Home Concentration of blow-up solutions for the Gross-Pitaveskii equation
Article Open Access

Concentration of blow-up solutions for the Gross-Pitaveskii equation

  • Shihui Zhu EMAIL logo
Published/Copyright: April 12, 2024
Download Article (PDF)
Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

We consider the blow-up solutions for the Gross-Pitaveskii equation modeling the attractive Boes-Einstein condensate. First, a new variational characteristic is established by computing the best constant of a generalized Gagliardo-Nirenberg inequality. Then, a lower bound on blow-up rate and a new concentration phenomenon of blow-up solutions are obtained in the L 2 supercritical case. Finally, in the L 2 critical case, a delicate limit of blow-up solutions is analyzed.

Keywords: attractive Bose-Einstein condensate; Gross-Pitaveskii equation; blow-up solution; blow-up rate; concentration
MSC 2010: 35Q55; 81Q99

1 Introduction

Experimental realization of the Bose-Einstein condensate in ultra cold vapors of 7 Li atoms was found in Bradley et al. [9], which opened a new field in the study of macroscopic quantum phenomena. In the last two decades, the Bose-Einstein condensates have been intensively studied due to their various quantum effects on superfluidity, quantized vortices, etc. [3,19,36,48,53,57]. Essentially, the attractive Boes-Einstein condensate can be modeled by the Gross-Pitaveskii equation, after scaling transformation, having the following form:

(1.1) i ψ t = − 1 2 Δ ψ + ω 2 2 ∣ x ∣ 2 ψ − ∣ ψ ∣ p − 2 ψ , t ≥ 0 , x ∈ R D ,

where ψ = ψ ( t , x ) : R + × R D → C is the complex valued wave function and D are the spatial dimension; the parameter ω > 0 corresponds to the trapping frequency; ∣ x ∣ 2 is the self-trapping magnetic potential [9,57], where p = 4 and D = 2 . Then, more researchers focus on dynamical properties of solutions for equation (1.1) to study the Bose-Einstein condensate from the aspect of mathematics [1,5–8,10,13,15,20,33,34,38,45,47,50,58,59].

The attractive Bose-Einstein condensate are known to be metastable in spatially localized systems, provided that the number of condensed particles, say N , is below a critical value N c , while they are unstable for N ≥ N c , which corresponds to wave’s collapse. From the mathematical studies, the wave’s collapse is called blow-up, i.e., there exists a finite time T < + ∞ such that lim t → T ‖ ψ ( t , x ) ‖ Σ = + ∞ , where Σ ≔ { v ∈ H 1 ∣ ∣ x ∣ v ∈ L 2 } , with the inner product defined by ⟨ u , v ⟩ = ∫ ( ∇ u ∇ v ¯ + u v ¯ + ∣ x ∣ 2 u v ¯ ) d x [9,57]. To understand how dynamical properties of blow-up phenomenon for nonlinear Schrödinger equations is interesting and fascinating [2,14,16,18,22,35,40,44,56,63].

In the present article, we impose the initial data

(1.2) ψ ( 0 , x ) = ψ 0 ,

to equation (1.1) and study the dynamical properties of blow-up solutions to Cauchy problems (1.1) and (1.2). The mathematical aspect research for equation (1.1) dates its history back to 1989 in the study by Oh [43], and Oh established the local well-posedness in the natural energy space Σ : Let D ≥ 2 and 2 < p < 2 * . If ψ 0 ∈ Σ , then there exists an unique solution ψ ( t , x ) of Cauchy problem (1.1) and (1.2) such that ψ ( t , x ) ∈ C ( [ 0 , T ) ; Σ ) and either T = + ∞ (global existence) or 0 < T < + ∞ and lim t → T ‖ ψ ( t , x ) ‖ Σ = + ∞ (blow-up). Furthermore, for all t ∈ [ 0 , T ) , ψ ( t , x ) satisfies:

  1. Conservation of mass: M ( ψ ) = ∫ ∣ ψ ( t , x ) ∣ 2 d x = M ( ψ 0 ) .

  2. Conservation of energy:

    E ( ψ ) ≔ 1 2 ∫ ∣ ∇ ψ ( t , x ) ∣ 2 d x + ω 2 2 ∫ ∣ x ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x − 2 p ∫ ∣ ψ ( t , x ) ∣ p d x = E ( ψ 0 ) .

Although the existence of blow-up solutions and sharp criteria of blow-up and global existence are widely studied [5,13,57,58], exploring dynamical properties of blow-up solutions is an intriguing and challenging problem due to the harmonic potential ∣ x ∣ 2 .

Particularly, in the L 2 critical case: p = 2 + 4 D , Carles [11] constructed a crucial transformation of solutions between equation (1.1) and the canonical critical semilinear Schrödinger equation

(1.3) i u t = − 1 2 Δ u − ∣ u ∣ 4 D u , t ≥ 0 , x ∈ R D .

Thus, Li and Zhang [32] obtained the sharp upper and lower blow-up rates for equation (1.1) in terms of Merle and Raphaël’s blow-up rates for equation (1.3) in studies by Merle and Raphaël [40,41]. Zhu et al. [60] investigated the limiting profile of blow-up solutions in Σ . But in the L 2 supercritical case: p > 2 + 4 D , there is no transformation, the dynamical properties of blow-up solutions, including blow-up rate and concentration are interesting and extensively open.

On the other hand, the presence of the harmonic potential breaks scaling and translation symmetries, introducing new challenges into the analysis for equation (1.1). The well known results, including stability of standing waves, concentration for the nonlinear Schrödinger equations with a potential, have dramatically changed, comparing with the canonical nonlinear Schrödinger equation without any potential; see [5,11,20,43,57]. Particularly, our work also fits in the context of the recent work on dispersive equations in the presence of broken symmetries, which have attracted a great deal of interest in recent years; see [4,23–27,29,30,46,51,54].

Before summarizing our results in this article, we now introduce the following profile decomposition of bounded sequences in H ˙ s c ∩ H ˙ 1 , which is the main tool in this article.

Proposition 1.1

Let D ≥ 2 , s c = D 2 − 2 p − 2 , 2 < p < 2 * , and { v n } n = 1 + ∞ be a bounded sequence in H ˙ s c ∩ H ˙ 1 . Then there exist a subsequence of { v n } n = 1 + ∞ (still denoted { v n } n = 1 + ∞ ), a family { x n j } j = 1 + ∞ of sequences in R D and a sequence { V j } j = 1 + ∞ of functions in H ˙ s c ∩ H ˙ 1 such that the following properties:

  1. For every k ≠ j , ∣ x n k − x n j ∣ → + ∞ as n → + ∞ .

  2. For every l ≥ 1 and every x ∈ R D

    (1.4) v n ( x ) = ∑ j = 1 l V j ( x − x n j ) + v n l ( x ) ,

    where the remaining term v n l ≔ v n l ( x ) is small in the following sense:

    (1.5) lim l → + ∞ limsup n → + ∞ ‖ v n l ‖ r = 0 f o r e v e r y r ∈ ( p c , 2 * ) ,

where p c = ( p − 2 ) D 2 . Moreover, as n → + ∞ , we have

(1.6) ‖ ∇ v n ‖ 2 2 = ∑ j = 1 l ‖ ∇ V j ‖ 2 2 + ‖ ∇ v n l ‖ 2 2 + o ( 1 ) , ‖ v n ‖ H ˙ s c 2 = ∑ j = 1 l ‖ V j ‖ H ˙ s c 2 + ‖ v n l ‖ H ˙ s c 2 + o ( 1 ) ,

(1.7) ∑ j = 1 l V j ( x − x n j ) p p = ∑ j = 1 l ‖ V j ( x − x n j ) ‖ p p + o ( 1 ) ,

where o ( 1 ) = o n ( 1 ) → 0 as n → + ∞ .

The profile decomposition of bounded sequences in H ˙ s c ∩ H ˙ 1 , is first established in the author’s Ph.D thesis in the study by Zhu [62], to prove the existence of ground state solutions of (1.8) and find the best constant in (1.2) by the variational argument. We should point out that the profile decomposition of bounded sequences in H 1 was first proposed by Hmidi and Keraani [21]. In fact, the profile decomposition tool is less technical and simpler, and there are two main advantages of the profile decomposition: one is that the decomposition form of bounded sequences is given, and it can the aim functionals. The other is that the decomposition is almost orthogonal, and the norms of bound sequences have similar decomposition. The profile decomposition argument has been successfully applied to find the best constant of generalized Gagliardo-Nirenberg inequalities [21,61,63].

Then, in the L 2 supercritical case: p > 2 + 4 D , we introduce a new stationary equation:

(1.8) − 1 2 Δ W + p − 2 4 ( − Δ ) s c W − ∣ W ∣ p − 2 W = 0 , W ∈ H ˙ s c ∩ H ˙ 1 ,

where s c = D 2 − 2 p − 2 , and ( − Δ ) s c is the pseudo-differential operator defined by

( − Δ ) s c v ( x ) = 1 ( 2 π ) D ∫ R D e i x ⋅ ξ ∣ ξ ∣ 2 s c v ^ ( ξ ) d ξ = ℱ − 1 [ ∣ ξ ∣ 2 s c ℱ [ v ( x ) ] ( ξ ) ] .

We obtain the following new sharp Gagliardo-Nirenberg inequality, which is sharp in the sense: let v = W in (1.9), then “ ≤ ” is “=” by the Pohozaev’s identities for equation (1.8).

Theorem 1.2

Let D ≥ 2 and s c = D 2 − 2 p − 2 . If 2 < p < 2 * , then ∀ v ∈ H ˙ s c ∩ H ˙ 1 ,

(1.9) ‖ v ‖ p p ≤ p 4 ‖ v ‖ H ˙ s c ‖ W ‖ H ˙ s c p − 2 ‖ ∇ v ‖ 2 2 ,

where W is a ground state solution of (1.8).

