A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
-
Lu Shun Wang
und
Xiao Long Yang
Abstract
In this article, we study a Hardy-Sobolev critical elliptic system involving coupled perturbation terms:
where
1 Introduction and main result
In this article, we consider a Hardy-Sobolev critical elliptic system involving coupled perturbation terms:
where
Systems of type (1.1) originate from searching standing wave solutions of the following time-dependent Schrödinger systems:
where
where
with
Another motivation on the study of system (1.1) is that it is related to a model describing the dynamics of elliptic galaxies in astrophysics (see [8,9,14,23,25]).
Liu in [21] studied (1.3) with
In this case, (1.3) became into the elliptic system of two equations:
where
Later on, Liu and Liu in [22] considered the Hardy-Sobolev critical elliptic system:
where
In [29], Tian and Zhang investigated a doubly coupled system:
where
In [6], Benmouloud et al. studied a doubly critical elliptic system:
where
In [32], Yang obtained a nontrivial weak solution
where
where
The works we have just mentioned earlier is far from being complete. For more information about global compactness results on scalar equations, please refer to [5,10,12,15,16,17, 18,28] and so on. When it comes to systems, one can refer to [4,6,13,21,22,32] and the references therein.
The main purpose of this article is to establish a global compactness result (i.e., a complete description for the Palais-Smale sequences of the corresponding energy functional) and some existence result for the doubly coupled system (1.1). Observing that (1.1) contains both the Hardy-Sobolev critical exponent and the nonlinear coupling exponent, it is very different from the well-studied systems (1.4)–(1.8). To our best knowledge, (1.1) has not been studied before.
Throughout this article, we assume that
(V)
(Q)
For fixed
We say
Clearly, a critical point of
If
The limit system of (1.1) involving Sobolev subcritical terms is
and the associated energy functional is
Define
where
for
In particular,
The limit system of (1.1) containing Hardy-Sobolev critical terms is expressed as follows:
and the associated energy functional is expressed as follows:
Solutions of system (1.15) are closely related to the scalar equation:
In [23], Mancinia et al. proved that all the positive solutions of (1.17) are of the form
where
In the same way as Theorem 5 of [3] or Lemma 2.2 of [4], we can show that any ground-state solution of system (1.15) is of the following form:
for
and
Let
Our first main result is the refined Sobolev inequality involving partly weighted Morrey norm (see (1.26) below), which is an efficient tool for further use in the study of (1.1).
Theorem 1.1
Let
where
Remark 1
Inequality (1.22) is a nontrivial extension of the inequality obtained by Palatucci and Pisante, see Theorem 2 of [27].
Benefiting from Theorem 1.1, we then obtain a global compactness result for (1.1).
Theorem 1.2
Let
and
as
In particular, if
Remark 2
Because of the lower order terms
Usually, the scaling rates and the translation elements generated by
Armed with Theorem 1.2, we finally derive the following existence result for (1.1).
Theorem 1.3
Let
then (1.1) admits a nontrivial solution
Moreover, we have
or
Remark 3
Theorem 1.3 partly extends the results of [15, 18] on scalar equations to a doubly coupled system. We also observe that (1.6) contains both the linear and nonlinear coupling exponents, while (1.1) involves two nonlinear coupling parameters. Therefore, the study of (1.1) is more complicated than that of (1.6).
The proof of Theorem 1.1 relies on a norm inequality for Riesz potentials on weighted Lebesgue space, see Lemma 3.1. This idea originates from [27]. However, the coupled term
The main difficulties in proving Theorem 1.2 are that the energy functional of (1.1) contains both the Hardy-Sobolev critical term
Theorem 1.3 is proved by using the Mountain Pass lemma (see [2]) and the global compactness result (see Theorem 1.2). We can check that the energy functional
The rest of the article is organized as follows: in Section 2, we present some preliminary results. In Section 3, we prove Theorems 1.1–1.2. In Section 4, we prove Theorem 1.3.
Notations: Letting
where
where
If
For simplicity, (PS) sequence denotes “Palais-Smale” sequence in short. We use “
2 Preliminaries
In this section, we present some preliminary results.
