Necessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrödinger Equation

Transcription

1 Necessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrödinger Equation Pascal Bégout aboratoire Jacques-ouis ions Université Pierre et Marie Curie Boîte Courrier 187, place Jussieu 755 Paris Cedex 05, FRANCE begout@ann.jussieu.fr Abstract In this paper, we consider the nonlinear Schrödinger equation with the super critical power of nonlinearity in the attractive case. We give a sufficient condition and a necessary condition to obtain global or blowing up solutions. These conditions coincide in the critical case, thereby extending the results of Weinstein [6, 7]. Furthermore, we improve a blow-up condition. 1 Introduction and notations We consider the following nonlinear Schrödinger equation, i u t + u + λ u α u = 0, t, x) T, T ) R N, u0) = ϕ, in R N, 1.1) where λ R, 0 α < 0 α < if N = 1) and ϕ a given initial data. N It is well-known that for every ϕ H 1 R N ), 1.1) has a unique solution u C T, T ); H 1 R N )) which satisfies the blow-up alternative and the conservation of charge and energy. In other words, if T < then lim t T ut) H 1 =. In the same way, if T < then lim t T ut) H 1 =. And for all t T, T ), ut) = ϕ and Eut)) = Eϕ), where Eϕ) def = 1 ϕ λ α+ ϕ α+ α+. If ϕ X def = H 1 R N ) x ; dx) then u C T, T ); X). Moreover, if λ 0, if α < N AMS Subject Classifications: 35Q55 or if 1

2 ϕ H 1 is small enough then T = T = and u R;H 1 ) <. Finally, is α N then there exist initial values ϕ H 1 R N ) such that the corresponding solution of 1.1) blows up in finite time. See Cazenave [9], Ginibre and Velo [11, 1, 13, 1], Glassey [15], Kato [17]. In the attractive and critical case λ > 0 and α = N ), there is a sharp condition to obtain global solutions see Weinstein [6, 7]). It is given in terms of the solution of a related elliptic problem. But in the super critical case α > N ), we only know that there exists ε > 0 sufficiently small such that if ϕ H 1 ε, then the corresponding solution is global in time. In this paper, we try to extend the results of Weinstein [6, 7] to the super critical case α > N. As we will see, we are not able to establish such a result, but we can give two explicit real values functions γ and r with 0 < γ < r such that if ϕ γ ϕ ), then the corresponding solution is global in time. Furthermore, for every a, b) 0, ) 0, ) such that a > r b), there exists ϕ a,b H 1 R N ) with ϕ a,b = a and ϕ a,b = b such that the associated solution blows up in finite time for both t < 0 et t > 0 see Theorem.1 below). Despite of the fact we do not obtain a sharp condition since γ < r ), we recover the results of Weinstein [6, 7] as α N. Setting A = {ϕ H 1 R N ); ϕ γ ϕ )}, it follows that for every ϕ A, the corresponding solution of 1.1) is global in time and uniformly bounded in H 1 R N ). It is interesting to note that A is an unbounded subset of H 1 R N ) as for the case α = N. We also improve some results about blow-up Theorems.1 and.). This paper is organized as follows. In Section, we give a sufficient blow-up condition. In Section 3, we recall the best constant in a Gagliardo-Nirenberg s inequality. In Section, we give the main result of this paper, that is necessary conditions and sufficient conditions to obtain global solutions. In Section 5, we prove the result given in Section. The following notations will be used throughout this paper. = N j=1 x j and we denote by B0, R), for R > 0, the ball of R N of center 0 with radius R. For 1 p, we design by p R N ) = p R N ; C), with norm. p, the usual ebesgue spaces and by H 1 R N ) = H 1 R N ; C), with norm. H 1, the Sobolev space. For k N {0} and 0 < γ < 1, we denote by C k,γ R N ) = C k,γ R N ; C) the Hölder spaces and we introduce the Hilbert space X = { ψ H 1 R N ; C); ψ X < } with norm ψ X = ψ H 1 R N ) + R N x ψx) dx. For a normed functional space E 1 loc RN ; C), we denote by E rad the space of functions f E such that f is spherically symmetric. E rad is endowed with the norm of E. Finally, C are auxiliary positive constants.