It is well known that the blow-up theory for equation (1.3) in H 1 has been developed during the last two decades by the energy arguments and variational arguments; see previous studies [12,28,40,57] and the references therein. Essentially, the blow-up theory is connected to the standing wave through the following Gagliardo-Nirenberg inequality proposed by Weinstein in [55]. It reads that for 2 < p < 2 * , ∀ v ∈ H 1 ,

(1.10) ‖ v ‖ p p ≤ C opt ‖ v ‖ 2 2 p − p D + 2 D 2 ‖ ∇ v ‖ 2 ( p − 2 ) D 2 ,

where C opt = p 2 p − p D + 2 D 2 ( p − 2 ) D − 4 4 [ ( p − 2 ) D ] ( p − 2 ) D 4 ‖ R ‖ 2 p − 2 and R is the unique positive radially symmetric solution of − 1 2 Δ R + R − ∣ R ∣ p − 2 R = 0 , whose existence and uniqueness are proved in the studies by Kwong [31] and Strauss [49], respectively. When p = 2 + 4 D , one can use (1.10) to control the nonlinear term in the energy E ( ψ ) by ‖ ∇ ψ ‖ 2 2 , and there is a balance between the kinetic energy ‖ ∇ ψ ‖ 2 2 and the nonlinear potential energy ‖ ψ ‖ 2 + 4 D 2 + 4 D . Basing on this observation, a lot of results on the blow-up solutions for (1.1) in this case ( p = 2 + 4 D ) can be obtained by the variational structures generated by (1.10), such as limiting profile, L 2 -concentration, and blow-up rate [40–42,55,56]. But for the case: p > 2 + 4 D (also called L 2 -supercritical case), the aforementioned control fails, which is the main difficulty to study the dynamical properties of blow-up solutions for nonlinear Schrödinger equations with L 2 -supercritical nonlinearity.

Finally, as an application of the new variational characteristic generated by the generalized Gagliardo-Nirenbergy inequality (1.9), we find the following blow-up rate and concentration phenomenon of blow-up solutions to Cauchy problems (1.1) and (1.2).

Theorem 1.3

Let D ≥ 2 , s c = D 2 − 2 p − 2 , and 2 + 4 D < p < 2 * . Assume that ψ 0 ∈ Σ and ψ ( t , x ) is the corresponding blow-up solution of Cauchy problems (1.1) and (1.2), where 0 < T < + ∞ is the blow-up time. Then, we have the followings:

  1. For all 0 < t < T , there exists a positive constant C such that

    (1.11) ‖ ∇ ψ ( t ) ‖ L x 2 ≥ C ( T − t ) 2 p − ( p − 2 ) D 4 ( p − 2 ) .

  2. If ψ ( t , x ) stratifies sup t ∈ [ 0 , T ) ‖ ψ ( t ) ‖ H ˙ s c < + ∞ , then there exists y ( t ) ∈ R D such that

    (1.12) liminf t ∈ [ 0 , T ) ‖ ψ ( t , x ) ‖ H ˙ s c ( ∣ x − y ( t ) ∣ ≤ ( T − t ) 1 2 − ) ≥ ‖ W ‖ H ˙ s c ,

    where W is a ground state solution of (1.8). Here, we denote 1 2 − be any real number 1 2 − ε with any sufficiently small ε > 0 .

In particular, when p = 2 + 4 D , we see that s c = 0 . Then, the rate of H ˙ s c -concentration phenomenon in (1.12) will recover Merle and Tsutsumi’s concentration results [42,52] for the canonical semilinear Schrödinger equation (1.3): there exists x ( t ) ∈ R D such that

(1.13) liminf t ∈ [ 0 , T ) ∫ ( ∣ x − x ( t ) ∣ ≤ ( T − t ) 1 2 − ) ∣ u ( t , x ) ∣ 2 d x ≥ ∫ R D ∣ Q ∣ 2 d x ,

where Q ( x ) is the unique positive radially symmetric solution of

(1.14) − 1 2 Δ Q + Q − ∣ Q ∣ 4 D Q = 0 , Q ∈ H 1 .

To our surprise, the concentration rate of blow-up solutions in (1.12) is the same to (1.13), and it does not changed with respect to p .

This article is organized as follows. In Section 2, we give a review of Cauchy problems (1.1) and (1.2), and state some preliminaries. In Section 3, by establishing the profile decomposition of bounded sequences in H ˙ s c ∩ H ˙ 1 , we give the proof of Theorem 1.2. In Section 4, we give the proof of Theorem 1.3. In Section 5, we further investigate the limiting profile of blow-up solutions in the L 2 critical case.

2 Review of the Cauchy problem

In this article, we denote L q ( R D ) , ‖ ⋅ ‖ L q ( R D ) , H 1 ( R D ) and ∫ R D ⋅ d x by L q , ‖ ⋅ ‖ L q , H 1 and ∫ ⋅ d x , respectively. q ′ is the conjugate of real number q which satisfies the condition 1 q + 1 q ′ = 1 . ℜ z and ℑ z are the real part and imaginary part of the complex number z , respectively. z ¯ denotes the complex conjugate of the complex number z .

First, we review the well-posedness of Cauchy problems (1.1) and (1.2). Let H = 1 2 Δ + ω 2 2 ∣ x ∣ 2 be the corresponding Schrödinger operator with a harmonic potential in R D . Then, Cauchy problems (1.1) and (1.2) have the following equivalent integral form:

(2.1) ψ ( t ) = S ( t ) ψ 0 − i ∫ 0 t S ( t − τ ) ∣ ψ ∣ p − 2 ψ ( τ ) d τ ,

where S ( t ) ≔ e i t H is the propagator of H . Fujiwara [17] found that the corresponding Schrödinger kernel k ( t , x , y ) to H has the following explicit format:

k ( t , x , y ) = − i 2 π sin t D 2 e i ( ∣ x ∣ 2 + ∣ y ∣ 2 ) cos t − 2 x y 4 sin t ω 2 .

With this result, one can easily deduce that t ↦ S ( t ) is a strongly continuous from L q ′ to L q for q ≥ 2 , and when 0 < t < δ ,

‖ S ( t ) v ‖ q ≤ ( 2 π ∣ t ∣ ) D q − D 2 ‖ v ‖ q ′ .

Thus, one can obtain the following Strichartz estimate for the Schrödinger operator H . A pair is called L 2 -admissible, if 2 ≤ q , r ≤ + ∞ , ( q , r ) ≠ ( 2 , + ∞ ) , and 2 q = D 2 − D r . Define the Strichartz norms: ‖ v ‖ L t q L x r ≔ ∫ R ( ( ∫ R D ∣ v ( t , x ) ∣ r d x ) q r d t ) 1 q and ‖ v ‖ L t ∞ L x 2 ≔ ( ∫ R D ∣ v ( t , x ) ∣ 2 d x ) 1 2 d t . We can see the following Strichartz estimates.

Lemma 2.1

Let H = − 1 2 Δ + ω 2 2 ∣ x ∣ 2 . If ( q , r ) is L 2 -admissible, then we have the following estimates:

(2.2) ‖ e i t H v ‖ L t q L x r ≤ C ‖ v ‖ L x 2 ,

(2.3) ∫ 0 t e i ( t − τ ) H ∣ v ∣ p − 2 v d τ L t q L x r ≤ C ‖ ∣ v ∣ p − 2 v ‖ L t q ′ L x r ′ ,

(2.4) ∫ + ∞ − ∞ e i ( t − τ ) H ∣ v ∣ p − 2 v d τ L x 2 ≤ C ‖ ∣ v ∣ p − 2 v ‖ L t q ′ L x r ′ .

Oh [43] established the local well-posedness for a class of nonlinear Schrödinger equations with a general potential including the harmonic potential. Here, we review the following the local well-posedness of Cauchy problems (1.1) and (1.2) in Σ ; see the study by Cazenave [12] for a review.

Proposition 2.2

Let D ≥ 2 and 2 < p < 2 * . If ψ 0 ∈ Σ , then there exists an unique solution ψ ( t , x ) of Cauchy problems (1.1) and (1.2) such that ψ ( t , x ) ∈ C ( [ 0 , T ) ; Σ ) and either T = + ∞ (global existence) or 0 < T < + ∞ and lim t → T ‖ ψ ( t , x ) ‖ Σ = + ∞ (wave’s collapse). Furthermore, for all t ∈ [ 0 , T ) , ψ ( t , x ) satisfies:

  1. Conservation of mass: M ( ψ ( t , x ) ) = M ( ψ 0 ) .

  2. Conservation of energy: E ( ψ ( t , x ) ) = E ( ψ 0 ) .

Lemma 2.3

Let D ≥ 2 , 2 + 4 D ≤ p < 2 * , and ψ 0 ∈ Σ . Suppose that ψ ( t , x ) ∈ C ( [ 0 , T ) ; Σ ) is the solution of Cauchy problems (1.1) and (1.2). Then, J ( t ) ≔ ∫ ∣ x ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x < + ∞ , and there exists a positive constant C 0 such that for all times t ,

(2.5) J ( t ) ≤ C 0 < + ∞ .

Particularly, when p = 2 + 4 D , there exists y 1 ∈ R D such that for all 0 < T * ≤ T ,

(2.6) lim t → T * ∫ ∣ ψ ( t , x ) ∣ 2 x d x = − ∫ ∣ Q ( x ) ∣ 2 d x y 1 ,

where Q ( x ) is the unique positive radially symmetric solution of equation (1.14).

Proof

From the local well-posedness, ψ 0 ∈ Σ implies that J ( t ) < + ∞ is well-defined and continuous for all times t . After some computations, we have d d t J ( t ) = 2 ℑ ∫ x ⋅ ∇ ψ ( t ) ψ ( t ) ¯ d x ,

d 2 d t 2 J ( t ) = 4 E ( ψ 0 ) − 4 ω 2 J ( t ) − ( p − 2 ) D − 4 p ∫ ∣ ψ ( t ) ∣ p d x ≤ 4 E ( ψ 0 ) .

After integrating, we deduce that (2.5) holds for all times t .

When p = 2 + 4 D , we obtain that J ( t ) has the following explicit form:

(2.7) J ( t ) = J ( 0 ) − E ( ψ 0 ) ω 2 cos 2 ω t + J ′ ( 0 ) 2 ω sin 2 ω t + E ( ψ 0 ) ω 2 .

For any x 0 ∈ R D , by setting g ( t ) ≔ ∫ x 0 ⋅ x ∣ ψ ( t , x ) ∣ 2 d x , we have g ′ ( t ) = ℑ ∫ x 0 ⋅ ∇ ψ ( t ) ψ ( t ) ¯ d x and g ″ ( t ) = − ω 2 ∫ x 0 ⋅ x ∣ ψ ( t ) ∣ 2 d x . Then, g ( t ) = g ( 0 ) cos ω t + g ′ ( 0 ) ω sin ω t . Since x 0 ∈ R D is arbitrary, we deduce that for all times 0 < t < T

∫ x ∣ ψ ( t , x ) ∣ 2 d x = ∫ x ∣ ψ 0 ∣ 2 d x cos ω t + ℑ ∫ ∇ ψ 0 ψ 0 ¯ d x ω sin ω t .

