Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
-
Jie Yang
and
Haibo Chen
Abstract
We study the following fractional Kirchhoff-Choquard equation:
where
1 Introduction
In this article, we study the following fractional Kirchhoff-Choquard equation:
where
is the Riesz potential. The operator
where
if
When
Choquard used (1.1) to describe electrons trapped in their own holes. To some extent, it is similar to the Hartree Fock theory of single component plasma in 1976, as shown in [16].
The more general form of (1.1) is as follows:
When
There exists
There exists
When
It originates from the study of fractional Laplacian with nonlocal Hartree-type nonlinearities. For the case
There exists
Let
where
There exists
The fractional Kirchhoff-type problem has been widely studied in recent years, a problem naturally arises: can we obtain a ground-state solution to problem (KC) involving the fractional upper H-L-S critical exponent? In this article, attention is paid to this case and a positive answer is provided. We assume that
there exist
Because of the emergence of convolution non-linearity, the methods used in [20] cannot be directly applied to obtain the desired solution of equation (KC). On the other hand, due to the variable potential
Throughout the article, we assume that
We remark that in order to apply Jeanjean’s monotonicity technique [13] to equation (KC), the corresponding limit problem plays an important role. Hence, we investigate the limit problem
Due to the fact that the Nehari-type monotonicity condition of
where
We will obtain the next result.
Theorem 1.1
Let
The proof of Theorem 1.1 was enlightened by He and Rădulescu [12]. The author studied the existence of positive solutions for the fractional Choquard equation with pure critical nonlinearity
They proved that in the case of small perturbations of the potential, the above equation has at least one positive solution. However, we note that due to the appearance of the Kirchhoff term and the fact that the nonlinear term is a more general function with a critical index rather than a power function, the ideas of He and Rădulescu [12] cannot be immediately applied to the proof of Theorem 1.1. We need to make some precise estimates of this problem. for example, see Lemma 3.5.
When
where
On this basis, we consider equation (KC) with general potential. Next is the second result.
Theorem 1.2
Let
Remark 1.1
Motivated by [6], we apply Jeanjean’s monotonicity technique [13] to prove Theorem 1.2. Compared with [6], the presence of fractional Laplacian results in some methods in [6] that not being applicable to our problem. The major difficulty is to construct a bounded (PS) sequence and prove that the sequence weakly converges to a critical point. By proving a global compactness lemma and utilizing Theorem 1.1, we can overcome the aforementioned difficulties.
Now, to obtain more information about solutions to equation (KC), we study the regularity of ground-state solutions.
Theorem 1.3
Let
If
If
Throughout the article, we use the following notations:
For any
The rest of this article is organized as follows. In Section 2, we state some preliminary notations and the main lemmas. In Section 3, we study the limit problem. The proof of Theorem 1.2 is given in Section 4. In Section 5, we study the regularity of weak solutions to equation (KC).
2 Preliminaries
Let
be the Hilbert space endowed with an equivalent norm
where
and
where
To prove Theorem 1.2, we consider the next functional
Obviously,
We define the Pohozaev manifold
where
is the Pohozaev identity associated with equation (KC). Throughout this article, we always assume that
Lemma 2.1
[15] Let
Lemma 2.2
[15] Let
where
It follows from Lemma 2.2 that for any
Combining the Hardy-Littlewood-Sobolev inequality with Sobolev’s inequality, for any
Let
and
Then,
and it is achieved by
where
and
By
Lemma 2.3
The following conclusions hold for F:
For any
For all
For all
Proof
(i) It follows immediately from
(ii) From (i), Lemma 2.1, and the Sobolev embedding theorem, we can conclude that
(iii) From
Now, we recall a fractional version of Lions vanishing lemma [26].
Lemma 2.4
[26] Assume that
for some
At this point, let us introduce a nonlocal version of the Brezis-Lieb lemma [6, Lemma 2.2].
Lemma 2.5
[6, Lemma 2.2] Assume that
Let
The next two lemmas are useful in proving the splitting property of the energy functional.
