Home Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
Article
Licensed
Unlicensed Requires Authentication

Dynamics of radial threshold solutions for generalized energy-critical Hartree equation

  • Xuemei Li , Chenxi Liu , Xingdong Tang and Guixiang Xu EMAIL logo
Published/Copyright: January 13, 2025
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 5

Abstract

In this paper, we study long time dynamics of radial threshold solutions for the focusing, generalized energy-critical Hartree equation and classify all radial threshold solutions. The main arguments are the spectral theory of the linearized operator, the modulational analysis and the concentration compactness rigidity argument developed by T. Duyckaerts and F. Merle to classify all threshold solutions for the energy critical NLS and NLW in [T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 2009, 6, 1787–1840, T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP 2008 2008, Art ID rpn002], later by D. Li and X. Zhang in [D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal. 256 2009, 6, 1928–1961, D. Li and X. Zhang, Dynamics for the energy critical nonlinear wave equation in high dimensions, Trans. Amer. Math. Soc. 363 2011, 3, 1137–1160] in higher dimensions. The new ingredient here is to solve the nondegeneracy of positive bubble solutions with nonlocal structure in H ˙ 1 ⁢ ( ℝ N ) (i.e. the spectral assumption in [C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, Forum Math. 27 2015, 1, 373–447]) by the nondegeneracy result of positive bubble solution in L ∞ ⁢ ( ℝ N ) in [X. Li, C. Liu, X. Tang and G. Xu, Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations, preprint 2023, https://arxiv.org/abs/2304.04139] and the Moser iteration method in [S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of ℝ n , Appunti. Sc. Norm. Super. Pisa (N. S.) 15, Edizioni della Normale, Pisa, 2017], which is related to the spectral analysis of the linearized operator with nonlocal structure, and plays a key role in the construction of the special threshold solutions, and the classification of all threshold solutions.

Keywords: Concentration compactness rigidity argument; Hartree equation; modulation analysis; nondegeneracy of ground state; spectral theory; uniqueness
MSC 2020: 35Q55; 35Q41

Communicated by Christopher D. Sogge

Funding source: National Key Research and Development Program of China

Award Identifier / Grant number: 2020YFA0712900

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12001284

Award Identifier / Grant number: 12371240

Award Identifier / Grant number: 12431008

Funding statement: The authors were partly supported by National Key Research and Development Program of China (No. 2020YFA0712900) and by National Natural Science Foundation of China (No. 12001284, No. 12371240, No. 12431008).

A Nondegeneracy of energy solution of equation (1.5)

From Remark 1.5 (b) and (c), it suffices to show L ∞ -regularity of H ˙ 1 ⁢ ( ℝ N ) solution of equation (1.5) to prove Proposition 1.4. Now we use the Moser iteration method in [16] to show this result in this appendix.

Recalling equation (1.5), energy solution u ∈ H ˙ 1 ⁢ ( ℝ N ) satisfies

(A.1) - Δ ⁢ u = ( p - 1 ) ⁢ ( I λ * W p ) ⁢ W p - 2 ⁢ u + p ⁢ [ I λ * ( W p - 1 ⁢ u ) ] ⁢ W p - 1 .

Let β ≥ 1 , T > 0 . We define the cutoff function

(A.2) φ ⁢ ( t ) = { β ⁢ T β - 1 ⁢ ( t - T ) + T β if  ⁢ t ≥ T , | t | β if  - T ≤ t ≤ T , β ⁢ T β - 1 ⁢ ( T - t ) + T β if  ⁢ t ≤ - T .

Since φ is convex and Lipschitz, we have

(A.3) ( - Δ ) s ⁢ φ ⁢ ( u ) ≤ φ ′ ⁢ ( u ) ⁢ ( - Δ ) s ⁢ u , s ∈ [ 0 , 1 ] ,

in the weak sense. By equation (A.1), we get

∫ ℝ N ( - Δ ⁢ u ) ⁢ φ ⁢ ( u ) ⁢ φ ′ ⁢ ( u ) ⁢ 𝑑 x = ( p - 1 ) ⁢ ∫ ℝ N ( I λ * W p ) ⁢ W p - 2 ⁢ u ⁢ φ ⁢ ( u ) ⁢ φ ′ ⁢ ( u ) ⁢ 𝑑 x
+ p ⁢ ∫ ℝ N [ I λ * ( W p - 1 ⁢ u ) ] ⁢ W p - 1 ⁢ φ ⁢ ( u ) ⁢ φ ′ ⁢ ( u ) ⁢ 𝑑 x .

Using (A.3) and integration by parts, we get

∫ ℝ N ( - Δ ⁢ u ) ⁢ φ ⁢ ( u ) ⁢ φ ′ ⁢ ( u ) ⁢ 𝑑 x = ∫ ℝ N | ∇ ⁡ u | 2 ⁢ ( φ ′ ⁢ ( u ) ) 2 ⁢ 𝑑 x + ∫ ℝ N | ∇ ⁡ u | 2 ⁢ φ ⁢ ( u ) ⁢ φ ′′ ⁢ ( u ) ⁢ 𝑑 x .

By (A.2), we know that the last integral is nonnegative, therefore, we have

∫ ℝ N | ∇ ⁡ u | 2 ⁢ ( φ ′ ⁢ ( u ) ) 2 ⁢ 𝑑 x ≤ ( p - 1 ) ⁢ ∫ ℝ N ( I λ * W p ) ⁢ W p - 2 ⁢ | u | ⁢ | φ ⁢ ( u ) ⁢ φ ′ ⁢ ( u ) | ⁢ 𝑑 x
+ p ⁢ ∫ ℝ N [ I λ * ( W p - 1 ⁢ | u | ) ] ⁢ W p - 1 ⁢ φ ⁢ ( u ) ⁢ | φ ′ ⁢ ( u ) | ⁢ 𝑑 x .

