Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities

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1 Physica D 175 3) Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities Gadi Fibich a, Xiao-Ping Wang b, a School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel b Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 6 March ; received in revised form 1 July ; accepted 5 July Communicated by I. Gabitov Abstract We show that the ground-state solitary waves of the critical nonlinear Schrödinger equation iψ t t, r)+ ψ+vɛr) ψ /d ψ = in dimension d are orbitally stable as ɛ ifv)v ) ) <G d [V )], where G d is a constant that depends only on d. Elsevier Science B.V. All rights reserved. Keywords: Solitary waves; Nonlinear Schrödinger equation; Inhomogeneous nonlinearities; Stability 1. Introduction The critical nonlinear Schrödinger equation iψ t t, x) + ψ + ψ /d ψ =, ψ, x) = ψ x), x R d, 1.1) models the propagation of intense laser beams in a homogeneous bulk medium with a Kerr nonlinearity. It is well known that solutions of 1.1) can become singular in finite time if ψ N c, where ψ = ψ dx is the input beam power, and N c, the critical power for singularity formation, is a constant which depends only on d. The critical power N c, thus, sets an upper limit on the amount of power ψ ) that can be propagated with a single beam. The critical NLS 1.1) admits solitary waves ψ = e it R x) whose power is exactly equal to the critical power, i.e., R N c [17]. These solitary waves are, however, strongly unstable. As a result, it is not possible to realize stable high-power propagation in a homogeneous bulk media. A few years ago, it was suggested that stable high-power propagation can be achieved in plasma by sending a preliminary laser beam that creates a channel with a reduced electron density, and thus reduces the nonlinearity inside the channel [,8]. Under these conditions, beam propagation can be modeled, in the simplest case, by the inhomogeneous nonlinear Schrödinger equation iψ t + ψ + Vɛx) ψ /d ψ =, Corresponding author. address: mawang@ust.hk X.-P. Wang). 1.) /3/$ see front matter Elsevier Science B.V. All rights reserved. PII: S )66-7

2 G. Fibich, X.-P. Wang / Physica D 175 3) where Vɛx) is proportional to the electron density and ɛ is a small parameter. It is possible to set the experimental system so that both the potential V and the initial condition ψ are radially symmetric, i.e., V = Vr)and ψ = ψ r), where r = x. In this case, the equation for ψ is iψ t t, r) + ψ + Vɛr) ψ /d ψ =, = r + d 1 r r. 1.3) Existence and nonexistence of blowup solutions of 1.) were studied by Merle for certain types of inhomogeneities [1]. These results imply that a necessary condition for blowup in the radially symmetric case 1.3) is that ψ N c/v d/ ). For comparison, in the absence of the preliminary beam V V ) and the critical power is N c /V d/ ). We thus see that it is possible to raise the critical power for blowup by lowering the magnitude of the nonlinearity near the origin. In particular, when V) = all solutions of 1.3) exist globally. The solitary waves of 1.3) are given by ψ = e it φ r), where φ is the solution of φ r) φ + Vɛr)φ /d+1 =, φ ) =, φ ) =. 1.) The following theorem gives the existence of positive ground-state) solitary waves. Theorem 1. Let <Vr)<Cand let >. Then, 1) there exists a positive solution to 1.). ) the positive solution is unique when ɛ is small enough. See Section 3 for the proof of part 1) and Section for the proof of part ). We note that existence of solutions of 1.) was proved in [15] in the framework of a more general equation. Our proof is, however, considerably simpler because of radial symmetry. The method used in the existence proof was originally due to Strauss [1] and to Berestycki and Lions [1]. When the inhomogeneity is induced by the preliminary laser beam, Vr)increases monotonically from V) to V ). In that case, lim ɛ φ = N c V d/ ) > N c V d/ ) = lim ɛ φ. 1.5) Therefore, it is reasonable to assume that when <ɛ 1, the power of the solitary waves will be below the critical power for blowup N c /V d/ ). In that case, one can expect the solitary waves to be stable, because solitary waves in NLS equations are typically unstable if and only if a small perturbation can lead to singularity formation. Surprisingly, however, our results show that monotonicity of V is not the correct condition for stability. Throughout this paper, we make the following assumptions on V : V>, V C L, V i) r) C e r for i = 1,, 3,, 1.6) where V i) is the ith derivative of V. We note that these assumptions are consistent with the electron density induced by the preliminary beam. The natural definition of stability of solitary waves is the one of orbital stability. Definition. Let φ be a solution of 1.). We say that ψr,t) = e it φ r) is an orbitally stable solution of 1.3) if ɛ >, δ> such that for any ψr,) H 1 R n ) which satisfies inf θ ψr,) e iθ φ H 1 <δ, the corresponding solution ψr,t) of 1.3) satisfies sup inf ψr,t) e iθ φ H 1 <ɛ. t θ