3 Blow-up The first two results are an improvement of a blow-up condition see Glassey [15], Ogawa and Tsutsumi [3]). We know that if a solution has a negative energy, then it blows up in finite time. We extend this result for any nontrivial solution with nonpositive energy. Theorem.1. et λ > 0, N < α < < α < if N = 1) and ϕ X, ϕ 0. If Eϕ) 0 N then the corresponding solution u C T, T ); X) of 1.1) blows up in finite time for both t > 0 and t < 0. In other words, T < and T <. Theorem.. et λ > 0, N, N < α < N < α if N = ) and ϕ H1 rad RN ), ϕ 0. If Eϕ) 0 then the corresponding solution u C T, T ); H 1 R N )) of 1.1) blows up in finite time for both t > 0 and t < 0. In other words, T < and T <. Remark.3. When Eϕ) = 0, the conclusion of Theorems.1 and. is false for α = N. Indeed, let ϕ X rad, ϕ 0, be a solution of ϕ + ϕ = λ ϕ N ϕ, in R N. Then Eϕ) = 0 from 3.5) but ut, x) = ϕx)e it is the solution of 1.1) and so T = T =. Similar results exist for the critical case. See Nawa [19, 1]. It is shown that if ϕ H 1 R N ) satisfies Eϕ) < ϕ, iϕ) ϕ, when N = 1, or if Eϕ) < 0, when N, then the corresponding solution of 1.1) blows up in finite time or grows up at infinity, the first case always occurring when N = 1. Here,, ) denotes the scalar product in R N ). See also Nawa [0, ]. Note that in the case N = 1, the result of Nawa [1] slightly improves that of Ogawa and Tsutsumi [], since it allows to make blow-up some solution with nonnegative energy. Proof of Theorem.1. We argue by contradiction. Set for every t T, T ), ht) = xut). Then h C T, T ); R) and t T, T ), h t) = Eϕ) ) ut).1) Glassey [15]). Since Eϕ) 0, we have by Gagliardo-Nirenberg s inequality Proposition 3.1) and conservation of energy and charge, ut) λ α+ ut) α+ α+ t T, T ). Since α > N obtain and ϕ 0, we deduce that inf t T, T ), h t) C. C ut), for every ut) > 0 and with.1), we t T,T ) 3

4 So, if T = or if T = then there exists S T, T ) with S large enough such that hs) < 0 which is absurd since h > 0. Hence the result. Proof of Theorem.. For Ψ W, R N ; R), Ψ 0, we set t T, T ), V t) = Ψx) ut, x) dx. We know that there exists Ψ W, R N ; R), Ψ 0, such that V C T, T ); R) and R N t T, T ), V t) Eϕ) ) ut), see the proof of Theorem.7 of Cazenave [8] and Remark.13 of this reference). We conclude in the same way that for Theorem.1. 3 Sharp estimate In this section, we recall the sharp estimate in a Gagliardo-Nirenberg s inequality Proposition 3.1) and a result concerning the ground states. et λ > 0, ω > 0 and 0 < α < 0 < α < if N = 1). We consider the following elliptic N equations. { R + R = R α R, in R N, R H 1 R N 3.1) ; R), R 0, { Φ + ωφ = λ Φ α Φ, in R N, Φ H 1 R N ; R), Φ 0. 3.) It is well-known that the equation 3.) possesses at less one solution ψ. Furthermore, each solution ψ of 3.) satisfies ψ C,γ R N ) W 3,p R N ), γ 0, 1), p [, ), ψx) Ce δ x, for all x R N, where C and δ are two positive constants which do not depend on x, multi-index. Finally, ψ satisfies the following identities. lim x Dβ ψx) = 0, β ψ = ω αn ) ψ, 3.3) ψ α+ ωα + ) = α+ λ αn )) ψ, 3.) ψ α+ α + ) = α+ λ ψ. 3.5)

