Regularity for critical fractional Choquard equation with singular potential and its applications
-
Senli Liu
Abstract
We study the following fractional Choquard equation
where
1 Introduction and main results
In this article, we focus our attention on the following equation:
where
where
Equation (1.1) comes from the research of solitary wave solution for the focusing time-dependent Hartree equation:
where
Mathematically, when
which was introduced by Pekar [35]. The early results of existence and symmetry for (1.3) are credited to Lieb and Lions. Lieb [24] established the existence and uniqueness, up to translations, of ground state solutions. Lions [26] studied the existence of a sequence of radially symmetric normalized solutions. Subsequently, equation (1.1) with
When
where
The study of problems involving singular potential have a long history, some first results can be found in the previous studies [3,4,16,39,40]. Recently, Feng and Su [20] extended the results of Badiale and Rolando [3] to a fractional equation as follows:
where
there exists an
By virtue of [38, Compactness lemma] and Lemma 2.1, the authors established the following continuous and compactness embeddings results, which are essential to study the functional framework by variational methods.
Proposition 1.1
[20] Assume that
Furthermore, the embeddings are compact if
and
To overcome the noncompactness embedding problems, Feng and Su [20] first proposed a compactness lemma, which can be regarded as a contrapositive proposition of the famous Lions vanishing lemma [42].
Proposition 1.2
[20] Assume that
Then the sequence
By applying Propositions 1.1 and 1.2 and the Nehari manifold method, the authors obtained the existence of ground state solution for equation (1.5).
Inspired by the aforementioned works, especially t-th study by Feng and Su [20], we find a problem left whether the fractional Choquard equation involving the embedding top and bottom indices and a singular potential possesses nontrivial solutions. Further, can we obtain some regularity results of these solutions? To respond these questions, we first introduce some preliminaries, which will be of use.
The following well-known Hardy-Littlewood-Sobolev inequality is crucial to handle our problem.
Proposition 1.3
[25] Let
If
We handle equation (1.1) in the space
endowed with the norm
Equation (1.1) is variational, the energy functional can be defined by
It follows from Propositions 1.1 and 1.3 that
We say
Usually, we know it is very tough to study the regularity of weak solutions. Furthermore, up to our knowledge, no results have been reported regarding the regularity of weak solution to equation (1.1). In this study, our goal is to fill this gap. The first result is based on the following assumtion:
In this direction, our first main result can be stated as follows.
Theorem 1.1
Let
where
Remark 1.1
For equation (1.1) with
To prove Theorem 1.1, following the ideas of [33], we first establish the nonlocal Brézis-Kato’s type regularity estimate with singular potential (Lemma 2.4). With the help of radial inequality (Lemma 2.1), the Hardy inequality [37], and Lemma 2.4, we increase the integrability of the weak solution of equation (1.1). Subsequently, by means of a new Moser iteration method, we prove that any weak solution
The second aim of this article is to give an application of Theorem 1.1. For this purpose, we study the following fractional Choquard equation:
Our second main result can be stated as follows.
Theorem 1.2
Let
Remark 1.2
When
There exists
There exists
As mentioned earlier, we know that conditions
Moreover, for
Remark 1.3
It follows from Proposition 1.3 that
and
In view of Proposition 1.1, the compactness of the following embeddings have disappeared
which set an obstacle to verify the compactness of the corresponding functional. Thus, we call
We outline our strategy in the proof of Theorem 1.2. By Propositions 1.1 and 1.3, the corresponding functional
Obviously,
To overcome the lack of compactness caused by the embedding top and bottom indicies, we consider the following subcritical perturbed equation:
where
By using the fact of Theorem 1.1, we know that all the weak solutions of equation (
We define
and
The following two inequalities are direct conclusions of Proposition 1.1 and the Sobolev embedding theorem:
and
where
In terms of the properties of
The remainder of this article is as follows. In Section 2, we study the regularity of weak solution of equation (1.1) and establish the Pohožaev identity. In Section 3, we devote to investigate the existence of ground state solution of equation (
2 Regularity of solutions and the Pohožaev identity
In this section, we study the regularity of solutions and Pohožaev identity of equation (1.1).
