The dynamics of the 3D radial NLS with the combined terms
aa r X i v : . [ m a t h . A P ] N ov THE DYNAMICS OF THE 3D RADIAL NLS WITH THECOMBINED TERMS
CHANGXING MIAO, GUIXIANG XU, AND LIFENG ZHAO
Abstract.
In this paper, we show the scattering and blow-up result of the radialsolution with the energy below the threshold for the nonlinear Schr¨odinger equation(NLS) with the combined terms iu t + ∆ u = −| u | u + | u | u (CNLS)in the energy space H ( R ). The threshold is given by the ground state W for theenergy-critical NLS: iu t + ∆ u = −| u | u . This problem was proposed by Tao, Visanand Zhang in [37]. The main difficulty is the lack of the scaling invariance. Illuminatedby [17], we need give the new radial profile decomposition with the scaling parameter,then apply it into the scattering theory. Our result shows that the defocusing, ˙ H -subcritical perturbation | u | u does not affect the determination of the threshold of thescattering solution of (CNLS) in the energy space. Introduction
We consider the dynamics of the radial solutions for the nonlinear Schr¨odinger equa-tion (NLS) with the combined nonlinearities in H ( R ) ( iu t + ∆ u = f ( u ) + f ( u ) , , ( t, x ) ∈ R × R ,u (0) = u ( x ) ∈ H ( R ) . (1.1)where u : R × R C and f ( u ) = −| u | u , f ( u ) = | u | u . As we known, f has the˙ H -critical growth, f has the ˙ H -subcritical growth.The equation has the following mass and Hamiltonian quantities M ( u )( t ) = 12 Z R | u ( t, x ) | dx ; E ( u )( t ) = Z R |∇ u ( t, x ) | dx + F ( u ( t )) + F ( u ( t ))where F ( u ( t )) = − Z R | u ( t, x ) | dx, F ( u ( t )) = 14 Z R | u ( t, x ) | dx. They are con-served for the sufficient smooth solutions of (1.1).
Mathematics Subject Classification.
Primary: 35L70, Secondary: 35Q55.
Key words and phrases.
Blow up; Dynamics; Nonlinear Schr¨odinger Equation; Scattering; ThresholdEnergy.
In [37], Tao, Visan and Zhang made the comprehensive study of iu t + ∆ u = | u | u + | u | u in the energy space. They made use of the interaction Morawetz estimate establishedin [6] and the stability theory for the scattering solution. Their result is based onthe scattering result of the defocusing, energy-critical NLS in the energy space, whichis established by Bourgain [3, 4] for the radial case, I-team [7], Ryckman-Visan [34]and Visan [38] for the general data. Since the classical interaction Morawetz estimatein [6] fails for (1.1), Tao, et al., leave the scattering and blow-up dichotomy of (1.1)below the threshold as an open problem in [37]. For other results, please refer to[15, 16, 30, 31, 32, 39, 40].For the focusing, energy-critical NLS iu t + ∆ u = −| u | u. (1.2)Kenig and Merle first applied the concentration compactness in [2, 21, 22] into thescattering theory of the radial solution of (1.2) in [19] with the energy below that ofthe ground state of − ∆ W = | W | W. (1.3)In this paper, we will also make use of the concentration compactness argument and thestability theory to study the dichotomy of the radial solution of (1.1) with the energybelow the threshold, which will be shown to be the energy of the ground state W for(1.2). For the applications of the concentration compactness in the scattering theoryand rigidity theory of the critical NLS, NLW, NLKG and Hartree equations, please see[8, 9, 10, 11, 12, 13, 17, 20, 23, 24, 25, 26, 27, 28, 29].We now show the differences between (1.1) and (1.2). On one hand, there is anexplicit solution W for (1.2), which is the ground state of (1.3) and does not scatter.The threshold of the scattering solution of (1.2) is determined by the energy of W .While for (1.1), there is no such explicit solution, whose energy is the threshold of thescattering solution of (1.1). We need look for a mechanism to determine the thresholdof the scattering solution of (1.1). It turns out that the constrained minimization of theenergy as (1.5) is appropriate . On the other hand, for (1.2), it is ˙ H -scaling invariant,which gives us many conveniences, especially in the nonlinear profile decompositionabout (1.2). While for (1.1), it is the lack of scaling invariance. We need give the new The similar constrained minimization of the energy as (1.5) is not appropriate for the focusing per-turbation: iu t + ∆ u = −| u | u − | u | u , since the threshold m in this way equals to 0 and it is not thedesired result. LS WITH THE COMBINED TERMS 3 profile decomposition with the scaling parameter of (1.1) in H ( R ), take care of therole of the scaling parameter in the linear and nonlinear profile decompositions, thenapply them into the scattering theory.Now for ϕ ∈ H , we denote the scaling quantity ϕ λ , − by ϕ λ , − ( x ) = e λ ϕ ( e λ x ) . We denote the scaling derivative of E by K ( ϕ ) K ( ϕ ) = L E ( ϕ ) := ddλ (cid:12)(cid:12)(cid:12) λ =0 E ( ϕ λ , − ) = Z R (cid:18) |∇ ϕ | − | ϕ | + 64 | ϕ | (cid:19) dx, (1.4)which is connected with the Virial identity, and then plays the important role in theblow-up and scattering of the solution of (1.1).Now the threshold m is determined by the following constrained minimization of theenergy E ( ϕ ) m = inf { E ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) = 0 } . (1.5)Since we consider the ˙ H -critical growth with the ˙ H -subcritical perturbation, we willuse the modified energy later E c ( u ) = Z R (cid:18) |∇ u ( t, x ) | − | u ( t, x ) | (cid:19) dx. As the nonlinearity | u | u is the defocusing, ˙ H -subcritical perturbation, one thinkthat the focusing, ˙ H -critical term plays the decisive role of the threshold of the scatter-ing solution of (1.1) in the energy space. The first result is to characterize the thresholdenergy m as following Proposition 1.1.
There is no minimizer for (1.5) . But for the threshold energy m , wehave m = E c ( W ) , where W ∈ ˙ H ( R ) is the ground state of the massless equation − ∆ W = | W | W. As the dynamics of the solution of (1.1) with the energy less than the threshold m ,the conjecture is In fact, the following minimization of the static energyinf { M ( ϕ ) + E ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) = 0 } also equals to m . MIAO, XU, AND ZHAO
Conjecture 1.2.
Let u ∈ H ( R ) with E ( u ) < m, (1.6) and u be the solution of (1.1) and I be its maximal interval of existence. Then (a) If K ( u ) ≥ , then I = R , and u scatters in both time directions as t → ±∞ in H ; (b) If K ( u ) < , then u blows up both forward and backward at finite time in H . In this paper, we verify the conjecture in the radial case.
Theorem 1.3.
Conjecture 1.2 holds whenever u is spherically symmetric. Remark 1.4.
Our consideration of the radial case is based on the following facts:(1) It is an open problem that the scattering result of (1.2) in dimension three,except for the radial case in [19]. Our result is based on the correspondingscattering result of (1.2).(2) It seems to be hard to lower the regularity of the critical element to L ∞ ˙ H s forsome s < L , which is used to control the spatialcenter function x ( t ) of the critical element. Remark 1.5.
