On well posedness for the inhomogeneous nonlinear Schrödinger equation
aa r X i v : . [ m a t h . A P ] J un ON WELL POSEDNESS FOR THE INHOMOGENEOUSNONLINEAR SCHR ¨ODINGER EQUATION
CARLOS M. GUZM ´AN
Abstract.
The purpose of this paper is to study well-posedness of the initialvalue problem (IVP) for the inhomogeneous nonlinear Schr¨odinger equation(INLS) iu t + ∆ u + λ | x | − b | u | α u = 0 , where λ = ± α , b > H s ( R N ), with 0 ≤ s ≤ Introduction
In this work, we study the initial value problem (IVP), also called the Cauchyproblem, for the inhomogenous nonlinear Schr¨odinger equation (INLS) (cid:26) i∂ t u + ∆ u + λ | x | − b | u | α u = 0 , t ∈ R , x ∈ R N ,u (0 , x ) = u ( x ) , (1.1)where u = u ( t, x ) is a complex-valued function in space-time R × R N , λ = ± α, b >
0. The equation is called “focusing INLS” when λ = +1 and “defocusingINLS” when λ = − i∂ t u + ∆ u + K ( x ) | u | α u = 0 . This model has been investigated by several authors, see, for instance, Merle [24]and Rapha¨el-Szeftel [25], for k < K ( x ) < k with k , k >
0, and Fibich-Wang[11], for K ( ǫ | x | ) with ǫ small and K ∈ C ( R N ) ∩ L ∞ ( R N ). However, in these works K ( x ) is bounded which is not verified in our case.Our main goal here is to establish local and global results for the Cauchy problem(1.1) in H s ( R N ), with 0 ≤ s ≤ C (cid:0) [ − T, T ]; H s ( R N ) (cid:1) such that the operator defined by G ( u )( t ) = U ( t ) u + iλ Z t U ( t − t ′ ) | x | − b | u ( t ′ ) | α u ( t ′ ) dt ′ , (1.2)where U ( t ) denotes the solution to the linear problem i∂ t u + ∆ u = 0, with initialdata u , is stable and contractive in this space. Thus by the contraction mappingprinciple we obtain a unique fixed point. The fundamental tool to prove these results are the classic Strichartz estimates satisfied by the solution of the linearSchr¨odinger equation.Notice that if u ( t, x ) is solution of (1.1) so is u δ ( t, x ) = δ − bα u ( δ t, δx ), withinitial data u ,δ ( x ) for all δ >
0. Computing the homogeneus Sobolev norm we get k u ,δ k ˙ H s = δ s − N + − bα k u k ˙ H s . Hence, the scale-invariant Sobolev norm is H s c ( R N ) with s c = N − − bα (criticalSobolev index). Note that, if s c = 0 (alternatively α = − bN ) the problem is knownas the mass-critical or L -critical; if s c = 1 (alternatively α = − bN − ) it is calledenergy-critical or ˙ H -critical, finally the problem is known as mass-supercriticaland energy-subcritical if 0 < s c <
1. On the other hand, the inhomogeneousnonlinear Schr¨odinger equation has the following conserved quantities:
M ass ≡ M [ u ( t )] = Z R N | u ( t, x ) | dx = M [ u ] (1.3)and Energy ≡ E [ u ( t )] = 12 Z R N |∇ u ( t, x ) | dx − λα + 2 (cid:13)(cid:13) | x | − b | u | α +2 (cid:13)(cid:13) L x = E [ u ] . The well-posedness theory for the INLS equation (1.1) was studied for many au-thors in recent years. Let us briefly recall the best results available in the literature.Cazenave [2] studied the well-posedness in H ( R N ) using an abstract theory. To dothis, he analyzed (1.1) in the sense of distributions, that is, i∂ t u +∆ u + | x | − b | u | α u =0 in H − ( R N ) for almost all t ∈ I . Therefore, using some results of Functional Anal-ysis and Semigroups of Linear Operators, he proved that it is appropriate to seeksolutions of (1.1) satisfying u ∈ C (cid:0) [0 , T ); H ( R N ) (cid:1) ∩ C (cid:0) [0 , T ); H − ( R N ) (cid:1) for some T > . It was also proved that for the defocusing case ( λ = −
1) any local solution of theIVP (1.1) with u ∈ H ( R N ) extends globally in time.Other authors like Genoud-Stuart [13] (see also references therein) also studied thisproblem for the focusing case ( λ = 1). Using the abstract theory developed byCazenave [2], they showed that the IVP (1.1) is locally well-posed in H ( R N ) if0 < α < ∗ , where 2 ∗ := (cid:26) − bN − N ≥ , ∞ N = 1 , . (1.4)Recently, using some sharp Gagliardo-Nirenberg inequalities, Genoud [12] and Farah[10] extended for the focusing INLS equation (1.1) some global well-posedness re-sults obtained, respectively, by Weinstein [27] for the L -critical NLS equation andby Holmer-Roudenko [18] for the L -supercritical and H -subcritical case. Theseauthors proved that the solution u of the Cauchy problem (1.1) is globally definedin H ( R N ) quantifying the smallness condition in the initial data.However, the abstract theory developed by Cazenave and later used by Genoud-Stuart [13] to show well-posedness for (1.1), does not give sufficient tools to studyother interesting questions, for instance, scattering and blow up investigated byKenig-Merle [21], Holmer-Roudenko-Duyckaerts [9] and others, for the NLS equa-tion. To study these problems, the authors rely on the Strichartz estimates for NLSequation and the classical fixed point argument combining with the concentration-compactness and rigidity technique. N WELL POSEDNESS FOR THE INLS EQUATIONS 3
Inspired by these papers and working toward the proof of scattering and blowup for the INLS equation, we show the well-posedness for the IVP (1.1) using theclassic Strichartz estimates and the contraction mapping principle.Applying this technique in the case b = 0 (classical nonlinear Schr¨odinger equa-tion (NLS)), the IVP (1.1) has been extensively studied over the three decades.The L -theory was obtained by Y. Tsutsumi [26] in the case 0 < α < N . The H -subcritical case was studied by Ginibre-Velo [15]-[16] and Kato [19] (these pa-pers also consider nonlinearities much more general than a pure power). Later,Cazenave-Weissler [4] treated the L -critical case and the H -critical case.We summarize the well known well-posedness theory for the NLS equation inthe following theorem (we refer, for instance, to Linares-Ponce [22] for a proof ofthese results). Theorem 1.1.
Consider the Cauchy problem for the NLS equation ((1.1) with b = 0) . Then, the following statements hold (1) If < α < N , then the IVP (1.1) is locally and globally well posed in L ( R N ) . Moreover if α = N , it is globally well posed in L ( R N ) for smallinitial data. (2) The IVP (1.1) with b = 0 is locally well posed in H ( R N ) if < α ≤ N − for N ≥ or < α < + ∞ , for N = 1 , . Also, it is globally well-posed in H ( R N ) if (i) λ < , (ii) λ > and < α < N , (iii) λ > , N < α < N − and small initial data, (iv) λ > , α = N − and small initial data. In addition, Cazenave-Weissler [5] and recently Cazenave-Fang-Han [3] showed thatthe IVP for the NLS is locally well posed in H s ( R N ) if 0 < α ≤ N − s and 0 < s < N , moreover the local solution extends globally in time for small initial data.Our main interest in this paper is to prove similar results for the INLS equation.To this end, we divide in two parts.The first part is devoted to study the local theory of the IVP (1.1). We startconsidering the local well-posedness in L ( R N ). Theorem 1.2.
Let < α < − bN and < b < min { , N } , then for all u ∈ L ( R N ) there exist T = T ( k u k L , N, α ) > and a unique solution u of (1.1) satisfying u ∈ C (cid:0) [ − T, T ]; L ( R N ) (cid:1) ∩ L q (cid:0) [ − T, T ]; L r ( R N ) (cid:1) , for any ( q, r ) L -admissible. Moreover, the continuous dependence upon the initialdata holds. It is worth to mention that the last theorem is an extension of a result byTsutsumi [26] (which asserts local well-posedness for the NLS equation, (1.1) with b = 0, when 0 < α < N ) to the INLS model.Next, we treat the local well-posedness in H s ( R N ) for 0 < s ≤
1. Before statingthe theorem, we define the following numbers e (cid:26) N N = 1 , , , N ≥ α s := (cid:26) − bN − s s < N , + ∞ s = N . (1.5) C. M. GUZM´AN
Theorem 1.3.
Assume < α < α s , < b < e and max { , s c } < s ≤ min { N , } .If u ∈ H s ( R N ) then there exist T = T ( k u k H s , N, α ) > and a unique solution u of (1.1) with u ∈ C (cid:0) [ − T, T ]; H s ( R N ) (cid:1) ∩ L q (cid:0) [ − T, T ]; H s,r ( R N ) (cid:1) for any ( q, r ) L -admissible. Moreover, the continuous dependence upon the initialdata holds. Remark 1.4.
Observe that α < − bN − s is equivalent to s c < s . On the other hand,if < α < − bN then s c < , for this reason we add the restriction s > max { , s c } in the above statement. As an immediate consequence of the Theorem 1.3, we have that the IVP (1.1)is locally well-posed in H ( R N ). Corollary 1.5.
Assume N ≥ , < α < ∗ and < b < e . If u ∈ H ( R N ) thenthe initial value problem (1.1) is locally well-posed and u ∈ C (cid:0) [ − T, T ]; H ( R N ) (cid:1) ∩ L q (cid:0) [ − T, T ]; H ,r ( R N ) (cid:1) , for any ( q, r ) L -admissible. Remark 1.6.
One important difference of the previous results and its its counter-part for the NLS model (see Theorem 1.1-(2)) is that we do not treat the criticalcase here, i.e. α = − bN − s with ≤ s ≤ and N ≥ . It is still an open problem. In the second part, we consider the global well-posedness of the Cauchy problem(1.1). We begin with a global result in L ( R N ) which is an immediate consequenceof Theorem 1.2. Theorem 1.7. If < α < − bN and < b < min { , N } , then for all u ∈ L ( R N ) the local solution u of the IVP (1.1) extends globally with u ∈ C (cid:0) R ; L ( R N ) (cid:1) ∩ L q (cid:0) R ; L r ( R N ) (cid:1) , for any ( q, r ) L -admissible. In the sequel we establish a small data global theory for the INLS model (1.1).
Theorem 1.8.
Let − bN < α < α s with < b < e , s c < s ≤ min { N , } and u ∈ H s ( R N ) . If k u k H s ≤ A then there exists δ = δ ( A ) such that if k U ( t ) u k S ( ˙ H sc ) < δ ,then the solution of (1.1) is globally defined. Moreover, k u k S ( ˙ H sc ) ≤ k U ( t ) u k S ( ˙ H sc ) and k u k S ( L ) + k D s u k S ( L ) ≤ c k u k H s . Remark 1.9.
Note that in the last result we don’t need the condition s > max { , s c } as in Theorem 1.3, since α > − bN implies s c > . Remark 1.10.
Also note that by the Strichartz estimates (2.10) , the condition k U ( t ) u k S ( ˙ H sc ) < δ is automatically satisfied if k u k ˙ H sc ≤ δc . A similar small data global theory for the NLS model can be found in Cazenave-Weissler [6], Holmer-Roudenko [18] and Guevara [17]. A consequence of the Theo-rem 1.8 is the following global well-posed result in H ( R N ). N WELL POSEDNESS FOR THE INLS EQUATIONS 5
Corollary 1.11.