2.1 The nonlocal Brézis-Kato’s type regularity estimate with singular potential
Lemma 2.1
[6] Let
Lemma 2.2
Let
where
Proof
Combining Lemma 2.1 and the Hardy inequality [37], we can derive
The proof is completed.□
Lemma 2.3
[33] Let
Assume that
Then, for any
Lemma 2.4
Let
Proof
Let
(i) Let
(ii) Let
(iii) Let
(iv) Let
Combining (i)–(iv), we can show
Since
The proof is completed.□
Remark 2.1
If
Lemma 2.5
Let
Then,
Proof
By applying Lemma 2.4 with
We choose sequences
Define a sequence of continuous coercive bilinear functional
From the Lax-Milgram theorem, it is easy to show there exists a unique solution
where
We next claim that
By this and recalling (2.2) and the Hölder and Young inequalities, we find that
Thus, we obtain
which implies that
Hence,
Since equation (2.3) admits a unique solution, which immediately yields that
For
We first claim that
and
which implies that
Let
Consider the functions
For fixed
Analogously, we can show that the aforementioned expressions are true for any
By using (2.6), there holds
Taking
If
Putting together (2.8) with (2.9), we obtain
where
By
where we have
By using the Hölder inequality and
Inserting the latter relation into (2.10) and combining the Sobolev embedding theorem, we have
By
where
So, we obtain
2.2 Regularity of weak solutions and the Pohožaev identity
Lemma 2.6
Let
then it follows that
where
Proof
By a straightforward computation, one easily sees that the inequality (2.12) is equal to
Then, we derive
We set
Case 1. If
We first compare
According to the range of
For
For
In the following, we will work out
and we set
It is not difficult to figure out the aforementioned equation, namely,
from which we infer
Therefore, we obtain
We next compare
As mentioned earlier, we know that
which indicates
Case 2. If
We first compare
By the monotonicity of function
one has
Combined the above two cases and our assumptions on
Lemma 2.7
[25] Let
where
Lemma 2.8
Assume that all the conditions described in Theorem
1.1
are satisfied. Let
Proof
By using Lemma 2.5, one has
We choose
where
Similarly, we obtain
By combining
Putting together with (2.15)–(2.17), one obtains
The proof is completed.□
Lemma 2.9
Assume that all the conditions described in Theorem
1.1
are satisfied. Let
For
Proof
We first claim that
and
from which one at once deduces that
Similar to (2.5)–(2.7), we obtain
By
The proof is completed.□
Thanks to Lemmas 2.8 and 2.9, we can finally prove Theorem 1.1.
Proof of Theorem 1.1
(i) (
Case 1.
In this case, we let
Step 1. It follows from Lemma 2.5 that
which yields
For
and
It is not difficult to see that
and
Moreover, we have
Hence, we obtain
For convenience, we set
According to
In view of (2.19), we can infer
Analogously, we obtain
Putting together (2.20), (2.21) with Lemma 2.9, then
Taking
By using Lemma 2.9 again, we can deduce
By
Let
Step 2. By Step 1, one obtains
and the definition of
Similar to Step 1, we just need to consider the case
Let us observe that
and
Taking into account (2.20), (2.21), (2.25), and (2.26), we deduce
which shows
Similarly, we have
Combining (2.24),
which gives
Let
Step 3. Iterating the above process, one sees that
Let
So, we obtain
Thus, the series
Case 2.
Step 1. Let
By the definition of
Let
By
Combined (2.30)–(2.31) with Lemma 2.9, then
Let
Since
Step 2. Let
Let
Combining Lemma 2.9 and (2.29), we observe
Let
In view of Young’s inequality, we obtain
where
Owing to
Then, there holds
from which we can deduce
Step 3. We iterate the above process and set
Thus, we have
Then, it follows that
which implies
By using (2.34), then
From (2.35) and (2.36), we deduce
It is not difficult to see that
Since
the series
Moreover, it follows that
Then, we know that the series
According to (2.38)–(2.40), we have
(ii) (Pohožaev identity) Combining Theorem 1.1 (i), Lemmas 2.8,2.9 and
where
where
Similar to [33, Theorem 3], we can prove the following Pohožaev identity
This ends the proof.□
3 Proof of Theorem 1.2
In this section, we will prove Theorem 1.2 by virtue of subcritical approximation arguments, the Pohožaev manifold method and the modified version of the generalized Lions-type theorem.