We can remove the radial assumption under the stronger constraint that M ( u ) + E ( u ) < m, which can help us to obtain the compactness of the critical element in L and control thespatial center function x ( t ) of the critical element. Of course, we need the precondition that the global wellposedness and scattering result of (1.2) holds for u ∈ ˙ H ( R ) with Z R (cid:16)(cid:12)(cid:12) ∇ u (cid:12)(cid:12) − (cid:12)(cid:12) u (cid:12)(cid:12) (cid:17) dx ≥ , Z R (cid:18) (cid:12)(cid:12) ∇ u (cid:12)(cid:12) − (cid:12)(cid:12) u (cid:12)(cid:12) (cid:19) dx LS WITH THE COMBINED TERMS 5 Remark 1.6. From the assumption in Theorem 1.3, we know that the solution startsfrom the following subsets of the energy space, K + = n ϕ ∈ H ( R ) (cid:12)(cid:12)(cid:12) ϕ is radial , E ( ϕ ) < m, K ( ϕ ) ≥ o , K − = n ϕ ∈ H ( R ) (cid:12)(cid:12)(cid:12) ϕ is radial , E ( ϕ ) < m, K ( ϕ ) < o . By the scaling argument, we know that K ± = ∅ (we can also know that K + = ∅ bythe small data theory). In fact, let χ ( x ) be a radial smooth cut-off function satisfying0 ≤ χ ≤ χ ( x ) = 1 for | x | ≤ χ ( x ) = 0 for | x | ≥ 2. If we take χ R ( x ) = χ ( x/R )and ϕ ( x ) = θλ − / χ R ( x/λ ) W ( x/λ ) , where θ, λ, R is determined later and the cutoff function χ R is not needed for dimension d ≥ W ∈ H . Then we have (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L = θ (cid:18)(cid:13)(cid:13) ∇ W (cid:13)(cid:13) L + Z (cid:16) ( χ R − (cid:12)(cid:12) ∇ W (cid:12)(cid:12) + |∇ χ R | | W | + 2 χ R ∇ χ R · W ∇ W (cid:17) dx (cid:19) , (cid:13)(cid:13) ϕ (cid:13)(cid:13) L = θ (cid:18)(cid:13)(cid:13) W (cid:13)(cid:13) L + Z ( χ R − | W | dx (cid:19) , (cid:13)(cid:13) ϕ (cid:13)(cid:13) L = λ · θ (cid:13)(cid:13) χ R W (cid:13)(cid:13) L , (cid:13)(cid:13) ϕ (cid:13)(cid:13) L = λ · θ (cid:13)(cid:13) χ R W (cid:13)(cid:13) L . Therefore, taking R sufficiently large, θ = 1 + ǫ and λ = ǫ , we have E ( ϕ ) = θ (cid:13)(cid:13) ∇ W (cid:13)(cid:13) L − θ (cid:13)(cid:13) W (cid:13)(cid:13) L + θ Z (cid:16) ( χ R − (cid:12)(cid:12) ∇ W (cid:12)(cid:12) + |∇ χ R | | W | + 2 χ R ∇ χ R · W ∇ W (cid:17) dx − θ Z ( χ R − | W | dx + λ · θ (cid:13)(cid:13) χ R W (cid:13)(cid:13) L = m − ǫ m + o ( ǫ ) ,K ( ϕ ) =2 θ (cid:13)(cid:13) ∇ W (cid:13)(cid:13) L − θ (cid:13)(cid:13) W (cid:13)(cid:13) L + 2 θ Z (cid:16) ( χ R − (cid:12)(cid:12) ∇ W (cid:12)(cid:12) + |∇ χ R | | W | + 2 χ R ∇ χ R · W ∇ W (cid:17) dx − θ Z ( χ R − | W | dx + λ · θ (cid:13)(cid:13) χ R W (cid:13)(cid:13) L = − ǫm + o ( ǫ ) . If taking ǫ < | ǫ | sufficient small, then we have ϕ ∈ K + ; If taking ǫ > ϕ ∈ K − . MIAO, XU, AND ZHAO Acknowledgements. The authors are partly supported by the NSF of China (No.10801015, No. 10901148, No. 11171033). The authors would like to thank ProfessorK. Nakanishi for his valuable communications. (cid:3) Preliminaries In this section, we give some notation and some wellknown results.2.1. Littlewood-Paley decomposition and Besov space. Let Λ ( x ) ∈ S ( R ) suchthat its Fourier transform e Λ ( ξ ) = 1 for | ξ | ≤ e Λ ( ξ ) = 0 for | ξ | ≥ 2. Then wedefine Λ k ( x ) for any k ∈ Z \{ } and Λ (0) ( x ) by the Fourier transforms: e Λ k ( ξ ) = e Λ (2 − k ξ ) − e Λ (2 − k +1 ξ ) , e Λ (0) ( ξ ) = e Λ ( ξ ) − e Λ (2 ξ ) . Let s ∈ R , 1 ≤ p, q ≤ ∞ . The inhomogeneous Besov space B sp,q is defined by B sp,q = (cid:26) f (cid:12)(cid:12) f ∈ S ′ , (cid:13)(cid:13)(cid:13) ks (cid:13)(cid:13) Λ k ∗ f (cid:13)(cid:13) L px (cid:13)(cid:13)(cid:13) l qk ≥ < ∞ (cid:27) , where S ′ denotes the space of tempered distributions. The homogeneous Besov space˙ B sp,q can be defined by˙ B sp,q = f (cid:12)(cid:12)(cid:12) f ∈ S ′ , X k ∈ Z \{ } qks (cid:13)(cid:13) Λ k ∗ f (cid:13)(cid:13) qL px + (cid:13)(cid:13) Λ (0) ∗ f (cid:13)(cid:13) L px (cid:13)(cid:13)(cid:13) q /q < ∞ . Linear estimates. We say that a pair of exponents ( q, r ) is Schr¨oidnger ˙ H s -admissible in dimension three if 2 q + 3 r = 32 − s and 2 ≤ q, r ≤ ∞ . If I × R is a space-time slab, we define the ˙ S ( I × R ) Strichartznorm by (cid:13)(cid:13) u (cid:13)(cid:13) ˙ S ( I × R ) := sup (cid:13)(cid:13) u (cid:13)(cid:13) L qt L rx ( I × R ) where the sup is taken over all L -admissible pairs ( q, r ). We define the ˙ S s ( I × R )Strichartz norm to be (cid:13)(cid:13) u (cid:13)(cid:13) ˙ S s ( I × R ) := (cid:13)(cid:13) D s u (cid:13)(cid:13) ˙ S ( I × R ) . We also use ˙ N ( I × R ) to denote the dual space of ˙ S ( I × R ) and˙ N k ( I × R ) := { u ; D k u ∈ ˙ N ( I × R ) } . By definition and Sobolev’s inequality, we have LS WITH THE COMBINED TERMS 7 Lemma 2.1. For any ˙ S function u on I × R , we have (cid:13)(cid:13) ∇ u (cid:13)(cid:13) L ∞ t L x + (cid:13)(cid:13) u (cid:13)(cid:13) L t ˙ B / / , ( I × R ) + (cid:13)(cid:13) u (cid:13)(cid:13) L ∞ t L x + (cid:13)(cid:13) u (cid:13)(cid:13) L t L x + (cid:13)(cid:13) u (cid:13)(cid:13) L t,x . (cid:13)(cid:13) u (cid:13)(cid:13) ˙ S . For any ˙ S / function u on I × R , we have (cid:13)(cid:13) u (cid:13)(cid:13) L ∞ t ˙ H / x + (cid:13)(cid:13) u (cid:13)(cid:13) L t ˙ B / / , ( I × R ) + (cid:13)(cid:13) u (cid:13)(cid:13) L ∞ t L x + (cid:13)(cid:13) u (cid:13)(cid:13) L t L / x + (cid:13)(cid:13) u (cid:13)(cid:13) L t,x . (cid:13)(cid:13) u (cid:13)(cid:13) ˙ S / . Now we state the standard Strichartz estimate. Lemma 2.2 ([5, 18, 36]) . Let I be a compact time interval, k ∈ [0 , , and let u : I × R → C be an ˙ S k solution to the forced Schr¨odinger equation iu t + ∆ u = F for a function F . Then we have (cid:13)(cid:13) u (cid:13)(cid:13) ˙ S k ( I × R ) . (cid:13)(cid:13) u ( t ) (cid:13)(cid:13) ˙ H k ( R d ) + (cid:13)(cid:13) F (cid:13)(cid:13) ˙ N k ( I × R ) , for any time t ∈ I . We shall also need the following exotic Strichartz estimate, which is important in theapplication of the stability theory. Lemma 2.3 ([14]) . For any F ∈ L t (cid:16) I ; ˙ B / / , (cid:17) , we have (cid:13)(cid:13)(cid:13)(cid:13)Z t e i ( t − s )∆ F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L t ˙ B / / , . (cid:13)(cid:13) F (cid:13)(cid:13) L t ˙ B / / , . Local wellposedness and Virial identity. Let ST ( I ) := L t ˙ B / / , ∩ L t L x ∩ L t ˙ B / / , ∩ L t,x ( I × R ) . By the definition of admissible pair, we know that L t ˙ B / / , ∩ L t L x is the ˙ H -admissible space, L t ˙ B / / , ∩ L t,x is the ˙ H / -admissible space. Now we have Theorem 2.4 ([37]) . Let u ∈ H , then for every η > , there exists T = T ( η ) suchthat if (cid:13)(cid:13) e it ∆ u (cid:13)(cid:13) ST ([ − T,T ]) ≤ η, then (1.1) admits a unique strong H x -solution u defined on [ − T, T ] . Let ( − T min , T max ) be the maximal time interval on which u is well-defined. Then, u ∈ S ( I × R d ) for everycompact time interval I ⊂ ( − T min , T max ) and the following properties hold:(1) If T max < ∞ , then (cid:13)(cid:13) u (cid:13)(cid:13) ST ((0 ,T max ) × R d ) = ∞ . MIAO, XU, AND ZHAO Similarly, if T min < ∞ , then (cid:13)(cid:13) u (cid:13)(cid:13) ST (( − T min , × R d ) = ∞ . (2) The solution u depends continuously on the initial data u in the followingsense: The functions T min and T max are lower semicontinuous from ˙ H x ∩ ˙ H / x to (0 , + ∞ ] . Moreover, if u ( m )0 → u in ˙ H x ∩ ˙ H / x and u ( m ) is the maximal solutionto (1.1) with initial data u ( m )0 , then u ( m ) → u in ST ( I × R ) and every compactsubinterval I ⊂ ( − T min , T max ) .Proof. The proof is based on the Strichartz estimate and exotic Strichartz estimate andthe following nonlinear estimates. (cid:13)(cid:13) | u | u (cid:13)(cid:13) L ˙ B / / , . (cid:13)(cid:13) u (cid:13)(cid:13) L t ˙ B / / , (cid:13)(cid:13) u (cid:13)(cid:13) L t,x , (cid:13)(cid:13) | u | u (cid:13)(cid:13) L ˙ B / / , . (cid:13)(cid:13) u (cid:13)(cid:13) L t ˙ B / / , (cid:13)(cid:13) u (cid:13)(cid:13) L t,x , (cid:13)(cid:13) | u | u (cid:13)(cid:13) L ˙ B / / , . (cid:13)(cid:13) u (cid:13)(cid:13) L t ˙ B / / , (cid:13)(cid:13) u (cid:13)(cid:13) L t L x , (cid:13)(cid:13) | u | u (cid:13)(cid:13) L ˙ B / / , . (cid:13)(cid:13) u (cid:13)(cid:13) L t ˙ B / / , (cid:13)(cid:13) u (cid:13)(cid:13) L t L / x . (cid:3) Lemma 2.5. Let φ ∈ C ∞ ( R ) , radially symmetric and u be the radial solution of (1.1) .Then we have ∂ t Z R φ ( x ) (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx = − ℑ Z R ∇ φ · ∇ ¯ u u dx∂ t Z R φ ( x ) (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx =4 Z R φ ′′ ( r ) (cid:12)(cid:12) ∇ u (cid:12)(cid:12) dx − Z R ∆ φ (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx − Z R ∆ φ (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx + Z R ∆ φ (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx, where r = | x | .Proof. By the simple computation, we have ∂ t Z R φ ( x ) (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx =4 Z R φ jk · ℜ ( u k u j ) dx − Z R ∆ φ · (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx − Z R ∆ φ · (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx + Z R ∆ φ · (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) dx. Then the result comes from the following fact ∂ jk φ ( x ) = φ ′′ ( r ) x j x k r + φ ′ ( r ) r (cid:16) δ jk − x j x k r (cid:17) holds for any radial symmetric function φ ( x ). (cid:3) Variational characterization. In this subsection, we give the threshold energy m (Proposition 1.1) by the variational method, and various estimates for the solutions LS WITH THE COMBINED TERMS 9 of (1.1) with the energy below the threshold. There is no the radial assumption on thesolution.We first give some notation before we show the behavior of K near the origin. Letus denote the quadratic and nonlinear parts of K by K Q and K N , that is, K ( ϕ ) = K Q ( ϕ ) + K N ( ϕ ) , where K Q ( ϕ ) = 2 Z R |∇ ϕ | dx, and K N ( ϕ ) = Z R (cid:18) − | ϕ | + 32 ϕ | (cid:19) dx . Lemma 2.6. For any ϕ ∈ H ( R ) , we have lim λ →−∞ K Q ( ϕ λ , − ) = 0 . (2.1) Proof. It is obvious by the definition of K Q . (cid:3) Now we show the positivity of K near 0 in the energy space. Lemma 2.7. For any bounded sequence ϕ n ∈ H ( R ) \{ } with lim n → + ∞ K Q ( ϕ n ) = 0 , then for large n , we have K ( ϕ n ) > . Proof. By the fact that K Q ( ϕ n ) → 0, we know that lim n → + ∞ (cid:13)(cid:13) ∇ ϕ n (cid:13)(cid:13) L = 0 . Then by theSobolev and Gagliardo-Nirenberg inequalities, we have for large n (cid:13)(cid:13) ϕ n (cid:13)(cid:13) L x . (cid:13)(cid:13) ∇ ϕ n (cid:13)(cid:13) L x = o ( (cid:13)(cid:13) ∇ ϕ n (cid:13)(cid:13) L ) , (cid:13)(cid:13) ϕ n (cid:13)(cid:13) L x . (cid:13)(cid:13) ϕ n (cid:13)(cid:13) L (cid:13)(cid:13) ∇ ϕ n (cid:13)(cid:13) L = o ( (cid:13)(cid:13) ∇ ϕ n (cid:13)(cid:13) L ) , where we use the boundedness of (cid:13)(cid:13) ϕ n (cid:13)(cid:13) L . Hence for large n , we have K ( ϕ n ) = Z R (cid:18) |∇ ϕ n | − | ϕ n | + 32 | ϕ n | (cid:19) dx ≈ Z R |∇ ϕ n | dx > . This concludes the proof. (cid:3) By the definition of K , we denote two real numbers by¯ µ = max { , , } = 6 , µ = min { , , } = 0 . Next, we show the behavior of the scaling derivative functional K . Lemma 2.8. For any ϕ ∈ H , we have (¯ µ − L ) E ( ϕ ) = Z R (cid:16)(cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx, L (¯ µ − L ) E ( ϕ ) = Z R (cid:16) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 12 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx. Proof. By the definition of L , we have L (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L = 4 (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L , L (cid:13)(cid:13) ϕ (cid:13)(cid:13) L = 12 (cid:13)(cid:13) ϕ (cid:13)(cid:13) L , L (cid:13)(cid:13) ϕ (cid:13)(cid:13) L = 6 (cid:13)(cid:13) ϕ (cid:13)(cid:13) L , which implies that(¯ µ − L ) E ( ϕ ) = 6 E ( ϕ ) − K ( ϕ ) = Z R (cid:16)(cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx, L (¯ µ − L ) E ( ϕ ) = L (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L + L (cid:13)(cid:13) ϕ (cid:13)(cid:13) L = Z R (cid:16) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 12 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx. This completes the proof. (cid:3) According to the above analysis, we will replace the functional E in (1.5) with apositive functional H , while extending the minimizing region from “ K ( ϕ ) = 0” to“ K ( ϕ ) ≤ H ( ϕ ) := (cid:18) − L ¯ µ (cid:19) E ( ϕ ) = Z R (cid:18) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 16 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx, then for any ϕ ∈ H \{ } , we have H ( ϕ ) > , L H ( ϕ ) ≥ . Now we can characterization the minimization problem (1.5) by use of H . Lemma 2.9. For the minimization m in (1.5) , we have m = inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) ≤ } = inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) < } . (2.2) Proof. For any ϕ ∈ H , ϕ = 0 with K ( ϕ ) = 0, we have E ( ϕ ) = H ( ϕ ), this implies that m = inf { E ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) = 0 }≥ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) ≤ } . (2.3)On the other hand, for any ϕ ∈ H , ϕ = 0 with K ( ϕ ) < 0, by Lemma 2.6, Lemma2.7 and the continuity of K in λ , we know that there exists a λ < K ( ϕ λ , − ) = 0 , then by L H ≥ 0, we have E ( ϕ λ , − ) = H ( ϕ λ , − ) ≤ H ( ϕ , − ) = H ( ϕ ) . LS WITH THE COMBINED TERMS 11 Therefore, inf { E ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) = 0 }≤ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) < } . (2.4)By (2.3) and (2.4), we haveinf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) ≤ }≤ m ≤ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) < } . In order to show (2.2), it suffices to show thatinf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) ≤ }≥ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) < } . (2.5)For any ϕ ∈ H , ϕ = 0 with K ( ϕ ) ≤ 0. By Lemma 2.8, we know that L K ( ϕ ) = ¯ µK ( ϕ ) − Z R (cid:16) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 12 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx < , then for any λ > K ( ϕ λ , − ) < , and as λ → H ( ϕ λ , − ) = Z R (cid:18) e λ (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + e λ (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx −→ H ( ϕ ) . This shows (2.5), and completes the proof. (cid:3) Next we will use the ( ˙ H -invariant) scaling argument to remove the L term (thelower regularity quantity than ˙ H ) in K , that is, to replace the constrained condition K ( ϕ ) < K c ( ϕ ) < 0, where K c ( ϕ ) := Z R (cid:0) |∇ ϕ | − | ϕ | (cid:1) dx. In fact, we have Lemma 2.10. For the minimization m in (1.5) , we have m = inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) < } = inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) ≤ } . Proof. Since K c ( ϕ ) ≤ K ( ϕ ), it is obvious that m = inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) < } ≥ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) < } . Hence in order to show the first equality, it suffices to show thatinf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) < }≤ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) < } . (2.6)To do so, for any ϕ ∈ H , ϕ = 0 with K c ( ϕ ) < 0, taking ϕ λ , − ( x ) = e λ ϕ ( e λ x ) , we have ϕ λ , − ∈ H and ϕ λ , − = 0 for any λ > 0. In addition, we have K ( ϕ λ , − ) = Z R (cid:18) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) − (cid:12)(cid:12) ϕ (cid:12)(cid:12) + 32 e − λ (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx −→ K c ( ϕ ) ,H ( ϕ λ , − ) = Z R (cid:18) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 16 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx = H ( ϕ ) , as λ → + ∞ . This gives (2.6), and completes the proof of the first equality.For the second equality, it is obvious thatinf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) < }≥ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) ≤ } , hence we only need to show thatinf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) < }≤ inf { H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K c ( ϕ ) ≤ } . (2.7)To do this, we use the ( L -invariant) scaling argument. For any ϕ ∈ H , ϕ = 0 with K c ( ϕ ) ≤ 0, we have ϕ λ , − ∈ H , ϕ λ , − = 0. In addition, by L K c ( ϕ ) = Z R (cid:16) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) − (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx = 4 K c ( ϕ ) − (cid:13)(cid:13) ϕ (cid:13)(cid:13) L < ,H ( ϕ λ , − ) = Z R (cid:18) e λ (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + e λ (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx, we have K c ( ϕ λ , − ) < λ > 0, and H ( ϕ λ , − ) → H ( ϕ ) , as λ → . This implies (2.7) and completes the proof. (cid:3) After these preparations, we can now make use of the sharp Sobolev constant in[1, 35] to compute the minimization m of (1.5), which also shows Proposition 1.1. LS WITH THE COMBINED TERMS 13 Lemma 2.11. For the minimization m in (1.5) , we have m = E c ( W ) . Proof. By Lemma 2.10, we have m = inf (cid:26) Z R (cid:0) |∇ ϕ | + | ϕ | (cid:1) dx (cid:12)(cid:12)(cid:12) ϕ ∈ H , ϕ = 0 , (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L ≤ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L (cid:27) ≥ inf (cid:26)Z R (cid:0) |∇ ϕ | + | ϕ | (cid:1) + 