Applying the weak Young inequality and noting that W = c ⁢ ( t t 2 + | x | 2 ) N - 2 2 ∈ L p ⁢ ( ℝ N ) , p > N - 2 N , we have

I λ * W p ∈ L ∞ ⁢ ( ℝ N ) .

Using that φ ⁢ ( u ) ⁢ φ ′ ⁢ ( u ) ≤ β ⁢ u 2 ⁢ β - 1 and the Hölder inequality, we have

∫ ℝ N ( I λ * W p ) ⁢ W p - 2 ⁢ | u | ⁢ | φ ⁢ ( u ) ⁢ φ ′ ⁢ ( u ) | ⁢ 𝑑 x ≤ C ⁢ β ⁢ ∫ ℝ N | u | 2 ⁢ β .

Similarly, we can get

∫ ℝ N [ I λ * ( W p - 1 ⁢ | u | ) ] ⁢ W p - 1 ⁢ φ ⁢ ( u ) ⁢ | φ ′ ⁢ ( u ) | ⁢ 𝑑 x ≤ C ⁢ β ⁢ ∫ ℝ N | u | 2 ⁢ β .

Thus, by the Sobolev inequality, we have

∥ φ ⁢ ( u ) ∥ L 2 * 2 ≤ ∥ φ ⁢ ( u ) ∥ H ˙ 1 2 = ∫ ℝ N | ∇ ⁡ u | 2 ⁢ ( φ ′ ⁢ ( u ) ) 2 ⁢ 𝑑 x ≤ C ⁢ β ⁢ ∫ ℝ N | u | 2 ⁢ β ,

where 2 * = 2 ⁢ N N - 2 . Let T → ∞ . We obtain

( ∫ ℝ N | u | 2 * ⁢ β ) 2 2 * ≤ C ⁢ β ⁢ ∫ ℝ N | u | 2 ⁢ β ,

where C changing from line to line, but independent of β. Therefore,

(A.4) ( ∫ ℝ N | u | 2 * ⁢ β ) 1 2 * ⁢ β ≤ ( C ⁢ β ) 1 2 ⁢ β ⁢ ( ∫ ℝ N | u | 2 ⁢ β ) 1 2 ⁢ β .

Because u ∈ H ˙ 1 ⁢ ( ℝ N ) , we know u ∈ L 2 ⁢ N N - 2 by the Sobolev inequality. Therefore, we can take β 1 = N N - 2 ≥ 1 . From now on, we can follow exactly the iteration argument. That is, we define β m + 1 , m ≥ 1 , so that

2 ⁢ β m + 1 = 2 * ⁢ β m .

Thus,

β m + 1 = ( 2 * 2 ) m ⁢ β 1 ,

and replacing in equation (A.4), we know that

( ∫ ℝ N | u | 2 ⁢ β m + 1 ) 1 2 ⁢ β m + 1 = ( ∫ ℝ N | u | 2 * ⁢ β m ) 1 2 * ⁢ β m ≤ ( C ⁢ β m ⁢ ∫ ℝ N | u | 2 ⁢ β m ) 1 2 ⁢ β m .

Defining C m + 1 := C ⁢ β m and

A m := ( ∫ ℝ N | u | 2 ⁢ β m + 1 ) 1 2 ⁢ β m + 1 ,

we conclude that there exists a constant C 0 > 0 independent of m, such that

A m + 1 ≤ ∏ k = 1 m C k 1 2 ⁢ β k ⁢ A 1 ≤ C 0 ⁢ A 1 .

Thus, we obtain

∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C 0 ⁢ A 1 < + ∞ .

This together with the result in [45] completes the proof of Proposition 1.4.

B Coercivity of Φ on H ⊥ ∩ H ˙ rad 1

In this appendix, we follow the argument in [51, Appendix C] and prove Proposition 2.13. We divide the proof into two steps.

Step 1: Nonnegative.

We claim that for any function h ∈ H ˙ 1 , ( h , W ) H ˙ 1 = 0 , there exists

Φ ⁢ ( h ) ≥ 0 .

Indeed, let

I ⁢ ( u ) = ∥ ∇ ⁡ u ∥ 2 2 ⁢ p ∥ ∇ ⁡ W ∥ 2 2 ⁢ p - ∬ ℝ N × ℝ N I λ ( x - y ) | u ( x ) | p | u ( y ) | p ) d x d y ∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( x ) p ⁢ W ⁢ ( y ) p ⁢ 𝑑 x ⁢ 𝑑 y .

By Proposition 1.1, we have

I ⁢ ( u ) ≥ 0   for all  ⁢ u ∈ H ˙ 1 .

Take h ∈ H ˙ 1 with ( W , h ) H ˙ 1 = 0 , α ∈ ℝ . We consider the expansion of I ⁢ ( W + α ⁢ h ) in α of order 2. Note that

∥ ∇ ⁡ ( W + α ⁢ h ) ∥ 2 2 ⁢ p = ∥ ∇ ⁡ W ∥ 2 2 ⁢ p ⁢ ( 1 + p ⁢ α 2 ⁢ ∥ h ∥ H ˙ 1 2 ∥ W ∥ H ˙ 1 2 + O ⁢ ( α 4 ) )