3 98 G. Fibich, X.-P. Wang / Physica D 175 3) Our stability proof follows [5,7,9,1]. We define d) = Eφ ) + Qφ ), 1.7) where Eu) = 1 u 1 Vɛr) u /d+, Qu) = 1 u. 1.8) /d + We recall that the generic condition for stability of solitary waves is d ) > [13,18]. We have the following lemma. Lemma 1. Let 1.6) hold. Then d ) > for ɛ sufficiently small if and only if V)V ) ) <G d [V )], 1.9) where G d = 6 d + d {r R /d+1 L 1 r R /d+1 )} r R /d+ 1.1) is a constant which depends only on d, Rr) is the ground-state 1 solution of R R + R /d+1 =, R ) = R ) =, 1.11) and L = 1 + /d + 1)R /d. We now state our main theorem which shows that the condition d ) > indeed implies stability. Theorem. Let 1.6) and 1.9) hold, let > and let φ be the ground-state solution of 1.). Then ψ = e it φ r) is an orbitally stable solution of 1.3) for ɛ sufficiently small. This result suggests that it may be possible to produce stable high-power beam propagation in plasma by sending a preliminary beam. In order to motivate the condition 1.9), we use perturbation analysis in Section to calculate the power of φ. Let φ r; ɛ) be the solution of 1.) and let ˆɛ = ɛ/. Then, we have, as ˆɛ, φ = 1 V d/ ) [ R ˆɛ + d) r R /d+ d[v)] G d [V )] V)V ) )) + Oˆɛ 6 ) ], 1.1) where R is the ground-state solution of 1.11) and G d is defined in 1.1). Thus, the stability condition 1.9) is also a necessary and sufficient condition for the power of the solitary waves to be below the critical power R /V d/ ). Indeed, from 1.8),.6) and 5.) we have that d ) = 1/) φ. Therefore, the stability condition d ) > is satisfied if and only if φ is monotonically increasing in hence monotonically decreasing in ɛ. The failure of the reasoning leading to the conclusion that monotonicity of V implies stability of solitary waves thus lies in the assumption that monotonicity of V implies that φ is monotonically decreasing in ɛ. Indeed, when V is monotonic then V ) >. In principle, this term would have given Oˆɛ ) contributions to φ, whereas V ) ) would only give Oˆɛ ) contributions. However, because the Oˆɛ ) terms due to V ) completely balance each other, stability is determined by both V ) and V ) ). 1 That is, the nontrivial solution with the smallest L norm.

