Strong instability of standing waves with negative energy for double power nonlinear Schrödinger equations
SStrong instability of standing waves with negative energy for double powernonlinear Schrödinger equations
Noriyoshi Fukaya and Masahito Ohta
Abstract
We study the strong instability of ground-state standing waves e iωt φ ω ( x ) for N -dimensional nonlinear Schrödinger equations with double power nonlinearity.One is L -subcritical, and the other is L -supercritical. The strong instability ofstanding waves with positive energy was proven by Ohta and Yamaguchi (2015). Inthis paper, we improve the previous result, that is, we prove that if ∂ λ S ω ( φ λω ) | λ =1 ≤
0, the standing wave is strongly unstable, where S ω is the action, and φ λω ( x ) := λ N/ φ ω ( λx ) is the L -invariant scaling. In this paper, we consider the nonlinear Schrödinger equation with double power non-linearity(NLS) i∂ t u = − ∆ u − a | u | p − u − b | u | q − u, ( t, x ) ∈ R × R N , where(1.1) N ∈ N , a > , b > , < p < N < q < N − , and u : R × R N → C is the unknown function of ( t, x ) ∈ R × R N . Here, 1 + 4 / ( N − ∞ if N = 1 or 2. Eq. (NLS) appears in various regions of mathematicalphysics (see [1, 6, 19] and references therein).The Cauchy problem for (NLS) is locally well-posed in the energy space H ( R N ) (see,e.g., [4, 9]), that is, for each u ∈ H ( R N ), there exist the maximal lifespan T max = Mathematics Subject Classification . 35Q55, 35B35
Key words and phrases . NLS, ground state, blowup a r X i v : . [ m a t h . A P ] J un N. Fukaya and M. Ohta T max ( u ) ∈ (0 , ∞ ] and a unique solution u ∈ C ([0 , T max ) , H ( R N )) of (NLS) with u (0) = u such that if T max < ∞ , then lim t % T max k∇ u ( t ) k L = ∞ . In the case T max < ∞ , wesay that the solution u ( t ) blows up in finite time . Moreover, (NLS) satisfies the twoconservation laws E ( u ( t )) = E ( u ) , k u ( t ) k L = k u k L for all t ∈ [0 , T max ), where E is the energy defined by E ( v ) = 12 k∇ v k L − ap + 1 k v k p +1 L p +1 − bq + 1 k v k q +1 L q +1 . Furthermore, if(1.2) u ∈ Σ := { v ∈ H ( R N ) | k xv k L < ∞ } , then the solution u ( t ) of (NLS) with u (0) = u belongs to C ([0 , T max ) , Σ) and satisfiesthe virial identity(1.3) d dt k xu ( t ) k L = 8 Q ( u ( t ))for all t ∈ [0 , T max ) (see [4, Section 6.5]), where v λ ( x ) = λ N/ v ( λx ) and Q ( v ) = ∂ λ S ω ( v λ ) | λ =1 (1.4) = k∇ v k L − aN ( p − p + 1) k v k p +1 L p +1 − bN ( q − q + 1) k v k q +1 L q +1 . Eq. (NLS) has standing wave solutions of the form e iωt φ ( x ), where ω > φ ∈ H ( R N ) is a nontrivial solution of the stationary equation(1.5) − ∆ φ + ωφ − a | φ | p − φ − b | φ | q − φ = 0 , x ∈ R N . Eq. (1.5) can be rewritten as S ω ( φ ) = 0, where S ω is the action defined by S ω ( v ) = E ( v ) + ω k v k L = 12 k∇ v k L + ω k v k L − ap + 1 k v k p +1 L p +1 − bq + 1 k v k q +1 L q +1 . It is known that if ω >
0, then (1.5) has ground state solutions, that is, the set G ω := { φ ∈ F ω | S ω ( φ ) ≤ S ω ( v ) for all v ∈ F ω } of nontrivial solutions to (1.5) with the minimal action is not empty (see, e.g., [3, 12, 20]),where F ω := { v ∈ H ( R N ) \ { } | S ω ( v ) = 0 } is the set of all nontrivial solutions of (1.5).The stability and instability of standing waves are defined as follows.trong instability of standing waves 3 Definition 1.1.