5 Such solutions are called bound states solutions. Furthermore, 3.) has a unique solution Φ satisfying the following additional properties. Φ S rad R N ; R); Φ > 0 over R N ; Φ is decreasing with respect to r = x ; for every multi-index β N N, there exist two constants C > 0 and δ > 0 such that for every x R N, Φx) + D β Φx) Ce δ x. Finally, for every solution ψ of 3.), we have Φ ψ. 3.6) Such a solution is called a ground state of the equation 3.). Equation 3.) is studied in the following references. Berestycki, Gallouët and Kavian [3]; Berestycki and ions [, 5] ; Berestycki, ions and Peletier [6]; Gidas, Ni and Nirenberg [10]; Jones and Küpper [16]; Kwong [18]; Strauss [5]. See also Cazenave [9], Section 8. Proposition 3.1. et 0 < α < 0 < α < if N = 1) and R be the ground state solution of N 3.1). Then the best constant C > 0 in the Gagliardo-Nirenberg s inequality, is given by f H 1 R N ), f α+ α+ C = α + ) C f αn ) αn ) ) f, 3.7) R α. 3.8) See Weinstein [6] for the proof in the case N. See also emma 3. of Cazenave [8] in the case α =. But for convenience, we give the proof. It makes use of a compactness result which is an N adaptation of the compactness lemma due to Strauss Strauss [5]). Proof of Proposition 3.1. We define for every f H 1 R N ), f 0, the functional Jf) = αn ) f f f α+ α+, and we set σ = inf Jf). Then σ 0, ) by 3.7). We have to show that σ = C 1 f H 1 where C \{0} is defined by 3.8). et f n ) n N H 1 R N ) be a minimizing sequence. et µ n = f n N, λ f n N n = f n f n and x R N, v n x) = µ n f n λ n x). Then v n = v n = 1 and Jf n ) = Jv n ) = v n α+) α+ n σ. et v n be the symmetrization of Schwarz of v n see Bandle [1]; Berestycki and ions [], Appendix A.III). Then Jv n) n σ and by compactness, vn l v as l in HwR 1 N ) and in particular, in α+ w Ω) for every 5

6 subset Ω R N ) and v n l α+ l v, for a subsequence v n l ) l v n) n and for some v H 1 rad RN ). Indeed, since v n) n N is bounded in H 1 rad RN ) and nonincreasing with respect to x, then l N and x R N, v n l x) C x N, where C > 0 does not depend on l and x Berestycki and ions [], Appendix A.II, Radial emma A.IV). From this and Hölder s inequality, we deduce that l N and R > 0, v n l α+ R N \B0,R)) CR α+), for a constant C > 0 which does not depend on l. Then R > 0, v α+ R N \B0,R)) lim inf l v n l α+ R N \B0,R)) CR α+). The strong convergence in α+ R N ) follows easily from the two above estimates and from the compact embedding H 1 B0, R)) α+ B0, R)), which holds for every R > 0. Since v n α+ = vn α+, it follows that v α+ α+ = σ 1 and then v 0. Thus, Jv) = σ and v = v = 1. It follows that w H 1 R N d ), dt Jv + tw) t=0 = 0. So v satisfies v + αn ) v = σ α+) v α v, in R N. Set ) 1 ) 1 α a =, b = and x R N, ux) = bvax). Then u Hrad 1 RN ) is a solution αn ) σα+) αn ) of 3.1) and Ju) = σ. By 3.3) 3.), we obtain Ju) = C 1 u α R α = σ and JR) = C 1 σ since R also satisfies 3.1)). Then u R and so with 3.6), u = R. Hence the result. Necessary condition and sufficient condition for global existence Theorem.1. et λ > 0, N < α < < α < if N = 1) and R be the ground state solution N of 3.1). We define for every a > 0, r a) = αn ) γ a) = 1. If ϕ H 1 R N ) satisfies ) αn )) ) λ 1 α αn ) α R a αn ),.1) ) αn )) r a)..) ϕ γ ϕ ),.3) then the corresponding solution u C T, T ); H 1 R N )) of 1.1) is global in time, that is T = T =, and the following estimates hold. ut) < t R, Eϕ), ut) < r 1 ϕ ), where r 1 is the function defined by.5). In particular, Eϕ) > ϕ. 6