Therefore, we can find y 1 ∈ R D such that (2.6) is true by the continuity of ψ ( t , x ) .□

3 A new variational structure

In the sequel, we set s c = D 2 − 2 p − 2 and p c = ( p − 2 ) D 2 . Theorem 1.2 is a consequence of the profile decomposition of the bounded sequences in Proposition 1.1. The author has proved Proposition 1.1 in his PhD thesis [62]. To keep the self-contained, here, we give the proof of Proposition 1.1 in detail.

Proof of Proposition 1.1

According to the fact that any bounded sequence { v n } n = 1 + ∞ in H ˙ s c ∩ H ˙ 1 has a weakly convergent subsequence, we shall use the weakly limit points as the profiles to decompose the sequence { v n } n = 1 + ∞ . Let v = { v n } , v l = { v n l } , and μ ( v ) be the set of functions obtained as weak limits of subsequences of the translated v n ( x + x n ) with { x n } n = 1 + ∞ in H ˙ s c ∩ H ˙ 1 . By denoting η ( v ) = sup { ‖ ∇ V ‖ 2 + ‖ V ‖ H ˙ s c ∣ V ∈ μ ( v ) } , we obtain η ( v ) ≤ limsup n → + ∞ { ‖ ∇ v n ‖ 2 + ‖ v n ‖ H ˙ s c } . Then, there exist a subsequence { V j } j = 1 + ∞ of μ ( v ) and a family { x n j } j = 1 + ∞ ⊂ R D satisfying k ≠ j , ∣ x n k − x n j ∣ → + ∞ as n → + ∞ . And up to extracting a subsequence, v n can be written as follows:

(3.1) v n ( x ) = ∑ j = 1 l V j ( x − x n j ) + v n l , η ( v l ) → 0 ( l → + ∞ ) ,

and also (1.6) are true. More precisely, the construction of { V j } j = 1 + ∞ are given in the following. Indeed, if η ( v ) = 0 , we can take V j = 0 for all j , otherwise, we choose V 1 ∈ μ ( v ) such that ‖ V 1 ‖ H ˙ s c + ‖ ∇ V 1 ‖ 2 ≥ 1 2 η ( v ) > 0 . By the definition of μ ( v ) , there exists a subsequence x n 1 of R D such that up to extracting a subsequence,

(3.2) v n ( x + x n 1 ) ⇀ V 1 ( x ) weakly in H ˙ s c ∩ H ˙ 1 .

Take the transformation v n 1 ( x ) = v n ( x ) − V 1 ( x − x n 1 ) . (3.2) implies that v n 1 ( x + x n 1 ) ⇀ 0 weakly in H ˙ s c ∩ H ˙ 1 . By Brézis and Lieb’s lemma, we obtain ‖ ∇ v n ‖ 2 2 = ‖ ∇ V 1 ‖ 2 2 + ‖ ∇ v n 1 ‖ 2 2 + o ( 1 ) , ‖ v n ‖ H ˙ s c 2 = ‖ V 1 ‖ H ˙ s c 2 + ‖ v n 1 ‖ H ˙ s c 2 + o ( 1 ) . Now, replace v n with v n 1 and repeat the same process. We can find V 2 ∈ μ ( v ) such that ‖ V 2 ‖ H ˙ s c + ‖ ∇ V 2 ‖ 2 ≥ 1 2 η ( v 1 ) > 0 and v n 1 ( x + x n 2 ) ⇀ V 2 ( x ) weakly in H ˙ s c ∩ H ˙ 1 . Take the transformation v n 2 ( x ) = v n 1 ( x ) − V 2 ( x − x n 2 ) . We have v n 2 ( x + x n 2 ) ⇀ 0 weakly in H ˙ s c ∩ H ˙ 1 . By Brézis and Lieb’s lemma, we obtain ‖ ∇ v n 1 ‖ 2 2 = ‖ ∇ V 2 ‖ 2 2 + ‖ ∇ v n 2 ‖ 2 2 + o ( 1 ) and ‖ v n 1 ‖ H ˙ s c 2 = ‖ V 2 ‖ H ˙ s c 2 + ‖ v n 2 ‖ H ˙ s c 2 + o ( 1 ) . Moreover, we claim that

(3.3) ∣ x n 1 − x n 2 ∣ → + ∞ as n → + ∞ .

Indeed, if (3.3) is not true, then v n 2 ( x + x n 2 ) = v n 1 ( x + x n 2 − x n 1 + x n 1 ) + V 2 ( x ) , which implies that V 2 = 0 by v n 1 ( ⋅ + x n 1 ) ⇀ 0 and v n 2 ( ⋅ + x n 2 ) ⇀ 0 in H ˙ s c ∩ H ˙ 1 . This is a contradiction, and (3.3) is true. Thus, an argument of iteration and orthogonal extraction allows us to construct the families { x n j } j = 1 + ∞ and { V j } j = 1 + ∞ satisfying the aforementioned claims. From the convergence of ∑ j = 1 l ( ‖ ∇ V j ‖ 2 2 + ‖ V j ‖ H ˙ s c 2 ) , we have ‖ ∇ V j ‖ 2 2 + ‖ V j ‖ H ˙ s c 2 → 0 as j → + ∞ . Then,

η ( v j ) ≤ 2 ( ‖ ∇ V j − 1 ‖ 2 + ‖ V j − 1 ‖ H ˙ s c ) → 0 as j → + ∞ .

Therefore, we prove that (1.6) and (3.1) are true.

Next, we shall prove that (1.5) for all r ∈ ( p c , 2 * ) . Let χ K ∈ S ′ ( R D ) be such that supp χ ˆ K ( ξ ) = { ξ ∈ R D ∣ 1 2 K ≤ ∣ ξ ∣ ≤ 2 K } , χ ˆ K = 1 on { ξ ∈ R D ∣ 1 K ≤ ∣ ξ ∣ ≤ K } , and 0 ≤ χ ˆ K ≤ 1 on supp χ ˆ K ( ξ ) , where ˆ denotes the Fourier transformation. We can decompose v n l :

(3.4) v n l = χ K * v n l + ( δ − χ K ) * v n l ,

where * is the convolution and δ is the Dirac function. It follows from the Sobolev embedding H ˙ s c ↪ L p c that

‖ ( δ − χ K ) * v n l ‖ p c ≤ C ∫ ∣ ξ ∣ ≤ 1 K ∣ ξ ∣ 2 s c ∣ v ˆ n l ( ξ ) ∣ 2 d ξ + ∫ ∣ ξ ∣ ≥ K ∣ ξ ∣ 2 s c ∣ v ˆ n l ( ξ ) ∣ 2 d ξ 1 2 .

Yielding by the assumption that { v n l } n = 1 + ∞ is bounded in H ˙ s c , we have ‖ ( δ − χ K ) * v n l ‖ p c → 0 as K → + ∞ . Similarly, by applying the Sobolev embedding H ˙ 1 ↪ L 2 * and the assumption that { v n l } n = 1 + ∞ is bounded in H ˙ 1 , we have ‖ ( δ − χ K ) * v n l ‖ 2 * → 0 as K → + ∞ . Moreover, from the Hölder interpolation inequality, we deduce that for all r ∈ ( p c , 2 * ) ,

(3.5) ‖ ( δ − χ K ) * v n l ‖ r ≤ ‖ ( δ − χ K ) * v n l ‖ p c θ ‖ ( δ − χ K ) * v n l ‖ 2 * 1 − θ → 0 as K → + ∞ ,

where 1 r = θ p c + 1 − θ 2 * and 0 < θ < 1 . For the term χ K * v n l , according to the definition of χ K , we see that for all r ∈ ( p c , 2 * ) ,

(3.6) ‖ χ K * v n l ‖ r ≤ ‖ χ K * v n l ‖ ∞ r − p c r ‖ χ K * v n l ‖ p c p c r ≤ C ‖ χ K * v n l ‖ ∞ r − p c r ‖ χ K * v n l ‖ H ˙ s c p c r .

In view of the definition of μ ( v ) , we see that

(3.7) limsup n → + ∞ ‖ χ K * v n l ‖ ∞ ≤ sup ∫ χ K ( − x ) V ( x ) d x ∣ V ∈ μ ( v ) .

By using the Parseval identity and the Hölder inequality, we deduce that

∫ χ K ( − x ) V ( x ) d x = ∫ ℱ − 1 [ χ K ( − x ) ] ℱ [ V ( x ) ] d ξ = ∫ χ K ^ ( ξ ) V ^ ( ξ ) d ξ ≤ C K D + 1 ‖ ∇ V ‖ 2 ≤ C K D + 1 η ( v l ) .

Take K = 1 η ( v l ) 1 D + 1 − ε with ε > 0 sufficiently small. We see that C K D + 1 η ( v l ) → 0 as l → + ∞ and ∫ χ K ( − x ) V ( x ) d x → 0 as l → + ∞ . Injecting this into (3.6), we deduce that

(3.8) ‖ χ K * v n l ‖ r ≤ C ‖ v n l ‖ 2 p c r ∫ χ K ( − x ) V ( x ) d x r − p c r → 0 as n → + ∞ , l → + ∞ .

Therefore, we can obtain (1.5) by collecting (3.4), (3.5), and (3.8).

Finally, we shall prove (1.7). Without loss of generality, we assume every V j is continuous and compactly supported. By the inequality: ∑ j = 1 l a j p − ∑ j = 1 l ∣ a j ∣ p ∣ ≤ C ∑ j ≠ k ∣ a j ∣ ∣ a k ∣ p − 1 for every p > 2 , we need to prove the mixed terms in ∑ j = 1 l V j ( x − x n j ) p p vanish. More precisely, we shall prove that for all j ≠ k , ∫ ∣ V j ∣ ∣ V k ∣ ∣ V m ∣ p − 2 d x → 0 as n → + ∞ . Indeed, by using the Hölder inequality and Sobolev embedding theorem, we obtain

(3.9) ∫ ∣ V j V k ∣ ∣ V m ∣ p − 2 d x ≤ ∫ ∣ V j V k ∣ p 2 d x 2 p ∫ ∣ V m ∣ p d x p − 2 p ≤ C ∫ ∣ V j V k ∣ p 2 d x 2 p .

By the orthogonality condition: for every k ≠ j , ∣ x n k − x n j ∣ → + ∞ as n → + ∞ , we have

(3.10) ∫ ∣ V j V k ∣ p 2 d x = ∫ ∣ V j ( y − ( x n j − x n k ) ) V k ( y ) ∣ p 2 d y → 0 as n → + ∞ .

Collecting (3.9) and (3.10), we prove that (1.7) is true. This completes the proof.□

At the end of this section, we shall apply the profile decomposition in Proposition 1.1 and the variational argument to finish the proof of Theorem 1.2.