Let φ ∈ F ω be a nontrivial solution of (1.5). • We say that the standing wave solution e iωt φ of (NLS) is stable if for each ε > δ > u ∈ H ( R N ) satisfies k u − φ k H < δ , then thesolution u ( t ) of (NLS) with u (0) = u exists globally in time and satisfiessup t ≥ inf ( θ,y ) ∈ R × R N k u ( t ) − e iθ φ ( · − y ) k H < ε. • We say that the standing wave solution e iωt φ of (NLS) is unstable if it is not stable. • We say that the standing wave solution e iωt φ of (NLS) is strongly unstable if foreach ε >
0, there exists u ∈ H ( R N ) such that k u − φ k H < δ , and the solution u ( t ) of (NLS) with u (0) = u blows up in finite time.In this paper, we study the strong instability of the standing wave solution e iωt φ ω for(NLS), where ω >
0, and φ ω ∈ G ω is a ground state.In the single power and L -critical or L -supercritical case when a = 0, b >
0, and1 + 4 /N ≤ q < / ( N − ω > q = 1 + 4 /N ), whereasin L -subcritical case when a > b = 0, and 1 < p < /N , Cazenave and Lions [5]proved that the standing wave is stable for any ω > ω >
0. In [15], he proved thatif ∂ λ S ω ( φ λω ) | λ =1 <
0, then the standing wave is unstable, where v λ ( x ) := λ N/ v ( λx ) isthe scaling, which does not change the L -norm. On the other hand, Fukuizumi [8]proved the stability of standing waves for sufficiently small ω >
0. See also [13, 14] forthe stability and instability in one dimensional case. The strong instability of standingwaves for sufficiently large ω was proven by Ohta and Yamaguchi [17]. In [17], theyproved the strong instability of standing waves with positive energy E ( φ ω ) > ∂ λ ˜ S ω ( φ λω ) | λ =1 ≤
0, then the standing waves is strongly unstable, where ˜ S ω is the corresponding action. This assumption is the same one as in Ohta [15]. More re-cently, Fukaya and Ohta [7] proved the strong instability of standing waves for nonlinearSchrödinger equation with an attractive inverse power potential(1.6) i∂ t u = − ∆ u − γ | x | α u − | u | q − u, ( t, x ) ∈ R × R N with γ >
0, 0 < α < min { , N } , and 1 + 4 /N < q < / ( N −
2) under thesame assumption ∂ λ ˜ S ω ( φ λω ) | λ =1 ≤ ∂ λ S ω ( φ λω ) | λ =1 ≤ Theorem 1.2.
Assume (1.1) , ω > , and that φ ω ∈ G ω satisfies ∂ λ S ω ( φ λω ) | λ =1 ≤ ,where φ λω ( x ) = λ N/ φ ω ( λx ) . Then the standing wave solution e iωt φ ω of (NLS) is stronglyunstable.Remark . In the case (1.1), E ( φ ω ) > ∂ λ S ω ( φ λω ) | λ =1 <
0. Indeed, let α = N ( p − / β = N ( q − /
2. Then since Q ( φ ω ) = ∂ λ S ω ( φ λω ) | λ =1 = 0 and 0 < α < < β , we have ∂ λ S ω ( φ λω ) | λ =1 = k∇ φ ω k L − aα ( α − p + 1 k φ ω k p +1 L p +1 − bβ ( β − q + 1 k φ ω k q +1 L q +1 = ( α + 1) Q ( φ ω ) − αE ( φ ω ) − b ( β − β − α ) q + 1 k φ ω k q +1 L q +1 < . Therefore, Theorem 1.2 is an improvement of the result of Ohta and Yamaguchi [17].To prove Theorem 1.2, we introduce the set B ω := (cid:40) v ∈ H ( R N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ω ( v ) < S ω ( φ ω ) , k v k L ≤ k φ ω k L ,K ω ( v ) < , Q ( v ) < (cid:41) , where(1.7) K ω ( v ) := ∂ λ S ω ( λv ) | λ =1 = k∇ v k L + ω k v k L − a k v k p +1 L p +1 − b k v k q +1 L q +1 is the Nehari functional. Then we obtain the following blowup result. Theorem 1.4.