7 . For every a > 0 and for every b > 0 satisfying a > r b), there exists ϕ a,b H 1 R N ) with ϕ a,b = a and ϕ a,b = b such that the associated solution u a,b C T, T ); H 1 R N )) of 1.1) blows up in finite time for both t > 0 et t < 0. In other words, T < and T <. Furthermore, Eϕ a,b ) > 0 r b) < a < ρ b) and Eϕ a,b ) = 0 a = ρ b), where for every a > 0, Finally, Eϕ a,b ) < ϕ a,b. Remark.. et γ be the function defined by.). Set ) αn ) ρ a) = r a)..) A = {ϕ H 1 R N ); ϕ γ ϕ )}. By Theorem.1, for every ϕ A, the corresponding solution of 1.1) is global in time and uniformly bounded in H 1 R N ). It is interesting to note that A is an unbounded subset of H 1 R N ). So Theorem.1 gives a general result for global existence for which we can take initial values with the H 1 R N ) norm large as we want. Remark.3. et γ, r, and ρ be the functions defined respectively by.),.1) and.). It is clear that since α > N, γ, γ 1 0, ) to 0, ) and for every a > 0,, r, r 1, ρ and ρ 1 are decreasing and bijective functions from γ 1 a) = r 1 a) = ρ 1 a) = ) 1 r 1 a), αn ) ) r 1 a). ) 1 ) λ 1 α α R a αn ),.5) So the condition condition.3) is equivalent to the condition ϕ γ < r < ρ and γ 1 < r 1 < ρ 1. γ 1 ϕ ). Furthermore, Remark.. et γ, r, and ρ be the functions defined respectively by.),.1) and.). Then γ α N λ 1 α α R and r N λ 1 α α R and even, ρ N λ 1 α R ). So we obtain the sharp condition for global existence, ϕ < λ 1 α R which coincide with the results obtained by Weinstein [6, 7]. However, we do not know if γ or r are optimum. 7

8 5 Proof of Theorem.1 In order to prove the blowing up result of Theorem.1), we need of several lemmas. We follow the method of Berestycki and Cazenave [] see also Cazenave [7] and Cazenave [9], Section 8.). A priori, we would expect to use Theorem.1, that is to construct initial values in X with nonpositive energy, which is the case for α =. But it will not be enough because we have to make blow-up N some solutions whose the initial values have a positive energy. We define the following functionals and sets. et λ > 0, ω > 0, β > 0, 0 < α < N 0 < α < if N = 1) and ψ H 1 R N ). β α + ) ψ) = λ Qψ) = ψ Sψ) = 1 ψ ψ ψ α+ α+, if ψ 0, λ α + ) ψ α+ α+, λ α + ψ α+ α+ + ω ψ, Pβ, ψ)x) = β N ψβx), for almost every x R N, M = { ψ H 1 R N ); ψ 0 and Qψ) = 0 }, A = { ψ H 1 R N ); ψ 0 and ψ + ωψ = λ ψ α ψ, in R N}, G = {ψ A; φ A, Sψ) Sφ)}. Note that by the discussion at the beginning of Section 3 and 3.3) 3.6), M, A and G. emma 5.1. We have the following results. 1. β > 0, β β ψ), SPβ, ψ)) < SPβ ψ), ψ)).. The following equivalence holds. ψ G { ψ M, Sψ) = min φ M Sφ), 3. et m def = min φ M Sφ). Then φ H1 R N ) with Qφ) < 0, Qφ) Sφ) m. See Cazenave [9], emma 8..5 for the proof of 1; Proposition 8.. for the proof of ; Corollary 8..6 for the proof of 3. There is a mistake in the formula 8..) of this reference. Replace the expression λ u) nα = α+ u R ) u α+) 1 n R with λ u) nα n = α+) nα u ) u α+) 1 R n R. n The proof of 1 of Theorem.1 relies on the following lemma. 8

