses of Theorem 1.2 ( i ) and Theorem 1.1 ( i ) , we have A = Ø . Proof.
The procedure of proof is similar as Section § d ( x ) by D ( x ) and ∂ Ω by C . (cid:3) From Proposition 4.1, we can prove uniqueness part in Theorem 1.1 ( i )and Theorem 1.2 ( i ) .The final goal in this note is to consider the nonexistence of solutions ofproblem (1.8) under the hypotheses of Theorem 1.1 ( iii ) and Theorem 1.2( ii ). Proposition 4.2
Under the hypotheses of Theorem 1.1 ( iii ) and Theorem1.2 ( ii ) , we assume that U and U are both sub-solutions (or both super-solutions) of (1.8) satisfying that U = U = 0 in Ω c and < lim inf x ∈ Ω \C , x →C U ( x ) D ( x ) − τ ≤ lim sup x ∈ Ω \C , x →C U ( x ) D ( x ) − τ < lim inf x ∈ Ω \C , x →C U ( x ) D ( x ) − τ ≤ lim sup x ∈ Ω \C , x →C U ( x ) D ( x ) − τ < + ∞ , for τ ∈ ( − , . For the case τ p > τ − α , we further assume that ( i ) if U , U are sub-solutions, there exist C > and ˜ δ > , ( − ∆) α U ( x ) ≤ − CD ( x ) τ − α , x ∈ A ˜ δ \ C ; (4.6) or ( ii ) if U , U are super-solutions, there exist C > and ˜ δ > , ( − ∆) α U ( x ) ≥ CD ( x ) τ − α , x ∈ A ˜ δ \ C . (4.7)19 hen there doesn’t exist any solution u of (1.8) such that lim sup x ∈ Ω \C , x →C U ( x ) u ( x ) < < lim inf x ∈ Ω \C , x →C U ( x ) u ( x ) . (4.8) Proof.
The proof is similar as Proposition 6.1 in [3], noting again that weneed to replace d ( x ) by D ( x ) and ∂ Ω by C . (cid:3) With the help of Proposition 2.2, for given t > t >
0, we construct twosub-solutions (or both super-solutions) U and U of (1.8) such thatlim x ∈ Ω \C ,x →C U ( x ) D ( x ) − τ = t , lim x ∈ Ω \C ,x →C U ( x ) D ( x ) − τ = t . So what we have to do is to prove that for any t >
0, we can constructsuper-solution (sub-solution) of problem (1.8).
Proof of Theorem 1.1 ( iii ) and Theorem 1.2 ( ii ) . We divide our proofof the nonexistence results into several cases under the assumption p > Zone 1:
We consider τ ∈ ( τ ( α ) ,
0) and α ∈ (0 , α ) . By Proposition 2.2 ( ii ),there exists δ > − ∆) α V τ ( x ) ≥ C D ( x ) τ − α , x ∈ A δ \ C . (4.9)Since V τ is C in Ω \ C , then there exists C > | ( − ∆) α V τ ( x ) | ≤ C, x ∈ Ω \ A δ . (4.10)Let ¯ U := V τ + C ¯ V in R N \ C , then we have ¯ U > \ C ,( − ∆) α ¯ U ≥ \ C and ( − ∆) α ¯ U ( x ) ≥ C D ( x ) τ − α , x ∈ A δ \ C . Then, we have that t ¯ U is super-solution of (1.8) for any t >
0. Using Propo-sition 4.2, we see that there is no solution of (1.8) satisfying (1.14).
Zone 2:
We consider τ − α < τ p and τ ∈ ( ( − , , α ∈ [ α , , ( − , τ ( α )) , α ∈ (0 , α ) . Let us define W µ,t = tV τ − µ ¯ V in R N \ C , where t, µ >
0. By Proposition 2.2 ( i ), for x ∈ A δ \ C ,( − ∆) α W µ,t ( x ) + | W µ,t | p − W µ,t ( x ) ≤ − tC D ( x ) τ − α + t p D ( x ) τp . t >
0, there exists δ ∈ (0 , δ ], for all µ ≥ − ∆) α W µ,t ( x ) + | W µ,t | p − W µ,t ( x ) ≤ , A δ \ C . (4.11)To consider x ∈ Ω \ A δ , in fact, there exists C > t | ( − ∆) α V τ ( x ) | + t p V pτ ( x ) ≤ C , x ∈ Ω \ A δ and ( − ∆) α W µ,t ( x ) + | W µ,t | p − W µ,t ( x ) ≤ C t − µ, x ∈ Ω \ A δ For given t >
0, there exists µ ( t ) > − ∆) α W µ ( t ) ,t ( x ) + | W µ,t | p − W µ ( t ) ,t ( x ) ≤ , x ∈ Ω \ A δ . (4.12)Therefore, together with (4.11) and (4.12), for any given t >
0, there sub-solutions W µ ( t ) ,t of problem (1.8) and by Proposition 4.2, we see that thereis no solution u of (1.8) satisfying (1.14). Zone 3:
We consider τ − α > τ p and τ ∈ ( ( − , , α ∈ [ α , , ( − , τ ( α )) , α ∈ (0 , α ) . We denote that U µ,t = tV τ + µ ¯ V in R N \ C , where t, µ >
0. Here U µ,t > \ C . By Proposition 2.2 ( i ),( − ∆) α U µ,t ( x ) + U pµ,t ( x ) ≥ − CtD ( x ) τ − α + t p D ( x ) τp , x ∈ A δ \ C . For any fixed t >
0, there exists δ ∈ (0 , δ ], for all µ ≥ − ∆) α U µ,t ( x ) + U pµ,t ( x ) ≥ , x ∈ A δ \ C . (4.13)For x ∈ Ω \ A δ , we see that ( − ∆) α V τ is bounded and( − ∆) α U µ,t ( x ) + U pµ,t ( x ) ≥ − Ct + µ. For given t >
0, there exists µ ( t ) > − ∆) α U µ ( t ) ,t ( x ) + U pµ ( t ) ,t ( x ) ≥ , x ∈ Ω \ A δ . (4.14)21ombining with (4.13) and (4.14), we have that for any t >
0, there exists µ ( t ) > − ∆) α U µ ( t ) ,t ( x ) + U pµ ( t ) ,t ( x ) ≥ , x ∈ Ω \ C . Therefore, for any given t >
0, there is a super-solution U µ ( t ) ,t of problem (1.8)and by Proposition 4.2, we see that there is no solution of (1.8) satisfying(1.14).We see that Zones 1 and 2 cover Theorem 1.1 part ( iii ) a) since τ > − α/ ( p − . From Zones 1, 2 and 3 we cover Theorem 1.1 part ( iii ) b)since τ ( α ) > α/ ( p − iii ) c) of Theorem 1.1, since τ ( α ) < α/ ( p − (cid:3) Acknowledgements . The authors thanks Peter Bates for proposing theproblem. H.C. was partially supported by Conicyt Ph.D. scholarship. P.F.was partially supported by Fondecyt
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