and

∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ | W ⁢ ( x ) + α ⁢ h ⁢ ( x ) | p ⁢ | W ⁢ ( y ) + α ⁢ h ⁢ ( y ) | p ⁢ 𝑑 x ⁢ 𝑑 y
  = ∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( x ) p ⁢ W ⁢ ( y ) p ⁢ 𝑑 x ⁢ 𝑑 y
  + p ⁢ α ⁢ ∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ ( W ⁢ ( x ) p ⁢ W ⁢ ( y ) p - 1 ⁢ h 1 ⁢ ( y ) + W ⁢ ( y ) p ⁢ W ⁢ ( x ) p - 1 ⁢ h 1 ⁢ ( x ) ) ⁢ 𝑑 x ⁢ 𝑑 y
  + α 2 ∬ ℝ N × ℝ N I λ ( x - y ) [ ( p ⁢ ( p - 1 ) 2 W ( x ) p - 2 h 1 ( x ) 2 + p 2 W ( x ) p - 2 h 2 ( x ) 2 ) W ( y ) p
           + ( p ⁢ ( p - 1 ) 2 ⁢ W ⁢ ( y ) p - 2 ⁢ h 1 ⁢ ( y ) 2 + p 2 ⁢ W ⁢ ( y ) p - 2 ⁢ h 2 ⁢ ( y ) 2 ) ⁢ W ⁢ ( x ) p
           + p 2 W ( x ) p - 1 h 1 ( x ) W ( y ) p - 1 h 1 ( y ) ] d x d y + O ( α 3 )
  = ∬ ℝ N × ℝ N I λ ( x - y ) W ( x ) p W ( y ) p d x d y ( 1 + α 2 ∥ W ∥ H ˙ 1 2 ∬ ℝ N × ℝ N I λ ( x - y ) [ p ( p - 1 ) W ( x ) p - 2 h 1 ( x ) 2 W ( y ) p
+ p W ( x ) p - 2 h 2 ( x ) 2 W ( y ) p + p 2 W ( x ) p - 1 h 1 ( x ) W ( y ) p - 1 h 1 ( y ) ] d x d y + O ( α 3 ) ) ,

where we use the facts that

∫ ℝ N ( I λ * W p ) ⁢ W ⁢ ( y ) p - 1 ⁢ h 1 ⁢ ( y ) ⁢ 𝑑 y = ∫ ℝ N - Δ ⁢ W ⁢ ( y ) ⋅ h 1 ⁢ ( y ) ⁢ d ⁢ y = 0 ,
∫ ℝ N ( I λ * W p ) ⁢ W ⁢ ( x ) p - 1 ⁢ h 1 ⁢ ( x ) ⁢ 𝑑 x = ∫ ℝ N - Δ ⁢ W ⁢ ( x ) ⋅ h 1 ⁢ ( x ) ⁢ d ⁢ y = 0 ,
∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( x ) p - 2 ⁢ h 1 ⁢ ( x ) 2 ⁢ 𝑑 x = ∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( y ) p - 2 ⁢ h 1 ⁢ ( y ) 2 ⁢ 𝑑 x ,
∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( x ) p ⁢ W ⁢ ( y ) p ⁢ 𝑑 x ⁢ 𝑑 y = ∫ ℝ N | ∇ ⁡ W | 2 ⁢ 𝑑 x .

Therefore we have

I ( W + α h ) = p ⁢ α 2 ∥ W ∥ H ˙ 1 2 ( ∥ h ∥ H ˙ 1 2 - ∬ ℝ N × ℝ N I λ ( x - y ) [ ( p - 1 ) W ( x ) p - 2 h 1 ( x ) 2 W ( y ) p
    + W ( x ) p - 2 h 2 ( x ) 2 W ( y ) p + W ( x ) p - 1 h 1 ( x ) W ( y ) p - 1 h 1 ( y ) ] d x d y ) + O ( α 3 ) .

Since I ⁢ ( W + α ⁢ h ) ≥ 0 for all α ∈ ℝ , we have

2 Φ ( h ) = ∫ ℝ N | ∇ h | 2 d x - ∬ ℝ N × ℝ N I λ ( x - y ) [ ( p - 1 ) W ( x ) p - 2 h 1 ( x ) 2 W ( y ) p
+ W ( x ) p - 2 h 2 ( x ) 2 W ( y ) p + W ( x ) p - 1 h 1 ( x ) W ( y ) p - 1 h 1 ( y ) ] d x d y
≥ 0 .

Step 2: Coercivity.

We show that there exists a constant c * > 0 such that for any radial function h ∈ H ⊥

Φ ⁢ ( h ) ≥ c * ⁢ ∥ h ∥ H ˙ 1 2 .

Note that Φ ⁢ ( h ) = Φ 1 ⁢ ( h 1 ) + Φ 2 ⁢ ( h 2 ) , where

Φ 1 ⁢ ( h 1 ) := 1 2 ⁢ ∫ ℝ N ( L + ⁢ h 1 ) ⁢ h 1 ⁢ 𝑑 x
= 1 2 ⁢ ∫ ℝ N | ∇ ⁡ h 1 | 2 ⁢ 𝑑 x - p - 1 2 ⁢ ∫ ℝ N ( I λ * W p ) ⁢ W ⁢ ( x ) p - 2 ⁢ h 1 ⁢ ( x ) 2 ⁢ 𝑑 x - p 2 ⁢ ∫ ℝ N ( I λ * W p - 1 ⁢ h 1 ) ⁢ W ⁢ ( x ) ⁢ h 1 ⁢ ( x ) ⁢ 𝑑 x

and

Φ 2 ⁢ ( h 2 ) := 1 2 ⁢ ∫ ℝ N ( L - ⁢ h 2 ) ⁢ h 2 ⁢ 𝑑 x = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ h 2 | 2 ⁢ 𝑑 x - 1 2 ⁢ ∫ ℝ N ( I λ * W p ) ⁢ W ⁢ ( x ) p - 2 ⁢ h 2 ⁢ ( x ) 2 ⁢ 𝑑 x .

By Step 1, L + is nonnegative on { W } ⊥ (in the sense of H ˙ 1 ) and L - is nonnegative. We will deduce the coercivity by the compactness argument in [51, 67].

We first show that there exists a constant c such that for any radial real-valued H ˙ 1 -function h 1 ∈ { W , W ~ } ⊥ (in the sense of H ˙ 1 ),

Φ 1 ⁢ ( h 1 ) ≥ c ⁢ ∥ h 1 ∥ H ˙ 1 2 .

Assume that the above inequality does not hold. Then there exists a sequence of real-valued radial H ˙ 1 -functions { f n } n such that

(B.1) f n ∈ H ⊥ , lim n → + ∞ ⁡ Φ 1 ⁢ ( f n ) = 0 , ∥ f n ∥ H ˙ 1 = 1 .