Proof of Theorem 1.2

First, we define the variational problem and study the properties of the corresponding minimizers. Define

(3.11) J ⋆ ≔ inf { J ( v ) ∣ v ∈ H ˙ s c ∩ H ˙ 1 } where J ( v ) ≔ ‖ v ‖ H ˙ s c p − 2 ‖ ∇ v ‖ 2 2 ‖ v ‖ p p .

By the assumption: 2 < p < 2 * , we obtain p c = ( p − 2 ) D 2 < p < 2 * . Let θ = p − 2 p . Then, by the Hölder interpolation estimate and Sobolev embedding, we deduce that

(3.12) ‖ v ‖ p p ≤ C ‖ v ‖ p c θ p ‖ v ‖ 2 * ( 1 − θ ) p ≤ C ‖ v ‖ H ˙ s c p − 2 ‖ ∇ v ‖ 2 2 .

Hence, the functional J ( v ) has a positive lower bound, and the variational problem (3.11) is well-defined. Now, we investigate the Euler-Lagrange equation to variational problem (3.11). If V is the minimizer of J ( v ) , then ∀ φ ∈ C 0 ∞ ( R D ) , we have d d ε J ( V + ε φ ) ∣ ε = 0 = 0 , which implies that V satisfies

(3.13) p − 2 2 ‖ V ‖ H ˙ s c p − 4 ‖ ∇ V ‖ 2 2 ( − Δ ) s c V − ‖ V ‖ H ˙ s c p − 2 Δ V − J ⋆ p 2 ∣ V ∣ p − 2 V = 0 .

Furthermore, from (3.13), we see that any minimizer of J ( v ) is a corresponding solution of equation (1.8). Since any smaller H ˙ s c -norm solution would correspond to a lower value of J ( v ) , the Pohozhaev identities show that it is in fact a minimal H ˙ s c -norm solution of equation (1.8). Therefore, to prove the existence of a ground state, it suffices to prove the existence of a minimizer for J ( v ) .

Next, we use Lemma 1.1 to prove the existence of a minimizer for J ( v ) , which implies the existence of a ground state solution to equation (1.8). Without loss of generality, we can inquire the minimizing sequence { v n } n = 1 + ∞ ⊂ H ˙ s c ∩ H ˙ 1 to variational problem (3.11) satisfying

(3.14) ‖ v n ‖ H ˙ s c = 1 , ‖ ∇ v n ‖ 2 = 1 and J ( v n ) = 1 ‖ v n ‖ p p → J ⋆ as n → + ∞ .

Indeed, if a minimizing sequence { v n } n = 1 + ∞ ⊂ H ˙ s c ∩ H ˙ 1 does not satisfy (3.14), then one can take the following transformation: v n λ , μ = μ v n ( λ x ) , where λ = ( ‖ v n ‖ H ˙ s c ‖ ∇ v n ‖ 2 ) 1 1 − s c and μ = ‖ v n ‖ H ˙ s c D − 2 2 ( 1 − s c ) ‖ ∇ v n ‖ 2 D − 2 s c 2 ( 1 − s c ) . The new sequence { v n λ , μ } n = 1 + ∞ is such that ‖ v n λ , μ ‖ H ˙ s c = 1 , ‖ ∇ v n λ , μ ‖ 2 = 1 , and J ( v n λ , μ ) = J ( v n ) . Hence, it follows from (3.14) that { v n } n = 1 + ∞ is bounded in H ˙ s c ∩ H ˙ 1 . By applying Lemma 1.1, we see that v n ( x ) can be decomposed by v n ( x ) = ∑ j = 1 l V j ( x − x n j ) + r n l ( x ) . From (3.14) and (1.6), we deduce that

(3.15) ∑ j = 1 l ‖ V n j ‖ H ˙ s c 2 ≤ ‖ v n ‖ H ˙ s c 2 = 1 , ∑ j = 1 l ‖ ∇ V n j ‖ 2 2 ≤ ‖ ∇ v n ‖ 2 2 = 1 ,

where V n j = V j ( x − x n j ) . As 2 < p < 2 * , we deduce that p c < p < 2 * and r n l satisfies lim l → + ∞ limsup n → + ∞ ∫ ∣ r n l ∣ p d x = 0 . From (1.7), we have lim n → + ∞ ∣ ∫ ∑ j = 1 l V j ( x − x n j ) p d x − ∑ j = 1 l ∫ ∣ V j ∣ p d x ∣ = 0 . Thus, we obtain the following estimate.

(3.16) ‖ v n ‖ p p → ∑ j = 1 l ‖ ∣ V j ∣ ∣ p p → 1 J ⋆ as n → + ∞ , l → + ∞ .

On the other hand, for all j ≥ 1 , we deduce that J ⋆ ‖ V j ‖ p p ≤ ( sup { ‖ V j ‖ H ˙ s c ∣ j ≥ 1 } ) p − 2 ‖ ∇ V j ‖ 2 2 . Since ∑ j ‖ V j ‖ H ˙ s c 2 is convergent, there exists j 0 ≥ 1 such that ‖ V j 0 ‖ H ˙ s c = sup { ‖ V j ‖ H ˙ s c ∣ j ≥ 1 } , and we have J ⋆ ( ∑ j = 1 l ‖ V j ‖ p p ) ≤ ‖ V j 0 ‖ H ˙ s c p − 2 ( ∑ j = 1 l ‖ ∇ V j ‖ 2 2 ) . Taking n → + ∞ and l → + ∞ , we can obtain ‖ V j 0 ‖ H ˙ s c p − 2 ≥ 1 by (3.16) and ∑ j = 1 l ‖ ∇ V j ‖ 2 2 ≤ 1 . But from (3.15), we obtain ‖ V j ‖ H ˙ s c ≤ 1 for every j ≥ 1 . Hence, there exists only one term V j 0 ≠ 0 such that ‖ V j 0 ‖ H ˙ s c = 1 , ‖ ∇ V j 0 ‖ 2 = 1 and ‖ V j 0 ‖ p p = 1 J ⋆ . Therefore, we prove that V j 0 is the minimizer of J ( v ) . It follows from (3.13) that V j 0 is the solution of

(3.17) p − 2 2 ( − Δ ) s c V j 0 − Δ V j 0 − J ⋆ p 2 ∣ V j 0 ∣ p − 2 V j 0 = 0 .

Because J ⋆ is a fixed real number, after rescaling, we can check that (3.17) and (1.8) are equivalent. Hence, the existence of ground state solutions of (3.17) implies that of (1.8).

Finally, to give the exact expression of J ⋆ by W , we take V j 0 = a W ( x + x 0 ) with a ∈ C * and x 0 ∈ R D , where ∣ a ∣ p − 2 = 4 J ⋆ p , where W is a ground state solution of (1.8). Since ‖ V j 0 ‖ H ˙ s c = 1 , we obtain ‖ V j 0 ‖ H ˙ s c 2 = ∣ a ∣ 2 ‖ W ‖ H ˙ s c 2 = 1 , which implies J ⋆ = 4 p ‖ W ‖ H ˙ s c p − 2 . This completes the proof of Theorem 1.2.□

Remark 3.1

In Theorem 1.2, we prove the existence of nontrivial solutions of (1.8), and by the rescaling trick, we can see that all solutions of (1.8) has the same H ˙ s c -norm, which is a fixed number. And we call the nontrivial solution of equation (1.8) with the same H ˙ s c -norm is a ground state solution. In particular, when s c = 0 , the new sharp Gagliardo-Nirenberg inequality (1.9) degenerates to inequality (1.10).

4 Blow-up rate and concentration

By the Strichartz estimates in Lemma 2.1 generated by the Schrödinger operator with a harmonic potential: e i t H , where H = − 1 2 Δ + ω 2 2 ∣ x ∣ 2 , we obtain the following lower bound of blow-up rate for Cauchy problems (1.1) and (1.2), in terms of Cazenave’s arguments in [12].

Proof of (i) in Theorem 1.3

First, define the Schrödinger operator with a harmonic potential: e i t H , where H = − 1 2 Δ + ω 2 2 ∣ x ∣ 2 . It follows from the local well-posedness in Section 2 that Cauchy problems (1.1) and (1.2) are equivalent to

(4.1) ψ ( t , x ) = e i t H ψ 0 − i ∫ 0 t e i ( t − τ ) H ∣ ψ ∣ p − 2 ψ ( τ ) d τ .

Let 2 + 4 D ≤ p < 2 * . It is easy to check that ( q , r ) = ( 4 p ( p − 2 ) D , p ) is L 2 -admissible, and the corresponding conjugate pair: ( q ′ , r ′ ) = ( 4 p ( 4 − D ) p + 2 D , p p − 1 ) . By applying the Strichartz estimates (2.2)–(2.4) in Lemma 2.1, we deduce that

(4.2) ‖ ∇ ψ ‖ L t ∞ ( ( t , τ ) ; L x 2 ) + ‖ ∇ ψ ‖ L t 4 p ( p − 2 ) D ( ( t , τ ) ; L x p ) ≤ C ‖ ∇ ψ 0 ‖ L x 2 + C ‖ ∇ ( ∣ ψ ∣ p − 2 ψ ) ‖ L t 4 p ( 4 − D ) p + 2 D ( ( t , τ ) ; L x p p − 1 ) .

Then, by the conservation of energy, there exists C 0 defined in (2.5), and C > 0 such that

(4.3) ‖ ψ ‖ L x p p ≤ p ω 2 4 C 0 − p 2 E ( ψ 0 ) + p 4 ‖ ∇ ψ ‖ L x 2 2 ≤ C ( 1 + ‖ ∇ ψ ‖ L x 2 ) 2 .

By applying the Leibniz’s law and Young inequality, we deduce that for all t ∈ [ 0 , T ) ,

(4.4) ‖ ∇ ( ∣ ψ ∣ p − 2 ψ ) ‖ L x p p − 1 ≤ C ‖ ∣ ψ ∣ p − 2 ∣ ∇ ψ ∣ ‖ L x p p − 1 ≤ C ‖ ψ ‖ L x p p − 2 ‖ ∇ ψ ‖ L x p ≤ C ( 1 + ‖ ∇ ψ ‖ L x 2 ) 2 ( p − 2 ) p ‖ ∇ ψ ‖ L x p .