Assume (1.1) , ω > , and that φ ω ∈ G ω satisfies ∂ λ S ω ( φ λω ) | λ =1 ≤ . Let u ∈ B ω ∩ Σ . Then the solution u ( t ) of (NLS) with u (0) = u blows up in finite time. Theorem 1.2 follows from Theorem 1.4 because the scaling of the ground state φ λω belongs to B ω ∩ Σ for all λ > Q ( v ) / ≤ S ω ( v ) − S ω ( φ ω ) for all v ∈ B ω (Lemma 2.1). Then by using the conservation laws, the variational characterization ofthe ground state by the Nehari functional, and the key estimate, we show the invarianceof B ω under the flow of (NLS) (Lemma 2.2). Combining the virial identity with the keyestimate, finally, we can obtain blowup of solutions to (NLS) with initial data belongingto B ω ∩ Σ by the classical argument as in Berestycki and Cazenave [2].We prove the key estimate Q/ ≤ S ω − S ω ( φ ω ) on B ω following the proof of the sameestimate for (1.6) in [7, Lemma 3.2]. The proof relies on the variational characterizationof the ground state by the Nehari functional S ω ( φ ω ) = inf { S ω ( v ) | v ∈ H ( R N ) \ { } , K ω ( v ) = 0 } trong instability of standing waves 5and the property of the graph of the function λ S ω ( v λ ). Note that the graph of S ω ( v λ )for (NLS) has the same property as that for (1.6). In the case of (1.6), since the action˜ S ω can be expressed by using the Nehari functional ˜ K ω ( v ) := ∂ λ ˜ S ω ( λv ) | λ =1 as(1.8) ˜ S ω ( v ) = 12 ˜ K ω ( v ) + ( q − q + 1) k v k q +1 L q +1 , the above variational characterization can be written by using L q +1 -norm. Therefore, in[7], they used not only the action but also L q +1 -norm effectively.On the other hand, in the case of (NLS), the action S ω cannot be expressed as (1.8)because (NLS) has double power nonlinearity. Due to this fact, we can not directlyapply the proof in [7]. However, in this case, we see that the action can be expressed as S ω ( v ) = 12 K ω ( v ) + 12 F ( v ) , where F ( v ) = a ( p − p + 1) k v k p +1 L p +1 + b ( q − q + 1) k v k q +1 L q +1 . Therefore, we can use F instead of L q +1 -norm. By applying the argument in [7] using F ,although the calculation processes differ from that in [7], we can prove the key estimateabove.We finally remark that in fact, the assumption ∂ λ S ω ( φ λω ) | λ =1 ≤ ∂ λ S ω ( φ λω ) | λ =1 ≤ ∂ λ S ω ( φ λω ) | λ =1 ≤
0, then the solution of (NLS) with u (0) = u ∈ B ω ∩ Σblows up in finite time. In Section 3, we prove the strong instability of standing wavesby using Theorem 1.4.