Lemma 2.6
[6, Lemma 2.6] Let
Lemma 2.7
Let
For any
Assume
For any
where
If we further assume that
for any
Proof
It can be proved similarly to [6, Lemma 2.5] and [24, Lemma 2.5]; thus, we omit the details for simplicity.□
The next result is a splitting property of the nonlocal energy functional for the fractional Kirchhoff-Choquard equation in
Lemma 2.8
Let
for any
Proof
Let
Note that for any
Step 1. We are going to show that for any
Recalling that
Then, for
Together with Lemma 2.1, (2.8), we derive that
uniformly for any
On the one hand, according to Lemma 2.7 (i) with
For
for any
for any
which yields
for any
Since
which together with [24, Lemma 2.6] implies
and Hölder’s inequality, we infer that
for any
Step 2. We claim
for any
for any
which implies
Together with Lemma 2.1 and (2.8), we deduce that
for any
By a similar measure, one obtains
for any
for any
Next, we show that
and
for any
Together with (2.11), we infer that (2.12) holds. By a similar argument as (2.12), we can prove (2.13). This completes the proof.□
3 Proof of Theorem 1.1
In this section, we consider the limit equation associated with equation (KC):
and denote the energy functional by
We define
where
Lemma 3.1
Proof
Set
Since
Remark 3.1
By direct calculations, we see that
Lemma 3.2
For any
Proof
Let
By
Next, we show
and
which is a contradiction and the proof is complete.□
In light of Lemmas 2.3 and 3.1, we obtain that
Lemma 3.3
The functional
there exist
there exists
Proof
(i) From Lemma 2.3(ii) and Sobolev’s embedding theorem, we obtain
which implies that there exist
(ii) It follows immediately from Lemma 3.1.□
We now define
where
Similar to the arguments of [31, Chapter 4], we obtain
Lemma 3.4
There exists
Proof
From (3.1), Lemma 2.3 (ii) and (iii), and the Sobolev embedding theorem, we deduce
which implies
for any
Lemma 3.5
Let
Proof
Let
where
where
By direct calculation, we conclude that
Together with Lemma 2.1, we see that
and
Meanwhile,
where
and
By computing directly, we obtain
and
By (3.6)–(3.8), we deduce that
Similarly, we note that
where
Therefore, we have
recalling that
It is clear that
Claim. There exist
By (3.3)–(3.5), (3.11), and
which implies that
Combining (3.2), (3.3), and (3.9), the above inequality yields
To find the estimate of
By (2.5), (3.3), and (3.9), we obtain
Using (3.2), (3.10), and (3.12), we conclude that
noting the fact
which implies that the desired result holds.□
Lemma 3.6
Let
Proof
Let
Hence,
Therefore,
We claim that there exists
If not, by Lemma 2.4, we derive
Let
Noting that
we can infer that
which implies that
Similarly,
It follows from (3.14) that
Set
From (3.15) and (2.4), we obtain
which yields
According to
which contradicts with Lemma 3.5. Thus, the Claim holds true.
Let
and
Passing to a subsequence, there exists
and
Define
Note that
From the weak lower-semi-continuity of the norm, we have
We now claim
By Lemma 3.2, there exists
and
which yields a contradiction. Then,
which is a contradiction. Thus,
In the same way as [19, Lemma 2.6], we can deduce the following lemma.
Lemma 3.7
If
4 Proof of Theorem 1.2
Since
Theorem 4.1
[13] Let X be a Banach space and
where
where
Then, for almost every
In order to explore Theorem 4.1 for
Then, set
where
as
We can easily verify all the conditions of Theorem 4.1.
Lemma 4.1
The functional
there exists
the map
By Theorem 1.1, we deduce that for any
admits a ground-state solution
such that
and
where
and
Lemma 4.2
Let
Proof
Let
The proof is completed.□
From Lemma 4.1, we obtain that for almost every
Next, we aim to show the convergence of the above sequence
It follows from
For any
and
They are corresponding energy functionals of the following problems:
Next, we give the global compactness lemma, which plays a crucial role in getting compactness.
Lemma 4.3
Let
Proof
Since
Then, we obtain
Hence, there exists
Using standard arguments, we can obtain that
By
Set
Indeed, assume by contradiction, for any
Then, Lemma 2.4 yields that
and
By Lemmas 2.5 and 2.8, we obtain
and
Set
Then, by (4.6) and (2.4), we deduce that
which yields
Together with (4.5) and (4.6), we derive that
This is a contradiction. Then, (4.3) holds.
Then, combining with the boundedness of
It follows from
From (2.1), we obtain
Recalling
which yields
together with (4.7), we obtain
From
We assume that
Thus, by Hölder’s inequality, we obtain
Combining (4.9) and (4.8), we obtain
Next, we will prove
In view of
Using Brezis-Lieb lemma [3] and Lemma 2.5, we obtain (4.10) holds.