Now, by injecting (4.3) and (4.4) into the last term in (4.2), we obtain that for all t ∈ [ 0 , T ) ,

(4.5) ‖ ∇ ( ∣ ψ ∣ p − 2 ψ ) ‖ L t 4 p ( 4 − D ) p + 2 D ( ( t , τ ) ; L x p p − 1 ) ≤ C ( 1 + ‖ ∇ ψ ‖ L t ∞ ( ( t , τ ) ; L x 2 ) ) 2 ( p − 2 ) p ∫ t τ 1 ⋅ ‖ ∇ ψ ( τ ) ‖ L x p 4 p ( 4 − D ) p + 2 D d τ ( 4 − D ) p + 2 D 4 p ≤ C ( 1 + ‖ ∇ ψ ‖ L t ∞ ( ( t , τ ) ; L x 2 ) ) 2 ( p − 2 ) p ( τ − t ) 2 p − ( p − 2 ) D 2 p ‖ ∇ ψ ‖ L t 4 p ( p − 2 ) D ( ( t , τ ) ; L x p ) ≤ C ( τ − t ) 2 p − ( p − 2 ) D 2 p 1 + ‖ ∇ ψ ‖ L t ∞ ( ( t , τ ) ; L x 2 ) + ‖ ∇ ψ ‖ L t 4 p ( p − 2 ) D ( ( t , τ ) ; L x p ) 3 p − 4 p .

Finally, we denote F t ( τ ) ≔ 1 + ‖ ∇ ψ ‖ L t ∞ ( ( t , τ ) ; L x 2 ) + ‖ ∇ ψ ‖ L t 4 p ( p − 2 ) D ( ( t , τ ) ; L x p ) . Substitute (4.5) into (4.2). There exists a positive constant K 0 > 0 such that

(4.6) F t ( τ ) ≤ K 0 ( 1 + ‖ ∇ ψ ( t ) ‖ L x 2 ) + K 0 ( τ − t ) 2 p − ( p − 2 ) D 2 p F t 3 p − 4 p ( τ ) ,

for any 0 < t < τ < T < + ∞ . Since ψ ≔ ψ ( t , x ) is the blow-up solution to Cauchy problems (1.1) and (1.2) and 0 < t < T < + ∞ is the blow-up time, we see that lim τ → T F t ( τ ) = + ∞ . On the other hand, F t ( τ ) is continuous and nondecreasing function on ( t , T ) , and as τ → t , we have F t ( τ ) → 1 + ‖ ∇ ψ ( t ) ‖ L x 2 . Then, for the constant K 0 > 0 in (4.6), we can find a τ 0 ∈ ( t , T ) satisfying F t ( τ 0 ) = ( K 0 + 1 ) ( 1 + ‖ ∇ ψ ( t ) ‖ L x 2 ) . Now, take τ = τ 0 in (4.6), yields

1 + ‖ ∇ ψ ( t ) ‖ L x 2 = F t ( τ 0 ) − K 0 ( 1 + ‖ ∇ ψ ( t ) ‖ L x 2 ) ≤ K 0 ( τ 0 − t ) 2 p − ( p − 2 ) D 2 p ( K 0 + 1 ) 3 p − 4 p ( 1 + ‖ ∇ ψ ( t ) ‖ L x 2 ) 3 p − 4 p ≤ ( K 0 + 1 ) 4 p − 4 p ( T − t ) 2 p − ( p − 2 ) D 2 p ( 1 + ‖ ∇ ψ ( t ) ‖ L x 2 ) 3 p − 4 p ,

is true for all 0 < t < T < + ∞ and so (1.11) is true.□

To prove (ii) in Theorem 1.3, we prove the following refined compactness results, which is the applications of the profile decomposition of bounded sequences in H ˙ s c ∩ H ˙ 1 and the new sharp Gagliardo-Nirenberg inequality (1.9).

Proposition 4.1

Let D ≥ 2 , s c = D 2 − 2 p − 2 , and 2 + 4 D < p < 2 * . If { v n } n = 1 + ∞ be a bounded sequences in H ˙ s c ∩ H ˙ 1 such that

(4.7) limsup n → + ∞ ‖ ∇ v n ‖ 2 ≤ M a n d limsup n → + ∞ ‖ v n ‖ p ≥ m ,

then, there exists a sequence { x n } n = 1 + ∞ of R D , and V ( x ) ∈ H ˙ s c ∩ H ˙ 1 such that up to a subsequence

(4.8) v n ( x + x n ) ⇀ V ( x ) weakly i n H ˙ s c ∩ H ˙ 1 .

with ‖ V ‖ H ˙ s c p − 2 ≥ 4 ‖ W ‖ H ˙ s c p − 2 m p p M 2 and W is the ground state solution of equation (1.8).

Proof

By extracting a subsequence, we may replace limsup in the assumption in Proposition 4.1 by lim . According to the profile decomposition in Proposition 1.1, the sequence { v n } n = 1 + ∞ can be written up to a subsequence as follows:

(4.9) v n ( x ) = ∑ j = 1 l V j ( x − x n j ) + v n l ( x ) .

And (1.5), (1.6) and (1.7) are true. Moreover, by (1.5), we deduce that

(4.10) m p ≤ limsup n → + ∞ ∑ j = 1 l V j ( x − x n j ) p + ‖ v n l ( x ) ‖ p p ≤ ∑ j = 1 l V j ( x − x n j ) p p as n → + ∞ , l → + ∞ .

It follows from (1.7) and (4.10) that m p ≤ ∑ j = 1 + ∞ ‖ V j ‖ p p + o ( 1 ) , where o ( 1 ) = o n ( 1 ) → 0 as n → + ∞ . By using the new Gagliardo-Nirenberg inequality (1.9), we deduce that

(4.11) m p ≤ ∑ j = 1 + ∞ ‖ V j ‖ p p + o ( 1 ) ≤ p 4 ‖ Q ‖ H ˙ s c p − 2 sup { ‖ V j ‖ H ˙ s c p − 2 , j ≥ 1 } ∑ j = 1 + ∞ ‖ ∇ V j ‖ 2 2 + o ( 1 ) .

From the hypothesis (4.7) and (1.6), we have ∑ j = 1 + ∞ ‖ ∇ V j ‖ 2 2 ≤ limsup n → ∞ ‖ ∇ v n ‖ 2 2 ≤ M 2 . Since the series ∑ j = 1 + ∞ ‖ V j ‖ H ˙ s c 2 is convergent, the supremum of { ‖ V j ‖ H ˙ s c p − 2 ; j ≥ 1 } is attained. Thus, by injecting these estimates into (4.11), we deduce that ‖ V j 0 ‖ H ˙ s c p − 2 = sup j ≥ 1 ‖ V j ‖ p c p − 2 ≥ 4 ‖ Q ‖ H ˙ s c p − 2 m p p M 2 . This implies that there exists a V j 0 ∈ H ˙ s c ∩ H ˙ 1 satisfying the lower bound. Next, we will prove that

(4.12) v n ( x + x n j 0 ) ⇀ V j 0 weakly in H ˙ s c ∩ H ˙ 1 ,

which implies the sequence { x n j 0 } n = 1 + ∞ and the function V j 0 now fulfill the condition of Proposition 4.1. Indeed, by a change of variables in (4.9), we have

(4.13) v n ( x + x n j 0 ) = V j 0 ( x ) + ∑ j ≠ j 0 V j ( x − x n j + x n j 0 ) + v ˜ n l ( x ) ,

where v ˜ n l ( x ) = v n l ( x + x n j 0 ) . By applying the pairwise orthogonality of the family { x n j } n = 1 + ∞ to (4.13), we see that as n → + ∞ , V j ( x − x n j + x n j 0 ) ⇀ 0 weakly in H ˙ s c ∩ H ˙ 1 for j ≠ j 0 . Denote v ˜ l to be the weak limit of v ˜ n l and take weak limit in (4.13) as n → + ∞ . We have

(4.14) v n ( x + x n j 0 ) ⇀ V j 0 + v ˜ l weakly in H ˙ s c ∩ H ˙ 1 .

Finally, by using (1.5), we deduce that ‖ v ˜ l ‖ p ≤ limsup n → + ∞ ‖ v ˜ n l ‖ p = limsup n → + ∞ ‖ v n l ‖ p → 0 , as l → + ∞ , and by the uniqueness of weak limits, we obtain: v ˜ l = 0 for l ≥ j 0 . Then, (4.12) is true, so is (4.8). This completes the proof.□

At the end of this section, we shall finish the proof of Theorem 1.3 by the refined compactness result in Proposition 4.1.

Proof of (ii) in Theorem 1.3

Let { t n } n = 1 + ∞ be an arbitrary sequence such that lim n → + ∞ t n = T , where T is the blow-up time. Since ψ ( t ) is the blow-up solutions to Cauchy problems (1.1) and (1.2) at the finite time 0 < T < + ∞ , we have lim n → + ∞ ‖ ∇ ψ ( t n ) ‖ 2 = + ∞ . Let the scaling sequence: ρ n ≔ ‖ ∇ W ‖ 2 1 1 − s c ‖ ∇ ψ ( t n ) ‖ 2 1 1 − s c → 0 , as n → + ∞ , where W is the ground state solution of equation (1.8). We take the scaling transformation V n = ρ n 2 p − 2 ψ ( t n , ρ n x ) . Then, for all n ≥ 1 , we have ‖ ∇ V n ‖ 2 = ρ n 1 − s c ‖ ∇ ψ ( t n ) ‖ 2 = ‖ ∇ W ‖ 2 and ‖ V n ‖ H ˙ s c = ‖ ψ ( t n ) ‖ H ˙ s c < + ∞ . Thus, { V n } n = 1 + ∞ is a bounded sequence in H ˙ s c ∩ H ˙ 1 . Substitute V n = ρ n 2 p − 2 ψ ( t n , ρ n x ) into E ( ψ ) .

1 2 ‖ ∇ ψ ( t n ) ‖ 2 2 − 2 p ‖ ψ ( t n ) ‖ p p = ρ n 2 ( 1 − s c ) E ( ψ 0 ) − ω 2 2 ∫ ∣ x ∣ 2 ∣ ψ ( t n , x ) ∣ 2 d x ,

which is nonpositive as n → + ∞ . Moreover, we obtain 1 2 ‖ ∇ W ‖ 2 2 = 1 2 ‖ ∇ ψ ( t n ) ‖ 2 2 ≤ 2 p ‖ ψ ( t n ) ‖ p p . Now, by applying Proposition 4.1 to the sequence { V n } n = 1 + ∞ with M 2 = ‖ ∇ W ‖ 2 2 , m p = p ‖ ∇ W ‖ 2 2 4 , there exists a subsequence { V n } n = 1 + ∞ and { y n } n = 1 + ∞ ⊂ R D such that

(4.15) V n ( x + y n ) ⇀ V ( x ) weakly in H s c with ‖ V ‖ H ˙ s c ≥ ‖ W ‖ H ˙ s c ,

where W is a ground state solution of equation (1.8), which implies ρ n 2 p − 2 ( − Δ ) s c 2 ψ ( t n , ρ n x ) ⇀ ( − Δ ) s c 2 V ( x ) weakly in L 2 . Thus, for every K > 0 ,

(4.16) liminf n → + ∞ ∫ ∣ x ∣ ≤ K ρ n 4 p − 2 ( − Δ ) s c 2 ψ ( t n , ρ n ( x + y n ) ) 2 d x ≥ ∫ ∣ x ∣ ≤ K ( − Δ ) s c 2 V 2 d x .