4 G. Fibich, X.-P. Wang / Physica D 175 3) We calculated numerically that in the physically relevant case d =, r R r dr 1.359, r R 3 L 1 r R 3 )r dr.1. Therefore, G Since G <, a necessary condition for stability is that V ) ) be negative! We recall that the NLS iψ t t, x) + ψ + ψ σ ψ =, ψ, x) = ψ x), x R d, 1.13) is called subcritical, critical, and supercritical when σd <, σd =, and σd >, respectively. It is well known that solutions of the NLS can become singular only in the critical and in the supercritical cases, and that the solitary wave solutions are stable only in the subcritical case. Indeed, there is a general rule that solitary waves of NLS equations are stable if and only if the equation does not admit blowup solutions. We see, thus, that when condition 1.9) is satisfied, Eq. 1.3) is an exception to this rule as it admits blowup solutions yet its waveguides are stable. Finally, we note that inhomogeneity of the nonlinearity is unlikely to affect the orbital stability of subcritical solitary waves of 1.13) or the strong instability of supercritical ones. 3 Indeed, d ) >, d ) = and d ) <, when the NLS 1.13) is subcritical, critical and supercritical, respectively. Our calculation see proof of Lemma 7) shows that the effect of inhomogeneity on d ) is Oɛ ). Therefore, stability can be affected by the inhomogeneity only in the critical case. The paper is organized as follows. In Section, we derive 1.1) which motivates our rigorous results. In Sections 3 and, we prove existence and some properties Theorem 1) of solitary wave solutions. Section 5 gives the proof of the stability results Theorem ).. Perturbation analysis In this section, we derive 1.1) by a perturbation analysis. Let φ = [/V )] d/ S r). Then, the equation for S is Sr; ɛ) S + Vˆɛr) V) S/d+1 ɛ =..1) When ˆɛr 1, we can expand Vˆɛr) V) 1 + a ˆɛ r + bˆɛ r + Oˆɛ 6 ),.) where a = V )/V) and b = V ) )/V). We look for a solution of.1) of the form S = R + a ˆɛ gr) +ˆɛ hr) + Oˆɛ 6 )..3) Therefore, ) ) S m = R m +ˆɛ amr m 1 g +ˆɛ mr m 1 m h + a R m g + Oˆɛ 6 ), and the equations for R, g, and h are 1.11), ) g g + d + 1 R /d g = r R /d+1,.) Another exception to this rule is the critical NLS on bounded domains []. 3 Strong instability of supercritical solitary waves was proved in [16].

5 1 G. Fibich, X.-P. Wang / Physica D 175 3) and ) ) d h h + d + 1 R /d h = a r + 1 R /d g br R /d+1, respectively. If we multiply.1) by R and integrate by parts we get that R S RS + VRS /d+1 =. If we substitute.) and.3) in this equation and collect terms, the Oˆɛ ) and Oˆɛ ) equations are ) R g Rg + d + 1 R /d+1 g = r R /d+,.5) and ) R h Rh + d + 1 R /d+1 h = d + 1 ) a R /d g a d + 1 r R /d+1 g b r R /d+..6) If we multiply 1.11) by S and integrate by parts we get that R S RS + SR /d+1 =. If we substitute.3) in this equation and collect terms, the Oˆɛ ) and Oˆɛ ) equations are R g Rg + R /d+1 g =.7) and R h Rh + R /d+1 h =..8) From.) we have that S + VS /d+ ˆɛ + + d /d + rv ˆɛr)S /d+ =..9) If we substitute.) and.3) in this equation and collect terms, the Oˆɛ ) and Oˆɛ ) equations are Rg = [ ) ] r R /d+ + + d d + R /d+1 g + r R /d+,.1) /d + and Rh + a g = + d br R /d+ + + d a r R /d+1 g + d d + d R/d+1 h + ) d d + 1 a R /d g,.11) respectively. If we subtract.7) from.5), we get that /d R /d+1 g = r R /d+. Substitution into.1) gives that Rg =..1)

