Standing waves for Choquard equation with noncritical rotation
-
Yicen Mao
Abstract
We investigate the existence and stability of standing waves with prescribed mass
1 Introduction
In last two decades, the impressive progress of cold atom physics [42] gave a new impetus to the theory of many-bosons systems: the study of Bose-Einstein condensate, we refer to [18]. In the mean-field approximation, Lewin et al. [26,27] described a Bose-Einstein condensate of nonrelativistic particles with rotation of the atoms by the following equation:
where
and
Then, we have the following useful identity:
Putting equations (1.1) and (1.2) into (
The rotation operator
If
If the interaction potential
An appealing aspect of cold atoms experiments is the possibility to tune the value and even the sign of
If one uses
Equation (
As mentioned above, if the interaction potential
The interaction potential
where
In this article, we consider the existence, stability, and mass collapse behavior results for equation (
The basic Sobolev space is
Our working space is defined as follows:
which is a Hilbert space with the inner product and norm
where “
with the norm
Clearly, the space
An important topic on equation (
There are two different approaches to study the existence of solutions to equation (1.3) using variational methods:
the frequency
the frequency
For case (i), the solutions to equation (1.3) can be obtained as the critical points of the following energy functional defined in
This case has attracted much attention in the last years; the existence and regularity of solutions with subcritical exponent was explored by Moroz and Van Schaftingen [38] and Cingolani and Tanaka [9], with critical exponent was studied by Gao and Yang [19], Su and Liu [47], and Giacomoni et al. [20], general nonlinearity was presented by Moroz and Van Schaftingen [39], and with double critical exponents was considered by Li and Ma [29]. For more results, we refer to [17,30,50].
For case (ii), the frequency
on the constraint
with Lagrange multiplier
If
If
Normalized solutions to equation (1.4) can be obtained by searching critical points of the following energy functional (set
on
For
To the best of our knowledge, there is no result concerning the existence, stability, and mass collapse results for equation (
1.1 Main results
We give the definition of a ground-state in the following sense.
Definition 1.1
We say that
First of all, we consider the mass subcritical case
is achieved.
Theorem 1.1
Let
There exists a couple of weak solution
We have
We know
where
There exists
where
Remark 1.1
Our method can be extended to Schrödinger equation (1.4). The results of the mass collapse behavior of standing waves
For
It follows from Theorem 1.1 that
Theorem 1.2
Let
Note that
where
For any fixed
where
Theorem 1.3
Let
where
For any
The lower and upper bounds of
Moreover, we know
where
In Theorem 1.3, we do not know whether
Theorem 1.4
Let
We know that
There exists
Remark 1.2
Our method can be extended to Schrödinger equation (1.2), we omit it. Compared with Luo and Yang [37], they restricted the parameter
We also study the stability of
Theorem 1.5
Let
For
Theorem 1.6
Let
Remark 1.3
For the existence of mountain pass-type solution, our method can be extended to Schrödinger equation (1.2), we omit it. Compared with Luo and Yang [37], they restricted the parameter
The article is organized as follows: in Section 2, we present some preliminaries; in Section 3, we prove the Pohozaev-type identity; in Section 4, we give the proof of Theorems 1.1 and 1.2; in Section 5, we prove Theorem 1.3; in Section 6, we present the proof of Theorems 1.4 and 1.5; in Section 7, we show Theorem 1.6.
2 Preliminaries
Lemma 2.1
[45] The following continuous and compactness embeddings hold
Lemma 2.2
[49, Gagliardo-Nirenberg inequality] Let
where
Lemma 2.3
[49] Let
with equality holding for the functions
Note that
We show the following result, which plays a key role in this article.
Lemma 2.4
If
Proof
It follows from Hölder’s, Cauchy-Schwarz’s, and Young’s inequalities that
for any
Applying equation (2.1) with
To prove the reverse inequality, we use equation (2.1) again
We choose
It follows from
The proof is complete.□
We recall the Hardy-Littlewood-Sobolev inequality as follows:
Lemma 2.5
[34] Let
If
3 Pohozaev-type identity
In this section, we establish the following Pohozaev-type identity:
Lemma 3.1
Let
Then, the Pohozaev-type identity
As a consequence, it satisfies
Proof
Multiplying equation (3.1) by
It is easy to see that
and
and
Calculating the first term on the right-hand side of equation (3.2), we know
which shows
Similarly,
and consequently, we obtain
Putting equations (3.3)–(3.6) into (3.2), one has
Note that
Combining equation (3.7) and (3.8), we obtain
which implies the Pohozaev-type identity
It is easy to check that
and
which induces
Inserting equation (3.10) into equation (3.9), we obtain the desired result.□
4 Proofs of Theorems 1.1 and 1.2
In this section, we prove Theorems 1.1 and 1.2, i.e., the existence, stability, and mass collapse results for mass subcritical case.
Lemma 4.1
Let
Proof
For any
Applying equation (4.1) and Lemma 2.4, for any
Note that
Lemma 4.2
Let
Proof
From equation (4.2), we know that
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1
We divide this proof into five steps.
Step 1. In this step, we show the existence of a normalized ground-state solution to equation (
The compact embedding of Lemma 2.1 shows that
Applying Lemma 2.4, one has that
which shows
and
Let
for all
Step 2. We estimate the upper bound of the Lagrange multiplier
and the corresponding eigenfunctions are given by Hermite functions (the eigenvalue is
Note that
which implies
Step 3. We study the mass collapse behavior of
where
Let
Then, we have
From Step 1 of Theorem 1.1, we know that there exists
Note that
It follows from
Then, we have the inequality (4.5).