Extracting a subsequence from { f n } , we may assume that

f n ⇀ f *   in  ⁢ H ˙ 1 .

The weak convergence of f n ∈ H ⊥ to f * implies that f * ∈ H ⊥ . In addition, by compactness, we have

∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( y ) p ⁢ W ⁢ ( x ) p - 2 ⁢ | f n ⁢ ( x ) | 2 ⁢ 𝑑 x ⁢ 𝑑 y → ∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( y ) p ⁢ W ⁢ ( x ) p - 2 ⁢ | f * ⁢ ( x ) | 2 ⁢ 𝑑 x ⁢ 𝑑 y ,
∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( x ) p - 1 ⁢ | f n ⁢ ( x ) | ⁢ W ⁢ ( y ) p - 1 ⁢ | f n ⁢ ( y ) | ⁢ 𝑑 x ⁢ 𝑑 y → ∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( x ) p - 1 ⁢ | f * ⁢ ( x ) | ⁢ W ⁢ ( y ) p - 1 ⁢ | f * ⁢ ( y ) | ⁢ 𝑑 x ⁢ 𝑑 y .

Thus, by the Fatou lemma, (B.1) and Step 1, we have

0 ≤ Φ 1 ⁢ ( f * ) ≤ lim inf n → + ∞ ⁡ Φ 1 ⁢ ( f n ) = 0 ,

where f * is the solution to the following minimizing problem:

0 = min f ∈ Ω \ { 0 } ⁡ ∫ ℝ N L + ⁢ f ⋅ f ⁢ 𝑑 x ∥ f ∥ H ˙ 1 2 , Ω = { f ∈ H ˙ rad 1 : ( f , W ) H ˙ 1 = ( f , W ~ ) H ˙ 1 = 0 } .

Thus for some Lagrange multipliers λ 0 , λ 1 , we have

L + ⁢ f * = λ 0 ⁢ Δ ⁢ W + λ 1 ⁢ Δ ⁢ W ~ .

Note that ( W , W ~ ) H ˙ 1 = 0 and L + ⁢ ( W ~ ) = 0 , we have

0 = ∫ ℝ N f * ⁢ L + ⁢ ( W ~ ) = ∫ ℝ N ( L + ⁢ f * ) ⁢ W ~ = λ 1 ⁢ ∥ W ~ ∥ H ˙ 1 2 ⟹ λ 1 = 0 .

Thus

L + ⁢ f * = λ 0 ⁢ Δ ⁢ W = - λ 0 ⁢ ( | x | - 4 * W 2 ) ⁢ W = λ 0 2 ⁢ L + ⁢ W .

By Null ⁡ ( L + ) = span ⁡ { W ~ } in Lemma 2.12, there exists μ 1 such that

f * = λ 0 2 ⁢ W + μ 1 ⁢ W ~ .

Using f * ∈ H ⊥ , we get μ 1 = 0 and λ 0 = 0 . Therefore, we have

f * = 0   and   f n ⇀ 0   in  ⁢ H ˙ 1 .

Now, by compactness, we have

∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( y ) p ⁢ W ⁢ ( x ) p - 2 ⁢ | f n ⁢ ( x ) | 2 ⁢ 𝑑 x ⁢ 𝑑 y → 0

and

∬ ℝ N × ℝ N I λ ⁢ ( x - y ) ⁢ W ⁢ ( x ) p - 1 ⁢ | f n ⁢ ( x ) | ⁢ W ⁢ ( y ) p - 1 ⁢ | f n ⁢ ( y ) | ⁢ 𝑑 x ⁢ 𝑑 y → 0 .

By Φ 1 ⁢ ( f n ) → 0 in (B.1), we get that

∥ ∇ ⁡ f n ∥ 2 → 0 ,

which contradicts ∥ f n ∥ H ˙ 1 = 1 in (B.1).

Using the same argument, we can show that there exists a constant c, such that for any real-valued radial H ˙ 1 -function h 2 ∈ { W } ⊥ , we have

Φ 2 ⁢ ( h 2 ) ≥ c ⁢ ∥ h 2 ∥ H ˙ 1 2 .

This completes the proof. ∎

C Spectral properties of the linearized operator

In this appendix, we follow the argument in [19, 18, 51] to give the proof of Proposition 2.14.

C.1 Existence, symmetry of the eigenfunctions

Note that

ℒ ⁢ v ¯ = - ℒ ⁢ v ¯ ,

so that if e 0 > 0 is an eigenvalue of ℒ with the eigenfunction 𝒴 + , then - e 0 is also an eigenvalue with eigenfunction 𝒴 - = 𝒴 ¯ + . Now we show the existence of 𝒴 + . Let 𝒴 1 = Re ⁡ 𝒴 + , 𝒴 2 = Im ⁡ 𝒴 + . It suffices to show that

(C.1) - L - ⁢ 𝒴 2 = e 0 ⁢ 𝒴 1 , L + ⁢ 𝒴 1 = e 0 ⁢ 𝒴 2 .

From the proof of the coercivity property of Φ on H ⊥ ∩ H ˙ rad 1 in Proposition 2.13, we know that L - on L 2 with domain H 2 is self-adjoint and nonnegative. By [34, Theorem 3.35, p. 281], it has a unique square root ( L - ) 1 2 with domain H 1 .

Assume that there exists a function f ∈ H rad 4 such that

𝒫 ⁢ f = - e 0 2 ⁢ f 1 , 𝒫 := ( L - ) 1 2 ⁢ ( L + ) ⁢ ( L - ) 1 2 .

Then taking

𝒴 1 := ( L - ) 1 2 ⁢ f , 𝒴 2 := 1 e 0 ⁢ ( L + ) ⁢ ( L - ) 1 2 ⁢ f

would yield a solution of (C.1), which implies the existence of the radial 𝒴 ± by the rotation invariance of the operator ℒ .