6 G. Fibich, X.-P. Wang / Physica D 175 3) If we subtract.8) from.6), we get that R /d+1 h = d + 1 ) a R /d g a d d + 1 r R /d+1 g b r R /d+. Substitution into.11) gives that Rh + a g = d + d b r R /d+ + a r R /d+1 g..13) Therefore, combining ] S = R + a ˆɛ Rg +ˆɛ [ Rh + a g + Oˆɛ 6 ), with.1) and.13) gives that [ d S = R +ˆɛ + d b r R /d+ + a Therefore, [ R ɛ = [V)] d/ R d V ) ) +ˆɛ + d V) ] r R /d+1 g + Oˆɛ 6 ). r R /d+ + [V )] V ) ) ] r R /d+1 g + Oˆɛ 6 ). Since g = L 1 r R /d+1 ), relation 1.1) follows. 3. Existence of a ground state In order to prove Theorem 1, we introduce the minimization problem M) = inf u Hradial 1 I u), subject to constraint Ku) = 1, where I u) = u + u, Ku) = 3.1) Vɛr) ur) /d+. 3.) Lemma. Let <Vr)<Cand let >. Then, the minimization problem 3.1) has a positive minimizer. Proof. Let u n be a minimizing sequence, i.e., I u n ) M) and Ku n ) = 1. We can assume that u n is positive. Since u n H 1 C uniformly in n, we have that u n u ɛ weakly in H 1 and thus I u ɛ ) lim n I u n ) = M). Because the embedding Hradial 1 Rd ) L /d+ is compact we have that u n u ɛ strongly in L /d+. Since, in addition, V is bounded, K 1/p u ɛ ) K 1/p u n ) = V 1/p u ɛ p V 1/p u n p V 1/p u ɛ u n ) p V 1/p u ɛ u n p, where p = /d +. Therefore, Ku ɛ ) = 1 and u ɛ is a positive minimizer of 3.1). Proof of 1) in Theorem 1. For clarity we write from now on φ instead of φ, except where we want to emphasize the parametric dependence on.

7 1 G. Fibich, X.-P. Wang / Physica D 175 3) The Euler Lagrange equation for the minimizer of 3.1) is u ɛ u ɛ + λv ɛr) u ɛ /d+1 =, 3.3) where λ is a Lagrange multiplier. Let φ = λ d/ u ɛ. Then, φ is a positive solution of 1.). 3.). Several technical lemmas In this section, we prove several technical results that are used in Section 5 and also part ) of Theorem 1. We first note that standard calculations show that solutions of 1.) satisfy the following identities: φ φ + Vɛr)φ /d+ =,.1) φ = + d Vɛr)φ /d+ + 1 /d + which are usually referred to as Pohozaev identities. ) d r dr Vɛr) φ /d+,.) Lemma 3. Let u ɛ be the minimizer of 3.1) and let φ be given by 3.). Then, I φ) = Kφ) = [M)] /d+1. In addition, when = 1 then φ H 1 = u ɛ d/+1. H 1.3).) Proof. The identity I φ) = Kφ) is simply.1). Since Ku ɛ ) = 1, it follows from 3.) that λ = [Kφ)] 1/d/+1)..5) Therefore, M) = I u ɛ ) = λ d/ I φ) = [Kφ)] 1//d+1) I φ) = [I φ)] 1//d+1), which leads to.3). Eq..) follows from.3) since I 1 ) = H 1. Let φr) = d/ R ɛ r)..6) Then, by 1.), R ɛ r) R ɛ + V ɛ r ) R /d+1 ɛ =..7) Lemma. Let P ɛ = R ɛ /. Then, V ɛ r ) R /d+1 ɛ P ɛ = d 8 ɛ 3/ rv ɛ r ) R /d+ ɛ..8)