Let N ≥ , − bN < α < ∗ with < b < e and u ∈ H ( R N ) .Assume k u k H ≤ A then there exists δ = δ ( A ) > such that if k U ( t ) u k S ( ˙ H sc ) < δ ,then there exists a unique global solution u of (1.1) such that k u k S ( ˙ H sc ) ≤ k U ( t ) u k S ( ˙ H sc ) and k u k S ( L ) + k∇ u k S ( L ) ≤ c k u k H . The rest of the paper is organized as follows. In section 2, we introduce somenotations and give a review of the Strichartz estimates. In section 3, we prove thelocal well-posedness results: Theorems 1.2 and 1.3. Finally, in Section 4, we provethe results concerning the global theory: Theorems 1.7 and 1.8.2.
Notation and preliminares
Let us start this section by introducing the notation used throughout the paper.We use c to denote various constants that may vary line by line. Let a set A ⊂ R N , A C = R N \ A denotes the complement of A . Given x, y ∈ R N , x.y denotes the innerproduct of x and y on R N .Let q, r ≥ T > s ∈ R , the mixed norms in the spaces L q [0 ,T ] L rx and L q [0 ,T ] H sx of f ( x, t ) are defined, respectively, as k f k L q ,T L rx = Z T k f ( t, . ) k qL rx dt ! q and k f k L q ,T H sx = Z T k f ( t, . ) k qH sx dt ! q with the usual modifications when q = ∞ or r = ∞ . In the case when I = [0 , T ]and we restrict the x -integration to a subset A ⊂ R N then the mixed norm will bedenoted by k f k L qI L rx ( A ) . Moreover, when f ( t, x ) is defined for every time t ∈ R weshall consider the notations k f k L qt L rx and k f k L qt H sx .For s ∈ R , J s and D s denote the Bessel and the Riesz potentials of order s ,given via Fourier transform by the formulas d J s f = (1 + | ξ | ) s b f and d D s f = | ξ | s b f , where the Fourier transform of f ( x ) is given by b f ( y ) = Z R N e − ix.ξ f ( x ) dx. On the other hand, we define the norm of the Sobolev spaces H s,r ( R N ) and˙ H s,r ( R N ), respectively, by k f k H s,r := k J s f k L r and k f k ˙ H s,r := k D s f k L r . If r = 2 we denote H s, simply by H s .Next, we recall some Strichartz type estimates associated to the linear Schr¨odingerpropagator. k f k L ∞ ,T = sup t ∈ [0 ,T ] | f ( t ) | . C. M. GUZM´AN
Strichartz type estimates.
We say the pair ( q, r ) is L -admissible or simplyadmissible par if they satisfy the condition q = N − Nr , where ≤ r ≤ NN − if N ≥ , ≤ r < + ∞ if N = 2 , ≤ r ≤ + ∞ if N = 1 . (2.1)We also called the pair ˙ H s -admissible if2 q = N − Nr − s, (2.2)where NN − s ≤ r ≤ (cid:16) NN − (cid:17) − if N ≥ , − s ≤ r ≤ (cid:16) ( − s ) + (cid:17) ′ if N = 2 , − s ≤ r ≤ + ∞ if N = 1 . (2.3)Here, a − is a fixed number slightly smaller than a ( a − = a − ε with ε > a + . Moreover ( a + ) ′ is the number suchthat 1 a = 1( a + ) ′ + 1 a + , (2.4)that is ( a + ) ′ := a + .aa + − a with a + . Finally we say that ( q, r ) is ˙ H − s -admissible if2 q = N − Nr + s, where (cid:16) NN − s (cid:17) + ≤ r ≤ (cid:16) NN − (cid:17) − if N ≥ , (cid:16) − s (cid:17) + ≤ r ≤ (cid:16) ( s ) + (cid:17) ′ if N = 2 , (cid:16) − s (cid:17) + ≤ r ≤ + ∞ if N = 1 . (2.5)Given s ∈ R , let A s = { ( q, r ); ( q, r ) is ˙ H s − admissible } and ( q ′ , r ′ ) is such that q + q ′ = 1 and r + r ′ = 1 for ( q, r ) ∈ A s . We define the following Strichartz norm k u k S ( ˙ H s ) = sup ( q,r ) ∈A s k u k L qt L rx and the dual Strichartz norm k u k S ′ ( ˙ H − s ) = inf ( q,r ) ∈A − s k u k L q ′ t L r ′ x . Note that, if s = 0 then A is the set of all L -admissible pairs. Moreover, if s = 0, S ( ˙ H ) = S ( L ) and S ′ ( ˙ H ) = S ′ ( L ). We just write S ( ˙ H s ) or S ′ ( ˙ H − s ) if themixed norm is evaluated over R × R N . To indicate a restriction to a time interval I ⊂ ( −∞ , ∞ ) and a subset A of R N , we will consider the notations S ( ˙ H s ( A ); I )and S ′ ( ˙ H − s ( A ); I ).We now list (without proving) some estimates that will be useful in our work. We included in the above definition the improvement, due to M. Keel and T. Tao [20], to thelimiting case for Strichartzs inequalities.
N WELL POSEDNESS FOR THE INLS EQUATIONS 7
Lemma 2.1. (Sobolev embedding)
Let s ∈ (0 , + ∞ ) and ≤ p < + ∞ . (i) If s ∈ (0 , Np ) then H s,p ( R N ) is continuously embedded in L r ( R N ) where s = Np − Nr . Moreover, k f k L r ≤ c k D s f k L p . (2.6)(ii) If s = N then H s ( R N ) ⊂ L r ( R N ) for all r ∈ [2 , + ∞ ) . Furthermore, k f k L r ≤ c k f k H s . (2.7) Proof.
See Bergh-L¨ofstr¨om [1, Theorem 6 . .
1] (see also Linares-Ponce [22, Theorem3 .
3] and Demenguel-Demenguel [8, Proposition 4.18]). (cid:3)
Remark 2.2.
Using ( i ) , with p = 2 , we have that H s ( R N ) , with s ∈ (0 , N ) , iscontinuously embedded in L r ( R N ) and k f k L r ≤ c k f k H s , (2.8) where r ∈ [2 , NN − s ] . Lemma 2.3. (Fractional product rule)
Let s ∈ (0 , and < r, r , r , p , p < + ∞ are such that r = r i + p i for i = 1 , . Then, k D s ( f g ) k L r ≤ c k f k L r k D s g k L p + c k D s f k L r k g k L p . Proof.
See Christ-Weinstein [7, Proposition 3 . (cid:3) Lemma 2.4. (Fractional chain rule)
Suppose G ∈ C ( C ) , s ∈ (0 , , and See Christ-Weinstein [7, Proposition 3 . (cid:3) The main tool to show the local and global well-posedness are the well-knownStrichartz estimates. See for instance Linares-Ponce [22] and Kato [19] (see alsoHolmer-Roudenko [18] and Guevara [17]). Lemma 2.5. The following statements hold. (i) (Linear estimates). k U ( t ) f k S ( L ) ≤ c k f k L , (2.9) k U ( t ) f k S ( ˙ H s ) ≤ c k f k ˙ H s . (2.10)(ii) (Inhomogeneous estimates). (cid:13)(cid:13)(cid:13)(cid:13)Z R U ( t − t ′ ) g ( ., t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) S ( L ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t U ( t − t ′ ) g ( ., t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) S ( L ) ≤ c k g k S ′ ( L ) , (2.11) (cid:13)(cid:13)(cid:13)(cid:13)Z t U ( t − t ′ ) g ( ., t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) S ( ˙ H s ) ≤ c k g k S ′ ( ˙ H − s ) . (2.12)The relations (2.11) and (2.12) will be very useful to perform estimates on thenonlinearity | x | − b | u | α u .We end this section with three important remarks. C. M. GUZM´AN Remark 2.6. Let F ( x, u ) = | x | − b | z | α z , where f ( z ) = | z | α z . The complex deriva-tive of f is f z ( z ) = α + 22 | z | α and f ¯ z ( u ) = α | z | α − z . For z, w ∈ C , we have f ( z ) − f ( w ) = Z h f z ( w + t ( z − w ))( z − w ) + f ¯ z ( w + t ( z − w ))( z − w ) i dt. Thus, | F ( x, z ) − F ( x, w ) | . | x | − b ( | z | α + | w | α ) | z − w | . (2.13) Remark 2.7. Let B = B (0 , 1) = { x ∈ R N ; | x | ≤ } and b > . If x ∈ B C then | x | − b < and so (cid:13)(cid:13) | x | − b f (cid:13)(cid:13) L rx ≤ k f k L rx ( B C ) + (cid:13)(cid:13) | x | − b f (cid:13)(cid:13) L rx ( B ) . The next remark provides a condition for the integrability of | x | − b on B and B C . Remark 2.8. We notice that if Nγ − b > then k| x | − b k L γ ( B ) < + ∞ , indeed Z B | x | − γb dx = c Z r − γb r N − dr = c r N − γb (cid:12)(cid:12) < + ∞ if N − γb > . Similarly, we have that k| x | − b k L γ ( B C ) is finite if Nγ − b < . Local well-posedness In this section we prove the local well-posedness results. The theorems followsfrom a contraction mapping argument based on the Strichartz estimates. First, weshow the local well-posedness in L ( R N ) (Theorem 1.2) and then in H s ( R N ) for0 < s ≤ L -Theory. We begin with the following lemma. It provides an estimate forthe INLS model nonlinearity in the Strichartz spaces. Lemma 3.1. Let < α < − bN and < b < min { , N } . Then, (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ; I ) ≤ c ( T θ + T θ ) k u k αS ( L ; I ) k v k S ( L ; I ) , (3.1) where I = [0 , T ] and c, θ , θ > .Proof. By Remark 2.7, we have (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ; I ) ≤ k| u | α v k S ′ ( L ( B C ); I ) + (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ( B ); I ) ≡ A + A . Note that in the norm A we don’t have any singularity, so we know that A ≤ cT θ k u k αS ( L ; I ) k v k S ( L ; I ) , (3.2)where θ > 0. See Kato [19, Theorem 0] (also see Linares-Ponce [22, Theorem 5 . . N WELL POSEDNESS FOR THE INLS EQUATIONS 9 On the other hand, we need to find an admissible pair to estimate A . In fact,using the H¨older inequality twice we obtain A ≤ (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L q ′ I L r ′ x ( B ) ≤ (cid:13)(cid:13)(cid:13) k| x | − b k L γ ( B ) k u k αL αr x k v k L rx (cid:13)(cid:13)(cid:13) L q ′ I ≤ k| x | − b k L γ ( B ) T q k u k αL αq I L αr x k v k L qI L rx ≤ T q k| x | − b k L γ ( B ) k u k αL qI L rx k v k L qI L rx , if ( q, r ) L -admissible and r ′ = γ + r + r q ′ = q + q + q q = αq , r = αr . (3.3)In order to have k| x | − b k L γ ( B ) < + ∞ we need Nγ > b , by Remark 2.8. Hence, inview of (3.3) ( q, r ) must satisfy ( Nγ = N − N ( α +2) r > b q = 1 − α +2 q . (3.4)From the first equation in (3.4) we have N − b − N ( α +2) r > 0, which is equivalent to α < r ( N − b ) − NN , (3.5)for r > NN − b . By hypothesis α < − bN , then setting r such that r ( N − b ) − NN = 4 − bN , we get r = − b +2 NN − b satisfying (3.5). Consequently, since ( q, r ) is L -admissiblewe obtain q = − b +2 NN . Next, applying the second equation in (3.4) we deduce1 q = 4 − b − αN − b + 2 N , which is positive by the hypothesis α < − bN . Thus, A ≤ cT θ k u k αS ( L ; I ) k v k S ( L ; I ) , where θ = q . Therefore, combining (3.2) and the last inequality we prove (3.1). (cid:3) Our goal now is to show Theorem 1.2. Proof of Theorem 1.2. We define X = C (cid:0) [ − T, T ]; L ( R N ) (cid:1) \ L q (cid:0) [ − T, T ]; L r ( R N ) (cid:1) , for any ( q, r ) L -admissible, and B ( a, T ) = { u ∈ X : k u k S ( L ;[ − T,T ]) ≤ a } , Since 0 < b < min { N, } the denominator of r is positive and r > NN − b . Moreover, by asimple computations we have 2 ≤ r ≤ NN − if N ≥ 3, and 2 ≤ r < + ∞ if N = 1 , 2, that is r satisfies (2.1). Therefore, the pair ( q, r ) above defined is L -admissible. where a and T are positive constants to be determined later. We follow the standardfixed point argument to prove this result. It means that for appropriate values of a , T we shall show that G defined in (1.2) defines a contraction map on B ( a, T ).Without loss of generality we consider only the case t > 0. Applying Strichartzinequalities (2.9) and (2.11), we have k G ( u ) k S ( L ; I ) ≤ c k u k L + c k| x | − b | u | α +1 k S ′ ( L ; I ) , (3.6)where I = [0 , T ]. Moreover, Lemma 3.1 yields k G ( u ) k S ( L ; I ) ≤ c k u k L + c ( T θ + T θ ) k u k α +1 S ( L ; I ) ≤ c k u k L + c ( T θ + T θ ) a α +1 , provided u ∈ B ( a, T ). Hence, k G ( u ) k S ( L ;[ − T,T ]) ≤ c k u k L + c ( T θ + T θ ) a α +1 . Next, choosing a = 2 c k u k L and T > ca α ( T θ + T θ ) < , (3.7)we conclude G ( u ) ∈ B ( a, T ).Now we prove that G is a contraction. Again using Strichartz inequality (2.11)and (2.13), we deduce k G ( u ) − G ( v ) k S ( L ; I ) ≤ c (cid:13)(cid:13) | x | − b ( | u | α u − | v | α v ) (cid:13)(cid:13) S ′ ( L ; I ) ≤ c (cid:13)(cid:13) | x | − b | u | α | u − v | (cid:13)(cid:13) S ′ ( L ; I ) + c (cid:13)(cid:13) | x | − b | v | α | u − v | (cid:13)(cid:13) S ′ ( L ; I ) ≤ c ( T θ + T θ ) k u k αS ( L ; I ) k u − v k S ( L ; I ) + c ( T θ + T θ ) k v k αS ( L ; I ) k u − v k S ( L ; I ) , where I = [0 , T ]. That is, k G ( u ) − G ( v ) k S ( L ; I ) ≤ c ( T θ + T θ ) (cid:16) k u k αS ( L ; I ) + k v k αS ( L ; I ) (cid:17) k u − v k S ( L ; I ) ≤ c ( T θ + T θ ) a α k u − v k S ( L ; I ) , provided u, v ∈ B ( a, T ). Therefore, the inequality (3.7) implies that k G ( u ) − G ( v ) k S ( L ;[ − T,T ]) ≤ c ( T θ + T θ ) a α k u − v k S ( L ;[ − T,T ]) < k u − v k S ( L ;[ − T,T ]) , i.e., G is a contraction on S ( a, T ).The proof of the continuous dependence is similar to the one given above andit will be omitted. (cid:3) H s -Theory. The aim of this subsection is to prove the local well-posedness in H s ( R N ) with 0 < s ≤ | x | − b | u | α u . First, we consider thenonlinearity in the space S ′ ( L ) and in the sequel in the space D − s S ′ ( L ), that is,we estimate the norm (cid:13)(cid:13) | x | − b | u | α u (cid:13)(cid:13) S ′ ( L ; I ) and (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) S ′ ( L ; I ) .We start this subsection with the following remarks. N WELL POSEDNESS FOR THE INLS EQUATIONS 11 Remark 3.2. Since we will use the Sobolev embedding (Lemma 2.1), we divide ourstudy in three cases: N ≥ and s < N ; N = 1 , and s < N ; N = 1 , and s = N .(see respectively Lemmas 3.4, 3.5 and 3.6 bellow). Remark 3.3. Another interesting remark is the following claim D s ( | x | − b ) = C N,b | x | − b − s . (3.8) Indeed, we use the facts d D s f = | ξ | s b f and \ ( | x | − β ) = C N,β | ξ | N − β for β ∈ (0 , N ) . Let f ( x ) = | x | − b , we have \ D s ( | x | − b ) = | ξ | s \ ( | x | − b ) = | ξ | s C N,β | ξ | N − b = C N,β | ξ | N − ( b + s ) . Since < b < e and < s ≤ min { N , } then < b + s < N , so taking β = s + b ,we get D s ( | x | − b ) = (cid:18) C N,β | y | N − ( b + s ) (cid:19) ∨ = C N,β | x | − b − s . Lemma 3.4. Let N ≥ and < b < e . If s < N and < α < − bN − s then thefollowing statements hold (i) (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ; I ) ≤ c ( T θ + T θ ) k D s u k αS ( L ; I ) k v k S ( L ; I ) (ii) (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) S ′ ( L ; I ) ≤ c ( T θ + T θ ) k D s u k α +1 S ( L ; I ) , where I = [0 , T ] and c, θ , θ > .Proof. (i) We divide the estimate in B and B C , indeed (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ; I ) ≤ (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ( B C ); I ) + (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ( B ); I ) ≡ B + B . First, we consider B . Let ( q , r ) L -admissible given by q = 4( α + 2) α ( N − s ) and r = N ( α + 2) N + αs . (3.9)If s < N then s < Nr and so using the Sobolev inequality (2.6) and the H¨olderinequality twice, we get B ≤ (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L q ′ I L r ′ x ( B C ) ≤ (cid:13)(cid:13)(cid:13) k| x | − b k L γ ( B C ) k u k αL αr x k v k L r x (cid:13)(cid:13)(cid:13) L q ′ I ≤k| x | − b k L γ ( B C ) (cid:13)(cid:13)(cid:13) k D s u k αL r x k v k L r x (cid:13)(cid:13)(cid:13) L q ′ I ≤k| x | − b k L γ ( B C ) T q k D s u k αL αq I L r x k v k L q I L r x = k| x | − b k L γ ( B C ) T q k D s u k αL q I L r x k v k L q I L r x , (3.10)where r ′ = γ + r + r q ′ = q + q + q q = αq , s = Nr − Nαr . (3.11) It is not difficult to check that q and r satisfy the conditions of admissible pair, see (2.1). In view of Remark 2.8 in order to show that the first norm in the right hand sideof (3.10) is bounded we need Nγ − b < 0. Indeed, (3.11) is equivalent to ( Nγ = N − Nr − Nαr + αs q = 1 − α +2 q , (3.12)which implies, by (3.9) Nγ = 0 and 1 q = 4 − α ( N − s )4 . (3.13)So Nγ − b < q > 0, by our hypothesis α < − bN − s . Therefore, setting θ = q we deduce B ≤ cT θ k D s u k αS ( L ; I ) k v k S ( L ; I ) . (3.14)We now estimate B . To do this, we use similar arguments as the ones inthe estimation of A in Lemma 3.1. It follows from H¨older’s inequality twice andSobolev embedding (2.6) that B ≤ (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L q ′ I L r ′ x ( B ) ≤ (cid:13)(cid:13)(cid:13) k| x | − b k L γ ( B ) k u k αL αr x k v k L rx (cid:13)(cid:13)(cid:13) L q ′ I ≤ (cid:13)(cid:13)(cid:13) k| x | − b k L γ ( B ) k D s u k αL rx k v k L rx (cid:13)(cid:13)(cid:13) L q ′ I ≤ k| x | − b k L γ ( B ) T q k D s u k αL αq I L rx k v k L qI L rx = k| x | − b k L γ ( B ) T q k D s u k αL qI L rx k v k L qI L rx if ( q, r ) L -admissible and the following system is satisfied r ′ = γ + r + r s = Nr − Nαr , s < Nr q ′ = q + q + q q = αq . (3.15)Similarly as in Lemma 3.1 we need to check that Nγ > b (so that k| x | − b k L γ ( B ) isfinite) and q > q, r ) L -admissible pair. From (3.15) thisis equivalent to ( Nγ = N − Nr − Nαr + αs > b q = 1 − α +2 q > . (3.16)The first equation in (3.16) implies that α < ( N − b ) r − NN − rs (assuming s < Nr ), thenlet us choose r such that ( N − b ) r − NN − rs = 4 − bN − s since, by our hypothesis α < − bN − s . Therefore r and q are given by r = 2 N [ N − b + 2(1 − s )] N ( N − s ) + 4 s − bN and q = 2[ N − b + 2(1 − s )] N − s , (3.17) It is easy to see that r > s < N and r < NN − if, and only if, b < 2. Thereforethe pair ( q, r ) given in (3.17) is L -admissible. N WELL POSEDNESS FOR THE INLS EQUATIONS 13 where we have used that ( q, r ) is a L -admissible pair to compute the value of q .Note that s < Nr if, and only if, b + 2 s − N < 0. Since s ≤ b < e N ≥ s < Nr holds. In addition, from the second equation of(3.16) and (3.17) we also have1 q = 4 − b − α ( N − s )2( N − b + 2 − s ) > , (3.18)since α < − bN − s .Hence, B ≤ cT θ k D s u k αS ( L ; I ) k v k S ( L ; I ) , (3.19)where θ is given by (3.18). Finally, collecting the inequalities (3.14) and (3.19) weobtain (i).(ii) Observe that (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) S ′ ( L ; I ) ≤ C + C , where C = (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) S ′ ( L ( B C ); I ) and C = (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) S ′ ( L ( B ); I ) . To estimate C we use the same admissible pair ( q , r ) used to estimate theterm B in item (i). Indeed, let C ( t ) = (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) L r ′ x ( B C ) then Lemma 2.3 (fractional product rule), Lemma 2.4 (fractional chain rule) andRemark 3.3 yield C ( t ) ≤k| x | − b k L γ ( B C ) k D s ( | u | α u ) k L βx + k D s ( | x | − b ) k L d ( B C ) k u k α +1 L ( α +1) ex ≤k| x | − b k L γ ( B C ) k u k ααr k D s u k L r x + k| x | − b − s k L d ( B C ) k D s u k α +1 L r x ≤k| x | − b k L γ ( B C ) k D s u k α +1 L r x + k| x | − b − s k L d ( B C ) k D s u k α +1 L r x , (3.20)where we also have used the Sobolev inequality (2.6) and (3.8). Moreover, we havethe following relations r ′ = γ + β = d + e β = r + r s = Nr − Nαr ; s < Nr s = Nr − N ( α +1) e which implies that ( Nγ = N − Nr − αNr + αs Nd = N − Nr − αNr + αs + s. (3.21)Note that, in view of (3.9) we have Nγ − b < Nd − b − s < 0. These relationsimply that k| x | − b k L γ ( B C ) and k| x | − b − s k L d ( B C ) are bounded quantities (see Remark2.8). Therefore, it follows from (3.20) that C ( t ) ≤ c k D s u k α +1 L r x . On the other hand, using q ′ = q + α +1 q and applying the H¨older inequality in thetime variable we conclude k C k L q ′ I ≤ cT q k D s u k α +1 L q I L r x , where q is given in (3.13). The estimate of C is finished since C ≤ k C k L q ′ I .We now consider C . Let C ( t ) = (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) L r ′ x ( B ) , we have C ≤k C k L q ′ I . Using the same arguments as in the estimate of C we obtain C ( t ) ≤ k| x | − b k L γ ( B ) k D s u k α +1 L rx + k| x | − b − s k L d ( B ) k D s u k α +1 L rx , (3.22)if (3.21) is satisfied replacing r by r (to be determined later), that is ( Nγ = N − Nr − αNr + αs Nd = N − Nr − αNr + αs + s. (3.23)In order to have that k| x | − b k L γ ( B ) and k| x | − b − s k L d ( B ) are bounded, we need Nγ > b and Nd > b + s , respectively, by Remark 2.8. Therefore, since the first equation in(3.23) is the same as the first one in (3.16), we choose r as in (3.17). So we get Nγ > b , which also implies that Nd − s > b . Finally, (3.22) and the H¨older inequalityin the time variable yield C ≤ cT q k D s u k α +1 L ( α +1) q I L rx = cT q k D s u k α +1 L qI L rx , (3.24)where 1 q ′ = 1 q + 1 q q = ( α + 1) q . (3.25)Notice that (3.25) is exactly to the second equation in (3.16), thus q > (cid:3) One important remark is that Lemma 3.4 only holds for N ≥ 3, since the admis-sible par ( q, r ) defined in (3.17) doesn’t satisfy the condition s < Nr , for N = 1 , Lemma 3.5. Let N = 1 , and < b < e . If s < N and < α < − bN − s then (i) (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ; I ) ≤ c ( T θ + T θ ) k D s u k αS ( L ; I ) k v k S ( L ; I ) (ii) (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) S ′ ( L ; I ) ≤ c ( T θ + T θ ) k D s u k α +1 S ( L ; I ) , where I = [0 , T ] and c, θ , θ > .Proof. (i) As before, we divide the estimate in B and B C . The estimate on B C is the same as the term B in Lemma 3.4-(i), since ( q , r ) given in (3.9) is L -admissible for s < N in all dimensions. Thus we only consider the estimate on B . N WELL POSEDNESS FOR THE INLS EQUATIONS 15 Indeed, set the L -admissible pair (¯ q, ¯ r ) = ( N − s , Ns ). We deduce from theH¨older inequality twice and Sobolev embedding (2.6) (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L ¯ q ′ I L ¯ r ′ x ( B ) ≤ (cid:13)(cid:13)(cid:13) k| x | − b k L γ ( B ) k u k αL αr x k v k L rx (cid:13)(cid:13)(cid:13) L q ′ I ≤ k| x | − b k L γ ( B ) T q k D s u k αL αq I L rx k v k L qI L rx = k| x | − b k L γ ( B ) T q k D s u k αL qI L rx k v k L qI L rx if ( q, r ) is L -admissible and the following system is satisfied r ′ = γ + r + r s = Nr − Nαr ; s < Nr q ′ = q + q + q q = αq . (3.26)Using the values of ¯ q and ¯ r given above, the previous system is equivalent to ( Nγ = N − b ) − s − Nr − α ( N − sr ) r + b q = − N − s − α +1 q . (3.27)From the first equation in (3.27) if α < r (4( N − b ) − s ) − NN − sr then Nγ > b , and so | x | − b ∈ L γ ( B ). Now, in view of the hypothesis α < − bN − s we set r such that r (4( N − b ) − s ) − N N − sr ) = 4 − bN − s , that is r = 4 N ( N − s + 4 − b )4 s (4 − b ) + ( N − s ) (4 N − b − s ) . (3.28)Note that, in order to satisfy the second equation in the system (3.26) we need toverify s < Nr . A simple calculation shows that it is true if, and only if, 4 b + 5 s < N and this is true since b < N and s < N .On the other hand, since we are looking for a pair ( q, r ) L -admissible one has q = 8( N − s + 4 − b )(8 − N + s )( N − s ) . (3.29)Finally, from (3.29) the second equation in (3.27) is given by1 q = (cid:18) − N + s (cid:19) (cid:18) − b − α ( N − s ) N − s + 4 − b (cid:19) . (3.30)which is positive, since α < − bN − s , s < N and N = 1 , B . Let D ( t ) = (cid:13)(cid:13) | x | − b | u | α u (cid:13)(cid:13) L ¯ r ′ x ( B ) . We claim that r satisfies (2.1). In fact, obviously r < + ∞ . Moreover r ≥ − N + s ≥ s > N = 1 , We use analogous arguments as the ones in the estimate of C in Lemma 3.4-(ii).Lemmas 2.3-2.4, the H¨older inequality, the Sobolev embedding (2.6) and Remark3.3 imply D ( t ) ≤k| x | − b k L γ ( B ) k D s ( | u | α u ) k L βx + k D s ( | x | − b ) k L d ( B ) k u k α +1 L ( α +1) ex ≤k| x | − b k L γ ( B ) k u k ααr k D s u k L rx + k| x | − b − s k L d ( B ) k D s u k α +1 L rx ≤k| x | − b k L γ ( B ) k D s u k α +1 L rx + k| x | − b − s k L d ( B ) k D s u k α +1 L rx , (3.31)where r ′ = γ + β = d + e β = r + r s = Nr − Nαr ; s < Nr s = Nr − N ( α +1) e , which is equivalent to Nγ = N − N ¯ r − ( α +1) Nr + αs Nd = N − N ¯ r − ( α +1) Nr + αs + s. (3.32)Hence, setting again (¯ q, ¯ r ) = ( N − s , Ns ) the first equation in (3.32) the same asthe first one in (3.27). Therefore choosing r as in (3.28) we have Nγ > b , which alsoimplies Nd > b + s . Therefore, it follows from Remark 2.8 and (3.31) that D ( t ) ≤ c k D s u k α +1 L rx . Since, q ′ = q + α +1 q (recall that q is given in (3.29)) and applying the H¨olderinequality in the time variable we conclude k D k L ¯ q ′ T ≤ cT q k D s u k α +1 L qT L rx , where q > (cid:3) We finish the estimates for the nonlinearity considering the case s = N . Notethat this case can only occur if N = 1 , 2, since here we are interested in local (andglobal) results in H s ( R N ) for max { , s c } < s ≤ min { N , } . Lemma 3.6. Let N = 1 , and < b < N . If s = N and < α < + ∞ then (i) (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ; I ) ≤ cT θ k u k αL ∞ I H sx k v k L ∞ I L x (ii) (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) S ′ ( L ; I ) ≤ cT θ k u k α +1 L ∞ I H sx , where I = [0 , T ] and c, θ > .Proof. (i) To this end we start defining the following numbers r = N ( α + 2) N − b and q = 4( α + 2) N α + 4 b , (3.33)it is easy to check that ( q, r ) is L -admissible.We divide the estimate in B and B C . We first consider the estimate on B .From H¨older’s inequality (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L r ′ x ( B ) ≤ k| x | − b k L γ ( B ) k u k αL αr x k v k L x , (3.34) N WELL POSEDNESS FOR THE INLS EQUATIONS 17 where 1 r ′ = 1 γ + 1 r + 12 . (3.35)In view of Remark 2.8 to show that | x | − b ∈ L γ ( B ), we need Nγ − b > 0. So, therelations (3.33) and (3.35) yield Nγ − b = α ( N − b )2( α + 2) − Nr . (3.36)If we choose αr ∈ (cid:16) N ( α +2) N − b , + ∞ (cid:17) then the right hand side of (3.36) is positive.Therefore, (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L r ′ x ( B ) ≤ c k u k αL αr x k v k L x . On the other hand, since N ( α +2) N − b > (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L r ′ x ( B ) ≤ c k u k αH s k v k L x . (3.37)Next, we consider the estimate on B C . Using the same argument as in the firstcase we get (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L r ′ x ( B C ) ≤ k| x | − b k L γ ( B C ) k u k αL αr x k v k L x , where the relations (3.35) and (3.36) hold. Thus, choosing αr ∈ (cid:16) , N ( α +2) N − b (cid:17) wehave that Nγ − b < 0, which implies | x | − b ∈ L γ ( B C ), by Remark 2.8. Therefore,again the Sobolev embedding (2.7) leads to (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L r ′ x ( B C ) ≤ c k u k αH sx k v k L x . Finally, it follows from the H¨older inequality in time variable, (3.37) and thelast inequality that (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L q ′ I L r ′ x ≤ cT θ k u k αL ∞ I H s k v k L ∞ I L x , (3.38)where θ = q ′ > 0, by (3.33).(ii) Similarly as in the proof of item (i) we begin setting r = N ( α + 2) N − b − s and q = 4( α + 2) αN + 2 b + 2 s . (3.39)Observe that, since s = N and 0 < b < N the denominator of r is a positivenumber. Furthermore it is easy to verify that ( q, r ) is L -admissible.First, we consider the estimate on B . Lemma 2.4 together with the H¨olderinequality and (3.8) imply E ( t ) ≤ k| x | − b k L γ ( B ) k D s ( | u | α u ) k L βx + k D s ( | x | − b ) k L d ( B ) k u k α +1 L ( α +1) ex ≤ k| x | − b k L γ ( B ) k u k αL αr x k D s u k L x + k| x | − b − s k L d ( B ) k u k α +1 L ( α +1) ex , where E ( t ) = (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) L r ′ x ( B ) and ( r ′ = γ + β = d + e β = r + , which implies ( Nγ = N − Nr − Nr Nd = N − Nr − Ne . (3.40)Now, we claim that k| x | − b k L γ ( B ) and k| x | − b − s k L d ( B ) are bounded quantities fora suitable choice of r and e . Indeed, using the value of r in (3.39), (3.40) and thefact that s = N we get Nγ − b = ( α +1)( N − b )2( α +2) − Nr Nd − b − s = ( α +1)( N − b )2( α +2) − Ne . (3.41)By Remark 2.8, if r , e > N ( α +2)( α +1)( N − b ) then the right hand side of both equationsin (3.41) are positive, so | x | − b ∈ L γ ( B ) and | x | − b − s ∈ L d ( B ). Hence E ( t ) ≤ c k u k αL αr x k D s u k L x + c k u k α +1 L ( α +1) ex . Choosing r and e as before, it is easy to see that αr > α + 1) e > 2, thuswe can use the Sobolev inequality (2.7) E ( t ) ≤ c k u k αH sx k D s u k L x + c k u k α +1 H sx ≤ c k u k α +1 H sx . (3.42)To complete the proof, we need to consider the estimate on B C . By the samearguments as before we have E ( t ) ≤ k| x | − b k L γ ( B C ) k u k αL αr x k D s u k L x + k| x | − b − s k L d ( B C ) k u k α +1 L ( α +1) ex , where E ( t ) = (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) L r ′ x ( B C ) and (3.41) holds. Similarly as in item (i),since Nα ( α +2)( α +1)( N − b ) , N ( α +2) N − b − s > 2, we can choose r and e such that αr ∈ (cid:18) , N α ( α + 2)( α + 1)( N − b ) (cid:19) and ( α + 1) e ∈ (cid:18) , N ( α + 2) N − b (cid:19) , and so we obtain from (3.41) that Nγ − b < Nd − b − s < 0. In other words, k| x | − b k L γ ( B C ) and k| x | − b − s k L d ( B C ) are bounded quantities for these choices of r and e (see Remark 2.8). In addition, by the Sobolev inequality (2.7) we conclude E ( t ) ≤ c k u k α +1 H sx . Finally, (3.42) and the last inequality lead to k (cid:13)(cid:13) D s ( | x | − b | u | α u ) (cid:13)(cid:13) L q ′ I L r ′ x ≤ cT q ′ k u k α +1 L ∞ I H sx , where q ′ > (cid:3) We now have all tools to prove the main result of this section, Theorem 1.3. Increasing the value of r if necessary. Notice that, since N = 1 , α > − bN we have2 Nα ( α + 2)( α + 1)( N − b ) > NαN − b > − b ) N − b > . N WELL POSEDNESS FOR THE INLS EQUATIONS 19 Proof of Theorem 1.3. We define X = C (cid:0) [ − T, T ]; H s ( R N ) (cid:1) \ L q (cid:0) [ − T, T ]; H s,r ( R N ) (cid:1) , for any ( q, r ) L -admissible, and k u k T = k u k S ( L ;[ − T,T ]) + k D s u k S ( L ;[ − T,T ]) . We shall show that G = G u defined in (1.2) is a contraction on the complete metricspace S ( a, T ) = { u ∈ X : k u k T ≤ a } with the metric d T ( u, v ) = k u − v k S ( L ;[ − T,T ]) , for a suitable choice of a and T .First, we claim that S ( a, T ) with the metric d T is a complete metric space.Indeed, the proof follows similar arguments as in [2] (see Theorem 1 . . S ( a, T ) ⊂ X and X is a complete space, itsuffices to show that S ( a, T ), with the metric d T , is closed in X . Let u n ∈ S ( a, T )such that d T ( u n , u ) → n → + ∞ , we want to show that u ∈ S ( a, T ). If u n ∈ C (cid:0) [ − T, T ]; H s ( R N ) (cid:1) (see the definition of S ( a, T )) we have, for almost all t ∈ [ − T, T ], u n ( t ) bounded in H s ( R N ) and so (since H s ( R N ) is reflexive) u n ( t ) ⇀ v ( t ) in H s ( R N ) and k v ( t ) k H s ≤ lim inf n → + ∞ k u n k H s ≤ a. (3.43)On the other hand, the hypothesis d T ( u n , u ) → u n → u in L qI L rx forall ( q, r ) L -admissible. Since ( ∞ , 2) is L -admissible we get u n ( t ) → u ( t ) in L ,for almost all t ∈ [ − T, T ]. Therefore, by uniqueness of the limit we deduce that u ( t ) = v ( t ). Also, we have from (3.43) k u ( t ) k H s ≤ a. That is, u ∈ C (cid:0) [ − T, T ]; H s ( R N ) (cid:1) . From similar arguments, if u n ∈ L q (cid:0) I ; H s,r ( R N ) (cid:1) we obtain u ∈ S ( a, I ). This completes the proof of the claim.Returning the proof of the theorem, it follows from the Strichartz inequalities(2.9) and (2.11) that k G ( u ) k S ( L ;[ − T,T ]) ≤ c k u k L + c k F k S ′ ( L ;[ − T,T ]) (3.44)and k D s G ( u ) k S ( L ;[ − T,T ]) ≤ c k D s u k L + c k D s F k S ′ ( L ;[ − T,T ]) , (3.45)where F ( x, u ) = | x | − b | u | α u . Similarly as in the proof of Theorem 1.2, withoutloss of generality we consider only the case t > 0. So, we deduce using Lemmas3.4-3.5-3.6 and (3.2) k F k S ′ ( L ; I ) ≤ c ( T θ + T θ ) k u k α +1 I and k D s F k S ′ ( L ; I ) ≤ c ( T θ + T θ ) k u k α +1 I . where I = [0 , T ] and θ , θ > 0. Hence, if u ∈ S ( a, T ) then k G ( u ) k T ≤ c k u k H s + c ( T θ + T θ ) a α +1 . Now, choosing a = 2 c k u k H s and T > ca α ( T θ + T θ ) < , (3.46) we obtain G ( u ) ∈ S ( a, T ). Such calculations establish that G is well defined on S ( a, T ).To prove that G is a contraction we use (2.13) and an analogous argument asbefore d T ( G ( u ) , G ( v )) ≤ c k F ( x, u ) − F ( x, v ) k S ′ ( L ;[ − T,T ]) ≤ c ( T θ + T θ ) ( k u k αT + k v k αT ) d T ( u, v ) , and so, taking u, v ∈ S ( a, T ) we get d T ( G ( u ) , G ( v )) ≤ c ( T θ + T θ ) a α d T ( u, v ) . Therefore, from (3.46), G is a contraction on S ( a, T ) and by the Contraction Map-ping Theorem we have a unique fixed point u ∈ S ( a, T ) of G . (cid:3) We finish this section noting that Corollary 1.5 follows directly from Theorem1.3. It is worth to mention that Corollary 1.5 only holds for N ≥ s ≤ min { N , } in Theorem 1.3.4. Global well-posedness This section is devoted to study the global well-posedness of the Cauchy problem(1.1). Similarly as the local theory we use the fixed point theorem to prove oursmall data results in H s ( R N ). We start with a global result in L ( R N ), which doesnot require any smallness assumption.4.1. L -Theory. The global well-posedness result in L ( R N ) (see Theorem 1.7)is an immediate consequence of Theorem 1.2. Indeed, using (3.7) we have that T ( k u k L ) = C k u k dL for some C, d > 0, then the conservation law (1.3) allows us toreapply Theorem 1.2 as many times as we wish preserving the length of the timeinterval to get a global solution.4.2. H s -Theory. In this subsection, we turn our attention to proof the Theorem1.8. Again the heart of the proof is to establish good estimates on the nonlin-earity F ( x, u ) = | x | − b | u | α u . First, we estimate the norm k F ( x, u ) k S ′ ( ˙ H − sc ) (seeLemma 4.1 below), next we estimate k F ( x, u ) k S ′ ( L ) (see Lemma 4.2) and finallywe consider the norm k D s F ( x, u ) k S ′ ( L ) (see Lemmas 4.3, 4.5 and 4.7).We begin defining the following numbers (depending only on N, α and b ) b q = 4 α ( α + 2 − θ ) α ( N α + 2 b ) − θ ( N α − b ) b r = N α ( α + 2 − θ ) α ( N − b ) − θ (2 − b ) (4.1)and e a = 2 α ( α + 2 − θ ) α [ N ( α + 1 − θ ) − b ] − (4 − b )(1 − θ ) b a = 2 α ( α + 2 − θ )4 − b − ( N − α , (4.2) N WELL POSEDNESS FOR THE INLS EQUATIONS 21 where θ > . It is easy to see that ( b q, b r ) L -admissible, ( b a, b r )˙ H s c -admissible and ( e a, b r ) ˙ H − s c -admissible. Moreover, we observe that1 b a + 1 e a = 2 b q . (4.3)Using the same notation of the previous section, we set B = B (0 , 1) and werecall that | x | − b ∈ L γ ( B ) if Nγ > b . Similarly, we have that | x | − b ∈ L γ ( B C ) if Nγ < b (see Remark 2.8).Our first result reads follows. Lemma 4.1. Let − bN < α < α s and < b < e . If s c < s ≤ min { N , } then thefollowing statement holds (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( ˙ H − sc ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k v k S ( ˙ H sc ) , (4.4) where c > and θ ∈ (0 , α ) is a sufficiently small number.Proof. The proof follows from similar arguments as the ones in the previous lemmas.We study the estimates in B and B C separately.We first consider the set B . From the H¨older inequality we deduce (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L b r ′ x ( B ) ≤ k| x | − b k L γ ( B ) k u k θL θr x k u k α − θL ( α − θ ) r x k v k L b rx = k| x | − b k L γ ( B ) k u k θL θr x k u k α − θL b rx k v k L b rx , (4.5)where 1 b r ′ = 1 γ + 1 r + 1 r + 1 b r and b r = ( α − θ ) r . (4.6)Now, we make use of the Sobolev embedding (Lemma 2.1), so we consider twocases: s = N and s < N . Case s = N . Since s ≤ min { N , } , we only have to consider the cases where( N, s ) is equal to (1 , ) or (2 , k| x | − b k L γ ( B ) boundedwe need Nγ > b . In fact, observe that (4.6) implies Nγ = N − N ( α + 2 − θ ) b r − Nr , and from (4.1) it follows that Nγ − b = θ (2 − b ) α − Nr . (4.7)Since α > − bN then Nα − b > 2, therefore choosing θr ∈ (cid:18) N α − b , + ∞ (cid:19) , (4.8) First note that, since θ > b q, b r, b a and e a are all positive numbers. Moreover, it is easy to see that b r satisfies (2.3). In fact b a can berewritten as b a = α +2 − θ − s c and since θ < α we have b a > − s c , which implies that b r < NN − , for N ≥ 3. We also note that b r ≤ (( − s c ) + ) ′ , for N = 2. Indeed, the last inequality is equivalentto ε b r < ( − s c ) + ( − s c ) (recall (2.4)) and this is true since ε > N = 1, we see that b r < ∞ . Finally, we have b r > NN − s c = Nα − b . Indeed, this is equivalent to( α + 2 − θ )(2 − b ) > α ( N − b ) − θ (2 − b ) ⇔ ( α + 2)(2 − b ) > α ( N − b ) ⇔ α < − bN − . So since α < − bN − s and s ≤ α < − bN − holds, consequently b r > NN − s c . Recall that s c is the critical Sobolev index given by s c = N − − bα . we get Nγ > b . Hence, inequality (4.5) and the Sobolev embedding (2.7) yield (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L b r ′ x ( B ) ≤ c k u k θH sx k u k α − θL b rx k v k L b rx . (4.9) Case s < N . Our goal here is to also obtain the inequality (4.9). Indeed wealready have the relation (4.7), then the only change is the choice of θr since wecan not apply the Sobolev embedding (2.7) when s < N . In this case we set θr = 2 NN − s , (4.10)so Nγ − b = θ ( s − s c ) > , that is, the quantity k| x | − b k L γ ( B ) is finite. Therefore by the Sobolev embedding(2.8) we obtain the desired inequality (4.9).Next, we consider the set B C . We claim that (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L b r ′ x ( B C ) ≤ c k u k θH sx k u k α − θL b rx k v k L b rx . (4.11)Indeed, Arguing in the same way as before we deduce (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L b r ′ x ( B C ) ≤ k| x | − b k L γ ( B C ) k u k θL θr x k u k α − θL b rx k v k L b rx , where the relation (4.7) holds. We first show that k| x | − b k L γ ( B C ) is finite for asuitable of r . Here we also consider two cases: s = N and s < N . In the first case,we choose r such that θr ∈ (cid:18) , N α − b (cid:19) (4.12)then, from (4.7), Nγ − b < 0, so | x | − b ∈ L γ ( B C ). Thus, by the Sobolev inequality(2.7) and using the last inequality we deduce (4.11). Now if s < N , choosing again θr as (4.12) one has Nγ − b < 