In this section, we prove Theorem 1.4. Throughout this section, we assume (1.1) and ω >
0. Recall that the ground state φ ω ∈ G ω satisfies K ω ( φ ω ) = 0 and the variationalcharacterization(2.1) S ω ( φ ω ) = inf { S ω ( v ) | v ∈ H ( R N ) \ { } , K ω ( v ) = 0 } (see, e.g., [11, 12]), where K ω is the Nehari functional defined in (1.7). N. Fukaya and M. OhtaFirstly, we prove the key lemma in the proof. Note that the action S ω is expressed as(2.2) S ω ( v ) = 12 K ω ( v ) + 12 F ( v ) , where F ( v ) = a ( p − p + 1 k v k p +1 L p +1 + b ( q − q + 1 k v k q +1 L q +1 . Therefore, the characterization (2.1) is rewritten as(2.3) S ω ( φ ω ) = 12 F ( φ ω ) = inf (cid:26) F ( v ) (cid:12)(cid:12)(cid:12)(cid:12) v = 0 , K ω ( v ) = 0 (cid:27) . Let α = N ( p − , β = N ( q − . Using this notation, we have S ω ( v λ ) = λ k∇ v k L + ω k v k L − aλ α p + 1 k v k p +1 L p +1 − bλ β q + 1 k v k q +1 L q +1 ,K ω ( v λ ) = λ k∇ v k L + ω k v k L − aλ α k v k p +1 L p +1 − bλ β k v k q +1 L q +1 ,N F ( v λ ) = aαλ α p + 1 k v k p +1 L p +1 + bβλ β q + 1 k v k q +1 L q +1 ,Q ( v ) = k∇ v k L − aαp + 1 k v k p +1 L p +1 − bβq + 1 k v k q +1 L q +1 ,∂ λ S ω ( v λ ) | λ =1 = k∇ v k L − aα ( α − p + 1 k v k p +1 L p +1 − bβ ( β − q + 1 k v k q +1 L q +1 , where v λ ( x ) = λ N/ v ( λx ). Note that by S ω ( φ ω ) = 0, K ω ( φ ω ) = h S ω ( φ ω ) , φ ω i = 0 , Q ( φ ω ) = h S ω ( φ ω ) , ∂ λ φ λω | λ =1 i = 0 . Lemma 2.1.
Assume that φ ω ∈ G ω satisfies ∂ λ S ω ( φ λω ) | λ =1 ≤ . Let v ∈ H ( R N ) satisfy v = 0 , k v k L ≤ k φ ω k L , K ω ( v ) ≤ , Q ( v ) ≤ . Then Q ( v )2 ≤ S ω ( v ) − S ω ( φ ω ) . Proof.
Since lim λ & K ω ( v λ ) = ω k v k L > K ω ( v ) ≤
0, there exists λ ∈ (0 ,
1] suchthat K ω ( v λ ) = 0. By the definition of the scaling v λ and (2.3), we have k v λ k L = k v k L ≤ k φ ω k L , (2.4) N F ( φ ω ) ≤ N F ( v λ ) = aαλ α p + 1 k v k p +1 L p +1 + bβλ β q + 1 k v k q +1 L q +1 . (2.5)trong instability of standing waves 7Now, we define f ( λ ) = S ω ( v λ ) − λ Q ( v )= ω k v k L − ap + 1 (cid:18) λ α − αλ (cid:19) k v k p +1 L p +1 − bq + 1 (cid:18) λ β − βλ (cid:19) k v k q +1 L q +1 . If we have f ( λ ) ≤ f (1), then by (2.1) and Q ( v ) ≤
0, we obtain(2.6) S ω ( φ ω ) ≤ S ω ( v λ ) ≤ S ω ( v λ ) − λ Q ( v ) ≤ S ω ( v ) − Q ( v )2 . This is the desired inequality.In what follows, we prove the inequality f ( λ ) ≤ f (1). This is equivalent to(2.7) ap + 1 k v k p +1 L p +1 ≤ bq + 1 · λ β − βλ − βαλ − λ α − α + 2 k v k q +1 L q +1 . Since(2.8) p + 1 α + 2 β = 2 N + 2 β + 2 α = q + 1 β + 2 α , we have K ω ( φ ω ) + 2 αβ ∂ λ S ω ( φ λω ) | λ =1 − (cid:18) αβ (cid:19) Q ( φ ω )= ω k φ ω k L − aαp + 1 (cid:18) p + 1 α + 2 β − − αβ (cid:19) k φ ω k p +1 L p +1 − bβq + 1 (cid:18) q + 1 β + 2 α − − αβ (cid:19) k φ ω k q +1 L q +1 = ω k φ ω k L − (cid:18) q + 1 β + 2 α − − αβ (cid:19) N F ( φ ω ) . Therefore, by K ω ( φ ω ) = Q ( φ ω ) = 0 and the assumption ∂ λ S ω ( φ λω ) | λ =1 ≤