8 G. Fibich, X.-P. Wang / Physica D 175 3) Proof. Differentiating.7) with respect to gives ) P ɛ P ɛ + d + 1 V ɛ r ) R /d ɛ P ɛ 1 ɛ 3/ rv Rɛ /d+1 =..9) If we multiply.9) by R ɛ and integrate, we have ) R ɛ P ɛ R ɛ P ɛ + d + 1 VR /d+1 ɛ P ɛ 1 ɛ 3/ If we multiply.7) by P ɛ and integrate, we have R ɛ P ɛ R ɛ P ɛ + VR /d+1 ɛ P ɛ =. rv R /d+ ɛ =. The difference of the last two equations gives.8). Let us define the linearized operator L ɛ on Hradial Rd ) by ) L ɛ = 1 + d + 1 V ɛ r ) R /d ɛ..1) We can rewrite.9) as L ɛ P ɛ ) = 1 ɛ 3/ rv Rɛ /d+1..11) It is well known that KerL ) is empty see e.g. [11]) and that L 1 is bounded. Therefore, there exists a constant C > such that L v C v..1) Lemma 5. Let R ɛ be the solution of.7), then a) R ɛ H 1 C uniformly as ɛ. b) lim ɛ V ɛ r ) R ɛ r) V)R r) =..13) c) Let >and let ɛ be sufficiently small. Then L ɛ is invertible and L 1 ɛ is bounded. d) Let R ɛ be a positive solution of.7). Then there exist positive constants ɛ, c and L such that R ɛ r) c e r/ for all <ɛ ɛ and for all r L. Proof. Without loss of generality, we can set = 1. From.) and.6) it follows that R ɛ H 1 = φ ɛ H 1 = u ɛ d/+1. Therefore, it is sufficient to show that u H 1 ɛ H 1 is uniformly bounded. To see that, let v ɛ = α ɛ u, where u is the minimizer of the minimization problem 3.1) when ɛ = and α ɛ is chosen so that Kv ɛ ) = 1. Therefore, for the minimizer u ɛ of 3.1) we have that u ɛ H 1 = I 1 u ɛ ) I 1 v ɛ ) = αɛ I 1u ). Since lim ɛ α ɛ = 1 it follows that u ɛ H 1 is uniformly bounded. That completes the proof of a). To prove b), we note from 1) and standard elliptic regularity theory [3], we have that R ɛ R weakly in H 1 and strongly in Cloc, where R is the unique solution of.7) for ɛ =. From the radial lemma of Strauss, we have that C R ɛ r d 1)/ R ɛ H 1 for r 1..1) Again, in light of a), we only need to prove.13) on a bounded domain which is now obvious.

9 1 G. Fibich, X.-P. Wang / Physica D 175 3) Remark. Our stability proof is limited to the case d because we rely on the uniform decay estimate.1) in the proof of b). We now proceed to prove c). Since L ɛ = L + /d + 1)V ɛr/ ))Rɛ V)R ), we have that ) L ɛ v, L ɛ v = L v, L v + L v, + ) d + 1 V + d + 1 ) V d + 1 V ɛ r ) R ɛ V)R ɛ r ) ) R ɛ V)R v ), L v ɛ r ) ) R ɛ V)R v, V ɛ r ) ) R ɛ V)R v ) L v d + 1 V ɛ r ) R ɛ V)R v L v ) d + 1 V ɛ r ) R ɛ V)R v. Using.1), we have that ) ) ) L ɛ v v [C d + 1 C + d + 1 V ɛ r ) R ɛ V)R Therefore, in light of b), when ɛ is sufficiently small there exists C 1 > such that L ɛ v C 1 v, from which c) follows. Next we prove d). Eq..7) can be rewritten as R ɛ = 1 V ) ɛr R /d ɛ ) R ɛ. From Lemma 5, inequality.1) and 1.6) it follows that there exist ɛ, L>such that for all <ɛ ɛ and for all r L, ) ɛr 1 V R /d ɛ 1. Let v ɛ r) = c e r/ R ɛ, where c is sufficiently large such that v ɛ L) >. Then, v ɛ = c e r/ R ɛ = c e r/ R ɛ 1 VR /d ɛ ) 1 c e r/ 1 R ɛ = 1 v ɛ. Therefore, from the maximum principle for exterior domains [3], we have that v ɛ for all r L, and thus that R ɛ r) c e r/. This completes the proof of the lemma. Proof of ) in Theorem 1. We prove uniqueness of positive solutions for.7), which is equivalent to 1.) up to a simple rescaling. Let R 1ɛ, R ɛ be two solutions of.7), i.e., R iɛ R iɛ + V ɛ r ) R /d+1 i ɛ =, i = 1,. ].