0. In addition, since α < − bN − s we obtain Nα − b < NN − s ,therefore the Sobolev inequality (2.8) implies (4.11). This completes the proof ofthe claim.Now, inequalities (4.9) and (4.11) yield (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L b r ′ x ≤ c k u k θH sx k u k α − θL b rx k v k L b rx (4.13)and the H¨older inequality in the time variable leads to (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L e a ′ t L b r ′ x ≤ c k u k θL ∞ t H sx k u k α − θL ( α − θ ) a t L b rx k v k L b at L b rx = c k u k θL ∞ t H sx k u k α − θL b at L b rx k v k L b at L b rx , where 1 e a ′ = α − θ b a + 1 b a . Since b a and e a defined in (4.2) satisfy the last relation we conclude the proof of(4.4). (cid:3) Lemma 4.2. Let − bN < α < α s and < b < e . If s c < s ≤ min { N , } then (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) S ′ ( L ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k v k S ( L ) , (4.14) where c > and θ ∈ (0 , α ) is a sufficiently small number. Recall that ( b a, b r ) is ˙ H s c -admissible and ( e a, b r ) is ˙ H − s c -admissible. N WELL POSEDNESS FOR THE INLS EQUATIONS 23 Proof. By the previous lemma we already have (4.13), then applying H¨older’s in-equality in the time variable we obtain (cid:13)(cid:13) | x | − b | u | α v (cid:13)(cid:13) L b q ′ t L b r ′ x ≤ c k u k θL ∞ t H sx k u k α − θL b at L b rx k v k L b qt L b rx , (4.15)since 1 b q ′ = α − θ b a + 1 b q (4.16)by (4.1) and (4.2). The proof is finished in view of ( b q, b r ) be L -admissible. (cid:3) We now estimate (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ) . To this end we divide our study inthree cases: N ≥ N = 3 and N = 1 , Lemma 4.3. Let N ≥ , < b < e and − bN < α < α s . If s c < s ≤ then thefollowing statement holds (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k D s u k S ( L ) , (4.17) where c > and θ ∈ (0 , α ) is a sufficiently small number.Proof. First note that we always have s < N in this lemma, since we are assuming N ≥ s c < s ≤ 1. Here, we also divide the estimate in B and B C separately.We begin estimating on B . The fractional product rule (Lemma 2.3) yields (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L b r ′ x ( B ) ≤ N ( t, B ) + N ( t, B ) , (4.18)where N ( t, B ) = (cid:13)(cid:13) | x | − b (cid:13)(cid:13) L γ ( B ) k D s ( | u | α u ) k L βx N ( t, B ) = (cid:13)(cid:13) D s ( | x | − b ) (cid:13)(cid:13) L d ( B ) k| u | α u k L ex and 1 b r ′ = 1 γ + 1 β = 1 d + 1 e . (4.19)It follows from the fractional chain rule (Lemma 2.4) and H¨older’s inequality that N ( t, B ) ≤ k| x | − b k L γ ( B ) k u k θL θr x k u k α − θL ( α − θ ) r x k D s u k L b rx = k| x | − b k L γ ( B ) k u k θL θr x k u k α − θL b rx k D s u k L b rx , (4.20)where 1 β = 1 r + 1 r + 1 b r and b r = ( α − θ ) r . (4.21)Notice that the right hand side of (4.20) is the same as the right hand side of (4.5),with v = D s u , so combining (4.19) and (4.21) we also have (4.6). Thus, arguing inthe same way as in Lemma 4.1 we obtain (recall that (4.9) also holds when s < N ) N ( t, B ) ≤ c k u k θH sx k u k α − θL b rx k D s u k L b rx . (4.22)On the other hand, we deduce from (3.8), H¨older’s inequality and the Sobolevemdebbing (2.6) N ( t, B ) ≤ k| x | − b − s k L d ( B ) k u k θL θr x k u k α − θL ( α − θ ) r x k u k L r x = k| x | − b − s k L d ( B ) k u k θL θr x k u k α − θL b rx k D s u k L b rx , (4.23)where ( e = r + r + r b r = ( α − θ ) r s = N b r − Nr with s < N b r , (4.24) which implies using (4.19) that Nd − s = N − N ( α + 2 − θ ) b r − Nr and so, by (4.1) Nd − b − s = θ (2 − b ) α − Nr . (4.25)Observe that the right hand side of (4.25) is the same as the right hand side of (4.7).Hence, choosing θr as in (4.10) (recall that s < N ) we have Nd − b − s > 0, so thequantity k| x | − b − s k L d ( B ) is bounded, by Remark 2.8. Now, the Sobolev embedding(2.8) and (4.23) imply that N ( t, B ) ≤ c k u k θH sx k u k α − θL b rx k D s u k L b rx . Therefore, the last inequality together with (4.22) lead to (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L b r ′ x ( B ) ≤ c k u k θH sx k u k α − θL b rx k D s u k L b rx . (4.26)Thus applying H¨older’s inequality in the time variable and recalling (4.16), (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L b q ′ t L b r ′ x ( B ) ≤ c k u k θL ∞ t H sx k u k α − θL b at L b rx k D s u k L b qt L b rx ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k D s u k S ( L ) . (4.27)Next we consider the norm (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L b r ′ x ( B C ) . Similarly as before, re-placing B by B C , we also get (4.20)-(4.21) and consequently by the proof of Lemma4.1 we have the inequality (4.22), that is N ( t, B C ) ≤ c k u k θH sx k u k α − θL b rx k D s u k L b rx . We also have (replacing B by B C ) N ( t, B C ) ≤ k| x | − b − s k L d ( B C ) k u k θL θr x k u k α − θL b rx k D s u k L b rx , where the relation (4.25) holds, thus setting θr = 2 we deduce Nd − b − s = − θs c < , which implies that | x | − b − s ∈ L d ( B C ), by Remark 2.8. Then the Sobolev embedding(2.8) yields N ( t, B C ) ≤ c k u k θH sx k u k α − θL b rx k D s u k L b rx . Therefore, (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L b r ′ x ( B C ) ≤ N ( t, B ) + N ( t, B C ) ≤ k u k θH sx k u k α − θL b rx k D s u k L b rx . Finally, using H¨older’s inequality in the time variable, the last inequality (re-calling (4.16)) and the relation (4.27) we get the estimate (4.17). (cid:3) Remark 4.4. Notice that Lemma 4.3 doesn’t hold in dimension three for every α < α s (recall (1.5) ). In fact, the condition s < N b r (used in (4.24) ) is only true for N ≥ . In the next lemma we consider the case N = 3 . N WELL POSEDNESS FOR THE INLS EQUATIONS 25 Before stating the lemma, we define the following numbers k = 4 α ( α + 1 − θ )4 − b − α p = 6 α ( α + 1 − θ )(4 − b )( α − θ ) + α (4.28)and l = 4 α ( α + 1 − θ ) α (3 α − b ) − θ (3 α − b ) , (4.29)where θ ∈ (0 , α ). It is not difficult to verify that ( l, p ) is L -admissible and ( k, p ) is˙ H s c -admissible .We also define m = 4 DD − ε n = 6 D D + ε (4.30)and a ∗ = 4 θ ε − D r ∗ = 6 αθ (4 − b ) θ − (2 + ε − D ) α , (4.31)where D = α − θ + µ with µ ∈ ( b, 1) and ε is a sufficiently small number such that ε < µ − b . Note that 2 < n < n satisfies the condition (2.1) for N = 3) and( m, n ) is L -admissible. Moreover, choosing θ = F α with F = − ε + µ − b − b we claimthat ( a ∗ , r ∗ ) is ˙ H s c -admissible. We first show that the denominators of a ∗ and r ∗ are positive numbers. Indeed2 + ε − D = 2 + ε − µ + F α − α = 2 + ε − µ − α (1 − F ) = 2 + ε − µ − α (cid:18) ε − µ − b (cid:19) , so by our hypothesis α < − b − s and since s ≤ ε − D > 0. We alsohave (using the value of F and the fact that D > µ )(4 − b ) θ − (2 + ε − D ) α = α ((4 − b ) F − − ε + D ) > (2( µ − b ) − ε ) , which is positive setting ε < µ − b .Next, we show that r ∗ satisfies the condition (2.3), with N = 3. Note that r ∗ canbe rewritten as r ∗ = αF µ − b − ε )+ α (1 − F ) . Hence, r ∗ < αF < µ − b − ε ) + α (1 − F ) ⇔ α < µ − b − ε )2 F − − b, which is true since α < − b − s and s ≤ 1. In addition, r ∗ > − s c = α − b is equivalentto (4 − b ) F > µ − b − ε ) + α (1 − F ) ⇔ α < − b. Finally, it is easy to see that ( a ∗ , r ∗ ) satisfy the condition (2.2). Lemma 4.5. Let N = 3 , − b < α < − b − s and < b < . If s c < s ≤ then thereexists µ ∈ ( b, such that (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) (cid:0) k D s u k S ( L ) + k u k S ( L ) (cid:1) + c k u k − µL ∞ t H sx k u k θS ( ˙ H sc ) k D s u k α − θ + µS ( L ) , (4.32) where c > , θ = αF with F = − ε + µ − b − b and ε > is a sufficiently small number. We see that α − b = − s c < p < 6, i.e., p satisfies the condition (2.3) (and therefore (2.1),since − s c > 2) for N = 3. It is easy to see that F ∈ (cid:0) , (cid:1) if ε < µ − b . Therefore, since θ = F α , we have θ < α . Proof. Observe that (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ) ≤ (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ( B )) + (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ( B C )) . Let A ⊂ R N that can be B or B C . Since (2 , 6) is L -admissible in 3D we have (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ( A )) ≤ (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ′ t L ′ x ( A ) . As before, applying the fractional product rule (Lemma 2.3) we have (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ′ x ( A ) ≤ M ( t, A ) + M ( t, A ) , (4.33)where M ( t, A ) = (cid:13)(cid:13) | x | − b (cid:13)(cid:13) L γ ( A ) k D s ( | u | α u ) k L βx , M ( t, A ) = (cid:13)(cid:13) D s ( | x | − b ) (cid:13)(cid:13) L d ( A ) k| u | α u k L ex and 16 ′ = 1 γ + 1 β = 1 d + 1 e . (4.34)Estimating M ( t, A ). It follows by the fractional chain rule (Lemma 2.4) andH¨older’s inequality that M ( t, A ) ≤ k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL ( α − θ ) r x k D s u k L px = k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL px k D s u k L px , (4.35)where 1 β = 1 r + 1 r + 1 p and p = ( α − θ ) r . (4.36)Combining (4.34) and (4.36) we obtain3 γ = 52 − r − α + 1 − θ ) p , which implies, by (4.28) 3 γ − b = θ (2 − b ) α − r . (4.37)In to order to show that k| x | − b k L γ ( A ) is finite we need to verify that γ − b > A = B and γ − b < A = B C , by Remark 2.8. Indeed if θr = − s , by (4.37),we have 3 γ − b = θ ( s − s c ) > θr = 2 then 3 γ − b = − θs c < . Therefore, the inequality (4.35) and the Sobolev embedding (2.8) yield M ( t, A ) ≤ c k u k θH sx k u k α − θL px k D s u k L px . (4.38)We now estimate M ( t, A ). Let A = B C , applying the H¨older inequality and(3.8) we have M ( t, B C ) ≤ k| x | − b − s k L d ( B C ) k u k θL θr x k u k α − θL ( α − θ ) r x k u k L px ≤ k| x | − b − s k L d ( B C ) k u k θL θr x k u k α − θL px k u k L px , where 1 e = 1 r + 1 r + 1 p and p = ( α − θ ) r . N WELL POSEDNESS FOR THE INLS EQUATIONS 27 The relation (4.34) and the last relation imply3 d = 52 − r − α + 1 − θ ) p . In view of (4.28) we deduce 3 d − b = θ (2 − b ) α − r . Setting θr = 2 we have d − b = − θs c , so d − b − s = − θs c − s < 0, i.e., | x | − b − s ∈ L d ( B C ). So, by the Sobolev inequality (2.8) M ( t, B C ) ≤ c k u k θH sx k u k α − θL px k u k L px . (4.39)We also deduce from the H¨older inequality, the Sobolev embedding (2.6) and(3.8) M ( t, B ) ≤ k| x | − b − s k L d ( B ) k u k θL θr x k u k α − θL ( α − θ ) r x k u k µL µr x k u k − µL (1 − µ ) r x ≤ k| x | − b − s k L d ( B ) k u k θL θr x k D s u k α − θL nx k D s u k µL nx k u k − µL (1 − µ ) r x = k| x | − b − s k L d ( B ) k u k θL r ∗ x k D s u k α − θ + µL nx k u k − µL (1 − µ ) r x , if the following system is satisfied e = r + r + r + r s = n − α − θ ) r s = n − µr r ∗ = θr . It follows from (4.34) and the previous system that3 d = 52 + sD − θr ∗ − Dn − r , (4.40)which implies by (4.30) and (4.31)3 d = 72 + sD − (2 − b ) θα − D − r , (4.41)where D = α − θ + µ . In view of Remark 2.8 to show that k| x | − b − s k L d ( B ) is boundedwe need d − b − s > 0. In fact, choosing (1 − µ ) r = − s we have3 d − b − s = 2 − b − α θ s ( α − θ ) − (2 − b ) θα = − α (cid:18) − − bα (cid:19) + θ (cid:18) − − bα (cid:19) + s ( α − θ )= ( s − s c )( α − θ ) , which is positive since s > s c . So | x | − b − s ∈ L d ( B ) and M ( t, B ) ≤ c k u k − µH sx k u k θL r ∗ x k D s u k α − θ + µL nx . (4.42)where we have used the Sobolev embedding (2.8).Therefore, combining (4.33), (4.38) with A = B C and (4.39) we obtain (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ′ x ( B C ) ≤ c k u k θH sx k u k α − θL px k D s u k L px + c k u k θH sx k u k α − θL px k u k L px . We can use the Sobolev embedding (2.6) since s ≤ < n . Moreover by (4.38) with A = B and (4.42) we have (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ′ x ( B ) ≤ c k u k θH sx k u k α − θL px k u k L px + c k u k − µH sx k u k θL r ∗ x k D s u k α − θ + µL nx . Finally, since 12 ′ = α − θk + 1 l and 12 ′ = θa ∗ + α − θ + µm , we can use H¨older’s inequality in the time variable in the last two inequalities toconclude (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ′ t L ′ x ( B C ) ≤ c k u k θL ∞ t H sx k u k α − θL kt L px (cid:16) k D s u k L lt L px + k u k L lt L px (cid:17) and (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ′ t L ′ x ( B ) ≤ c k u k θL ∞ t H sx k u k α − θL kt L px k D s u k L lt L px + c k u k − µL ∞ t H sx k u k θL a ∗ t L r ∗ x k D s u k α − θ + µL mt L nx , The proof is completed recalling that ( m, n ) and ( l, p ) are L -admissible as wellas ( k, p ) and ( a ∗ , r ∗ ) are ˙ H s c -admissible. (cid:3) Remark 4.6. It is worth to mention that in the previous lemma θ > is given by θ = F α and since F < , we only have that θ < α and it might be not true that θ is close to . Before proving our global well-posedness result, we finish estimating the norm (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ) in the dimensions N = 1 , Lemma 4.7. Let N = 1 , and − bN < α < α s with < b < e . If s c < s ≤ min { N , } then (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) S ′ ( L ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k D s u k S ( L ) + c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) , (4.43) where c > and θ ∈ (0 , α ) is a sufficiently small number.Proof. The proof follows from analogous arguments as the ones used in the previouslemmas. Let A ⊂ R N that can be B or B C and ( q, r ) any L -admissible pair. Bythe fractional product rule (Lemma 2.3) we get (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L r ′ x ( A ) ≤ P ( t, A ) + P ( t, A ) , (4.44)where P ( t, A ) = (cid:13)(cid:13) | x | − b (cid:13)(cid:13) L γ ( A ) k D s ( | u | α u ) k L βx , P ( t, A ) = (cid:13)(cid:13) D s ( | x | − b ) (cid:13)(cid:13) L d ( A ) k| u | α u k L ex (4.45)and 1 r ′ = 1 γ + 1 β = 1 d + 1 e . (4.46)To estimate P ( t, A ) and P ( t, A ), we consider three cases: N = 1 and s < ; N = 2 and s < N = 1 , s = N . N WELL POSEDNESS FOR THE INLS EQUATIONS 29 Case N = 1 and s < . We define the following numbers k ∗ = 4 α ( α + 1 − θ )(4 − b )( α − θ + 1) − α l ∗ = 4( α + 1 − θ ) α − θ p ∗ = 2( α + 1 − θ ) (4.47) q = 2 ααb + θ (2 − b ) , and r = 2 αα (1 − b ) − θ (4 − b ) . (4.48)It is straightforward to verify that, if θ > < b < implies that the denominators of q , r , k ∗ and l ∗ are allpositive numbers. Furthermore, ( q , r ), ( l ∗ , p ∗ ) are L -admissible and ( k ∗ , p ∗ ) is˙ H s c -admissible.First, we estimate P ( t, A ) with r = r . The fractional chain rule (Lemma 2.4)and H¨older’s inequality yield P ( t, A ) ≤ k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL ( α − θ ) r x k D s u k L p ∗ x = k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL p ∗ x k D s u k L p ∗ x , (4.49)where 1 β = 1 r + 1 r + 1 p ∗ and p ∗ = ( α − θ ) r . (4.50)This implies 1 γ − b = θ (2 − b ) α − r , (4.51)where we have used (4.46), (4.50), (4.47) and (4.48). Now, if A = B and setting θr = − s we get γ − b = θ ( s − s c ) > 0, furthermore, taking A = B C and choosing θr = 2 one has γ − b = − θs c < 0. Hence, from the Sobolev embedding (2.8) andRemark 2.8 P ( t, A ) ≤ c k u k θH sx k u k α − θL p ∗ x k D s u k L p ∗ x . (4.52)We now consider P ( t, A ) with r = r . It follows from (4.45) and (3.8) that P ( t, A ) ≤ k| x | − b − s k L d ( A ) k u k θ +1 L ( θ +1) ex k u k α − θL ∞ x (4.53)and by (4.46) 1 d − b = 12 + θ (2 − b ) α − e . (4.54)We claim that k| x | − b − s k L d ( A ) is a finite quantity for a suitable choice of e . If A = B we choose ( θ + 1) e = − s , and if A = B C we set ( θ + 1) e = 2. We obtain in thefirst case 1 d − b − s = θ ( s − s c ) > , and in the second case 1 γ − b − s = − θs c < . So, the Sobolev embedding (2.8), Remark 2.8 and (4.53) yield P ( t, A ) ≤ c k u k θ +1 H sx k u k α − θL ∞ x . Note that, r > N = 1). Moreover, since 0 < b < we have p ∗ ≥ − s c = α − b (see (2.2) for N = 1). Since θr ∈ [2 , − s ] in both cases. Therefore, relations (4.44), (4.52) and the last inequality with A = B and A = B C imply that (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L r ′ x ( B ) ≤ c k u k θH sx k u k α − θL p ∗ x k D s u k L p ∗ x + c k u k θ +1 H sx k u k α − θL ∞ x and (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L r ′ x ( B C ) ≤ c k u k θH sx k u k α − θL p ∗ x k D s u k L p ∗ x + c k u k θ +1 H sx k u k α − θL ∞ x . Finally since 1 q ′ = α − θk ∗ + 1 l ∗ we apply the H¨older inequality in the time variable to get (recalling ( l ∗ , p ∗ ) is L -admissible and ( k ∗ , p ∗ ) is ˙ H s c -admissible) (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L q ′ t L r ′ x ≤ c k u k θL ∞ t H sx k u k α − θL k ∗ t L p ∗ x k D s u k L l ∗ t L p ∗ x + c k u k θ +1 L ∞ t H sx k u k α − θL ( α − θ ) q ′ t L ∞ x ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k D s u k S ( L ) + c k u k θ +1 L ∞ t H sx k u k α − θS ( ˙ H sc ) . where we have used the fact that ( α − θ ) q ′ = − s c , by (4.48), and ( − s c , ∞ ) is˙ H s c -admissible. Case N = 2 and s < . We consider the following numbers e q = 2 αα [ b + 2 ε ( α − θ )] + θ (2 − b ) e r = 2 αα [1 − b − ε ( α − θ )] − θ (2 − b ) , (4.55) l = 2( α + 1 − θ )( α − θ )(1 − ε ) p = 2( α + 1 − θ )1 + 2 ε ( α − θ ) (4.56)and k = 2 α ( α + 1 − θ ) α [1 − b − ε ( α − θ )] + (2 − b )(1 − θ ) (4.57)Note that ( e q, e r ), ( l , p ) are L -admissible and ( k , p ) is ˙ H s c -admissible .Estimating P ( t, A ) (recall (4.45)-(4.46)) with r = e r . The fractional chain rule(Lemma 2.4) and H¨older’s inequality lead to P ( t, A ) ≤ k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL ( α − θ ) r x k D s u k L p x = k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL p x k D s u k L p x , (4.58)where 1 β = 1 r + 1 r + 1 p and p = ( α − θ ) r , (4.59) The hypothesis 0 < b < N with N = 2 guarantee that the denominators of e q , e r , k , l and p are all positive numbers. Moreover, e r > α ( b + 2 ε ( α − θ )) > − θ (2 − b ) whichis true, therefore e r satisfies (2.1) for N = 2. We claim that α − b = − s c ≤ p ≤ (( − s c ) + ) ′ . Indeed, the first inequality is equivalent to α (1 − b ) + (1 − θ )(2 − b ) ≥ εα ( α − θ ) which holds true since ε > εp ≤ ( − s c ) + ( − s c ) (recall (2.4)) can be verifiedfor ε > N WELL POSEDNESS FOR THE INLS EQUATIONS 31 so by the relations (4.46), (4.59), (4.56) and (4.55) one has2 γ − b = θ (2 − b ) α − r . (4.60)As in the previous case, if A = B we set θr = − s and then γ − b > 0. On theother hand, if A = B C , we set