16 (cid:0) |∇ ϕ | − | ϕ | (cid:1) dx (cid:12)(cid:12)(cid:12) ϕ ∈ H , ϕ = 0 , (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L ≤ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L (cid:27) where the equality holds if and only if the minimization is taken by some ϕ with (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L = (cid:13)(cid:13) ϕ (cid:13)(cid:13) L . Whileinf (cid:26)Z R |∇ ϕ | dx (cid:12)(cid:12) ϕ ∈ H , ϕ = 0 , (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L ≤ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L (cid:27) = inf (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L (cid:13)(cid:13) ϕ (cid:13)(cid:13) L ! / (cid:12)(cid:12)(cid:12) ϕ ∈ H , ϕ = 0 = inf (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L (cid:13)(cid:13) ϕ (cid:13)(cid:13) L ! (cid:12)(cid:12)(cid:12) ϕ ∈ H , ϕ = 0 = inf (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L (cid:13)(cid:13) ϕ (cid:13)(cid:13) L ! (cid:12)(cid:12)(cid:12) ϕ ∈ ˙ H , ϕ = 0 = 13 (cid:0) C ∗ (cid:1) − . where we use the density property H ֒ → ˙ H in the last second equality and that C ∗ isthe sharp Sobolev constant in R , that is, (cid:13)(cid:13) ϕ (cid:13)(cid:13) L x ≤ C ∗ (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L x , ∀ ϕ ∈ ˙ H ( R ) , and the equality can be attained by the ground state W of the following elliptic equation − ∆ W = | W | W. This implies that (cid:0) C ∗ (cid:1) − = E c ( W ). The proof is completed. (cid:3) After the computation of the minimization m in (1.5), we next give some variationalestimates. Lemma 2.12. For any ϕ ∈ H with K ( ϕ ) ≥ , we have Z R (cid:18) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 16 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx ≤ E ( ϕ ) ≤ Z R (cid:18) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 14 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx. (2.8) Proof. On one hand, the right hand side of (2.8) is trivial. On the other hand, by thedefinition of E and K , we have E ( ϕ ) = Z R (cid:18) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 16 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:19) dx + 16 K ( ϕ ) , which implies the left hand side of (2.8). (cid:3) At the last of this section, we give the uniform bounds on the scaling derivativefunctional K ( ϕ ) with the energy E ( ϕ ) below the threshold m , which plays an importantrole for the blow-up and scattering analysis in Section 3 and Section 6. Lemma 2.13. For any ϕ ∈ H with E ( ϕ ) < m .(1) If K ( ϕ ) < , then K ( ϕ ) ≤ − (cid:0) m − E ( ϕ ) (cid:1) . (2.9) (2) If K ( ϕ ) ≥ , then K ( ϕ ) ≥ min (cid:18) m − E ( ϕ )) , (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L + 12 (cid:13)(cid:13) ϕ (cid:13)(cid:13) L (cid:19) . (2.10) Proof. By Lemma 2.8, for any ϕ ∈ H , we have L E ( ϕ ) = ¯ µ L E ( ϕ ) − (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L − (cid:13)(cid:13) ϕ (cid:13)(cid:13) L . Let j ( λ ) = E ( ϕ λ , − ), then we have j ′′ ( λ ) = ¯ µj ′ ( λ ) − e λ (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L − e λ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L . (2.11) Case I: If K ( ϕ ) < 0, then by (2.1), Lemma 2.7 and the continuity of K in λ , thereexists a negative number λ < K ( ϕ λ , − ) = 0, and K ( ϕ λ , − ) < , ∀ λ ∈ ( λ , . By (1.5), we obtain j ( λ ) = E ( ϕ λ , − ) ≥ m . Now by integrating (2.11) over [ λ , Z λ j ′′ ( λ ) dλ ≤ ¯ µ Z λ j ′ ( λ ) dλ, which implies that K ( ϕ ) = j ′ (0) − j ′ ( λ ) ≤ ¯ µ ( j (0) − j ( λ )) ≤ − ¯ µ (cid:0) m − E ( ϕ ) (cid:1) , which implies (2.9). Case II: K ( ϕ ) ≥ 0. We divide it into two subcases: LS WITH THE COMBINED TERMS 15 When 2¯ µK ( ϕ ) ≥ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L . Since12 Z R (cid:12)(cid:12) ϕ (cid:12)(cid:12) dx = − K ( ϕ ) + Z R (cid:16) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 9 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx, then we have 2¯ µK ( ϕ ) ≥ − K ( ϕ ) + Z R (cid:16) (cid:12)(cid:12) ∇ ϕ (cid:12)(cid:12) + 9 (cid:12)(cid:12) ϕ (cid:12)(cid:12) (cid:17) dx, which implies that K ( ϕ ) ≥ (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L + 12 (cid:13)(cid:13) ϕ (cid:13)(cid:13) L . When 2¯ µK ( ϕ ) ≤ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L . By (2.11), we have for λ = 00 < µj ′ ( λ ) < e λ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L ,j ′′ ( λ ) = ¯ µj ′ ( λ ) − e λ (cid:13)(cid:13) ∇ ϕ (cid:13)(cid:13) L − e λ (cid:13)(cid:13) ϕ (cid:13)(cid:13) L ≤ − ¯ µj ′ ( λ ) . (2.12)By the continuity of j ′ and j ′′ in λ , we know that j ′ is an accelerating decreasing functionas λ increases until j ′ ( λ ) = 0 for some finite number λ > , λ ].By K ( ϕ λ , − ) = j ′ ( λ ) = 0 , we know that E ( ϕ λ , − ) ≥ m. Now integrating (2.12) over [0 , λ ], we obtain that − K ( ϕ ) = j ′ ( λ ) − j ′ (0) ≤ − ¯ µ (cid:0) j ( λ ) − j (0) (cid:1) ≤ − ¯ µ ( m − E ( ϕ )) . This completes the proof. (cid:3) Part I: Blow up for K − In this section, we prove the blow-up result of Theorem 1.3. We can also refer to[33]. Now let φ be a smooth, radial function satisfying ∂ r φ ( r ) ≤ φ ( r ) = r for r ≤ φ ( r ) is constant for r ≥ 3. For some R , we define V R ( t ) := Z R φ R ( x ) | u ( t, x ) | dx, φ R ( x ) = R φ (cid:18) | x | R (cid:19) . By Lemma 2.5, ∆ φ R ( r ) = 6 for r ≤ R, and ∆ φ R ( r ) = 0 for r ≤ R, we have ∂ t V R ( t ) = 4 Z R φ ′′ R ( r ) (cid:12)(cid:12) ∇ u ( t, x ) (cid:12)(cid:12) dx − Z R (∆ φ R )( x ) | u ( t, x ) | dx − Z R (∆ φ R ) | u ( t, x ) | dx + Z R (∆ φ R ) | u ( t, x ) | dx ≤ Z R (cid:18) |∇ u ( t ) | − | u ( t ) | + 32 | u ( t ) | (cid:19) dx + cR Z R ≤| x |≤ R (cid:12)(cid:12) u ( t ) (cid:12)(cid:12) dx + c Z R ≤| x |≤ R (cid:16)(cid:12)(cid:12) u ( t ) (cid:12)(cid:12) + (cid:12)(cid:12) u ( t ) (cid:12)(cid:12) (cid:17) dx. By the Gagliardo-Nirenberg and radial Sobolev inequalities, we have (cid:13)(cid:13) f (cid:13)(cid:13) L ( | x |≥ R ) ≤ cR (cid:13)(cid:13) f (cid:13)(cid:13) L ( | x |≥ R ) (cid:13)(cid:13) ∇ f (cid:13)(cid:13) L ( | x |≥ R ) , (cid:13)(cid:13) f (cid:13)(cid:13) L ∞ ( | x |≥ R ) ≤ cR (cid:13)(cid:13) f (cid:13)(cid:13) / L ( | x |≥ R ) (cid:13)(cid:13) ∇ f (cid:13)(cid:13) / L ( | x |≥ R ) . Therefore, by mass conservation and Young’s inequality, we know that for any ǫ > R such that ∂ t V R ( t ) ≤ K ( u ( t )) + ǫ (cid:13)(cid:13) ∇ u ( t, x ) (cid:13)(cid:13) L + ǫ . =48 E ( u ) − (cid:0) − ǫ (cid:1)(cid:13)(cid:13) ∇ u ( t ) (cid:13)(cid:13) L − (cid:13)(cid:13) u ( t ) (cid:13)(cid:13) L + ǫ (3.1)By K ( u ) < 0, mass and energy conservations, Lemma 2.13 and the continuity argument,we know that for any t ∈ I , we have K ( u ( t )) ≤ − m − E ( u ( t ))) < . By Lemma 2.9, we have m ≤ H ( u ( t )) < (cid:13)(cid:13) u ( t ) (cid:13)(cid:13) L . where we have used the fact that K ( u ( t )) < m = ( C ∗ ) − and the Sharp Sobolev inequality, we have (cid:13)(cid:13) ∇ u ( t ) (cid:13)(cid:13) L ≥ ( C ∗ ) − (cid:13)(cid:13) u ( t ) (cid:13)(cid:13) L > ( C ∗ ) − , which implies that (cid:13)(cid:13) ∇ u ( t ) (cid:13)(cid:13) L > m .In addition, by E ( u ) < m and energy conservation, there exists δ > E ( u ( t )) ≤ (1 − δ ) m . Thus, if we choose ǫ sufficiently small, we have ∂ t V R ( t ) ≤ − δ ) m − (cid:0) − ǫ (cid:1) m + ǫ ≤ − δ m, which implies that u must blow up at finite time. (cid:3) Perturbation theory In this part, we give the perturbation theory of the solution of (1.1) with the globalspace-time estimate. First we denote the space-time space ST ( I ) on the time interval LS WITH THE COMBINED TERMS 17 I by ST ( I ) := (cid:16) L t ˙ B / / , ∩ L t L x ∩ L t ˙ B / / , ∩ L t,x (cid:17) ( I × R ) ,ST ∗ ( I ) := (cid:16) L t ˙ B / / , ∩ L t ˙ B / / , (cid:17) ( I × R ) . The main result in this section is the following. Proposition 4.1. Let I be a compact time interval and let w be an approximate solutionto (1.1) on I × R in the sense that i∂ t w + ∆ w = −| w | w + | w | w + e for some suitable small function e . Assume that for some constants L, E > , we have (cid:13)(cid:13) w (cid:13)(cid:13) ST ( I ) ≤ L, (cid:13)(cid:13) w ( t ) (cid:13)(cid:13) H x ( R ) ≤ E for some t ∈ I . Let u ( t ) close to w ( t ) in the sense that for some E ′ > , we have (cid:13)(cid:13) u ( t ) − w ( t ) (cid:13)(cid:13) H x ≤ E ′ . Assume also that for some ε , we have (cid:13)(cid:13) e i ( t − t )∆ (cid:0) u ( t ) − w ( t ) (cid:1)(cid:13)(cid:13) ST ( I ) ≤ ε, (cid:13)(cid:13) e (cid:13)(cid:13) ST ∗ ( I ) ≤ ε, (4.1) where < ε ≤ ε = ε ( E , E ′ , L ) is a small constant. Then there exists a solution u to (1.1) on I × R with initial data u ( t ) at time t = t satisfying (cid:13)(cid:13) u − w (cid:13)(cid:13) ST ( I ) ≤ C ( E , E ′ , L ) ε, and (cid:13)(cid:13) u (cid:13)(cid:13) ST ( I ) ≤ C ( E , E ′ , L ) . Proof. Since w ∈ ST ( I ), there exists a partition of the right half of I at t : t < t < · · · < t N , I j = ( t j , t j +1 ) , I ∩ ( t , ∞ ) = ( t , t N ) , such that N ≤ C ( L, δ ) and for any j = 0 , , . . . , N − 1, we have (cid:13)(cid:13) w (cid:13)(cid:13) ST ( I j ) ≤ δ ≪ . (4.2)The estimate on the left half of I at t is analogue, we omit it.Let γ ( t, x ) = u ( t, x ) − w ( t, x ) ,γ j ( t, x ) = e i ( t − t j )∆ (cid:16) u ( t j , x ) − w ( t j , x ) (cid:17) , then γ satisfies the following difference equation iγ t + ∆ γ = O ( w γ + w γ + w γ + wγ + γ + w γ + wγ + γ ) − e, which implies that γ ( t ) = γ j ( t ) − i Z tt j e i ( t − s )∆ (cid:16) O ( w γ + w γ + w γ + wγ + γ + w γ + wγ + γ ) − e (cid:17) ds,γ j +1 ( t ) = γ j ( t ) − i Z t j +1 t j e i ( t − s )∆ (cid:16) O ( w γ + w γ + w γ + wγ + γ + w γ + wγ + γ ) − e (cid:17) ds. By Lemma 2.2, we have (cid:13)(cid:13) γ − γ j (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) ∩ L t,x ( I j ) + (cid:13)(cid:13) γ j +1 − γ j (cid:13)(cid:13) L t (cid:16) R ; ˙ B / / , (cid:17) ∩ L t,x ( R × R ) (4.3) . (cid:13)(cid:13) O ( w γ + w γ + w γ + wγ + γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w γ + wγ + γ ) (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) e (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) . (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) w (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) w (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) w (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) + (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; L x ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) (cid:13)(cid:13) w (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) w (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) + (cid:13)(cid:13) γ (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) (cid:13)(cid:13) γ (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) + (cid:13)(cid:13) e (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) . At the same time, by Lemma 2.3, we have (cid:13)(cid:13) γ − γ j (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) ∩ L t ( I j ; L x ) + (cid:13)(cid:13) γ j +1 − γ j (cid:13)(cid:13) L t (cid:16) R ; ˙ B / / , (cid:17) ∩ L t ( R ; L x ) (4.4) . (cid:13)(cid:13) O ( w γ + w γ + w γ + wγ + γ + w γ + wγ + γ ) (cid:13)(cid:13) L ( I j ; ˙ B / , ) + (cid:13)(cid:13) e (cid:13)(cid:13) L ( I j ; ˙ B / , ) . (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) LS WITH THE COMBINED TERMS 19 + (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) w (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) + (cid:13)(cid:13) γ (cid:13)(cid:13) L t,x ( I j ) (cid:13)(cid:13) γ (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) + (cid:13)(cid:13) e (cid:13)(cid:13) L ( I j ; ˙ B / // , ) . By the interpolation, we have (cid:13)(cid:13) f (cid:13)(cid:13) L (cid:16) I j ; L / x (cid:17) . (cid:13)(cid:13) f (cid:13)(cid:13) L t (cid:16) I j ; ˙ B / / , (cid:17) , (cid:13)(cid:13) f (cid:13)(cid:13) L t,x ( I j ) . (cid:13)(cid:13) f (cid:13)(cid:13) L t ( I j ; ˙ B / / , ) . Therefore, assuming that (cid:13)(cid:13) γ (cid:13)(cid:13) ST ( I j ) ≤ δ ≪ , ∀ j = 0 , , . . . , N − , (4.5)then by (4.2), (4.3) and (4.4), we have (cid:13)(cid:13) γ (cid:13)(cid:13) ST ( I j ) + (cid:13)(cid:13) γ j +1 (cid:13)(cid:13) ST ( t j +1 ,t N ) ≤ C (cid:13)(cid:13) γ j (cid:13)(cid:13) ST ( t j ,t N ) + ε, for some absolute constant C > 0. By (4.1) and iteration on j , we get (cid:13)(cid:13) γ (cid:13)(cid:13) ST ( I ) ≤ (2 C ) N ε ≤ δ , if we choose ε sufficiently small. Hence the assumption (4.5) is justified by continuityin t and induction on j . then repeating the estimate (4.3) and (4.4) once again, we canobtain the ST -norm estimate on γ , which implies the Strichartz estimate on u . (cid:3) Profile decomposition In this part, we will use the method in [2, 17, 21] to show the linear and nonlinearprofile decompositions of the sequences of radial, H -bounded solutions of (1.1), whichwill be used to construct the critical element (minimal energy non-scattering solution)and show its properties, especially the compactness. In order to do it, we now introducethe complex-valued function −→ v ( t, x ) by −→ v ( t, x ) = h∇i v ( t, x ) , v ( t, x ) = h∇i − −→ v ( t, x ) . Given ( t jn , h jn ) ∈ R × (0 , τ jn , T jn denote the scaled time drift, the scaling trans-formation, defined by τ jn = − t jn (cid:0) h jn (cid:1) , T jn ϕ ( x ) = 1( h jn ) / ϕ (cid:18) xh jn (cid:19) . We also introduce the set of Fourier multipliers on R . MC = { µ = F − e µ F | e µ ∈ C ( R ) , ∃ lim | ξ |→ + ∞ e µ ( ξ ) ∈ R } . Linear profile decomposition. In this subsection, we show the profile decompo-sition with the scaling parameter of a sequence of the radial, free Schr¨odinger solutionsin the energy space H ( R ), which implies the profile decomposition of a sequence ofradial initial data. Proposition 5.1. Let −→ v n ( t, x ) = e it ∆ −→ v n (0) be a sequence of the radial solutions of the free Schr¨odinger equation with bounded L norm. Then up to a subsequence, there exist K ∈ { , , , . . . , ∞} , radial functions { ϕ j } j ∈ [0 ,K ) ⊂ L ( R ) and { t jn , h jn } n ∈ N ⊂ R × (0 , satisfying −→ v n ( t, x ) = k − X j =0 −→ v jn ( t, x ) + −→ w kn ( t, x ) , (5.1) where −→ v jn ( t, x ) = e i ( t − t jn )∆ T jn ϕ j , and lim k → K lim n → + ∞ (cid:13)(cid:13) −→ w kn (cid:13)(cid:13) L ∞ t ( R ; B − / ∞ , ∞ ( R )) = 0 , (5.2) and for any Fourier multiplier µ ∈ MC , any l < j < k ≤ K and any t ∈ R , lim n → + ∞ (cid:18) log (cid:12)(cid:12)(cid:12)(cid:12) h jn h ln (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) t jn − t ln ( h ln ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = ∞ , (5.3)lim n → + ∞ (cid:10) µ −→ v ln ( t ) , µ −→ v jn ( t ) (cid:11) L x = lim n → + ∞ (cid:10) µ −→ v jn ( t ) , µ −→ w kn ( t ) (cid:11) L x = 0 . (5.4) Moreover, each sequence { h jn } n ∈ N is either going to or identically for all n . Remark 5.2. We call −→ v jn and −→ w kn the free concentrating wave and the remainder,respectively. From (5.4), we have the following asymptotic orthogonalitylim n → + ∞ (cid:13)(cid:13) µ −→ v n ( t ) (cid:13)(cid:13) L − k − X j =0 (cid:13)(cid:13) µ −→ v jn ( t ) (cid:13)(cid:13) L − (cid:13)(cid:13) µ −→ w kn ( t ) (cid:13)(cid:13) L ! = 0 . (5.5) Proof of Proposition 5.1. Let ν := lim n →∞ (cid:13)(cid:13) −→ v n (cid:13)(cid:13) L ∞ t B − / ∞ , ∞ = lim n →∞ sup ( t,x ) ∈ R × R ,k ≥ − k/ (cid:12)(cid:12) Λ k ∗ −→ v n ( t, x ) (cid:12)(cid:12) . If ν = 0, then we have done with K = 0. LS WITH THE COMBINED TERMS 21 Otherwise, ν = lim n →∞ (cid:13)(cid:13) −→ v n (cid:13)(cid:13) L ∞ t B − / ∞ , ∞ > 0. By the radial Gagliardo-Nirenberg inequalityand the Bernstein inequality, we havesup t ∈ R , | kx |≥ R,k ≥ − k/ (cid:12)(cid:12) Λ k ∗ −→ v n ( t, x ) (cid:12)(cid:12) . sup k ≥ k − k/ R (cid:13)(cid:13) Λ k ∗ −→ v n ( t, x ) (cid:13)(cid:13) / L ∞ t L x · (cid:13)(cid:13) ∇ Λ k ∗ −→ v n ( t, x ) (cid:13)(cid:13) / L ∞ t L x . sup k ≥ R (cid:13)(cid:13) −→ v n ( t, x ) (cid:13)(cid:13) L ∞ t L x . R . If taking R sufficiently large, we havesup t ∈ R , | k x |≥ R,k ≥ − k/ (cid:12)(cid:12) Λ k ∗ −→ v n ( t, x ) (cid:12)(cid:12) ≤ ν. thus, there exists a sequence ( t n , x n , k n ) with k n ≥ | k n x n | ≤ R such that forlarge n , 12 