By (1.11) and the definition of ρ n , we see that K ρ n < ( T − t ) 1 2 − , where we employing 1 1 − s c ⋅ 2 p − ( p − 2 ) D 4 ( p − 2 ) = 1 2 . After rescaling transform: z = ρ n ( x + y n ) in (4.16), we deduce that

liminf n → + ∞ sup y ∈ R D ∫ ∣ z − y ∣ ≤ ( T − t ) 1 2 − ( − Δ ) s c 2 ψ ( t n , z ) 2 d z ≥ ∫ ∣ x ∣ ≤ K ( − Δ ) s c 2 V 2 d x ,

Since { t n } n = 1 + ∞ is an arbitrary sequence approaching T , by letting K → + ∞ , we see that

(4.17) liminf t → T sup y ∈ R D ∫ ∣ x − y ∣ ≤ ( T − t ) 1 2 − ( − Δ ) s c 2 ψ ( t , x ) 2 d x ≥ ∫ ( − Δ ) s c 2 W 2 d x .

Furthermore, because the function y ↦ ∫ ∣ x − y ∣ ≤ ( T − t ) 1 2 − ∣ ψ ( t , x ) ∣ 2 d x is continuous and goes to 0 at infinity for every t ∈ [ 0 , T ) , there exists y ( t ) ∈ R D such that

sup y ∈ R D ∫ ∣ x − y ∣ ≤ ( T − t ) 1 2 − ∣ ψ ( t , x ) ∣ 2 d x = ∫ ∣ x − y ( t ) ∣ ≤ ( T − t ) 1 2 − ∣ ψ ( t , x ) ∣ 2 d x .

Then, we can obtain (1.12) by injecting the aforementioned identity into (4.17).□

5 Dirac function concentration

In this section, we will continuously investigate the dynamical properties of blow-up solutions to Cauchy problem (1.1) and (1.2) in the L 2 -critical case: p = 2 + 4 D . Zhu et al. [60] proved the following limiting result.

Lemma 5.1

Let D ≥ 2 and ψ 0 ∈ Σ . Suppose that M ( ψ 0 ) = ‖ Q ‖ 2 2 and ψ ( t , x ) ∈ C ( [ 0 , T ) ; Σ ) is the blow-up solution of Cauchy problems (1.1) and (1.2). Then there exist y ( t ) ∈ R D and γ ( t ) ∈ R such that

(5.1) λ D 2 ( t ) ψ ( t , λ ( t ) ( x + y ( t ) ) ) e i γ ( t ) → Q ( x ) strongly i n H 1 a s t → T ,

where λ ( t ) = ‖ ∇ Q ‖ 2 ‖ ∇ ψ ( t ) ‖ 2 and Q ( x ) is the unique positive radially symmetric solution of equation (1.14).

We remark that λ D 2 ( t ) ψ ( t , λ ( t ) ( x + y ( t ) ) ) e i γ ( t ) → Q ( x ) strongly in L 2 , as t → T . By taking z = λ ( t ) ( x + y ( t ) ) , from λ ( t ) → 0 as t → T , we deduce that

(5.2) liminf t → T ∫ ∣ z − y ( t ) ∣ < K ∣ ψ ( t , z ) ∣ 2 d z ≥ ∫ ∣ Q ( x ) ∣ 2 d x

is true for all K > 0 . This reveals that all mass of blow-up solutions concentrates at y ( t ) ∈ R D .

Here, by exploring the variational characteristic of (1.14) and the order of infinitesimal term J ( t ) ≔ ∫ ∣ x ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x as t goes to the blow-up time T , we obtain a delicate profile of blow-up solutions to Cauchy problems (1.1) and (1.2), as follows.

Theorem 5.2

Let D ≥ 2 and p = 2 + 4 D . Suppose that ψ 0 ∈ Σ , M ( ψ 0 ) = ‖ Q ‖ 2 2 , and ψ ( t , x ) ∈ C ( [ 0 , T ) ; Σ ) is the blow-up solution of Cauchy problems (1.1) and (1.2). Then, we have the followings:

  1. (5.3) ∣ ψ ( t , x ) ∣ 2 → ‖ Q ( x ) ‖ 2 2 δ x = y 1 , a s t → T ,

    where Q ( x ) is the unique positive radially symmetric solution of equation (1.14), δ x = y 1 is a Dirac function at the point y 1 , and y 1 ∈ R D is a fixed vector defined in Lemma 2.3.

  2. There exists C > 0 such that

    (5.4) ‖ ∇ ψ ( t , x ) ‖ 2 ≥ C T − t for a l l t ∈ [ 0 , T ) .

Proof

(i) It follows from (5.1) that

λ D ( t ) ∣ ψ ( t , λ ( t ) ( x + y ( t ) ) ) ∣ 2 → ∣ Q ( x ) ∣ 2 strongly in L 1 as t → T .

By the hypothesis: ‖ ∇ ψ ( t ) ‖ 2 → + ∞ as t → T , we deduce that

(5.5) ∣ ψ ( t , x + y ( t ) ) ∣ 2 → ‖ Q ( x ) ‖ 2 2 δ x = 0 as t → T .

Here, we use the definition of Dirac function δ x = 0 to obtain that (5.5) is true in distribution. From (2.5) in Lemma 2.3, for all t ∈ [ 0 , T ) , there exists a constant C 0 > 0 such that ∫ ∣ x ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x ≤ C 0 , and then

∣ y ( t ) ∣ 2 ∫ ∣ ψ ( t , x + y ( t ) ) ∣ 2 d x ≤ C ∫ ∣ y ( t ) ∣ 2 ∣ ψ ( t , x + y ( t ) ) ∣ 2 d x + C ∫ ∣ x ∣ 2 ∣ ψ ( t , x + y ( t ) ) ∣ 2 d x ≤ ∫ ∣ x + y ( t ) ∣ 2 ∣ ψ ( t , x + y ( t ) ) ∣ 2 d x ≤ C 0 ,

which implies that lim t → T ¯ ∣ y ( t ) ∣ ≤ C 0 ∕ C ‖ Q ( x ) ‖ 2 , and there exists an M 0 > 0 such that ∣ y ( t ) ∣ ≤ M 0 for all t ∈ [ 0 , T ) . Moreover, we claim that ∀ M > M 0

(5.6) lim t → T ∫ B ( 0 , M ) ∣ ψ ( t , x ) ∣ 2 x d x − ∫ ∣ Q ( x ) ∣ 2 d x y ( t ) = 0 ,

where B ( 0 , M ) = { x ∈ R D ∣ ∣ x ∣ ≤ M } . Indeed, we decompose

(5.7) ∫ B ( 0 , M ) ∣ ψ ( t , x ) ∣ 2 x d x = ∫ B ( 0 , M ) ∣ ψ ( t , x ) ∣ 2 ( x − y ( t ) ) d x + ∫ B ( 0 , M ) ∣ ψ ( t , x ) ∣ 2 y ( t ) d x = ∫ B ( − y ( t ) , M ) ∣ ψ ( t , y + y ( t ) ) ∣ 2 y d y + ∫ B ( − y ( t ) , M ) ∣ ψ ( t , y + y ( t ) ) ∣ 2 y ( t ) d y = ∫ B ( 0 , η ) ∣ ψ ( t , y + y ( t ) ) ∣ 2 y d y + ∫ B ( − y ( t ) , M ) \ B ( 0 , η ) ∣ ψ ( t , y + y ( t ) ) ∣ 2 y d y + ∫ B ( − y ( t ) , M ) ∣ ψ ( t , y + y ( t ) ) ∣ 2 y ( t ) d y .

Due to M > M 0 , there exists η > 0 (small enough), B ( 0 , η ) ⊂ B ( − y ( t ) , M ) . From (5.5) and (5.7), we can complete the proof of Claim (5.6). By (2.7), we deduce that for all t ∈ [ 0 , T ) , ∫ ∣ x ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x ≤ C 0 and then ∣ ∫ ∣ x ∣ ≥ M ∣ ψ ( t , x ) ∣ 2 x d x ∣ ≤ C 0 M . By combining this with (5.6), we obtain

(5.8) lim t → T ∫ ∣ ψ ( t , x ) ∣ 2 x d x − ( ∫ ∣ Q ( x ) ∣ 2 d x ) y ( t ) = 0 .

By comparing (2.6) with (5.8), we have lim t → T y ( t ) = − y 1 . Then, (5.5) becomes ∣ ψ ( t , x − y 1 ) ∣ 2 → ‖ Q ( x ) ‖ 2 2 δ x = 0 as t → T . This implies that (5.3) is true.

(ii) First, for any real number α ∈ R , we consider the new energy E ( ψ ( t ) e i α ∣ x + y 1 ∣ 2 ) . It follows from the assumption M ( ψ 0 ) = M ( ψ ( t ) ) = ‖ Q ‖ 2 2 and the sharp Gagliardo-Nirenberg inequality (1.10) that ‖ ψ ( t ) e i α ∣ x + y 1 ∣ 2 ‖ 2 = ‖ Q ‖ 2 and

E ( ψ ( t ) e i α ∣ x + y 1 ∣ 2 ) ≥ 1 2 ‖ ∇ ( ψ ( t ) e i α ∣ x + y 1 ∣ 2 ) ‖ 2 2 + ω 2 2 ‖ x ψ ( t ) ‖ 2 2 − 1 2 ‖ ψ ( t ) ‖ 2 ‖ Q ‖ 2 4 D ‖ ∇ ( ψ ( t ) e i α ∣ x + y 1 ∣ 2 ) ‖ 2 2 ≥ 0

for any real number α ∈ R . On the other hand, we can expand E ( ψ ( t ) e i α ∣ x + y 1 ∣ 2 ) and obtain

E ( ψ ( t ) e i α ∣ x + y 1 ∣ 2 ) = 2 α 2 ∫ ∣ x + y 1 ∣ 2 ∣ ψ ( t ) ∣ 2 d x − 2 α ℑ ∫ ( x + y 1 ) ∇ ψ ( t ) ¯ ψ ( t ) d x + E ( ψ 0 )

is nonnegative for any real number α ∈ R . Yield, we deduce that ∀ t ∈ [ 0 , T )

(5.9) ℑ ∫ ( x + y 1 ) ∇ ψ ( t ) ¯ ψ ( t ) d x 2 ≤ 2 E ( ψ 0 ) ∫ ∣ x + y 1 ∣ 2 ∣ ψ ( t ) ∣ 2 d x .