It suffices to show that the operator 𝒫 on L 2 with domain H rad 4 has a strictly negative eigenvalue. Since 𝒫 is a relatively compact, self-adjoint, perturbation of ( - Δ ) 2 , then by the Weyl theorem [33, 34], we know that

σ ess ⁢ ( 𝒫 ) = [ 0 , + ∞ ) .

Thus we only need to show that 𝒫 has at least one negative eigenvalue - e 0 2 .

Lemma C.1.

We have

σ - ⁢ ( 𝒫 ) := inf ⁡ { ( 𝒫 ⁢ f , f ) L 2 : f ∈ H rad 4 , ∥ f ∥ L 2 = 1 } < 0 .

Proof.

Note that

( 𝒫 ⁢ f , f ) L 2 = ( L + ⁢ F , F ) L 2 , F := ( L - ) 1 2 ⁢ f .

Thus it suffices to find F such that there exists g ∈ H rad 4 , F = ( Δ + V ) ⁢ g and

(C.2) ( L + ⁢ F , F ) L 2 < 0 ,

where V := ( I λ * W p ) ⁢ W p - 2 .

We divide the proof into two cases.

Case (1). Assume that N = 3 , 4 , in these cases W ∉ L 2 , we will use the localization method. Let W a := χ ⁢ ( x a ) ⁢ W ⁢ ( x ) , where χ is a smooth, radial positive function such that χ ⁢ ( r ) = 1 for r ≤ 1 and χ ⁢ ( r ) = 0 for r ≥ 2 . We first claim

(C.3) ∃ a > 0 , E a := ∫ L + ⁢ W a ⁢ W a < 0 .

Recall that - Δ ⁢ W = ( I λ * W p ) ⁢ W p - 1 . Thus

L + ⁢ W a = - ( p - 2 ) ⁢ ( I λ * W p ) ⁢ χ ⁢ ( x a ) ⁢ W p - 1 - p ⁢ [ I λ * ( χ ⁢ ( x a ) ⁢ W p ) ] ⁢ W p - 1 - 2 a ⁢ ∇ ⁡ χ ⁢ ( x a ) ⁢ ∇ ⁡ W - 1 a 2 ⁢ Δ ⁢ χ ⁢ ( x a ) ⁢ W .

Hence

∫ L + ⁢ W a ⁢ W a = - ( p - 2 ) ⁢ ∫ ( I λ * W p ) ⁢ χ 2 ⁢ ( x a ) ⁢ W p - p ⁢ ∫ [ I λ * ( χ ⁢ ( x a ) ⁢ W p ) ] ⁢ W p
- 2 a ⁢ ∫ ∇ ⁡ χ ⁢ ( x a ) ⁢ ∇ ⁡ W ⁢ W - 1 a 2 ⁢ ∫ Δ ⁢ χ ⁢ ( x a ) ⁢ W 2 .

Let

A := 2 a ⁢ ∫ ∇ ⁡ χ ⁢ ( x a ) ⁢ ∇ ⁡ W ⁢ W , B := 1 a 2 ⁢ ∫ Δ ⁢ χ ⁢ ( x a ) ⁢ W 2 .

According to the explicit expression (1.4) of W, W ≤ C ⁢ | x | - ( N - 2 ) and ∇ ⁡ W ≤ C ⁢ | x | - ( N - 1 ) at infinity, which gives | ( A ) | + | ( B ) | ≤ C a if N = 3 and | ( A ) | + | ( B ) | ≤ C a 2 if N = 4 . Hence (C.3).

Let us fix a such that (C.3) holds. Recall that W is not in L 2 . Thus Δ + V is a self-adjoint operator on L 2 , with domain H 2 , and without eigenfunction. In particular, the orthogonal of its range R ⁢ ( Δ + V ) is 0, and thus R ⁢ ( Δ + V ) is dense in L 2 . Let ϵ > 0 , and consider G ϵ ∈ H 2 such that

∥ ( Δ + V ) ⁢ G ϵ - ( Δ + V - 1 ) ⁢ W a ∥ L 2 ≤ ϵ .

Taking

F ϵ := ( L - + 1 ) - 1 ⁢ L - ⁢ G ϵ ,

we obtain ∥ ( Δ + V - 1 ) ⁢ ( F ϵ - W a ) ∥ L 2 ≤ ϵ which implies ∥ F ϵ - W a ∥ H 2 ≤ ϵ ⁢ ∥ ( Δ + V - 1 ) - 1 ∥ L 2 → L 2 . Hence for some constant C 0 , we have

| ∫ ℝ N L + ⁢ F ϵ ⋅ F ϵ - ∫ ℝ N L + ⁢ W a ⋅ W a | ≤ C 0 ⁢ ϵ .

As a consequence of (C.3), we get (C.2) for F = F ϵ , ϵ = - E a 2 ⁢ C 0 , which shows the claim in the case N = 3 , 4 .

Case (2). Assume now that N ≥ 5 , so that W ∈ L 2 and more generally in all space H s ⁢ ( ℝ N ) . In this case

Ran ( L - ) ⊥ = Null ( L - ) = span { W } .

Thus

(C.4) Ran ⁡ ( L - ) = { f ∈ L 2 : ( f , W ) L 2 = 0 } .

Note that L + is a self-adjoint compact perturbation of - Δ and

( L + ⁢ W , W ) L 2 = - ( 2 ⁢ p - 2 ) ⁢ ∫ | ∇ ⁡ W | 2 ⁢ 𝑑 x < 0 ,

which implies that L + has a negative eigenvalue. Let Z be the eigenfunction for this eigenvalue (it is radial by the minimax principle). Note that L + ⁢ W ~ = 0 . Then for any real number α, we have

(C.5) E 0 := ∫ ℝ d L + ⁢ ( Z + α ⁢ W ~ ) ⋅ ( Z + α ⁢ W ~ ) = ∫ ℝ d L + ⁢ Z ⋅ Z < 0 .