10 G. Fibich, X.-P. Wang / Physica D 175 3) We then have, L ɛ R 1 ɛ R ɛ ) =,.15) where we denote and L ɛ = 1 + V ɛ r ) WR 1ɛ,R ɛ ), R /d+1 1 ɛ R /d+1 ɛ = R 1ɛ R ɛ )W R 1ɛ,R ɛ ). Since the positive solution R of.7) is unique [6], then, as in Lemma 5b) R 1ɛ,R ɛ R uniformly. Therefore, ) WR 1ɛ,R ɛ ) d + 1 R /d as ɛ. As in Lemma 5c), we can show that L ɛ is invertible and L ɛ v C v, for ɛ small enough. By.15), this implies R 1ɛ = R ɛ for ɛ small enough. 5. Orbital stability Lemma 6. d) is differentiable and strictly increasing for >. Proof. Using 1.7), 1.8) and 3.), we have that d) = 1 I φ) 1 /d + Kφ). 5.1) Therefore, by.3) d) = 1 + d Kφ) = + d V ) ɛr R /d+ ɛ. 5.) Differentiating d) with respect to and using.8) gives that d ) = 1 VR /d+ ɛ + d ) d + 1/ ɛ rv Rɛ /d+. 5.3) Therefore, from.) we have that d ) = QR ɛ )>. 5.) Lemma 7. Let 1.6) hold. Then, d ) > for ɛ sufficiently small if and only if V ) ) <6 d + {r R /d+1 L 1 r R /d+1 )} d r R /d+ [V )].

11 16 G. Fibich, X.-P. Wang / Physica D 175 3) Proof. If we differentiate 5.3) with respect to, use.8) and.11) and expand V ɛr/ ) and V ɛr/ ) in a Taylor series in ɛ, we get that + d)d ) = d 8 ɛ 3/ rv Rɛ /d+ d 8 ɛ r V Rɛ /d+ + d ) d + ɛ 1/ rv Rɛ /d+1 P ɛ = d 8 ɛ 3/ rv Rɛ /d+ ) d + 1 ɛ + d [ d = ɛ 3 d 8 ɛ rv R /d+1 ɛ L 1 ɛ r V ) )R /d+ + d r V R /d+ ɛ rv R /d+1 ɛ ) r V )R /d+1 )L 1 r V )R /d+1 ) + FV ɛ,r ɛ )) Here FV ɛ,r ɛ ) is the remainder from the Taylor expansion. Using Lemma 5c) and d), it is easy to show that FV ɛ,r ɛ ) asɛ. Therefore, condition 1.9) implies that d ) > for ɛ sufficiently small. ]. Proof of Lemma 1. This follows from Lemma 7 and the rescaling R = V d/ )R. Lemma 8. Let d ) >. Then there exists δ = δ) > such that for all with <δ, d ) d) + d ) ) + 1 d ). Proof. Taylor expansion. Given a solution φ of 1.), we can define the set U,δ ={u H 1 radial Rd ), u φ H 1 <δ}. Since d) is monotonic Lemma 6), we can define the C 1 map ) : U,δ R + by ) 1 u) = d 1 + d Ku). 5.5) Lemma 9. Let d ) > for some >. Then there exists δ = δ) > such that for all u U,δ, Eu) Eφ) + u)[qu) Qφ)] 1 d ) u), where u) is defined in 5.5). Proof. From 1.8) and 3.), we have that Eu) + u)qu) = 1 I u)u) 1 Ku). 5.6) /d +