Next, let G ( t ) = ∫ ∣ x + y 1 ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x , where y 1 ∈ R D is defined in Lemma 2.3. We claim that

(5.10) lim t → T G ( t ) = lim t → T ∫ ∣ x + y 1 ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x = 0 ,

where T is the blow-up time. By comparing (2.6) with (5.8), we see that lim t → T y ( t ) = − y 1 . Next, we shall prove Claim (5.10) by the following two cases.

Case a: y 1 = 0 . For any ε > 0 , we decompose that for all t ∈ [ 0 , T )

∫ ∣ x ∣ 2 ∣ ψ ( t ) ∣ 2 d x = ∫ ∣ x ∣ ≤ A 1 ∣ x ∣ 2 ∣ ψ ( t ) ∣ 2 d x + ∫ A 1 < ∣ x ∣ ≤ A 2 ∣ x ∣ 2 ∣ ψ ( t ) ∣ 2 d x + ∫ ∣ x ∣ > A 2 ∣ x ∣ 2 ∣ ψ ( t ) ∣ 2 d x = I + I I + I I I ,

where A 1 and A 2 are two positive constants, which are determined below. For the first term I , by using ‖ ψ 0 ‖ 2 = ‖ Q ‖ 2 and taking A 1 ≤ ε 3 ‖ Q ‖ 2 , we deduce that, for all t ∈ [ 0 , T ) ,

(5.11) I = ∫ ∣ x ∣ ≤ A 1 ∣ x ∣ 2 ∣ ψ ( t ) ∣ 2 d x ≤ A 1 2 ‖ Q ‖ 2 2 ≤ ε 3 .

For I I I , it follows from ∫ ∣ x ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x ≤ C 0 in (2.5) (uniformly bounded) that there exists A 2 > 0 sufficiently large such that for all t ∈ [ 0 , T )

(5.12) I I I = ∫ ∣ x ∣ > A 2 ∣ x ∣ 2 ∣ ψ ( t ) ∣ 2 d x ≤ ε 3 .

For the second term I I , for the A 2 determined in (5.12), we deduce that for all t ∈ [ 0 , T ) ,

(5.13) I I ≤ A 2 2 ∫ ∣ x ∣ ≤ A 2 ∣ ψ ( t ) ∣ 2 d x − ∫ ∣ x ∣ ≤ A 1 ∣ ψ ( t ) ∣ 2 d x < ε 3 .

Here, by employing (5.7) and lim t → T y ( t ) = 0 , we remark that there exists τ 0 > 0 , when ∣ t − T ∣ < τ 0

∫ ∣ x ∣ ≤ A 1 ∣ ψ ( t ) ∣ 2 d x ≥ ∫ ∣ Q ∣ 2 d x − ε 3 A 2 2 .

Collecting (5.11), (5.12), and (5.13), we derive that for any ε > 0 , there exists τ 0 > 0 , when ∣ t − T ∣ < τ 0 , ∫ ∣ x ∣ 2 ∣ ψ ( t ) ∣ 2 d x < ε . That is, (5.10) is true.

Case b: y 1 ≠ 0 . For any ε > 0 , we have the following decomposition for all t ∈ [ 0 , T )

∫ ∣ x + y 1 ∣ 2 ∣ ψ ( t ) ∣ 2 d x = ∫ ∣ x + y 1 ∣ ≤ A 1 ∣ x + y 1 ∣ 2 ∣ ψ ( t ) ∣ 2 d x + ∫ A 1 < ∣ x + y 1 ∣ ≤ A 2 ∣ x + y 1 ∣ 2 ∣ ψ ( t ) ∣ 2 d x + ∫ ∣ x + y 1 ∣ > A 2 ∣ x + y 1 ∣ 2 ∣ ψ ( t ) ∣ 2 d x = I + I I + I I I .

For I and I I I , we can estimate them as in Case a. For sufficiently small A 1 and sufficiently large A 2 , we deduce that for all t ∈ [ 0 , T ) ,

(5.14) I ≤ ε 3 and I I I ≤ ε 3 .

For the second term I I , since lim t → T y ( t ) = − y 1 , there exists η > 0 sufficiently small such that B ( − y 1 , A 1 ) ⊃ B ( y ( t ) , A 1 2 ) ∪ B ( y ( t ) + y 1 , η ) . Then, for the A 2 determined in (5.14), we have

(5.15) I I ≤ A 2 2 ∫ A 1 < ∣ x + y 1 ∣ ≤ A 2 ∣ ψ ( t ) ∣ 2 d x ≤ A 2 2 ∫ ∣ x + y 1 ∣ < A 2 ∣ ψ ( t ) ∣ 2 d x − ∫ ∣ x − y ( t ) ∣ < A 1 2 ∣ ψ ( t ) ∣ 2 d x − ∫ ∣ y ( t ) + y 1 ∣ < η ∣ ψ ( t ) ∣ 2 d x ,

By using to (5.7) for y ( t ) ≠ 0 , we remark that there exists τ 0 > 0 , when ∣ t − T ∣ < τ 0

(5.16) ∫ ∣ x − y ( t ) ∣ ≤ A 1 2 ∣ ψ ( t ) ∣ 2 d x ≥ ∫ ∣ Q ∣ 2 d x − ε 6 A 2 2 .

From the continuity of integration and lim t → T y ( t ) = − y 1 , we deduce that when ∣ t − T ∣ < τ 0 ,

(5.17) ∫ ∣ y ( t ) + y 1 ∣ < η ∣ ψ ( t ) ∣ 2 d x ≤ ε 6 A 2 2 .

Substitute (5.16) and (5.17) into (5.15). We have

I I ≤ A 2 2 ∫ ∣ Q ∣ 2 d x − ∫ ∣ Q ∣ 2 d x + ε 6 A 2 2 + ε 6 A 2 2 = ε 3 .

By combining the aforementioned estimations with (5.14), we can deduce that (5.10) is true in this case.

Finally, we return the proof of (ii) in Theorem 5.2. After some computation, we deduce that

(5.18) G ′ ( t ) = 2 ℑ ∫ ( x + y 1 ) ∇ ψ ( t ) ψ ( t ) ¯ d x and G ″ ( t ) = 4 E ( ψ 0 ) − 4 ω 2 G ( t ) .

By combining (5.9) with (5.10), we deduce that lim t → T G ′ ( t ) = lim t → T G ( t ) = 0 . Then, by (5.18) and the classical analytic identity, we have

G ( t ) = G ( T ) + G ′ ( T ) ( t − T ) + ∫ t T G ″ ( T + t − s ) ( T − s ) d s ≤ 2 E ( ψ 0 ) ( T − t ) 2 .

Therefore, by the inequality: ‖ Q ‖ 2 2 = ‖ ψ ( t ) ‖ 2 2 = 1 D ∫ R D ∇ ⋅ ( x + y 1 ) ∣ ψ ( t ) ∣ 2 d x ≤ 2 D G ( t ) ‖ ∇ ψ ( t ) ‖ 2 , we deduce that

‖ ∇ ψ ( t ) ‖ 2 ≥ D ‖ Q ‖ 2 2 2 2 E ( ψ 0 ) ( T − t ) .

Then, (5.4) holds for all 0 < t < T . This completes the proof.□

Remark 5.3

Theorem 5.2 parallels the existing results in the study by Merle [39] for the minimal mass blow-up solutions to the canonical nonlinear Schrödinger equation (1.3). But, as we will see, our argument is by exploring the variational characteristic of (1.14) and the order of infinitesimal term J ( t ) ≔ ∫ ∣ x ∣ 2 ∣ ψ ( t , x ) ∣ 2 d x as the time t goes to the blow-up time T , and it is a very simple and direct way to study the delicate profile of blow-up solutions for the nonlinear Schrödinger equation with a potential, which has potential applications for the nonlinear Schrödinger equations without a translation from themselves to the canonical nonlinear Schrödinger equation (1.3). Of course, these results rely in an essential way on those existing results.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (Grant No. 12071323) and the Sichuan Sciences and Technology Program (Grant No. 2022NSFSC1852).

  2. Conflict of interest: The authors state no conflict of interest.

References

[1] P. Antonelli, P. Markowich, R. Obermeyer, J. Sierra, and C. Sparber, On a dissipative Gross-Pitaevskii-type model for exciton-polariton condensates, Nonlinearity 32 (2019), 4317–4345. 10.1088/1361-6544/ab2bc1Search in Google Scholar

[2] T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math. 53 (2013), 629–672. 10.1215/21562261-2265914Search in Google Scholar

[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269 (1995), 198–201. 10.1126/science.269.5221.198Search in Google Scholar PubMed

[4] M. Bai, J. Zhang, and S. Zhu, Small solitons and multisolitons in the generalized Davey-Stewartson system, Adv. Nonlinear Anal. 12 (2023), 1–41. 10.1515/anona-2022-0266Search in Google Scholar

[5] W. Bao and Y. Cai, Mathematical theory and Numerical methods for Bose-Einstein condensation, Kinetic Related Models 6 (2013), 1–135. 10.3934/krm.2013.6.1Search in Google Scholar

[6] W. Bao, N. B. Abdallah, and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal. 44 (2012), 1713–1741. 10.1137/110850451Search in Google Scholar

[7] J. Bellazzini and L. Jeanjean, On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal. 48 (2016), 2028–2058. 10.1137/15M1015959Search in Google Scholar

[8] J. Bellazzini and L. Forcella, Asymptotic dynamic for Dipolar Quantum Gases below the ground state energy threshold, J. Funct. Anal. 277 (2019), 1958–1998. 10.1016/j.jfa.2019.04.005Search in Google Scholar

[9] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number, Phys. Rev. Lett. 78 (1997), 985–989. 10.1103/PhysRevLett.78.985Search in Google Scholar

[10] C. Brennecke, The low energy spectrum of trapped bosons in the Gross-Pitaevskii regime, J. Math. Phys. 63 (2022), 051101. 10.1063/5.0089630Search in Google Scholar

[11] R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Mod. Meth. Appl. Sci. 12 (2002), 1513–1523. 10.1142/S0218202502002215Search in Google Scholar