Note that

( W ~ , W ) L 2 ≠ 0 .

We can choose the real number α 1 such that

( Z + α 1 ⁢ W ~ , W ) L 2 = 0 ,

which means that

( ( L - + 1 ) ⁢ ( Z + α 1 ⁢ W ~ ) , W ) L 2 = ( Z + α 1 ⁢ W ~ , ( L - + 1 ) ⁢ W ) L 2 = ( Z + α 1 ⁢ W ~ , W ) L 2 = 0 .

By (C.4), for any ϵ > 0 , there exists a function G ϵ ∈ H rad 2 such that

∥ L - ⁢ G ϵ - ( L - + 1 ) ⁢ ( Z + α 1 ⁢ W ~ ) ∥ L 2 < ϵ .

Taking

F ϵ := ( L - + 1 ) - 1 ⁢ L - ⁢ G ϵ ,

we obtain that

∥ ( L - + 1 ) ⁢ ( F ϵ - ( Z + α 1 ⁢ W ~ ) ) ∥ L 2 ≤ ϵ ,

which implies that

∥ F ϵ - ( Z + α 1 ⁢ W ~ ) ∥ H 2 ≤ ϵ ⁢ ∥ ( L - + 1 ) - 1 ∥ L 2 → L 2 .

Hence for some constant C 0 , we have

| ∫ ℝ d L + ⁢ F ϵ ⋅ F ϵ - ∫ ℝ d L + ⁢ ( Z + α 1 ⁢ W ~ ) ⋅ ( Z + α 1 ⁢ W ~ ) | ≤ C 0 ⁢ ϵ .

By (C.5), we have (C.2) for F = F ϵ , ϵ = - E 0 2 ⁢ C 0 . ∎

C.2 Decay of the eigenfunctions at infinity

By the bootstrap argument, we know that 𝒴 ± ∈ C ∞ ∩ H ∞ . In fact, we have 𝒴 ± ∈ 𝒮 . By (C.1), it suffices to show that 𝒴 1 ∈ 𝒮 . Note that 𝒴 1 satisfies that

( e 0 2 + Δ 2 ) ⁢ 𝒴 1 = - p ⁢ Δ ⁢ ( I λ * ( W p - 1 ⁢ 𝒴 1 ) ⁢ W p - 1 ) - p ⁢ ( I λ * W p ) ⁢ ( I λ * ( W p - 1 ⁢ 𝒴 1 ) ) ⁢ W 2 ⁢ p - 3
- ( p - 1 ) ⁢ Δ ⁢ ( ( I λ * W p ) ⁢ W p - 2 ⁢ 𝒴 1 )
- ( p - 1 ) ⁢ ( ( I λ * W p ) 2 ⁢ W 2 ⁢ p - 4 ⁢ 𝒴 1 ) - ( I λ * W p ) ⁢ W p - 2 ⁢ Δ ⁢ 𝒴 1 .

Thus,

( e 0 - Δ ) 2 ⁢ 𝒴 1 = - ( p - 1 ) ⁢ Δ ⁢ ( ( I λ * W p ) ⁢ W p - 2 ⁢ 𝒴 1 ) - ( p - 1 ) ⁢ ( ( I λ * W p ) 2 ⁢ W 2 ⁢ p - 4 ⁢ 𝒴 1 )
- p ⁢ Δ ⁢ ( I λ * ( W p - 1 ⁢ 𝒴 1 ) ⁢ W p - 1 ) - p ⁢ ( I λ * W p ) ⁢ ( I λ * ( W p - 1 ⁢ 𝒴 1 ) ) ⁢ W 2 ⁢ p - 3
- ( I λ * W p ) ⁢ W p - 2 ⁢ Δ ⁢ 𝒴 1 - 2 ⁢ e 0 ⁢ Δ ⁢ 𝒴 1 .

Because of the existence of the nonlocal interaction on the right-hand side, the decay estimate in [19, Section 7.2] does not work. From the Bessel potential theory in [62], we know that the integral kernel G of the operator ( e 0 - Δ ) - 2 is

G ⁢ ( x ) = 1 ( 4 ⁢ π ) 2 ⁢ ∫ 0 ∞ e - e 0 4 ⁢ π ⁢ δ ⁢ e - π δ ⁢ | x | 2 ⁢ δ - N + 4 2 ⁢ d ⁢ δ δ .

Hence we have:

  1. G ⁢ ( x ) = | x | - N + 4 γ ⁢ ( 4 ) + o ⁢ ( | x | - N + 4 ) , | x | → 0 ,

  2. there exists c > 0 such that

    G ⁢ ( x ) = o ⁢ ( e - c ⁢ | x | ) , | x | → + ∞ .

Then the conclusion follows by the analogue estimates in [26, 39].

References

[1] A. K. Arora, Scattering of radial data in the focusing NLS and generalized Hartree equations, Discr. Cont. Dyn. Syst. 39 (2019), no. 11, 6643–6668. 10.3934/dcds.2019289Search in Google Scholar

[2] A. K. Arora and S. Roudenko, Well-posedness and blow-up properties for the generalized Hartree equation, J. Hyperbolic Differ. Equ. 17 (2020), no. 4, 727–763. 10.1142/S0219891620500228Search in Google Scholar

[3] A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math. J. 71 (2022), no. 3, 619–672. Search in Google Scholar

[4] A. K. Arora and S. Roudenko, Threshold solutions for the Hartree equation, preprint (2022), https://arxiv.org/abs/2210.07344. Search in Google Scholar

[5] A. K. Arora, S. Roudenko and K. Yang, On the focusing generalized Hartree equation, Math. Appl. Sci. Eng. 1 (2020), no. 4, 383–402. 10.5206/mase/10855Search in Google Scholar

[6] K. Atkinson and W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Lecture Notes in Math. 2044, Springer, Heidelberg, 2012. 10.1007/978-3-642-25983-8Search in Google Scholar

