Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
-
Fumei Ye
Abstract
This article is devoted to studying the existence of positive solutions to the following fractional Choquard equation:
where
1 Introduction
In this article, we focus our attention on the following nonlinear fractional Choquard equation:
where
In particular, the fractional Laplace operator
where
An important issue concerning nonlinear evolution equations such as (1.1) is to study their standing wave solutions. A stationary wave solution is a solution of the form
In the case of
which has been used to model the dynamics of pseudo-relativistic boson stars in [10,12]. Moreover, the regularity, existence, nonexistence, symmetry, and decay properties of weak solutions for Problem (1.2) have been extensively studied. We refer the reader to [9,14,15,31] and references therein. On the other hand, it is worth noting that, if
Compared to the previous results on
where
and proved that (1.4) has no ground-state solutions. Based on the aforementioned facts, they analyzed the behavior of Palais-Smale sequences and showed a precise estimate of the higher levels of energy. Then, the existence of positive solutions for (1.4) can be obtained. Then, a great attention has been paid to Schrödinger equations or fractional Schrödinger equations in exterior domains. We refer the reader to [1,2,6,7,33] and the related results mentioned there. It is worth noting that there are few works concerning the existence of positive solutions for Choquard equations in exterior domains. Indeed, the corresponding results are presented in [5,8,17,34]. For example, the authors in [8] considered the following nonlinear Choquard equation in exterior domains:
where
where
Mainly motivated by [17,29], we first intend to establish the limit profiles and uniqueness of the radial ground-states for equation (1.6) as
Given this reality, in this article, we consider the relevant issues of Problem (1.7) instead of Problem (1.6). By establishing the convergence properties of Choquard term in (1.7), the following result can be obtained.
Theorem 1.1
Let
According to the aforementioned result, the following Choquard equation in exterior domains can be considered
where
Corollary 1.2
There exist
In what follows, we are particularly interested in the existence of positive solutions for Problem (1.8) with the fractional Hardy-Littlewood-Sobolev critical exponent. In this perspective, it is natural to consider the following problem:
where
To our knowledge, the uniqueness of solution for limit Problem (1.10) is not ensured. Meanwhile, due to the appearance of the fractional Hardy-Littlewood-Sobolev critical exponent, it gives rise to another difficulty: the lack of compactness of Sobolev embedding. Inspired by [2,8], the difficulties can be overcome by restricting the parameter
Theorem 1.3
Problem (1.9) has no ground-state solution.
Once Theorem 1.3 is established, in order to obtain the existence of positive solutions for Problem (1.9), it is necessary to observe higher levels of energy and study the behaviors of the Palais-Smale sequences as
Theorem 1.4
There exist
An outline of this article is as follows. The limit profiles and uniqueness for Problem (1.7) as
Throughout this article, we make use of the following notations:
The fractional Sobolev space
which is a Hilbert space endowed with the norm:
where
and endowed with the norm
where
For
For any
2 Limit profiles and uniqueness of limit equation
In this section, for
In particular, the existence of a ground-state solution to (2.1) when
i.e., Problem (1.7). Here,
Obviously,
In what follows, we insert the fractional concentration compactness principle established in [11, Lemma 2.2].
Lemma 2.1
Let
where
Next, we reveal the Hardy-Littlewood-Sobolev inequality and the estimate of the Riesz potential as in [20, Section 4.3].
Lemma 2.2
Let
Then, for any
Lemma 2.3
Let
Then, for
where
The following lemma gives an upper bound for the Riesz potential energy (see [28, Lemma 3.3]).
Lemma 2.4
Let
In particular, if the function
As a consequence of the aforementioned lemma, similar to [28, Lemma 3.4], we can prove the convergence results as follows.
Lemma 2.5
Let
If, moreover,
Proof
Given
For each
Now, we prove that for each
where the symbol
Moreover, if
where
for
Since
Next, we prove (2.3). We assume that, up to a subsequence,
for every
According to Lemma 2.5, the following result can be directly obtained.
Corollary 2.6
Let
In addition, for any
Now, we introduce the variational structure of Problem (1.7). Naturally, associated with Problem (1.7), the energy functional
It is standard to verify that
It is obvious that the ground-state energy level
Moreover,
where the function
It can be seen that for
The existence and properties of the ground-state to (2.5) can be deduced. Indeed, the solutions of Problem (2.5) and
are related to each other. It is well known that Problem (2.6) has a unique (up to symmetries) positive ground-state solution
then we infer that
Based on the previous arguments, the following theorem can be established, which is crucial for the proof of Theorem 1.1.