0, we obtain ω k φ ω k L ≤ (cid:18) q + 1 β + 2 α − − αβ (cid:19) N F ( φ ω ) . Combining (2.4) and (2.5) with this inequality, and using (2.8) again, it follows that(2.9) ω k v k L ≤ (cid:18) a + ap + 1 · β (2 α − αβ − (cid:19) λ α k v k p +1 L p +1 + (cid:18) b + bq + 1 · α (2 β − αβ − (cid:19) λ β k v k q +1 L q +1 . N. Fukaya and M. OhtaMoreover, by K ω ( v λ ) = 0, Q ( v ) ≤
0, and (2.9), we deduce a k v k p +1 L p +1 = λ − α k∇ v k L + λ − α ω k v k L − bλ β − α k v k q +1 L q +1 ≤ λ − α (cid:18) aαp + 1 k v k p +1 L p +1 + bβq + 1 k v k q +1 L q +1 (cid:19) + (cid:18) a + ap + 1 · β (2 α − αβ − (cid:19) k v k p +1 L p +1 + (cid:18) b + bq + 1 · α (2 β − αβ − (cid:19) λ β − α k v k q +1 L q +1 − bλ β − α k v k q +1 L q +1 = (cid:18) a + ap + 1 · β (cid:0) α − αβ − αβλ − α (cid:1)(cid:19) k v k p +1 L p +1 + bq + 1 · α (cid:16) (2 β − αβ − λ β − α + αβλ − α (cid:17) k v k q +1 L q +1 , and thus ap + 1 · β (cid:0) αβ + 4 − α − αβλ − α (cid:1) k v k p +1 L p +1 ≤ bq + 1 · α (cid:16) (2 β − αβ − λ β − α + αβλ − α (cid:17) k v k q +1 L q +1 . Since αβ + 4 − α − αβλ − α ≥ − α >
0, this is rewritten as(2.10) ap + 1 k v k p +1 L p +1 ≤ bq + 1 · β (2 β − αβ − λ β − α + αβ λ − α α ( αβ + 4 − α − αβλ − α ) k v k q +1 L q +1 . In view of (2.7) and (2.10), it suffices to show that β (2 β − αβ − λ β − α + αβ λ − α α ( αβ + 4 − α − αβλ − α ) ≤ λ β − βλ − βαλ − λ α − α + 2 . This inequality follows if we have g ( λ ) := α (2 λ β − βλ − β )( αβ + 4 − α − αβλ − α )( αλ − λ α − α + 2) λ β − α − β (2 β − αβ − − αβ λ β − ≥ λ ∈ (0 , λ % g ( λ ) = 0, it is enough to show that g ( λ ) ≤ λ ∈ (0 , g ( λ ) = aλ α − β +1 ( αλ − λ α − α + 2) · (cid:0) (2 − α )( β − − βλ − α + ( αβ − α + 4) λ − (cid:1) · (cid:0) α (2 − α ) λ β − αβ ( β − α ) λ + 2 β ( β − λ α − (2 − α )( β − β − α ) (cid:1) . trong instability of standing waves 9Now, we put h ( λ ) = (2 − α )( β − − βλ − α + ( αβ − α + 4) λ − . Since h (1) = 0 and for λ ∈ (0 , h ( λ ) = − αβ ( λ − − λ − α − ) − − α ) λ − ≤ , we have h ( λ ) ≥
0. Thus, we only have to show that g ( λ ) := 2 α (2 − α ) λ β − αβ ( β − α ) λ + 2 β ( β − λ α − (2 − α )( β − β − α ) ≤ λ ∈ (0 , g (1) = 0, it suffices to show that g ( λ ) = 2 αβλ α − (cid:0) (2 − α ) λ β − α − ( β − α ) λ − α + β − (cid:1) ≥ λ ∈ (0 , g ( λ ) := (2 − α ) λ β − α − ( β − α ) λ − α + β − ≥ . Since g (1) = 0, and g ( λ ) = − ( β − α )(2 − α ) λ − α (1 − λ β − ) ≤ λ ∈ (0 , g ( λ ) ≥ λ ∈ (0 , f ( λ ) ≤ f (1).Thus, the inequality (2.6) follows. This completes the proof.Next, we show that the set B ω is invariant under the flow of (NLS). Recall that thedefinition of B ω is given by B ω = (cid:40) v ∈ H ( R N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ω ( v ) < S ω ( φ ω ) , k v k L ≤ k φ ω k L ,K ω ( v ) < , Q ( v ) < (cid:41) . Lemma 2.2.