lim n →∞ (cid:13)(cid:13) −→ v n (cid:13)(cid:13) L ∞ t B − / ∞ , ∞ = 12 ν ≤ − k n / (cid:12)(cid:12) Λ k n ∗ −→ v n ( t n , x n ) (cid:12)(cid:12) . Now we define h n and ψ n by h n = 2 − k n ∈ (0 , 1] and −→ v n ( t n , x ) = ( T n ψ n ) ( x − x n ) = 1( h n ) / ψ n (cid:18) x − x n h n (cid:19) (5.6)= T n (cid:18) ψ n (cid:16) x − x n h n (cid:17)(cid:19) . Since (cid:13)(cid:13) ψ n (cid:13)(cid:13) L = (cid:13)(cid:13) T n ψ n (cid:13)(cid:13) L = (cid:13)(cid:13) −→ v n ( t n ) (cid:13)(cid:13) L ≤ C, then there exists some ψ ∈ L , suchthat, up to a subsequence, we have as n → + ∞ x n h n → x , and ψ n ⇀ ψ weakly in L . (5.7)On the other hand, if k n = 0, we have2 − k n / (cid:12)(cid:12) Λ k n ∗ −→ v n ( t n , x n ) (cid:12)(cid:12) = Z R Λ ( y ) 2 − k n / −→ v n (cid:16) t n , x n − y k n (cid:17) dy = Z R Λ ( y ) ψ n ( − y ) dy −→ Z R Λ ( y ) ψ ( − y ) dy . (cid:13)(cid:13) ψ (cid:13)(cid:13) L . By the same way, if k n ≥ 1, we have2 − k n / (cid:12)(cid:12) Λ k n ∗ −→ v n ( t n , x n ) (cid:12)(cid:12) = Z R Λ (0) ( y ) 2 − k n / −→ v n (cid:16) t n , x n − y k n (cid:17) dy = Z R Λ (0) ( y ) ψ n ( − y ) dy −→ Z R Λ (0) ( y ) ψ ( − y ) dy . (cid:13)(cid:13) ψ (cid:13)(cid:13) L . If h n → 0, then we take( t n , h n ) = ( t n , h n ) , ϕ ( x ) = ψ (cid:0) x − x (cid:1) , otherwise, up to a subsequence, we may assume that h n → h ∞ for some h ∞ ∈ (0 , t n , h n ) = ( t n , , ϕ ( x ) = 1( h ∞ ) / ψ (cid:18) xh ∞ − x (cid:19) , then T n (cid:18) ψ (cid:16) x − x n h n (cid:17)(cid:19) − T n ϕ ( x ) −→ L . (5.8)In addition, since −→ v n ( t n , x ) = ( T n ψ n ) ( x − x n ) is radial, so is ϕ ( x ).Let −→ v n ( t, x ) = e i ( t − t n )∆ T n ϕ , we define −→ w n by −→ v n ( t, x ) = −→ v n ( t, x ) + −→ w n ( t, x ) , (5.9)then by (5.7) and (5.8), we have( T n ) − −→ w n ( t n ) = ( T n ) − T n (cid:18) ψ n (cid:16) x − x n h n (cid:17)(cid:19) − ϕ ⇀ L , which implies that (cid:10) µ −→ v n ( t ) , µ −→ w n ( t ) (cid:11) = (cid:10) µ −→ v n ( t n ) , µ −→ w n ( t n ) (cid:11) = (cid:10) µ n ϕ , µ n ( T n ) − −→ w n ( t n ) (cid:11) −→ , where we used the conservation law in the first equality and the dominated convergencetheorem and µ n ( D ) = µ (cid:16) Dh n (cid:17) in the last equality. It is the decomposition for k = 1.Next we apply the above procedure to the sequence −→ w n in place of −→ v n , then eitherlim n →∞ (cid:13)(cid:13) −→ w n (cid:13)(cid:13) L ∞ t B − / ∞ , ∞ = 0 or we can find the next concentrating wave −→ v n and the remain-der −→ w n , such that for some ( t n , h n ) with h n ∈ (0 , 1] and radial function ϕ ∈ L ( R ), −→ w n ( t, x ) = −→ v n ( t, x )+ −→ w n ( t, x ) = e i ( t − t n )∆ T n ϕ ( x ) + −→ w n ( t, x ) , (5.10)and lim n → + ∞ (cid:13)(cid:13) −→ w n (cid:13)(cid:13) L ∞ t B − / ∞ , ∞ . (cid:13)(cid:13) ϕ (cid:13)(cid:13) L = (cid:13)(cid:13) −→ v n (cid:13)(cid:13) L , (5.11)( T n ) − −→ w n ( t n ) ⇀ L = ⇒ (cid:10) µ −→ v n ( t ) , µ −→ w n ( t ) (cid:11) −→ . Iterating the above procedure, we can obtain the decomposition (5.1). It remains toshow the properties (5.2), (5.3) and (5.4). LS WITH THE COMBINED TERMS 23 We first assume that (5.4) holds, then by (5.5) and the Cauchy criterion, we havelim n → + ∞ (cid:13)(cid:13) −→ w kn (cid:13)(cid:13) L ∞ t B − / ∞ , ∞ . (cid:13)(cid:13) ϕ k (cid:13)(cid:13) L = (cid:13)(cid:13) −→ v kn (cid:13)(cid:13) L −→ k → + ∞ . (5.12)which implies (5.2).Now we show (5.3) by contradiction. Suppose that (5.3) fails, then there exists aminimal ( l, j ) which violates (5.3). By extracting a subsequence, We may assume that h ln → h l ∞ and h ln /h jn and ( t ln − t jn ) / ( h ln ) all converge.Now consider (cid:0) T ln (cid:1) − −→ w l +1 n ( t ln ) = j X m = l +1 (cid:0) T ln (cid:1) − −→ v mn ( t ln ) + (cid:0) T ln (cid:1) − −→ w j +1 n ( t ln )= j X m = l +1 (cid:0) T ln (cid:1) − e i ( t ln − t mn )∆ T mn ϕ m + (cid:0) T ln (cid:1) − −→ w j +1 n ( t ln )= j − X m = l +1 S l,mn ϕ m + S l,jn ϕ j + (cid:0) T ln (cid:1) − −→ w j +1 n ( t ln ) , where S l,mn = (cid:0) T ln (cid:1) − e i ( t ln − t mn )∆ T mn = e i tln − tmn ( hln )2 ∆ (cid:0) T ln (cid:1) − T mn := e it l,mn ∆ T l,mn with the sequence t l,mn = t ln − t mn ( h ln ) , h l,mn = h mn h ln . (5.13)By the procedure of constructing (5.1), as n → + ∞ , we have (cid:0) T ln (cid:1) − −→ w l +1 n ( t ln ) ⇀ L , (cid:0) T jn (cid:1) − −→ w j +1 n ( t jn ) ⇀ L , and by the asymptotic orthogonality (5.3) between m and l with m ∈ [ l + 1 , j − S l,mn ϕ m ⇀ , ∀ m ∈ [ l + 1 , j − , and by the convergence of h ln /h jn and ( t ln − t jn ) / ( h ln ) , we have S l,jn ϕ j → S l,j ∞ ϕ j and (cid:0) T ln (cid:1) − −→ w j +1 n ( t ln ) = S l,jn (cid:0) T jn (cid:1) − −→ w j +1 n ( t jn ) ⇀ L . Then ϕ j = 0, it is a contradiction. Thus we obtain the orthogonality (5.3).Last we show (5.4). For j = l , we have (cid:10) µ −→ v ln ( t ) , µ −→ v jn ( t ) (cid:11) L x = (cid:10) µ −→ v ln (0) , µ −→ v jn (0) (cid:11) L x = D µe − it ln ∆ T ln ϕ l , µe − it jn ∆ T jn ϕ j E L x = D e − it ln ∆ T ln µ ln ϕ l , e − it jn ∆ T jn µ jn ϕ j E L x = D(cid:0) T jn (cid:1) − e i ( t jn − t ln )∆ T ln µ ln ϕ l , µ jn ϕ j E L x = * e i tjn − tln ( hjn )2 ∆ (cid:0) T jn (cid:1) − T ln µ ln ϕ l , µ jn ϕ j + L x = (cid:10) S j,ln µ ln ϕ l , µ jn ϕ j (cid:11) L x → n → + ∞ where e µ ln ( ξ ) = e µ (cid:0) ξ/h ln (cid:1) and we used the fact that S j,ln ⇀ L as n → + ∞ by (5.3). In addition, we have (cid:10) µ −→ v jn ( t ) , µ −→ w kn ( t ) (cid:11) L x = * µ −→ v jn ( t ) , µ (cid:16) −→ w j +1 n ( t ) − k − X m = j +1 −→ v mn ( t ) (cid:17)+ L x −→ n → + ∞ . This completes the proof of (5.4). (cid:3) After the orthogonality’s proof of the linear energy, we begin with the orthogonalanalysis for the nonlinear energy. Lemma 5.3. Let −→ v n be a sequence of the radial solutions of the free Schr¨odingerequation. Let −→ v n ( t, x ) = k − X j =0 −→ v jn ( t, x ) + −→ w kn ( t, x ) be the linear profile decomposition given by Proposition 5.1. Then we have lim k → K lim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ( v n (0)) − k − X j =0 M ( v jn (0)) − M ( w kn (0)) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , lim k → K lim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( v n (0)) − k − X j =0 E ( v jn (0)) − E ( w kn (0)) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , lim k → K lim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ( v n (0)) − k − X j =0 K ( v jn (0)) − K ( w kn (0)) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Proof. We can show that the quadratic terms in M , E and K have the orthogonaldecomposition by taking µ = h∇i and µ = |∇|h∇i in Remark 5.2, thus it suffices to showthat lim k → K lim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F i ( v n (0)) − X j 2, separately.First for i = 2, we compute as following, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F X j 0, we have h∇i − T jn ψ j → L p , ∀ ≤ p < , which implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F X j 0, and b ψ j = h∇i − ψ j if h jn ≡ 1, then we have b ψ j ∈ L x , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F X j 00 if h jn ≡ −→ , as n → + ∞ , which shows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F X j 1; otherwise , where h j,ln is determined by (5.13). By (5.3), we know that h j,ln → 0, therefore as n → + ∞ e ψ jn → b ψ j , a.e. x ∈ R , and e ψ jn → b ψ j , in L x , which implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F X j Lemma 5.4. Let k ∈ N and radial functions ϕ , . . . , ϕ k ∈ H ( R ) , m be determined by (1.5) . Assume that there exist some δ , ε > with ε < δ such that k X j =0 E ( ϕ j ) − ε ≤ E k X j =0 ϕ j ! < m − δ, and − ε ≤ K k X j =0 ϕ j ! ≤ k X j =0 K ( ϕ j ) + ε. Then ϕ j ∈ K + for all j = 0 , . . . , k .Proof. Suppose that K ( ϕ l ) < l . Then by Lemma 2.9, we have H ( ϕ l ) ≥ inf (cid:8) H ( ϕ ) | ϕ ∈ H ( R ) , ϕ = 0 , K ( ϕ ) ≤ (cid:9) = m. By the nonnegativity of H ( ϕ j ) for j ≥ 0, we have m ≤ H ( ϕ l ) ≤ k X j =0 H ( ϕ j ) = k X j =0 (cid:18) E ( ϕ j ) − K ( ϕ j ) (cid:19) ≤ E k X j =0 ϕ j ! + ε − K k X