As a consequence of Theorem 1.1, we introduce the following Pohožaev manifold
where
For each
Lemma 3.1
Assume that
Then
Proof
Obviously, we have
Since one can easily see that
and
From this, we conclude that
This implies that
Lemma 3.2
Assume that all conditions described in Theorem
1.2
are satisfied. Then, there exists a unique
Proof
We set
and
By
Lemma 3.3
Assume that all conditions described in Theorem 1.2 are satisfied. Then
where
and
Proof
To do this, we choose
By using Lemma 3.2, we can find
Consequently, one has
For this, there holds
Thus, we know that
Next, we verify
Letting
Obviously, the latter relation yields a contradiction with
Hence, we observe that there exists
The proof is completed.□
Lemma 3.4
Assume that all conditions described in Theorem
1.2
are satisfied. Then, there exists a unique
The proof is similar to Lemma 3.2. We omit any further detail.
Lemma 3.5
Assume that all conditions described in Theorem 1.2 are satisfied. Then
Proof
(i) By virtue of Lemma 3.4, we obtain
We claim that
From the latter expressions, it is clear that
which leads to
(ii) For
By using the Young inequality, one can see that
and
Let us define
By (3.5), (3.6) and the Lebesgue dominated convergence theorem, it is not difficult to see
for
Since by (3.4), we have
for
Lemma 3.6
Assume that all the conditions described in Theorem
1.2
are satisfied. Let
Proof
Clearly, we have
In terms of (3.3), it is standard to show
To finish the proof, it suffices to show, for
and
Since the proof of (3.8)–(3.12) are similar, we just show (3.8). For each
Making use of Young’s inequality, we obtain
As a result,
In light of
Recalling
For
From this, we obtain
It follows from Proposition 1.3 that the Riesz potential defines a linear continuous map from
Analogous to (3.14), we immediately have
Then, for each
We observe that
Considering (3.18) and (3.19), one can infer
From (3.13) and (3.20), we have that
Similarly, we can prove (3.9)–(3.12). Hence, for each
This completes the proof.□
The following is the modified version of the generalized Lions-type theorem. As we mentioned earlier, it plays a key role in recovering the compactness.
Proposition 3.1
Let
Then, we have
The proof is same as in [20, Theorem 1.3]. So we omit it here for the details.
We are finally ready to prove Theorem 1.2.
Proof of Theorem 1.2
We divide the proof into three steps.
Step 1. Let
We claim that
Hence, by Proposition 1.3 and the Hölder and Young inequalities, we observe that
and
and
and
and
and
Gathering together (3.22)–(3.28) and
It leads to a contradiction with (3.21). As a consequence of (3.21), we can notice
Step 2. Up to subsequences, we assume that there exists
We deduce from Lemma 3.6 that
In what follows, we distinguish three cases:
Case 1.
Case 2.
Case 3.
If Case 1 happens, exploiting (1.7) and (3.23)–(3.28), we see that
From this, we obtain
Combining the latter formula and Lemma 3.5 (ii), we immediately obtain
which yields a contradiction with Lemma 3.3. So, Case 1 does not happen.
If Case 2 happens, combining (1.8) and (3.23)–(3.28), we obtain
which implies
We obtain from this and Lemma 3.5 (ii) that
It gives a contradiction with Lemma 3.3. Hence, Case 2 does not happen.
Hence, we observe that only Case 3 happen. From Proposition 3.1, one has
Step 3. Finally, we claim
So, we have
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Funding information: J. Yang is supported by Natural Science Foundation of Hunan Province of China (Nos. 2023JJ30482 and 2022JJ30463), Research Foundation of Education Bureau of Hunan Province (22A0540).
-
Conflict of interest: The authors declare that they have no conflicts of interest to this work.
-
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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- 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”