Set
and
and
and
for any
Combining with (4.10), we obtain
From the fact
If
and thus Lemma 4.3 holds with
Since
In the following lemma, we conclude the convergence of the bounded
Lemma 4.4
Let
Proof
It follows from Theorem 4.1 and Lemma 4.1, for almost all
By Lemma 4.3, we can conclude that there exist
and
and
where
Note that
By (4.2), we have
From Lemma 3.2, there exists
From (4.17) and (4.18), if
which is a contradiction according to Lemma 4.2. Thus,
Remark 4.1
From the proof of Lemma 3.4, we see that the map
Proof of Theorem 1.2
Combining Theorem 4.1 and Lemma 4.1, for
Applying Lemma 4.4, for
Now, we need to show
Together with
Therefore by Theorem 4.1(iii), we obtain
and
which imply that
At last, we prove the existence of a ground-state solution to equation (KC). Set
It is easy to see that
which yields
5 Regularity of solutions
In this section, we prove the regularity of solutions to equation (KC). We present a preliminary result, which is crucial for the subsequent proof.
Lemma 5.1
[23] Let
If
then for every
Lemma 5.2
If
Proof
Set
Taking
In a similar vein, applying Lemma 5.1 with
with
From the Sobolev inequality, we conclude that for each
We can choose
This completes the proof.□
Lemma 5.3
If
Then,
Proof
Using Lemma 5.2 with
Take sequences
which is bilinear and coercive. It follows from the Minty-Browder theorem [8] that there exists a unique solution
where
Combining with (5.2),
which yields
This means
It follows from
Therefore,
so that
Set
which implies
For
We note that
and
Let
Denoting
It follows that
Taking
If
Combining with (5.8), recalling Lemma 2.1, we deduce that
where
Combining with (5.9), using the Sobolev embedding theorem, we conclude that
which yields
From
Taking
Then, we have
Lemma 5.4
Suppose that all conditions of Theorem
1.3
hold, and
Proof
Define
and
Therefore,
which implies
Proof of Theorem 1.3
For
and
which yields that
Similar to (5.6) and (5.7), we obtain
From
Now, we provide an
Case 1.
We will divide the proof into three steps.
Step 1. Let
According to the definition of
Let
By
The above two inequalities together with (5.12) imply that
Letting
Since
Step 2. Let
Choosing
It follows from (5.12) and (5.13) that
Letting
Noting that
Then, we can see that
which yields
Step 3. Iterating the above steps and setting
we obtain that
which implies that
Together with (5.16), we obtain that
It is easy to see that
Case 2.
In this case, we take
Step 1. It follows from Lemma 5.2 that
which implies
and
For any
and
Clearly,
and
If
Without loss of generality, we take
Noting that
Together with (5.18), we deduce that
In the same way, due to
Using (5.19), (5.20), and (5.12), we obtain that
Taking
which yields that
Recalling that
Particularly, taking
Step 2. From
and the definition of
Similar to step 1, we only need to present the case
Furthermore, through simple calculations, we can infer that
and
Similarly, we can see that
Combining (5.19), (5.20), (5.24), and (5.25), we conclude that
which yields
Similarly, we can see that
From (5.26),
which implies that
Particularly, taking
Step 3. By repeating the above process, for
It is easy to check that
Therefore, taking the limit as
Combining
where
Acknowledgements
The authors thank the anonymous referees for their valuable comments and nice suggestions to improve the results. J. Yang is supported by the Natural Science Foundation of Hunan Province of China (2023JJ30482, 2022JJ30463), Research Foundation of Education Bureau of Hunan Province (23A0558, 22A0540), and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.
-
Author contributions: All authors contributed equally to this work.
-
Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.
-
Data availability statement: Not applicable.
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- Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
- Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
- Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
- Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
- Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
- Generalized quasi-linear fractional Wentzell problems
- Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
- Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
- Boundedness and global stability in a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics
- Review Article
- Existence and stability of contact discontinuities to piston problems
Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
- Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
- Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
- Shape of extremal functions for weighted Sobolev-type inequalities
- One-dimensional boundary blow up problem with a nonlocal term
- Doubling measure and regularity to K-quasiminimizers of double-phase energy
- General solutions of weakly delayed discrete systems in 3D
- Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
- Optimal large time behavior of the 3D rate type viscoelastic fluids
- Local well-posedness for the two-component Benjamin-Ono equation
- Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
- Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
- Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
- On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
- Normal forms of piecewise-smooth monodromic systems
- Fractional Dirichlet problems with singular and non-locally convective reaction
- Sharp forced waves of degenerate diffusion equations in shifting environments
- Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
- Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
- Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
- Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
- Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
- Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
- Generalized quasi-linear fractional Wentzell problems
- Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
- Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
- Boundedness and global stability in a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics
- Review Article
- Existence and stability of contact discontinuities to piston problems