j =0 ϕ j ! + 16 ε ≤ m − δ + ε + 13 ε < m. It is a contradiction. Hence for any j ∈ { , . . . , k } , we have K ( ϕ j ) ≥ , which implies that E ( ϕ j ) = H ( ϕ j ) + 16 K ( ϕ j ) ≥ , and E ( ϕ j ) ≤ k X i =0 E ( ϕ i ) < m − δ + ε < m, which means that ϕ j ∈ K + for all j . (cid:3) According to the above results, we conclude as following. Proposition 5.5. Let −→ v n ( t, x ) be a sequence of the radial solutions of the free Schr¨odingerequation satisfying v n (0) ∈ K + and E ( v n (0)) < m. Let −→ v n ( t, x ) = k − X j =0 −→ v jn ( t, x ) + −→ w kn ( t, x ) , be the linear profile decomposition given by Proposition 5.1. Then for large n and all j < K , we have v jn (0) ∈ K + , w Kn (0) ∈ K + , and lim k → K lim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ( v n (0)) − X j After the linear profile decomposition of asequence of initial data in the last subsection, we now show the nonlinear profile de-composition of a sequence of radial solutions of (1.1) with the same initial data in theenergy space H ( R ). First we introduce some notation h∇i jn = q(cid:0) h jn (cid:1) − ∆ , h∇i j ∞ = q(cid:0) h j ∞ (cid:1) − ∆ . Now let v n ( t, x ) be a sequence of radial solutions for the free Schr¨odinger equationwith initial data in K + , that is, v n ∈ H ( R ) is radial and( i∂ t + ∆) v n = 0 , v n (0) ∈ K + . LS WITH THE COMBINED TERMS 31 Let −→ v n ( t, x ) = h∇i v n ( t, x ) , then by Proposition 5.1, we have a sequence of the radial, free concentrating wave −→ v jn ( t, x ) with −→ v jn ( t jn ) = T jn ϕ j , v jn (0) ∈ K + for j = 0 , . . . , K , such that −→ v n ( t, x ) = k − X j =0 −→ v jn ( t, x ) + −→ w kn ( t, x ) = k − X j =0 e i ( t − t jn )∆ T jn ϕ j + −→ w kn = k − X j =0 T jn e i (cid:18) t − tjn ( hjn )2 (cid:19) ∆ ϕ j + −→ w kn . Now for any concentrating wave −→ v jn , j = 0 , . . . , K , we undo the group action, i.e.,the scaling transformation T jn , to look for the linear profile V j . Let −→ v jn ( t, x ) = T jn −→ V j (cid:18) t − t jn ( h jn ) (cid:19) , then we have ( i∂ t + ∆) −→ V j = 0 , −→ V j (0) = ϕ j . Now let u jn ( t, x ) be the nonlinear solution of (1.1) with initial data v jn (0), that is( i∂ t + ∆) −→ u jn ( t, x ) = h∇i f ( h∇i − −→ u jn ) + h∇i f ( h∇i − −→ u jn ) , −→ u jn (0) = −→ v jn (0) = T jn −→ V j ( τ jn ) , u jn (0) ∈ K + , where τ jn = − t jn / ( h jn ) . In order to look for the nonlinear profile −→ U j ∞ associated to theradial, free concentrating wave ( −→ v jn ; h jn , t jn ), we also need undo the group action. Wedenote −→ u jn ( t, x ) = T jn −→ U jn (cid:18) t − t jn ( h jn ) (cid:19) , then we have( i∂ t + ∆) −→ U jn = (cid:16) h∇i jn (cid:17) f (cid:18)(cid:16) h∇i jn (cid:17) − −→ U jn (cid:19) + h jn · (cid:16) h∇i jn (cid:17) f (cid:18)(cid:16) h∇i jn (cid:17) − −→ U jn (cid:19) , −→ U jn ( τ jn ) = −→ V j ( τ jn ) . Up to a subsequence, we may assume that there exist h j ∞ ∈ { , } and τ j ∞ ∈ [ −∞ , ∞ ]for every j = { , . . . , K } , such that h jn → h j ∞ , and τ jn → τ j ∞ . As n → + ∞ , the limit equation of −→ U jn is given by( i∂ t + ∆) −→ U j ∞ = (cid:16) h∇i j ∞ (cid:17) f (cid:18)(cid:16) h∇i j ∞ (cid:17) − −→ U j ∞ (cid:19) + h j ∞ · (cid:16) h∇i j ∞ (cid:17) f (cid:18)(cid:16) h∇i j ∞ (cid:17) − −→ U j ∞ (cid:19) , −→ U j ∞ ( τ j ∞ ) = −→ V j ( τ j ∞ ) ∈ L ( R ) . Let b U j ∞ := (cid:16) h∇i j ∞ (cid:17) − −→ U j ∞ , then ( i∂ t + ∆) b U j ∞ = f (cid:16) b U j ∞ (cid:17) + h j ∞ · f (cid:16) b U j ∞ (cid:17) , (5.17) b U j ∞ ( τ j ∞ ) = (cid:16) h∇i j ∞ (cid:17) − −→ V j ( τ j ∞ ) . (5.18)The unique existence of a local radial solution −→ U j ∞ around τ j ∞ is known in all cases,including h j ∞ = 0 and τ j ∞ = ±∞ . −→ U j ∞ on the maximal existence interval is called thenonlinear profile associated with the radial, free concentrating wave ( −→ v jn ; h jn , t jn ).The nonlinear concentrating wave u j ( n ) associated with ( −→ v jn ; h jn , t jn ) is defined by −→ u j ( n ) ( t, x ) = T jn −→ U j ∞ (cid:18) t − t jn ( h jn ) (cid:19) , then we have( i∂ t + ∆) −→ u j ( n ) = vuut |∇| + h j ∞ h jn ! f vuut |∇| + h j ∞ h jn ! − −→ u j ( n ) + h j ∞ h jn · vuut |∇| + h j ∞ h jn ! f vuut |∇| + h j ∞ h jn ! − −→ u j ( n ) = h∇i j ∞ f (cid:18)(cid:16) h∇i j ∞ (cid:17) − −→ u j ( n ) (cid:19) + h j ∞ · h∇i j ∞ f (cid:18)(cid:16) h∇i j ∞ (cid:17) − −→ u j ( n ) (cid:19) , −→ u j ( n ) (0) = T jn −→ U j ∞ ( τ jn ) , which implies that (cid:13)(cid:13) −→ u j ( n ) (0) − −→ u jn (0) (cid:13)(cid:13) L = (cid:13)(cid:13) T jn −→ U j ∞ ( τ jn ) − T jn −→ V j ( τ jn ) (cid:13)(cid:13) L = (cid:13)(cid:13) −→ U j ∞ ( τ jn ) − −→ V j ( τ jn ) (cid:13)(cid:13) L ≤ (cid:13)(cid:13) −→ U j ∞ ( τ jn ) − −→ U j ∞ ( τ j ∞ ) (cid:13)(cid:13) L + (cid:13)(cid:13) −→ V j ( τ jn ) − −→ V j ( τ j ∞ ) (cid:13)(cid:13) L → . LS WITH THE COMBINED TERMS 33 We denote −→ u j ( n ) = h∇i u j ( n ) . If h j ∞ = 1, we have h jn ≡ 1, then u j ( n ) ∈ H ( R ) is radial and satisfies( i∂ t + ∆) u j ( n ) = f ( u j ( n ) ) + f ( u j ( n ) ) . If h j ∞ = 0, then u j ( n ) ∈ H ( R ) is radial and satisfies( i∂ t + ∆) u j ( n ) = |∇|h∇i f (cid:18) h∇i|∇| u j ( n ) (cid:19) . Let u n be a sequence of (local) radial solutions of (1.1) with initial data in K + at t = 0, and let v n be the sequence of the radial, free solutions with the same initial data.We consider the linear profile decomposition given by Proposition 5.1 −→ v n ( t, x ) = k − X j =0 −→ v jn ( t, x ) + −→ w kn ( t, x ) , −→ v jn ( t jn ) = T jn ϕ j , v jn (0) ∈ K + . With each free concentrating wave {−→ v jn } n ∈ N , we associate the nonlinear concentratingwave {−→ u j ( n ) } n ∈ N . A nonlinear profile decomposition of u n is given by −→ u In the nonlinear profile decomposition (5.19) . Suppose that for each j < K , we have (cid:13)(cid:13) b U j ∞ (cid:13)(cid:13) ST j ∞ ( R ) + (cid:13)(cid:13) −→ U j ∞ (cid:13)(cid:13) L ∞ t L x ( R ) < ∞ . Then for any finite interval I , any j < K and any k ≤ K , we have lim n → + ∞ (cid:13)(cid:13) u j ( n ) (cid:13)(cid:13) ST ( I ) . (cid:13)(cid:13) b U j ∞ (cid:13)(cid:13) ST j ∞ ( R ) , (5.20)lim n → + ∞ (cid:13)(cid:13) u Proof of (5.20) . By the definitions of u j ( n ) and b U j ∞ , we know that u j ( n ) ( t, x ) = h∇i − −→ u j ( n ) ( t, x ) = h∇i − T jn −→ U j ∞ (cid:18) t − t jn ( h jn ) (cid:19) = h∇i − T jn h∇i j ∞ b U j ∞ (cid:18) t − t jn ( h jn ) (cid:19) = h jn T jn h∇i j ∞ h∇i jn b U j ∞ (cid:18) t − t jn ( h jn ) (cid:19) . For the case h j ∞ = 1, we have u j ( n ) ( t, x ) = b U j ∞ ( t − t jn , x ), hence (5.20) is trivial. For thecase h j ∞ = 0, by the above relation between u j ( n ) and b U j ∞ , we have (cid:13)(cid:13) u j ( n ) (cid:13)(cid:13) (cid:16) L t ˙ B / / , ∩ L t L x (cid:17) ( I × R ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) |∇|h∇i jn b U j ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:16) L t ˙ B / / , ∩ L t L x (cid:17) ( R × R ) . (cid:13)(cid:13) b U j ∞ (cid:13)(cid:13) (cid:16) L t ˙ B / / , ∩ L t L x (cid:17) ( R × R ) , and (cid:13)(cid:13) u j ( n ) (cid:13)(cid:13) L t ˙ B / / , ( I × R ) . | I | (cid:13)(cid:13) u j ( n ) (cid:13)(cid:13) L t ˙ B / / , . | I | ( h jn ) (cid:13)(cid:13) b U j ∞ (cid:13)(cid:13) L t ˙ B / , → , (cid:13)(cid:13) u j ( n ) (cid:13)(cid:13) L t,x ( I × R ) . | I | (cid:13)(cid:13) u j ( n ) (cid:13)(cid:13) L t L x . | I | ( h jn ) (cid:13)(cid:13) b U j ∞ (cid:13)(cid:13) L t ˙ B , → . where we use the fact that the boundedness of b U j ∞ in L t ˙ B / / , ∩ L t L x ∩ L ∞ ˙ H impliesits boundedness in L t ˙ B / , ∩ L t ˙ B , by (5.17). LS WITH THE COMBINED TERMS 35 Proof of (5.21) . We estimate the left hand side of (5.21) by (cid:13)(cid:13) u 0, we havelim n → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j After this preliminaries, we now show that −→ u Let u n be a sequence of local, radial solutions of (1.1) around t = 0 in K + satisfying M ( u n ) < ∞ , lim n →∞ E ( u n ) < m. Suppose that in the nonlinear profile decomposition (5.19) , every nonlinear profile b U j ∞ has finite global Strichartz and energy norms we have (cid:13)(cid:13) b U j ∞ (cid:13)(cid:13) ST j ∞ ( R ) + (cid:13)(cid:13) −→ U j ∞ (cid:13)(cid:13) L ∞ t L x ( R ) < ∞ . Then u n is bounded for large n in the Strichartz and the energy norms lim n →∞ (cid:13)(cid:13) u n (cid:13)(cid:13) ST ( R ) + (cid:13)(cid:13) −→ u n (cid:13)(cid:13) L ∞ t L x ( R ) < ∞ . Proof. We only need to verify the condition of Proposition 4.1. Note that u In this subsection, by the profile decompositionand the stability theory of the scattering solution of (1.1), we show the existence of thecritical element, which is the radial, energy solution of (1.1) with the smallness energy E ∗ and infinite Strichartz norm.By the definition of E ∗ and the fact that E ∗ < m , there exist a sequence of radialsolutions { u n } n ∈ N of (1.1) in K + , which have the maximal existence interval I n andsatisfy that M ( u n ) < ∞ , E ( u n ) → E ∗ < m, (cid:13)(cid:13) u n (cid:13)(cid:13) ST ( I n ) → + ∞ , as n → + ∞ , then we have (cid:13)(cid:13) u n (cid:13)(cid:13) H < ∞ by Lemma 2.12. By the compact argument (profile decom-position) and the stability theory, we can show that Theorem 6.1. Let u n be a sequence of radial solutions of (1.1) in K + on I n ⊂ R satisfying M ( u n ) < ∞ , E ( u n ) → E ∗ < m, (cid:13)(cid:13) u n (cid:13)(cid:13) ST ( I n ) → + ∞ , as n → + ∞ . Then there exists a global, radial solution u c of (1.1) in K + satisfying E ( u c ) = E ∗ < m, K ( u c ) > , (cid:13)(cid:13) u c (cid:13)(cid:13) ST ( R ) = ∞ . In addition, there are a sequence t n ∈ R and radial function ϕ ∈ L ( R ) such that, upto a subsequence, we have as n → + ∞ , (cid:13)(cid:13)(cid:13)(cid:13) |∇|h∇i (cid:16) −→ u n (0 , x ) − e − it n ∆ ϕ ( x ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) L → . (6.2) Proof. By the time translation symmetry of (1.1), we can translate u n in t such that0 ∈ I n for all n . Then by the linear and nonlinear profile decomposition of u n , we have e it ∆ −→ u n (0 , x ) = X j LS WITH THE COMBINED TERMS 41 (1) Every radial solution of (1.1) in K + with the energy less than E ∗ has globalfinite Strichartz norm by the definition of E ∗ .(2) Lemma 5.7 precludes that all the nonlinear profiles −→ U j ∞ have finite globalStrichartz norm.we deduce that there is only one radial profile and E ( u n ) (0)) → E ∗ , u n ) (0) ∈ K + , (cid:13)(cid:13) b U ∞ (cid:13)(cid:13) ST ∞ ( I ) = ∞ , (cid:13)(cid:13) w n (cid:13)(cid:13) L ∞ t ˙ H x → . If h n → 0, then b U ∞ = |∇| − −→ U ∞ solves the ˙ H -critical NLS( i∂ t + ∆) b U ∞ = f ( b U ∞ )and satisfies E c (cid:16) b U ∞ ( τ ∞ ) (cid:17) = E ∗ < m, K c (cid:16) b U ∞ ( τ ∞ ) (cid:17) ≥ , (cid:13)(cid:13) b U ∞ (cid:13)(cid:13) (cid:16) L t ˙ B / / , ∩ L t L x (cid:17) ( I × R ) = ∞ . However, it is in contradiction with Kenig-Merle’s result in [19]. Hence h n ≡ 1, whichimplies (6.2).Now we show that b U ∞ = h∇i − −→ U j ∞ is a global solution, which is the consequence ofthe compactness of (6.2). Suppose not, then we can choose a sequence t n ∈ R whichapproaches the maximal existence time. Since b U ∞ ( t + t n ) satisfies the assumption ofthis theorem, then applying the above argument to it, we obtain that for some ψ ∈ L and another sequence t ′ n ∈ R , as n → + ∞ (cid:13)(cid:13)(cid:13)(cid:13) |∇|h∇i (cid:16) −→ U ∞ ( t n ) − e − it ′ n ∆ ψ ( x ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) L → . (6.3)Let −→ v ( t ) := e it ∆ ψ. For any ε > 0, there exist δ > I = [ − δ, δ ] such that (cid:13)(cid:13) h∇i − −→ v ( t − t ′ n ) (cid:13)(cid:13) ST ( I ) ≤ ε, which together with (6.3) implies that for sufficiently large n (cid:13)(cid:13) h∇i − e it ∆ −→ U ∞ ( t n ) (cid:13)(cid:13) ST ( I ) ≤ ε. If ε is small enough, this implies that the solution b U ∞ exists on [ t n − δ, t n + δ ] for large n by the small data theory. This contradicts the choice of t n . Hence b U ∞ is a globalsolution and it is just the desired critical element u c . By Proposition 1.1, we know that K ( u c ) > (cid:3) By the global L t,x estimate of solution u of (1.2), we can obtain the global L qt ˙ W ,rx estimate of u forany Schr¨odinger L -admissible pair ( q, r ). Compactness of the critical element. In order to preclude the critical element,we need obtain some useful properties about the critical element. In the followingsubsections, we establish some properties about the critical element by its minimalenergy with infinite Strichartz norm, especially its compactness and its consequence.Since (1.1) is symmetric in t , we may assume that (cid:13)(cid:13) u c (cid:13)(cid:13) ST (0 , + ∞ ) = ∞ , (6.4)we call it a forward critical element. Proposition 6.2. Let u c be a forward critical element. Then the set { u c ( t, x ); 0 < t < ∞} is precompact in ˙ H s for any s ∈ (0 , .Proof. By the conservation of the mass, it suffices to prove the precompactness of u c ( t n ) } in ˙ H for any positive time t , t , . . . . If t n converges, then it is trivial from the continuityin t .If t n → + ∞ . Applying Theorem 6.1 to the sequence of solutions −→ u c ( t + t n ), we getanother sequence t ′ n ∈ R and radial function ϕ ∈ L such that |∇|h∇i (cid:16) −→ u c ( t n , x ) − e − it ′ n ∆ ϕ ( x ) (cid:17) → L . (1) If t ′ n → −∞ , then we have (cid:13)(cid:13) h∇i − e it ∆ −→ u c ( t n ) (cid:13)(cid:13) ST (0 , + ∞ ) = (cid:13)(cid:13) h∇i − e it ∆ ϕ (cid:13)(cid:13) ST ( − t ′ n , + ∞ ) + o n (1) → . Hence u c can solve (1.1) for t > t n with large n globally by iteration with smallStrichartz norms, which contradicts (6.4).(2) If t ′ n → + ∞ , then we have (cid:13)(cid:13) h∇i − e it ∆ −→ u c ( t n ) (cid:13)(cid:13) ST ( −∞ , = (cid:13)(cid:13) h∇i − e it ∆ ϕ (cid:13)(cid:13) ST ( −∞ , − t ′ n ) + o n (1) → u c can solve (1.1) for t < t n with large n with vanishing Strichartz norms,which implies u c = 0 by taking the limit, which is a contradiction.Thus t ′ n is bounded, which implies that t ′ n is precompact, so is u c ( t n , x ) in ˙ H . (cid:3) As a consequence, the energy of u c stays within a fixed radius for all positive time,modulo arbitrarily small rest. More precisely, we define the exterior energy by E R ( u ; t ) = Z | x |≥ R (cid:16)(cid:12)(cid:12) ∇ u ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) + (cid:12)(cid:12) u ( t, x ) (cid:12)(cid:12) (cid:17) dx for any R > 0. Then we have LS WITH THE COMBINED TERMS 43 Corollary 6.3. Let u c be a forward critical element. then for any ε , there exist R ( ε ) > such that E R ( u c ; t ) ≤ εE ( u c ) , for any t > . Death of the critical element. We are in a position to preclude the soliton-likesolution by a truncated Virial identity. Theorem 6.4. The critical element u c of (1.1) cannot be a soliton in the sense ofTheorem 6.1.Proof. We still drop the subscript c . Now let φ be a smooth, radial function satisfying0 ≤ φ ≤ φ ( x ) = 1 for | x | ≤ 1, and φ ( x ) = 0 for | x | ≥ 2. For some R , we define V R ( t ) := Z R φ R ( x ) | u ( t, x ) | dx, φ R ( x ) = R φ (cid:18) | x | R (cid:19) . On one hand, we have ∂ t V R ( t ) = 4 ℑ Z R φ ′ (cid:18) | x | R (cid:19) x · ∇ u ( t, x ) u ( t, x ) dx. Therefore, we have (cid:12)(cid:12) ∂ t V R ( t ) (cid:12)(cid:12) . R (6.5)for all t ≥ R > ∂ t V R ( t ) = 4 Z R φ ′′ R ( r ) (cid:12)(cid:12) ∇ u ( t, x ) (cid:12)(cid:12) dx − Z R (∆ φ R )( x ) | u ( t, x ) | dx − Z R (∆ φ R )( x ) | u ( t, x ) | dx + Z R (∆ φ R )( x ) | u ( t, x ) | dx =4 Z R (cid:18) |∇ u ( t, x ) | − | u ( t, x ) | + 32 | u ( t, x ) | (cid:19) dx + O Z | x |≥ R (cid:0) |∇ u ( t, x ) | + | u ( t, x ) | + | u ( t, x ) | (cid:1) dx + (cid:18)Z R ≤| x |≤ R | u ( t, x ) | dx (cid:19) / ! =4 K ( u ( t )) + O Z | x |≥ R (cid:0) |∇ u ( t, x ) | + | u ( t, x ) | (cid:1) dx + (cid:18)Z R ≤| x |≤ R | u ( t, x ) | dx (cid:19) / ! . 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Box 8009, Beijing, China, 100088, E-mail address : miao [email protected] Guixiang XuInstitute of Applied Physics and Computational Mathematics,P. O. Box 8009, Beijing, China, 100088, E-mail address : xu [email protected] Lifeng ZhaoUniversity of Science and Technology of China,Hefei, China, E-mail address ::