9 emma 5.. et I R, be an open interval, t 0 I, p > 1, a > 0, b > 0 and Φ CI; R + ). We set, x 0, fx) = a x + bx p, x = bp) 1 p 1 and b = p 1 p x. Assume that Φt 0) < x, a b and that f Φ > 0. Then, t I, Φt) < x. Proof. Since Φt 0 ) < x and Φ is a continuous function, there exists η > 0 with t 0 η, t 0 +η) I such that, t t 0 η, t 0 + η), Φt) < x. If Φt ) = x for some t I, then f Φt ) = fx) = a b 0. But f Φ > 0. Then, t I, Φt) < x. The proof of of Theorem.1 makes use the following lemma. emma 5.3. et λ > 0, ω > 0 and N < α < < α < if N = 1). We set for every β > 0 N and for every ψ H 1 R N ), ϕ β = Pβ, ψ). et u β C T, T ); H 1 R N )) be the solution of 1.1) with initial value ϕ β. Then we have, ψ G, β > 1, T < and T <. Proof. et ψ G. By 3.5), we have β ϕ β ) α + ) = λ Qϕ β ) = β ψ β ψ β ψ α+ α+, λ ) α + ) β ψ α+, α+ λ α + ) ψ α+ = ψ α+. ) So, β ϕ β ) = β, Qϕ β ) = β ψ β 1 and β ψ) = 1. From these three last equalities, from 1 and of emmas 5.1 and by conservation of charge and energy, we have β > 1, Qϕ β ) < 0, 5.1) β 1, Sϕ β ) < Sψ) m, 5.) β > 0, t T, T ), Su β t)) = Sϕ β ). 5.3) By continuity of u β, by 5.1) 5.3) and from 3 of emma 5.1, we have for every β > 1, t T, T ), Qu β t)) Sϕ β ) m < 0. 5.) Set t T, T ), ht) = xu β t). Then we have by Glassey [15], h C T, T ); R) and t T, T ), h t) = 8 u β t) λ α+ u βt) α+ α+ 8Qu β t)). So with 5.), t T, T ), h t) 8Sϕ β ) m) < 0, 9

10 for every β > 1. It follows that T < and T <. Hence the result. Proof of Theorem.1. We proceed in two steps. Step 1. We have 1. et C be the constant defined by 3.8). b = λ α+ C ϕ αn ) We set I = T, T ), t 0 = 0, p =, a = ϕ,, x = bp) 1 p 1, b = p 1 p x, t I, Φt) = ut) and for any x 0, fx) = a x + bx p. Then by conservation of energy, by Proposition 3.1 and by conservation of charge, we have t I, ut) = Eϕ) + λ α + ut) α+ α+ < ϕ + λ α + C ϕ αn ) ut) ). And so, t I, a ut) +b ut) ) p > 0, that is f Φ > 0. Furthermore, Φt 0 ) a b < x. Indeed, by Remark.3, we have Φt 0 ) b ϕ γ 1 ϕ ) ϕ γ ϕ ). So by emma 5., Φt) < x [r 1 ϕ )], t I. Thus, I = R and for every t R, t R, Eϕ) 1 ut) < r 1 ϕ ). It follows from conservation of charge and energy, 3.7), 3.8), and the above inequality, that ut) λ ) ut) Hence 1. Step. We have. = 1 ut) 1 > 1 ut) 1 = ut). α + C ϕ αn ) [ r 1 ) ϕ ) ut) 1 ] et R be the ground state solution of 3.1). et first remark from the assumptions and from Remark.3, we have b > r 1 a). We set ) ν = [r 1 a)] N αn ) ) N a N R 1, ) α ω = [r 1 a)] αn ) a = λ 1 α R a 1, and for every x R N, ψx) = νr ωx). Then ψ S rad R N ) A. Since R satisfies 3.1) 3.6), it follows that ψ G. Furthermore, ψ = a and ψ = r 1 a). et β = b 10 r 1 a) > 1. Set for every