Let u ∈ B ω . Then the solution u ( t ) of (NLS) with u (0) = u belongs to B ω for all t ∈ [0 , T max ) .Proof. Since S ω and k · k L are the conserved quantities of (NLS), we have S ω ( u ( t )) = S ω ( u ) < S ω ( φ ω ) and k u ( t ) k L = k u k L ≤ k φ ω k L for all t ∈ [0 , T max ). Therefore, by(2.1), we have K ω ( u ( t )) = 0 for all t ∈ [0 , T max ). Moreover, by K ω ( u ) < u ( t ), we obtain K ω ( u ( t )) < t ∈ [0 , T max ). Finally, weshow that Q ( u ( t )) < t ∈ [0 , T max ). If not, there exists t ∈ (0 , T max ) such that Q ( u ( t )) = 0. Then by Lemma 2.1 and S ω ( u ( t )) < S ω ( φ ω ), we have Q ( u ( t )) <
0. Thisis a contradiction. This completes the proof.Finally, we prove the blowup result.
Proof of Theorem 1.4.
By the virial identity (1.3), Lemmas 2.1 and 2.2, and the conser-vation of S ω , we have d dt k xu ( t ) k L = 8 Q ( u ( t )) ≤ (cid:0) S ω ( u ( t )) − S ω ( φ ω ) (cid:1) = 16 (cid:0) S ω ( u ) − S ω ( φ ω ) (cid:1) < t ∈ [0 , T max ). This implies T max < ∞ . This completes the proof.0 N. Fukaya and M. Ohta In this section, we prove Theorem 1.2 using Theorem 1.4. Throughout this section, weimpose the assumption of Theorem 1.2.We remark that S ω ( v λ ) = 12 K ω ( v λ ) + 12 F ( v λ )= λ k∇ v k L + ω k v k L − aλ α p + 1 k v k p +1 L p +1 − bλ β q + 1 k v k q +1 L q +1 ,Q ( v λ ) = λ∂ λ S ω ( v λ ) ,Q ( φ ω ) = ∂ λ S ω ( φ λω ) | λ =1 = 0 , ∂ λ S ω ( φ λω ) | λ =1 ≤ . Lemma 3.1.
Assume that φ ω ∈ G ω satisfies ∂ λ S ω ( φ λω ) | λ =1 ≤ . Then φ λω ∈ B ω for all λ > .Proof. By the definition of the scaling λ v λ , we have k φ λω k L = k φ ω k L for all λ > ∂ λ S ω ( φ λω ) | λ =1 = 0 and ∂ λ S ω ( φ λω ) | λ =1 ≤
0, in view of the graph of λ S ω ( φ λω ), wesee that S ω ( φ λω ) < S ω ( φ ω ) and Q ( φ λω ) = λ∂ λ S ω ( φ λω ) < λ >
1. Finally, we obtain K ω ( φ λω ) = 2 S ω ( φ λω ) − F ( φ λω ) < S ω ( φ ω ) − F ( φ ω ) = 0for all λ >
1. This completes the proof.Now, we prove our main theorem.
Proof of Theorem 1.2.
By an analogous argument in the proof of [4, Theorem 8.1.1], wesee that φ ω decays exponentially. This implies φ ω ∈ Σ, where Σ is the weighted spacedefined in (1.2). Therefore, combining this with Lemma 3.1, we have φ λω ∈ B ω ∩ Σ for all λ >
1. Thus, Theorem 1.4 implies that for any λ >
1, the solution u ( t ) of (NLS) with u (0) = φ λω blows up in finite time. Moreover, we obtain φ λω → φ ω in H ( R N ) as λ & e iωt φ ω of (NLS) is strongly unstable. Acknowledgements
The first author was supported by Grant-in-Aid for JSPS Fellows 18J11090. The secondauthor was supported by JSPS KAKENHI Grant Numbers 18K03379 and 26247013.