[7] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976), no. 4, 573–598. 10.4310/jdg/1214433725Search in Google Scholar

[8] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131–175. 10.1353/ajm.1999.0001Search in Google Scholar

[9] C. Bardos, L. Erdős, F. Golse, N. Mauser and H.-T. Yau, Derivation of the Schrödinger-Poisson equation from the quantum N-body problem, C. R. Math. Acad. Sci. Paris 334 (2002), no. 6, 515–520. 10.1016/S1631-073X(02)02253-7Search in Google Scholar

[10] C. Bardos, F. Golse and N. J. Mauser, Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), no. 2, 275–293. 10.4310/MAA.2000.v7.n2.a2Search in Google Scholar

[11] L. Campos, L. G. Farah and S. Roudenko, Threshold solutions for the nonlinear Schrödinger equation, Rev. Mat. Iberoam. 38 (2022), no. 5, 1637–1708. 10.4171/rmi/1337Search in Google Scholar

[12] D. Cao and Q. Guo, Divergent solutions to the 5D Hartree equations, Colloq. Math. 125 (2011), no. 2, 255–287. 10.4064/cm125-2-10Search in Google Scholar

[13] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, American Mathematical Society, Providence, 2003. 10.1090/cln/010Search in Google Scholar

[14] T. Cazenave, D. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 1, 135–147. 10.1016/j.anihpc.2010.11.005Search in Google Scholar

[15] F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, Springer Monogr. Math., Springer, New York, 2013. 10.1007/978-1-4614-6660-4Search in Google Scholar

[16] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of ℝ n , Appunti. Sc. Norm. Super. Pisa (N. S.) 15, Edizioni della Normale, Pisa, 2017. 10.1007/978-88-7642-601-8Search in Google Scholar

[17] L. Du and M. Yang, Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst. 39 (2019), no. 10, 5847–5866. 10.3934/dcds.2019219Search in Google Scholar

[18] T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP 2008 (2008), Art ID rpn002. Search in Google Scholar

[19] T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 (2009), no. 6, 1787–1840. 10.1007/s00039-009-0707-xSearch in Google Scholar

[20] T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam. 26 (2010), no. 1, 1–56. 10.4171/rmi/592Search in Google Scholar

[21] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in ℝ , Acta Math. 210 (2013), no. 2, 261–318. 10.1007/s11511-013-0095-9Search in Google Scholar

[22] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. 10.1002/cpa.21591Search in Google Scholar

[23] R. L. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy–Littlewood–Sobolev inequality, Spectral Theory, Function Spaces and Inequalities, Oper. Theory Adv. Appl. 219, Birkhäuser/Springer, Basel (2012), 55–67. 10.1007/978-3-0348-0263-5_4Search in Google Scholar

[24] R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (2) 176 (2012), no. 1, 349–381. 10.4007/annals.2012.176.1.6Search in Google Scholar

[25] J. Fröhlich, S. Graffi and S. Schwarz, Mean-field- and classical limit of many-body Schrödinger dynamics for bosons, Comm. Math. Phys. 271 (2007), no. 3, 681–697. 10.1007/s00220-007-0207-5Search in Google Scholar

[26] J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Effective dynamics for boson stars, Nonlinearity 20 (2007), no. 5, 1031–1075. 10.1088/0951-7715/20/5/001Search in Google Scholar

[27] P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, École Polytechnique, Palaiseau (1997), 1–11, Exp. No. 4. Search in Google Scholar

[28] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z. 170 (1980), no. 2, 109–136. 10.1007/BF01214768Search in Google Scholar

[29] J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, Nonlinear Wave Equations (Providence 1998), Contemp. Math. 263, American Mathematical Society, Providence (2000), 29–60. 10.1090/conm/263/04190Search in Google Scholar

[30] L. Grafakos, Classical Fourier Analysis, 2nd ed., Grad. Texts in Math. 249, Springer, New York, 2008. 10.1007/978-0-387-09432-8Search in Google Scholar

[31] C. Guzmán and C. Xu, The energy-critical inhomogeneous generalized Hartree equation in 3D, preprint (2023), https://arxiv.org/abs/2305.00972. Search in Google Scholar

[32] K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265–277. 10.1007/BF01646348Search in Google Scholar

[33] P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Appl. Math. Sci. 113, Springer, New York, 1996. 10.1007/978-1-4612-0741-2Search in Google Scholar

[34] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1984. Search in Google Scholar

[35] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675. 10.1007/s00222-006-0011-4Search in Google Scholar

[36] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), no. 2, 147–212. 10.1007/s11511-008-0031-6Search in Google Scholar

[37] R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, Lecture notes, Clay Mathematics Institute Summer School, Zürch, 2008. Search in Google Scholar

[38] J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for L 2 -critical Hartree NLS, Ann. Henri Poincaré 10 (2009), no. 6, 1159–1205. 10.1007/s00023-009-0010-2Search in Google Scholar

[39] J. Krieger, Y. Martel and P. Raphaël, Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math. 62 (2009), no. 11, 1501–1550. 10.1002/cpa.20292Search in Google Scholar

[40] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE 2 (2009), no. 1, 1–27. 10.2140/apde.2009.2.1Search in Google Scholar

[41] D. Li, C. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Differential Equations 246 (2009), no. 3, 1139–1163. 10.1016/j.jde.2008.05.013Search in Google Scholar

[42] D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal. 256 (2009), no. 6, 1928–1961. 10.1016/j.jfa.2008.12.007Search in Google Scholar

[43] D. Li and X. Zhang, On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation, Adv. Math. 220 (2009), no. 4, 1171–1192. 10.1016/j.aim.2008.10.013Search in Google Scholar

[44] D. Li and X. Zhang, Dynamics for the energy critical nonlinear wave equation in high dimensions, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1137–1160. 10.1090/S0002-9947-2010-04999-2Search in Google Scholar

[45] X. Li, C. Liu, X. Tang and G. Xu, Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations, preprint (2023), https://arxiv.org/abs/2304.04139. Search in Google Scholar