Theorem 2.7
Fix
The proof is based on the following lemmas.
Lemma 2.8
There holds
where
Proof
For
Taking the infimum with respect to
Moreover, we suppose
Since
then Inequality (2.7) ensures that the ground-state solution
The proof is complete.□
Next, let
Lemma 2.9
There exists a positive radial solution
Proof
Since
Then, we see from Lemma 2.8 that
From this and Corollary 2.6, it can be inferred that
which implies
Hence, we obtain a uniform lower bound for
Lemma 2.10
It results that
Namely,
Proof
According to Corollary 2.6 and Lemmas 2.8 and 2.9, one sees immediately that
Proof of Theorem 2.7
As a conclusion of Lemmas 2.9 and 2.10, the proof is complete.□
The following lemma is a fractional version of [29, Lemma 5.1], and its proof is presented in [17, Lemma 3.7]
Lemma 2.11
Let
For
Lemma 2.12
The operator
Proof
Let
Note that
Now, we claim that
Indeed, in view of Lemma 2.5, we can conclude that
On the other hand, it is easy to verify that
Denoting
Then, one can repeat the same arguments earlier to see that
Lemma 2.13
Let
Proof
Note that
Since
is compact from
is compact from
Proof of Theorem 1.1
Assume, by contradiction, that there exist sequences
3 Nonexistence result and technical lemmas
In this section, we will give some results that are crucial to prove main theorems. To this end, we choose
Some necessary notations are as follows: let us denote by
Meanwhile, we need to consider the following functionals:
and
which correspond to the limit Problems (1.10) and (1.8), respectively. Using standard arguments, it is easy to prove that the functionals
and
Furthermore, we define the following minimization problems:
and
Obviously, the functional
where
On the other hand, it is easy to check that the functionals
Lemma 3.1
There holds
Proof
Letting
We can take
On the other hand, set
In view of (3.1), for every
Hence,
Due to
Combining (3.2) and
Taking the limit, one has
where
and
Hence,
which indicates that
The proof follows.□
Now, define
which is well-defined (see [15, Section 2] for instance). Then, the following result holds.
Lemma 3.2
There exists
Proof
Let
Note that
for
Lemma 3.3
Let
Proof
Assume, by contradiction, that Lemmas 2.1–2.2 state that
which guarantees that
According to the definition of
which contradicts Lemma 3.2.□
Now, we present the existence result for Problem (1.10).
Proposition 3.4
There exists
where
Proof
According to the mountain pass theorem [32], we can deduce that
Combining with (3.5) and
which implies that
On the one hand, if
Let us consider the sequence
Then,
which ensures that
Hence, we have
On the other hand, if
Without loss of generality, the ground-state solution of limit Problem (1.10) is denoted by
Furthermore, from the fractional Polya-Szegö inequality in [25] and formula (A.11) in [30], we have
By Riesz’s rearrangement inequality (see [20, Theorem 3.4]), there hold
and
Thus,
for some
as desired.□
In what follows, let
Lemma 3.5
There holds
Proof
Define
where
where
Now, we prove that
i.e.,
Note that
Then, we define
Since
In addition, a direct application of the mean value theorem gives
Consequently, the Lebesgue’s dominated convergence theorem leads to
On the other hand, let
and
Using the Lebesgue’s dominated convergence theorem again, we have
Hence, we can conclude that (3.9) holds.
Finally, we claim that
Once the claim holds, then
which, together with (3.8), signifies that
Thus, the fact that
By virtue of the definition of the
Obviously,
Proof of Theorem 1.3
If there exists
Without loss of generality, it can be supposed
which imply that
Arguing as [6, Lemma 3.1], one can have the following compactness lemma.