References [1] I. V. Barashenkov, A. D. Gocheva, V. G. Makhankov, and I. V. Puzynin,
Stabilityof the soliton-like “bubbles” , Phys. D (1989), 240–254.trong instability of standing waves 11[2] H. Berestycki and T. Cazenave, Instabilité des états stationaires dans les équationsde Schrödinger et de Klein–Gordon non linéaires , C. R. Acad. Sci. Paris Sér. I Math. (1981), 489–492.[3] H. Berestycki and P.-L. Lions,
Nonlinear scalar field equations, I, Existence of aground state , Arch. Rational Mech. Anal. (1983), 313–345.[4] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathe-matics, 10. New York University, Courant Institute of Mathematical Sciences, NewYork; American Mathematical Society, Providence, RI, 2003.[5] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinearSchrödinger equations , Comm. Math. Phys. (1982), 549–561.[6] G. Fibich, The nonlinear Schrödinger equation: Singular solutions and optical col-lapse, Applied Mathematical Sciences, , Springer, Cham, 2015.[7] N. Fukaya and M. Ohta, Strong instability of standing waves for nonlinearSchrödinger equations with inverse power potential , preprint, arXiv:1804.02127.[8] R. Fukuizumi,
Remarks on the stable standing waves for nonlinear Schrödingerequations with double power nonlinearity , Adv. Math. Sci. Appl. (2003),549–564.[9] T. Kato, On nonlinear Schrödinger equations , Ann. Inst. H. Poincaré Phys. Théor. (1987), 113–129.[10] S. Le Coz, A note on Berestycki–Cazenave’s classical instability result for nonlinearSchrödinger equations , Adv. Nonlinear Stud. (2008), 455–463.[11] S. Le Coz, Standing waves in nonlinear Schrödinger equations , Analytical and Nu-merical Aspects of Partial Differential Equations, de Gruyter, Berlin, (2009), 151–192.[12] P.-L. Lions,
The concentration-compactness principle in the calculus of variations.The locally compact case, II , Ann. Inst. H. Poincaré Anal. Non Linéaire (1984),223–283.[13] M. Maeda, Stability and instability of standing waves for 1-dimensional nonlinearSchrödinger equation with multiple-power nonlinearity , Kodai Math. J. (2008),263–271.[14] M. Ohta, Stability and instability of standing waves for one-dimensional nonlinearSchrödinger equations with double power nonlinearity , Kodai Math. J. (1995),68–74.2 N. Fukaya and M. Ohta[15] M. Ohta, Instability of standing waves for the generalized Davey–Stewartson system ,Ann. Inst. H. Poincaré Phys. Théor. (1995), 69–80.[16] M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equationswith harmonic potential , Funkcial. Ekvac. (2018), 135–143.[17] M. Ohta and T. Yamaguchi, Strong instability of standing waves for nonlinearSchrödinger equations with double power nonlinearity , SUT J. Math. (2015),49–58.[18] M. Ohta and T. Yamaguchi, Strong instability of standing waves for nonlinearSchrödinger equations with a delta potential , Harmonic analysis and nonlinear par-tial differential equations, 79–92, RIMS Kôkyûroku Bessatsu,
B56 , Res. Inst. Math.Sci. (RIMS), Kyoto, 2016.[19] C. Sulem and P-L. Sulem, The nonlinear Schrödinger equation: Self-focusing andwave collapse, Applied Mathematical Sciences, , Springer-Verlag, New York,1999.[20] W. A. Strauss,
Existence of solitary waves in higher dimensions , Comm. Math.Phys. (1977), 149–162.[21] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates ,Comm. Math. Phys. (1982/83), 567–576.[22] J. Zhang, Cross-constrained variational problem and nonlinear Schrödinger equa-tion , Foundations of computational mathematics (Hong Kong, 2000), 457–469,World Sci. Publ., River Edge, NJ, 2002.Noriyoshi FukayaDepartment of Mathematics, Graduate School of Science, Tokyo University of Science,1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
E-mail address : Masahito OhtaDepartment of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
E-mail address ::