First, we recall the following Hardy-Sobolev inequalities:
Lemma 2.1
([8], Theorem 2.1) Let
We also need two important lemmas: the concentration-compactness principle and a variant of the “Vanishing” lemma.
Lemma 2.2
([33], Lemma 2.1) Let
Then, up to a subsequence, one of the following two possibilities must occur:
(Vanishing): for all
(Nonvanishing): there exist
Lemma 2.3
Let
then
Proof
The proof is motivated by Lemma 1.21 in [30]. Letting
where
Covering
Next, we prove several useful facts about the functionals
Lemma 2.4
Let
Proof
From
where we use condition (V) in the last inequality. Consequently, we have
It follows immediately that
Lemma 2.5
Let
Proof
Since
and
Observe that
Obviously,
which gives
and
Therefore, we have
The proof is ended.□
Lemma 2.6
Let
Proof
Let
Taking
This implies that
By standard elliptic regularity theory, we have
Lemma 2.7
Let
Proof
It is obvious that
For
In parallel with (2.8), we have
From
In view of the aforementioned facts, we can use Hölder’s inequality with
Consequently, (2.7) and (2.10) indicate that
Similarly, we also deduce that
By using (2.8), (2.9), (2.11), and (2.12), we have
In the same way as (2.4), we have
From the Brézis-Lieb type Lemma (see [13], Lemma 2.3), we have
We end the proof by observing that
3 The proofs of Theorems 1.1–1.2
In this section, we first prove the improved Sobolev inequality (1.22), and see Theorem 1.1. Then, we adopt (1.22) to derive a global compactness result for (1.1), and see Theorem 1.2.
We need a lemma, which is vital for the proof of Theorem 1.1.
Lemma 3.1
([24], Theorem D) Suppose that
for all cubes
where
In [32], Yang only gives an outline on the proof of (1.22). Actually, it is particularly difficult to deal with the general case
Proof of Theorem 1.1
For any
Thus, the following inequality holds
where
Case (i):
Take
For any cubes
Second, we verify condition (3.4). From (3.5), it is sufficient to verify
Since
By using Hölder’s inequality and Hardy’s inequality (see Lemma 2.1), we have
From
where
Therefore,
For any
Then, for any
where
Case (ii):
As a consequence of Theorem 1.1, we obtain a useful proposition.
Proposition 3.2
Let
for some constant C, then there exist, up to a subsequence, a family of positive numbers
with
Proof
Taking
where
where
then
where
as
Since
Finally, (3.9) gives
so we have
Benefiting from Proposition 3.2, we can exclude the blowing up bubbles originated from the Hardy-Sobolev critical terms. Moreover, we see that any nonnegative Palais-Smale sequence of
Proposition 3.3
Let
then there exist, up to a subsequence, a family of positive numbers
with
is a Palais-Smale sequence of I at level
Proof
(i) We prove the existence of
in
Similar to (3.10), we can prove that there exists
We claim that
If
If
This proves
(ii) We prove that
then we have
Furthermore, we deduce from
In the same way, we also have
By using (3.12), we have
Then, we learn from (3.13), (3.14), and (3.15) that
A change of variable in (3.16) implies that
Consequently,
(iii) We claim that
Observing that
and it is sufficient to prove
In view of this, we have
In particular, we see that
(iv) Conclusion. By the Brézis-Lieb type Lemma (see [13], Lemma 2.3) and the weak convergence of
Next, we consider the blowing up bubbles caused by the unbounded domains. We have the following.
Proposition 3.4
Let
then there exist, up to a subsequence, a family of
In addition,
is a Palais-Smale sequence of I at level
Proof
(i) We prove the existence of
In a similar fashion to (2.1), we deduce from
Consequently, there exists
By using Lemma 2.2, up to a subsequence, two possible cases must occur:
Vanishing: for all
Nonvanishing: there exist
(3.18)
as
then we derive from
From (3.18), we obtain
This implies that
In the last inequality, we use the fact that
Consequently, we have
(ii) We prove that
By using
Recall that
and
That is,
In the same way, we can prove that
Observing that
Hence,
(iii) We claim that
Observing that
we obtain
(iv) Conclusion. From (3.19), we have
This implies that
At the end of this section, we use Propositions 3.3–3.4 to prove Theorem 1.2.