[12] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, AMS, NYU, CIMS, 2003. 10.1090/cln/010Search in Google Scholar

[13] G. G. Chen and J. Zhang, Sharp threshold of global existence for nonlinear Gross-Pitaevskii equation in RN, IMA J. Appl. Math. 71 (2006), 232–240. 10.1093/imamat/hxh095Search in Google Scholar

[14] Y. Cho, T. Ozawa, and C. B. Wang, Finite time blowup for the fourth-order NLS, Bull. Korean Math. Soc. 53 (2016), 615–640. 10.4134/BKMS.2016.53.2.615Search in Google Scholar

[15] Y. B. Deng, Y. J. Guo, and L. Lu, On the collapse and concentration of Bose-Einstein condensates with inhomogeneous attractive interactions, Calc. Var. Partial Differential Equations 54 (2015), 99–118. 10.1007/s00526-014-0779-9Search in Google Scholar

[16] G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse, Springer, New York, 2015. 10.1007/978-3-319-12748-4Search in Google Scholar

[17] D. Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J. 47 (1980), 559–600. 10.1215/S0012-7094-80-04734-1Search in Google Scholar

[18] D. Fujiwara and T. Ozawa, Lifespan of strong solutions to the periodic nonlinear Schrödinger equation without gauge invariance, J. Evol. Equ. 17 (2017), 1023–1030. 10.1007/s00028-016-0364-0Search in Google Scholar

[19] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Bose-Einstein condensation of chromium, Phys. Rev. Lett. 94 (2005), 160401. 10.1103/PhysRevLett.94.160401Search in Google Scholar PubMed

[20] Y. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), 141–156. 10.1007/s11005-013-0667-9Search in Google Scholar

[21] T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not. 46 (2005), 2815–2828. 10.1155/IMRN.2005.2815Search in Google Scholar

[22] J. Holmer, R. Platte, and S. Roudenko, Blow-up criteria for the 3D cubic nonlinear Schrödinger equation, Nonlinearity 23 (2010), 977–1030. 10.1088/0951-7715/23/4/011Search in Google Scholar

[23] Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal. 15 (2016), no. 5, 1571–1601. 10.3934/cpaa.2016003Search in Google Scholar

[24] A. D. Ionescu and B. Pausader, Global well-posedness of the energy-critical defocusing NLS on R×T3, Commun. Math. Phys. 312 (2012), no. 3, 781–831. 10.1007/s00220-012-1474-3Search in Google Scholar

[25] A. D. Ionescu and B. Pausader, The energy-critical defocusing NLS on T3, Duke Math. J. 161 (2012), no. 8, 1581–1612. 10.1215/00127094-1593335Search in Google Scholar

[26] A. D. Ionescu, B. Pausader, and G. Staffilani, On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE 5 (2012), no. 4, 705–746. 10.2140/apde.2012.5.705Search in Google Scholar

[27] C. Jao, The energy-critical quantum harmonic oscillator, Commun. Partial Differ. Equ. 41 (2016), no. 1, 79–133. 10.1080/03605302.2015.1095767Search in Google Scholar

[28] 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), 645–675. 10.1007/s00222-006-0011-4Search in Google Scholar

[29] R. Killip, T. Oh, O. Pocovnicu, and M. Visan, Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R3, Arch. Ration. Mech. Anal. 225 (2017), no. 1, 469–548. 10.1007/s00205-017-1109-0Search in Google Scholar

[30] R. Killip, M. Visan, and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, Am. J. Math. 138 (2016), no. 5, 1193–1346. 10.1353/ajm.2016.0039Search in Google Scholar

[31] M. K. Kwong, Uniqueness of positive solutions of Δu−u+up=0 in Rn, Arch. Rational. Mech. Anal. 105 (1989), 243–266. 10.1007/BF00251502Search in Google Scholar

[32] X. G. Li and J. Zhang, Limit behavior of blow-up solutions for critical nonlinear Schrödinger equation with harmonic potential, Differential Integral Equations 19 (2006), 761–771. 10.57262/die/1356050348Search in Google Scholar

[33] G. D. Li, Y. Y. Li, and C. L. Tang, Infinitely many radial and non-radial sign-changing solutions for Schrödinger equations, Adv. Nonlinear Anal. 11 (2022), 907–920. 10.1515/anona-2021-0221Search in Google Scholar

[34] J. J. Liu and C. Ji, Concentration results for a magnetic Schrödinger-Poisson system with critical growth, Adv. Nonlinear Anal. 10 (2021), 775–798. 10.1515/anona-2020-0159Search in Google Scholar

[35] B. Liu, L. Ma, and J. Wang, Blow up threshold for the Gross-Pitaevskii system with trapped dipolar quantum gases, ZAMM-Z. Angew. Math. Mech. 96 (2016), 344–360. 10.1002/zamm.201400189Search in Google Scholar

[36] M. Lu, N. Q. Burdick, S.-H. Youn, and B. L. Lev, A strongly dipolar Bose-Einstein condensate of dysprosium, Phys. Rev. Lett. 107 (2011), 190401. 10.1103/PhysRevLett.107.190401Search in Google Scholar PubMed

[37] L. Ma and B.-W. Schulze, Blow-up theory for the coupled L2-critical nonlinear Schrödinger system in the plane, Milan J. Math. 78 (2010), 591–601. 10.1007/s00032-010-0131-6Search in Google Scholar

[38] L. Ma and C. Pei, The threshold for the focusing Gross-Pitaevskii equation with trapped dipolar quantum gases, J. Math. Anal. Appl. 381 (2011), 240–246. 10.1016/j.jmaa.2011.02.031Search in Google Scholar

[39] F. Merle, Determination of blow-up solutionswith minimal mass for nonlinear Schrödinger equations withcritical power, Duke Math. J. 69 (1993), no. 2, 427–454. 10.1215/S0012-7094-93-06919-0Search in Google Scholar

[40] F. Merle and P. Raphaël, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math. 16 (2005), 157–222. 10.4007/annals.2005.161.157Search in Google Scholar

[41] F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the L2-critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), 37–90. 10.1090/S0894-0347-05-00499-6Search in Google Scholar

[42] F. Merle and Y. Tsutsumi, L2 concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations 84 (1990), 205–214. 10.1016/0022-0396(90)90075-ZSearch in Google Scholar

[43] Y. G. Oh, Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equations with potentials, J. Differential Equations 81 (1989), 255–274. 10.1016/0022-0396(89)90123-XSearch in Google Scholar

[44] J. J. Pan and J. Zhang, Blow-up solutions with minimal mass for nonlinear Schrödinger equation with variable potential, Adv. Nonlinear Anal. 11 (2022), 58–71. 10.1515/anona-2020-0185Search in Google Scholar

[45] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291. 10.1007/BF00946631Search in Google Scholar

[46] G. Rozenblum, Discrete spectrum of zero order pseudodifferential operators, Opuscula Math. 43 (2023), 247–268. 10.7494/OpMath.2023.43.2.247Search in Google Scholar

[47] T. Saanouni, A note on coupled nonlinear Schroödinger equations, Adv. Nonlinear Anal. 3 (2014), 247–269. 10.1515/anona-2014-0015Search in Google Scholar

[48] C. A. Sackett, J. M. Gerton, M. Welling, and R. G. Hulet, Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions, Phys. Rev. Lett. 82 (1999), 876–879. 10.1103/PhysRevLett.82.876Search in Google Scholar

[49] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55 (1977), 149–162. 10.1007/BF01626517Search in Google Scholar

[50] C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse, Springer, New York, 1999. Search in Google Scholar

[51] L. Tavares and J. Sousa, Solutions for a nonhomogeneous p & q-Laplacian problem via variational methods and sub-supersolution technique, Opuscula Math. 43 (2023), 603–613. 10.7494/OpMath.2023.43.4.603Search in Google Scholar

[52] Y. Tsutsumi, Rate of L2 concentration of blow up solutions for thenonlinear Schrödinger equation with critical power, Nonlinear Anal. 15 (1990), 719–724. 10.1016/0362-546X(90)90088-XSearch in Google Scholar

[53] M. Wadati and T. Tsurumi, Critical number of atoms for the magnetically trapped Bose-Einstein condensate with negative s-wave scattering length, Phys. Lett. A 247 (1998), 287–293. 10.1016/S0375-9601(98)00583-0Search in Google Scholar

[54] C. Wang and J. Sun, Normalized solutions for the p-Laplacian equation with a trapping potential, Adv. Nonlinear Anal. 12 (2023), 1–14. 10.1515/anona-2022-0291Search in Google Scholar

[55] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567–576. 10.1007/BF01208265Search in Google Scholar

[56] M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11 (1986), 545–565. 10.1080/03605308608820435Search in Google Scholar

[57] J. Zhang, Stability of attractive Bose-Einstein condensate, J. Statist. Phys. 101 (2000), 731–746. 10.1023/A:1026437923987Search in Google Scholar

[58] J. Zhang, Sharp threshold for blowup and global existence innonlinear Schrödinger equation under a harmonic potential, Comm. Partial Differential Equations 30 (2005), 1429–1443. 10.1080/03605300500299539Search in Google Scholar

[59] X. Zhang, B. L. Zhang, and M. Q. Xiang, Ground states for fractional Schroödinger equations involving a critical nonlinearity, Adv. Nonlinear Anal. 5 (2016), 293–314. 10.1515/anona-2015-0133Search in Google Scholar

[60] S. H. Zhu, J. Zhang, and X. G. Li, Limiting profile of blow-up solutions for the Gross-Pitaevskii equation, Science China 52 (2009), 1017–1030. 10.1007/s11425-008-0140-xSearch in Google Scholar

[61] S. H. Zhu, J. Zhang, and H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differential Equations 7 (2010), 187–205. 10.4310/DPDE.2010.v7.n2.a4Search in Google Scholar

[62] S. H. Zhu, Dynamical Properties of Blow-up Solutions to Nonlinear Schrödinger Equations, Ph.D. thesis, Sichuan University, 2011. Search in Google Scholar

[63] S. H. Zhu, On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations 261 (2016), 1506–1531. 10.1016/j.jde.2016.04.007Search in Google Scholar
Received: 2023-05-19
Revised: 2024-02-24
Accepted: 2024-03-01
Published Online: 2024-04-12

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
  3. Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
https://doi.org/10.1515/anona-2024-0007
Keywords for this article
attractive Bose-Einstein condensate; Gross-Pitaveskii equation; blow-up solution; blow-up rate; concentration
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
  3. Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
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 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anona-2024-0007/html
Scroll to top button