11 x R N, ϕ a,b x) = ϕ β x) = Pβ, ψ)x). In particular, ϕ a,b S rad R N ) and ϕ a,b satisfies ϕ a,b + ωβ ϕ a,b = λβ ϕ a,b α ϕ a,b, in R N. Denote u a,b C T, T ); H R N ) X rad ) the solution of 1.1) with initial value ϕ a,b. Then by emma 5.3, T < and T <. Moreover, ϕ a,b = a, ϕ a,b = b and by 3.5), By Remark.3, it follows that Eϕ a,b ) = 1 ϕ a,b λ α + ϕ a,b α+ α+ = 1 ϕ a,b λ α + β ψ α+ α+ ) = ϕ a,b = ϕ a,b β β ). Eϕ a,b ) 0 β ) b ) r 1 a) ρ 1 a) a ρ b). Hence the result. Acknowledgments The author would like to thank his thesis adviser, Professor Thierry Cazenave, for his suggestions and encouragement. References [1] C. Bandle. Isoperimetric inequalities and applications. Pitman, ondon, [] H. Berestycki and T. Cazenave. Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci. Paris Sér. I Math., 939):89 9, [3] H. Berestycki, T. Gallouët, and O. Kavian. Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math., 975): , [] H. Berestycki and P.-. ions. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal., 8):313 35, [5] H. Berestycki and P.-. ions. Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal., 8):37 375, [6] H. Berestycki, P.-. ions, and. A. Peletier. An ODE approach to the existence of positive solutions for semilinear problems in R N. Indiana Univ. Math. J., 301):11 157, [7] T. Cazenave. Stability and instability of stationary states in nonlinear Schrödinger equations. In Contributions to nonlinear partial differential equations I, volume 89 of Research Notes in Mathematics, pages Pitman, Boston, Mass.,

12 [8] T. Cazenave. Blow-up and scattering in the nonlinear Schrödinger equation, volume 30 of Textos de Métodos Matemáticos. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, second edition, [9] T. Cazenave. An introduction to nonlinear Schrödinger equations, volume 6 of Textos de Métodos Matemáticos. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, third edition, [10] B. Gidas, W. M. Ni, and. Nirenberg. Symmetry of positive solutions of nonlinear elliptic equations in R N. In Mathematical analysis and applications, Part A, pages Academic Press, New York, [11] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, and 3. Ann. Inst. H. Poincaré Sect. A N.S.), 83):87 316, [1] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal., 31):1 3, [13] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. II. Scattering theory, general case. J. Funct. Anal., 31):33 71, [1] J. Ginibre and G. Velo. The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non inéaire, ):309 37, [15] R. T. Glassey. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys., 189): , [16] C. Jones and T. Küpper. On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal., 17): , [17] T. Kato. On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor., 61):113 19, [18] M. K. Kwong. Uniqueness of positive solutions of u u + u p = 0 in R N. Arch. Rational Mech. Anal., 1053):3 66, [19] H. Nawa. Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger. In R. Caflisch and G. Papanicolaou, editors, Singularities in Fluids, Plasmas and Optics, NATO ASI series Kluwer, pages Dordrecht, [0] H. Nawa. Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity. J. Math. Soc. Japan, 6): , 199. [1] H. Nawa. imiting profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity. Proc. Japan Acad. Ser. A Math. Sci., 7310): , [] H. Nawa. Asymptotic and limiting profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power. Comm. Pure Appl. Math., 5):193 70, [3] T. Ogawa and Y. Tsutsumi. Blow-up of H 1 solution for the nonlinear Schrödinger equation. J. Differential Equations, 9): , [] T. Ogawa and Y. Tsutsumi. Blow-up of H 1 solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity. Proc. Amer. Math. Soc., 111):87 96, [5] W. A. Strauss. Existence of solitary waves in higher dimensions. Comm. Math. Phys., 55):19 16, [6] M. I. Weinstein. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87): ,

13 [7] M. I. Weinstein. On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Comm. Partial Differential Equations, 115):55 565,