12 G. Fibich, X.-P. Wang / Physica D 175 3) In addition, from 5.) and 5.5), we have that Ku) = d + )du)) = Kφ u) ). Since φ u) is a minimizer of I u) u) subject to the constraint Ku) = Kφ u) ), we have that I u) u) I u) φ u) ). Therefore, using Lemma 8 and 5.1), Eu) + u)qu) 1 I 1 u)φ u) ) /d + Kφ u)) = du)) d) + d )u) ) + 1 d ) u). From 5.), we have that d ) = Qφ). Therefore, using 1.7), Eu) + u)qu) Eφ) + Qφ) + Qφ)[u) ] + 1 d ) u) = Eφ) + u)qφ) + 1 d ) u). Proof of Theorem. Assume that ψ is unstable. From the definition of stability δ > and initial data ψ k ) U,1/k) such that sup t> inf θ ψ k t) e iθ φ H 1 δ, where ψ k t) is the solution of refeq13 with initial data ψ k ). Let t k be the first time at which inf ψ kt k ) e iθ φ H 1 = δ. θ Let us denote Φ k r) = ψ k t k ). Since Eψt)) and Qψt)) are conserved in t and continuous in ψ, then 5.7) EΦ k ) Eφ) = Eψ k )) Eφ), QΦ k ) Qφ) = Qψ k )) Qφ), as k. 5.8) Let δ be small enough so that Lemma 9 applies. We then have that EΦ k ) Eφ) + k [QΦ k ) Qφ)] 1 d ) k, k = Φ k ). 5.9) From 5.7), we have that Φ k H 1 C uniformly. Since u) is a continuous map, k is uniformly bounded in k. Therefore, by 5.8), ask the left-hand side of 5.9) goes to zero. Since d ) > Lemma 1), this implies that lim k k =. Hence, using 5.) and 5.5), lim KΦ k) = lim d + )d k) = d + )d) = Kφ). 5.1) k k Using this and 5.6) and 5.8), we have that I Φ k ) = [EΦ k ) + QΦ k )] + d d + KΦ k) = [Eφ) + Qφ)] + d d + KΦ k) + [EΦ k ) Eφ)] + [QΦ k ) Qφ)] d + )d). Since d + )d) = I φ) see.3) and 5.)), we see that I Φ k ) I φ). Let v k = [KΦ k )] 1//d+) Φ k. Then Kv k ) = 1 and by.3) I v k ) = [KΦ k )] 1//d+1) I Φ k ) [Kφ)] 1//d+1) I φ) = M). Hence, {v k } is a minimizing sequence of 3.1). By uniqueness of the minimizer u ɛ Theorem 1), there exists a sequence {θ k } such that lim k v k e iθ ku ɛ H1 =. Using this and 3.),.5) and 5.1), we get that Φ k e iθ k φ H 1 = [KΦ k )] 1//d+) v k e iθ k [KΦ)] 1//d+) u ɛ H 1, which is in contradiction with 5.7).

13 18 G. Fibich, X.-P. Wang / Physica D 175 3) Acknowledgements We thank C.F. Gui for some helpful discussions. We also thank the referees for helpful suggestions. This work is supported by an RGC Competitive Earmarked Research Grant HKUST 6176/99P. The work of G. Fibich is also partially supported by Grant No. 311 from the United States Israel Binational Science Foundation BSF), Jerusalem, Israel. References [1] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal ) [] G. Fibich, F. Merle, Self-focusing on bounded domains, Physica D 155 1) [3] D. Gilbert, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, [] T.S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana J. Phys. 55 ) [5] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal ) [6] M.K. Kwong, Uniqueness of positive solutions of u u + u p = inr n, Arch. Ration. Mech. Anal ) [7] S. Levandosky, Stability and instability of fourth-order solitary waves, J. Dynam. Differential Equations ) [8] C.S. Liu, V.K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas 1 199) [9] Y. Liu, X.P. Wang, Nonlinear stability of solitary waves of a generalized Kadomtsev Petviashvili equation, Commun. Math. Phys ) [1] F. Merle, Nonexistence of minimal blow-up solutions of equations iu t = u kx) u /N u in R N, Ann. Inst. Henry Poincare Phys. Theorique ) [11] W.M. Ni, I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J ) [1] J. Shatah, Stable Klein Gordon equations, Commun. Math. Phys ) [13] J. Shatah, W.A. Strauss, Instability of nonlinear bound states, Commun. Math Phys ) [1] W.A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys ) [15] X.F. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal ) [16] K. Wang, On the strong instability of standing wave solutions of the inhomogeneous nonlinear Schrödinger equations, M.Phil. Thesis, HKUST, 1. [17] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys ) [18] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math )