[46] X. Li, X. Tang and G. Xu, Nondegeneracy of the positive solutions for critical nonlinear Hartree equation in ℝ 6 , Proc. Amer. Math. Soc. 150 (2022), no. 12, 5203–5216. Search in Google Scholar

[47] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. 10.2307/2007032Search in Google Scholar

[48] E. H. Lieb and M. Loss, Analysis, Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. Search in Google Scholar

[49] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys. 112 (1987), no. 1, 147–174. 10.1007/BF01217684Search in Google Scholar

[50] Y. Liu and J. Wei, Nondegeneracy, Morse index and orbital stability of the KP-I lump solution, Arch. Ration. Mech. Anal. 234 (2019), no. 3, 1335–1389. 10.1007/s00205-019-01413-5Search in Google Scholar

[51] C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, Forum Math. 27 (2015), no. 1, 373–447. 10.1515/forum-2011-0087Search in Google Scholar

[52] C. Miao, G. Xu and J.-U. Yang, Global well-posedness for the defocusing Hartree equation with radial data in ℝ 4 , Commun. Contemp. Math. 22 (2020), no. 2, Article ID 1950004. 10.1142/S0219199719500044Search in Google Scholar

[53] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Funct. Anal. 253 (2007), no. 2, 605–627. 10.1016/j.jfa.2007.09.008Search in Google Scholar

[54] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl. (9) 91 (2009), no. 1, 49–79. 10.1016/j.matpur.2008.09.003Search in Google Scholar

[55] C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math. 114 (2009), no. 2, 213–236. 10.4064/cm114-2-5Search in Google Scholar

[56] C. Miao, G. Xu and L. Zhao, On the blow-up phenomenon for the mass-critical focusing Hartree equation in ℝ 4 , Colloq. Math. 119 (2010), no. 1, 23–50. 10.4064/cm119-1-2Search in Google Scholar

[57] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in ℝ 1 + n , Comm. Partial Differential Equations 36 (2011), no. 5, 729–776. 10.1080/03605302.2010.531073Search in Google Scholar

[58] K. Nakanishi, Energy scattering for Hartree equations, Math. Res. Lett. 6 (1999), no. 1, 107–118. 10.4310/MRL.1999.v6.n1.a8Search in Google Scholar

[59] P. Raphaël and J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc. 24 (2011), no. 2, 471–546. 10.1090/S0894-0347-2010-00688-1Search in Google Scholar

[60] T. Saanouni and C. Xu, Scattering theory for a class of radial focusing inhomogeneous Hartree equations, Potential Anal. 58 (2023), no. 4, 617–643. 10.1007/s11118-021-09952-xSearch in Google Scholar

[61] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), no. 3, 569–615. 10.1103/RevModPhys.52.569Search in Google Scholar

[62] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar

[63] Q. Su and Z. Zhao, Dynamics of subcritical threshold solutions for energy-critical NLS, Dyn. Partial Differ. Equ. 20 (2023), no. 1, 37–72. 10.4310/DPDE.2023.v20.n1.a3Search in Google Scholar

[64] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. 10.1007/BF02418013Search in Google Scholar

[65] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Reg. Conf. Ser. Math. 106, American Mathematical Society, Providence, 2006. 10.1090/cbms/106Search in Google Scholar

[66] G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., Grad. Stud. Math. 157, American Mathematical Society, Providence, 2014. Search in Google Scholar

[67] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. 10.1137/0516034Search in Google Scholar

[68] K. Yang, S. Roudenko and Y. Zhao, Stable blow-up dynamics in the L 2 -critical and L 2 -supercritical generalized Hartree equation, Stud. Appl. Math. 145 (2020), no. 4, 647–695. 10.1111/sapm.12328Search in Google Scholar

[69] T. Zhou, Threshold solutions for the focusing generalized Hartree equations, preprint (2022), https://arxiv.org/abs/2204.01567. 10.2139/ssrn.4425320Search in Google Scholar
Received: 2024-06-27
Revised: 2024-11-03
Published Online: 2025-01-13
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Rings of differential operators on (k,s)-th Tjurina algebras of singularities
  3. Cokernels of random matrix products and flag Cohen–Lenstra heuristic
  4. Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
  5. The L p -L q compactness of commutators of oscillatory singular integrals
  6. Determination of a pair of newforms from the product of their twisted central values
  7. Metrical properties of exponentially growing partial quotients
  8. Nonlinear operations and factorizations on a class of affine modulation spaces
  9. On algebraic degrees of certain exponential sums over finite fields
  10. The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
  11. Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
  12. Modular representations of GL2(𝔽𝑞) using calculus
  13. Bounding the number of p'-degree characters from below
  14. Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space
  15. Duality theorems for polyanalytic functions
  16. Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)
  17. Cylindrical ample divisors on Du Val del Pezzo surfaces
https://doi.org/10.1515/forum-2024-0301
Keywords for this article
Concentration compactness rigidity argument; Hartree equation; modulation analysis; nondegeneracy of ground state; spectral theory; uniqueness

Articles in the same Issue

  1. Frontmatter
  2. Rings of differential operators on (k,s)-th Tjurina algebras of singularities
  3. Cokernels of random matrix products and flag Cohen–Lenstra heuristic
  4. Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
  5. The L p -L q compactness of commutators of oscillatory singular integrals
  6. Determination of a pair of newforms from the product of their twisted central values
  7. Metrical properties of exponentially growing partial quotients
  8. Nonlinear operations and factorizations on a class of affine modulation spaces
  9. On algebraic degrees of certain exponential sums over finite fields
  10. The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
  11. Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
  12. Modular representations of GL2(𝔽𝑞) using calculus
  13. Bounding the number of p'-degree characters from below
  14. Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space
  15. Duality theorems for polyanalytic functions
  16. Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)
  17. Cylindrical ample divisors on Du Val del Pezzo surfaces
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 6.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2024-0301/html
Scroll to top button