θr = 2 and then γ − b < 0. Hence, the Sobolevembedding (2.8) and Remark 2.8 yield P ( t, A ) ≤ c k u k θH sx k u k α − θL p x k D s u k L p x . (4.61)Next we estimate P ( t, A ) with with r = e r . An application of the H¨olderinequality together with (4.45) and (3.8) imply P ( t, A ) ≤ k| x | − b − s k L d ( A ) k u k θ +1 L ( θ +1) r x k u k α − θL ( α − θ ) r x ≤ k| x | − b − s k L d ( A ) k u k θ +1 L ( θ +1) r x k u k α − θL εx , where 1 e = 1 r + 1 r , ( α − θ ) r = 1 ε . (4.62)We deduce from (4.62) and (4.46)2 d = 2 − e r − r − ε ( α − θ )= 1 + b + θ (2 − b ) α − r , (4.63)where we have used (4.55). In addition, if A = B and ( θ + 1) r = − s we get2 d − b − s = θ ( s − s c ) > , likewise if A = B C and ( θ + 1) r = 2, we have2 d − b − s = − θs c − s < . Thus P ( t, A ) ≤ c k u k θ +1 H sx k u k α − θL εx , where we have used the Sobolev inequality (2.8) and and Remark 2.8.Hence, by the relations (4.44), (4.61) and the last inequality (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L e rx ≤ c k u k θH sx k u k α − θL p x k D s u k L p x + c k u k θ +1 H sx k u k α − θL εx . Finally, from (4.55) and (4.57) 1 e q ′ = α − θk + 1 l , so applying the H¨older inequality in the time variable we deduce (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L e q ′ t L e r ′ x ≤ c k u k θL ∞ t H sx k u k α − θL k t L p x k D s u k L l t L p x + c k u k θ +1 L ∞ t H sx k u k α − θL ( α − θ ) e q ′ t L εx ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k D s u k S ( L ) + c k u k θ +1 L ∞ t H sx k u k α − θS ( ˙ H sc ) . where we have used the fact that ( α − θ ) e q ′ = α − b − εα and (cid:16) α − b − εα , ε (cid:17) is ˙ H s c -admissible. Case N = 1 , and s = N . As before, we start defining the following numbers¯ a = 2( α + 1 − θ )2 − s c ¯ q = 2( α + 1 − θ )2 + s c ( α − θ ) (4.64)¯ r = 2 N ( α + 1 − θ ) N ( α + 1 − θ ) − s c ( α − θ ) − k = 2( α + 1 − θ ) α − θ )(1 − s c ) − s c ¯ l = 2( α + 1 − θ ) α − θ )(1 − s c ) + s c (( α + 1 − θ ) − 1) (4.66)¯ p = 2 N ( α + 1 − θ ) ( N − s c )( α + 1 − θ ) − α − θ )(1 − s c ) + 2 s c . (4.67)It is not difficult to check that (¯ q, ¯ r ) and (¯ l, ¯ p ) L -admissible and (¯ a, ¯ r ), (¯ k, ¯ p ) ˙ H s c -admissible. First, we estimate P ( t, A ) with r = ¯ r . The fractional chain rule (Lemma 2.4)and H¨older’s inequality lead to P ( t, A ) ≤ k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL ( α − θ ) r x k D s u k L ¯ px = k| x | − b k L γ ( A ) k u k θL θr x k u k α − θL ¯ px k D s u k L ¯ px , (4.68)where 1 β = 1 r + 1 r + 1¯ p and ¯ p = ( α − θ ) r , (4.69)and so combining (4.46), (4.69) (4.65) and (4.67) we obtain Nγ − b = N − b − Nr − N ¯ r − N ( α + 1 − θ )¯ p = N − b − Nr − (cid:18) ( α + 1 − θ )( N − s c ) + N − − s c )2 (cid:19) = θ (2 − b ) α − Nr . (4.70)In order to have that the first norm in the right hand side of (4.68) is finite, weneed to verify Nγ − b > A = B and Nγ − b < A = B C for suitable choices of r . To this end, we set r such that θr > N α (2 − b ) (when A = B ) and 2 < θr < N α (2 − b ) (when A = B C ) (4.71) It is easy to see that the denominators of ¯ a and ¯ q are positive numbers (since s c < α > θ ). Furthermore, the denominators of ¯ r, ¯ k, ¯ l and ¯ p are also positive numbers for θ > b < N . We also have ¯ r, ¯ p ≥ NN − s c = Nα − b . Indeed ¯ r = N ( α +1 − θ ) N − b − θ ( N − s c ) ≥ Nα − b ⇔ α (4 − N ) + (1 − θ )(4 − b ) > − θα ( N − s c ) which is true since N = 1 , θ < p ≥ Nα − b is equivalent to 2( α − θ )(4 − b − α ( N − ≥ Nα − (4 − b ) so α (2(4 − b ) − N − α ( N − − b ) ≥ θ (4 − b − α ( N − , this is true since θ small enough, N = 1 , b < N . N WELL POSEDNESS FOR THE INLS EQUATIONS 33 Hence, the Sobolev embedding (2.7) and (4.68) yield P ( t, A ) ≤ c k u k θH sx k u k α − θL ¯ px k D s u k L ¯ px . (4.72)We now consider P ( t, A ) with r = ¯ r . By the H¨older inequality and (4.45) P ( t, A ) ≤ k| x | − b − s k L d ( A ) k u k θ +1 L ( θ +1) r x k u k α − θL ( α − θ ) r x = k| x | − b − s k L d ( A ) k u k θ +1 L ( θ +1) r x k u k α − θL ¯ rx , (4.73)where 1 e = 1 r + 1 r and ¯ r = ( α − θ ) r . (4.74)The relations (4.46) and (4.74) as well as ¯ r defined in (4.65), yield (recall s = N ) Nd − b − s = N − b − s − Nr − N ( α + 1 − θ )¯ r = N − b ) − Nr − N ( α + 1 − θ )2 + s c ( α − θ )= θ (2 − b ) α − Nr . (4.75)We see that the right hand side of (4.75) is equal to the right hand side of (4.70),so choosing r as in (4.71) and again applying the Sobolev inequality (2.7), weconclude P ( t, A ) ≤ c k u k θ +1 H sx k u k α − θL ¯ rx . The inequalities (4.44), (4.72) and the last inequality imply that (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ¯ r ′ x ≤ c k u k θH sx k u k α − θL ¯ px k D s u k L ¯ px + c k u k θ +1 H sx k u k α − θL ¯ rx . Since 1¯ q ′ = α − θ ¯ k + 1¯ l we can apply the H¨older inequality in the time variable to deduce (cid:13)(cid:13) D s (cid:0) | x | − b | u | α u (cid:1)(cid:13)(cid:13) L ¯ q ′ t L ¯ r ′ x ≤ c k u k θL ∞ t H sx k u k α − θL ¯ kt L ¯ px k D s u k L ¯ lt L ¯ px + c k u k θ +1 L ∞ t H sx k u k α − θL ( α − θ )¯ q ′ t L ¯ rx ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k D s u k S ( L ) + c k u k θ +1 L ∞ t H sx k u k α − θL ¯ at L ¯ rx , where in the last equality we have used the fact that ¯ a = ( α − θ )¯ q ′ . This completesthe proof since (¯ a, ¯ r ) ˙ H s c -admissible. (cid:3) The next result follows directly from Lemmas 4.3, 4.5 and 4.7. Corollary 4.8. Assume − bN < α < α s and < b < e . If s c < s ≤ min { N , } then following statement hold: k D s F k S ′ ( L ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) (cid:0) k D s u k S ( L ) + k u k S ( L ) + k u k L ∞ t H sx (cid:1) + c k u k − µL ∞ t H sx k u k θS ( ˙ H sc ) k D s u k α − θ + µS ( L ) , where F ( x, u ) = | x | − b | u | α u . Now, we have all the tools to prove the Theorem 1.8. Similarly as in the localtheory, we use the contraction mapping principle. Proof of Theorem 1.8. First, we define B = { u : k u k S ( ˙ H sc ) ≤ k U ( t ) u k S ( ˙ H sc ) and k u k S ( L ) + k D s u k S ( L ) ≤ c k u k H s } . We prove that G = G u defined in (1.2) is a contraction on B equipped with themetric d ( u, v ) = k u − v k S ( L ) + k u − v k S ( ˙ H sc ) . Indeed, we deduce by the Strichartz inequalities (2.9), (2.10), (2.11) and (2.12) k G ( u ) k S ( ˙ H sc ) ≤ k U ( t ) u k S ( ˙ H sc ) + c k F k S ′ ( ˙ H − sc ) (4.76) k G ( u ) k S ( L ) ≤ c k u k L + c k F k S ′ ( L ) (4.77)and k D s G ( u ) k S ( L ) ≤ c k D s u k L + c k D s F k S ′ ( L ) , (4.78)where F ( x, u ) = | x | − b | u | α u . On the other hand, it follows from Lemmas 4.1 and4.2 together with Corollary 4.8 that k F k S ′ ( ˙ H − sc ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k u k S ( ˙ H sc ) k F k S ′ ( L ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k u k S ( L ) and k D s F k S ′ ( L ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) (cid:0) k D s u k S ( L ) + k u k S ( L ) + k u k L ∞ t H sx (cid:1) + c k u k − µL ∞ t H sx k u k θS ( ˙ H sc ) k D s u k α − θ + µS ( L ) . Combining (4.76)-(4.78) and the last inequalities, we get for u ∈ B k G ( u ) k S ( ˙ H sc ) ≤k U ( t ) u k S ( ˙ H sc ) + c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k u k S ( ˙ H sc ) ≤k U ( t ) u k S ( ˙ H sc ) + 2 α +1 c θ +1 k u k θH s k U ( t ) u k α − θ +1 S ( ˙ H sc ) . In addition, setting X = k D s u k S ( L ) + k u k S ( L ) + k u k L ∞ t H sx k G ( u ) k S ( L ) + k D s G ( u ) k S ( L ) ≤ c k u k H s + c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) X + c k u k − µL ∞ t H sx k u k θS ( ˙ H sc ) k D s u k α − θ + µS ( L ) ≤ c k u k H s + 2 α +2 c θ +2 k u k θ +1 H s k U ( t ) u k α − θS ( ˙ H sc ) +2 α +1 c α − θ +2 k u k α − θ +1 H s k U ( t ) u k θS ( ˙ H sc ) , where we have have used the fact that X ≤ c k u k H s since u ∈ B .Now if k U ( t ) u k S ( ˙ H sc ) < δ with δ ≤ min ( α − θ r c θ +1 α +1 A θ , α − θ r c θ +1 α +2 A θ , θ r c α − θ +1 α +1 A α − θ ) , (4.79)where A > k u k H s ≤ A , we get k G ( u ) k S ( ˙ H sc ) ≤ k U ( t ) u k S ( ˙ H sc ) and k G ( u ) k S ( L ) + k D s G ( u ) k S ( L ) ≤ c k u k H s , that is G ( u ) ∈ B . N WELL POSEDNESS FOR THE INLS EQUATIONS 35 To complete the proof we show that G is a contraction on B . From (2.13) andrepeating the above computations one has k G ( u ) − G ( v ) k S ( ˙ H sc ) ≤ c k F ( x, u ) − F ( x, v ) k S ( ˙ H − sc ) ≤ c (cid:13)(cid:13) | x | − b | u | α | u − v | (cid:13)(cid:13) S ( ˙ H − sc ) + (cid:13)(cid:13) | x | − b | v | α | u − v | (cid:13)(cid:13) S ( ˙ H − sc ) ≤ c k u k θL ∞ t H sx k u k α − θS ( ˙ H sc ) k u − v k S ( ˙ H sc ) + c k v k θL ∞ t H sx k v k α − θS ( ˙ H sc ) k u − v k S ( ˙ H sc ) which implies, taking u, v ∈ B k G ( u ) − G ( v ) k S ( ˙ H sc ) ≤ c (2 c ) θ k u k θH s α − θ k U ( t ) u k α − θS ( ˙ H sc ) k u − v k S ( ˙ H sc ) = 2 α +1 c θ +1 k u k θH s k U ( t ) u k α − θS ( ˙ H sc ) k u − v k S ( ˙ H sc ) . By similar arguments we also obtain k G ( u ) − G ( v ) k S ( L ) ≤ α +1 c θ +1 k u k θH s k U ( t ) u k α − θS ( ˙ H sc ) k u − v k S ( L ) . 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Comm.Math. Phys. , 87(4):567–576, 1982/83. CARLOS M. GUZM ´ANDepartment of Mathematics, University Federal of Minas Gerais, BRAZIL E-mail address ::