It is easy to see that
which gives
Inequality (4.5) implies that
It follows from equation (4.4) that
Then, by using the monotone bounded convergence theorem, we know that the limitation
Then, there exist
Step 4. In this step, we estimate the mass collapse behavior of
where
where
Note that
Thus,
which gives, by using
We show that
This is a contradiction. Hence,
Hence,
It follows from Lemma 2.4, (4.1), (4.3), and
which gives
Thus, one has
and
Combining the above two results, we have
Then, we obtain the desired result
Step 5. In this step, we study the mass collapse behavior of
By using the Sandwich theorem, we know that
This shows
Then
It follows from Lemma 3.1 that
which shows
Combining equations (4.10) and (4.11), we have
We obtain the desired result.□
Lemma 4.3
Let
Proof
Let
for some constant
By Lemma 2.4, equation (4.12), and the energy conservation, for all
Taking
which gives
From above, we know
Clearly, the blow-up cannot occur at finite time.□
Proof of Theorem 1.2
By way of contradiction, we assume that the set of ground-state
and a sequence of times
where
Without loss of generality, we may assume that
and energy conservation
Consequently, the continuity in time implies that
This is a contradiction.□
5 Proof of Theorem 1.3
In this section, we prove Theorem 1.3. We show that
Lemma 5.1
For any
Proof
Let
On the other hand, let
Set
and
For any
which gives
Lemma 5.2
Let
such that for any
where
Proof
First, we show
Clearly, for any
Note that
Then, we have
This shows that for any
For any
where we restrict
From Lemma 5.1, one has
For any
Note that
Moreover, if
We choose
Then, for all
The proof is completed.□
Proof of Theorem 1.3
We divide this proof into three steps.
Step 1. In this step, we show the existence of a local minimizer to equation (
Then,
Hence, we know
Applying Lemma 2.4, one has that
which shows
and
Hence, any minimizing sequence of
Let
which gives
where
Step 2. We estimate the bound of the Lagrange multiplier
Note that
which implies
On the other hand, by applying Lemma 2.4 and equation (4.1), one deduces
which gives
where
Step 3. In this step, we estimate the mass collapse behavior of
where
Note that
which gives
Thus, one has
and
Combining the above two results, we have
Then, we obtain the desired result
6 Proof of Theorems 1.4 and 1.5
In this section, we prove Theorems 1.4 and 1.5.
Proof of Theorem 1.4
We show that
Note that
for some
It follows from equation (6.1),
This shows that
Moreover, let
This shows that
Step 3. We study the asymptotic behavior of
where
Let
and
which gives
Then, we have
From Step 1 of Theorem 1.3, we know that there exists
Note that
It follows from
Then, we have the inequality (6.3).
It is easy to see that
which gives
It follows from equation (6.4) and Theorem 1.3 (ii) that
This shows that
Then, there exist
Step 4. We study the asymptotic behavior of
From equation (4.1) and above result, one has
By using the Sandwich theorem, we know that
This shows
Then,
It follows from Lemma 3.1 that
which shows
Combining equations (6.5) and (6.6), we have
We obtain the desired result.□
Lemma 6.1
Let
Proof
Let
where
It follows from Lemma 2.4 and equations (4.1) and (6.7) that
for some constant
By Lemma 2.4 and equation (4.1), and the mass and energy conservations, for all
where
where
From
We now show that
which gives
From equations (6.7) and (6.8), one deduces that
However,
We are now in a position to prove Theorem 1.5.
Proof of Theorem 1.5
We prove that
and a sequence of times
where
Without loss of generality, we may assume that
and energy conservation
We need to show that
This gives that
This is a contradiction, since the minimizer of
This is a contradiction.□
7 Proof of Theorem 1.6
In this section, we show Theorem 1.6, i.e., the existence of mountain pass-type solution. First, we construct a min-max class by using the result of Theorem 1.3.
Lemma 7.1
Let
Then, there exists a nonempty min-max class
where
Proof
Let
Set
and
and by using equation (7.1),
and by applying equation (7.2),
It follows from equations (7.3) and (7.4) and
These show that
Set
and a min-max value
It follows from
The proof is completed.□
Second, we introduce an auxiliary functional
Define a set of paths
and a min-max value
We need to show that
satisfy
Denote
Lemma 7.2
Let
Then, there exists a pair of
for all
Proof
The proof is similar to that of [23, Lemma 2.3].□
Definition 7.1
We say the
We construct a
Lemma 7.3
Let
Proof
For each
Note that
It follows from Lemma 7.2 that there exists a sequence
for all
It is easy to see that
From
We now show that, for
For any
Note that
For
From equation (3), we obtain
Then,
The proof is completed.□
Lemma 7.4
Let
Proof
The proof is similar to [6, Lemma 3], we omit it.□
Lemma 7.5
Let
up to a subsequence,
up to a subsequence,
up to a subsequence,
Proof
It follows from Lemma 7.4 that
Then, we have
for any
where
Thus, (3) is proved.
In order to prove equation (2), we need to show that
This shows that
Thus, equation (2) is proved.
We show that
which shows
Note that
Moreover, combining equations (2) and (3), we have equation (4).
Finally, by using (2)–(4), we have
According to equation (7.6) and Lemma 2.4, one has
Then, equation (1) is proved.□
We are now in a position to show Theorem 1.6.
Proof of Theorem 1.6
From Lemma 7.3, we know that there exists a
and
The proof is completed.□
-
Funding information: This paper is supported by the National Natural Science Foundation of China (Grant No. 12101006).
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. The authors contributed equally to this study.
-
Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.
-
Data availability statement: No data were used for the research described in the article.
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- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