Proof of Theorem 1.2
Let
Denote
We have
Since
If
Step 1. We exclude the blowing up bubbles originated from the Hardy-Sobolev critical terms. In a similar fashion as Lemma 2.5, we can assume that
then
in
is a Palais-Smale sequence of
If there still exists
then we repeat the previous argument. Since
the iterations must stop after finite times. Hence, there exist a positive constant
with
Step 2. We exclude the blowing up bubbles caused by the unbounded domains. Suppose there exists
then we deduce from Proposition 3.4 that, there exist, up to a subsequence, a family of
with
is a Palais-Smale sequence of
If there still exist
then one can repeat Step 2 for finite times (
with
as
The fact
So we have
We end the proof by combining (3.27) and (3.30) with (3.31).□
4 The proof of Theorem 1.3
In this section, we shall use Theorem 1.2 to prove Theorem 1.3, i.e., the existence of a nontrivial weak solution to equation (1.1). This relies on the following lemma.
Lemma 4.1
(Mountain pass lemma, [2]) Let
There exist
There exists a
where
Then,
We now use Lemma 4.1 to prove the following proposition.
Proposition 4.2
Let
where
Proof
We now verify the conditions of Lemma 4.1. Similar to Lemma 2.1 of [4], we have
By using (1.20) and (4.2), we have
Since
so
we derive that
Let
and
where
Clearly, we have
For the case of
From (1.19), we rewrite
for
Letting
then
Substituting
From (4.5), we see that
On the one hand, we learn from (4.5), (4.6), and
On the other hand, (4.5), (4.6), and (4.7) indicate that
By using (4.7) and (4.8), it holds that
for
for
For the case of
Notice that
Letting
then we have
Direct calculation indicates that
Then
Observing that
From (4.10) and the assumptions that
This proves (4.9).
From (4.4) and (4.9), we have
The proof is ended.□
Proof of Theorem 1.3
Let
Lemma 2.4 implies that
Consequently, it must be that
Since
or
and we can prove that
If
Acknowledgment
The authors would like to thank Professor Gongbao Li for helpful suggestions on the present article.
-
Funding information: This work was supported by National Natural Science Foundation of China (Grant No. 11901531) and the excellent doctorial dissertation cultivation grant (Grant No. 30106220328) from Central China Normal University.
-
Conflict of interest: The authors state no 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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- Periodic solutions to a class of distributed delay differential equations via variational methods
- Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
- Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
- Global Sobolev regular solution for Boussinesq system
- Normalized solutions for the p-Laplacian equation with a trapping potential
- Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
- Blow-up for compressible Euler system with space-dependent damping in 1-D
- High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
- On the dynamics of grounded shallow ice sheets: Modeling and analysis
- A survey on some vanishing viscosity limit results
- Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
- Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
- Front propagation in a double degenerate equation with delay
- Positive solutions for a class of singular (p, q)-equations
- Higher integrability for anisotropic parabolic systems of p-Laplace type
- The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
- On a system of multi-component Ginzburg-Landau vortices
- Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
- Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
- On double phase Kirchhoff problems with singular nonlinearity
- Estimates for eigenvalues of the Neumann and Steklov problems
- Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
- Dirichlet problems involving the Hardy-Leray operators with multiple polars
- Incompressible limit for compressible viscoelastic flows with large velocity
- Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
- Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
- Noncoercive parabolic obstacle problems
- Touchdown solutions in general MEMS models
- Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
- Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
- Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
- Symmetries of Ricci flows
- Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
- On the topological gradient method for an inverse problem resolution
- Supersolutions to nonautonomous Choquard equations in general domains
- Uniform complex time heat Kernel estimates without Gaussian bounds
- Global existence for time-dependent damped wave equations with nonlinear memory
- Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
- Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
- Lamé system with weak damping and nonlinear time-varying delay
- Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
- Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
- Boundary regularity results for minimisers of convex functionals with (p, q)-growth
- Parametric singular double phase Dirichlet problems
- Special Issue on Nonlinear analysis: Perspectives and synergies
- Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
- Identification of discontinuous parameters in double phase obstacle problems
- Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
- On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
- On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
- The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
- Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
- On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
- Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