Lemma 3.6
Let
where
and
Based on the fact of Theorem 1.3, we can construct some energy-level interval close to
Proposition 3.7
For every
Proof
Assume, by contradiction, that there exist
with
and we suppose that for every
Firstly, we claim that
which ensures the claim holds. Moreover, we have the following estimate:
Next, we claim that there exist
Indeed, if the case does not occur, then for every
which, together with Lemma 2.1, yields that
and then,
Thus, using
It follows from (3.4) that
Combining (3.14), one has
which implies that
In view of (3.15) and (3.16), we deduce that
Furthermore, there holds
which contradicts with (3.12). Thus, the claim is true. Moreover, up to a subsequence of
Therefore, combining with (3.11) and (3.17), for every
Let us denote the sequences
and
for
Namely,
Now, we show that
Therefore,
which is absurd. Then, by means of Lemma 3.6, there exist
Finally, we claim that
which ensures that
which is a contradiction.□
As a consequence of Proposition 3.7, we can prove the following result.
Proposition 3.8
For every
Proof
Let
where
Hence, there is a sequence
where
Note that
Thus, we can assume that
that
4 Proof of Theorem 1.4
In this section, we focus our main attention to establish the proof of Theorem 1.4. First, we give some properties of the levels
Lemma 4.1
There holds
Proof
For every
which yields that
and
It follows that
We claim that
This contradiction validates the claim. Hence,
In addition, since
which, together with (4.2), yields that
Therefore,
The proof follows.□
Let us define the barycentric function
where
for some
and
It is clearly seen that
Lemma 4.2
There holds
Proof
For every
Moreover, we consider
Thus, it suffices to prove that the reverse inequality. Let
Since
which ensures that
Combining the aforementioned arguments and
which, together with (4.3), states that
For further discussions, we shall define
where
Then, it is necessary to consider the following minimization problems:
Lemma 4.3
There holds
Proof
It is obvious that
According to the Ekeland variational principle, we assume that
Letting
Combining Lemma 3.6 and the uniqueness of ground-state solutions of Problem (1.7), we infer that there exists a sequence
We consider the sets
Here, we have used the radial lemma and the mean value theorem. Furthermore, similar to [33, Lemma 3.2], we can deduce that
which, together with (4.4) and the Cauchy-Schwarz inequality, states that
where
Lemma 4.4
One has
Proof
Let
which indicates that
Since
i.e.,
Therefore, we can infer that
Hence,
Moreover, we easily see from the Lemma 4.3 that
as desired.□
Let
For every
where
Lemma 4.5
The following statements hold:
For all
For each
Proof
(1) Similar to [1, Lemma 4.1], it is easy to check that (1) holds, so we omit the details.
(2) In view of Lemma 2.2, we have
which ensures that
uniformly in
(3) We claim that
Indeed, letting
We see from
and hence,
Furthermore,
and
Then,
Note that
Since
Therefore,
uniformly in
(4) For each
and
Consequently,
The following lemma is a direct conclusion of Lemma 4.5(3).
Lemma 4.6
Fixed
for all
In what follows, we fix
Lemma 4.7
There exist
Proof
(1) By means of
Combining the aforementioned inequality, Lemma 3.5, and Theorem 1.3, one has
Moreover, in view of Lemmas 4.1, 4.2, and 4.4, for
Furthermore, by virtue of Item-(4) of Lemma 4.5, there exists
The previous estimates lead that (1) holds.
(2) The proof is similar to [1, Lemma 4.3], so we omit it.□
Now, we define the following sets:
and
Then, the next result can be found in [1, Lemma 5.1], whose arguments is standard, so it will be omitted.
Lemma 4.8
If
Finally, we establish the existence of positive solutions for Problem (1.9).
Proof of Theorem 1.4
Let us define the level
By means of Lemma 4.8, for every
for all
On the other hand, choose
Noting that for every
Letting
Thus, setting
then there exists
In what follows, we claim that for each
Assume, by contradiction, that there exists
Applying the deformation lemma [32, Lemma 5.15], there exists
and
where
According to the definition of
which states that
i.e.,
Therefore,
In addition, since
then
i.e.,
Combining the definition of
which, together with (4.5), signifies that
This leads to an obvious contradiction. Hence, there exists
with
To sum up, there exists
Acknowledgements
The authors would like to thank the anonymous referee for several valuable suggestions and comments that helped to improve this article.
-
Funding information: This project was supported by the National Natural Science Foundation of China (No.12371120), Natural Science Foundation of Chongqing, China (No. CSTB2023NSCQ-BHX0226), and the special subsidy from Chongqing human resources and Social Security Bureau.
-
Author contributions: F. Ye wrote the manuscript; S. Yu contributed the analyses; C-L. Tang contributed to the conception of the study.
-
Conflict of interest: The authors do not have conflict of interest.
-
Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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- 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
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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”
