Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
-
Yunyun Hu
Abstract
In this article, we establish some reverse weighted Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. We then show the existence of extremal functions for the above inequalities by combining the subcritical approach and the renormalization method.
1 Introduction
Stein and Weiss [34] established the following weighted Hardy-Littlewood-Sobolev (HLS) inequality:
where
It is known as the Stein-Weiss inequality. A fundamental task in understanding Stein-Weiss inequality (1.1) is to study the sharp constant and existence of extremals. For
The Stein-Weiss inequality (1.1) contains the following HLS inequality (
where
The HLS inequality was intensively studied since it plays, important role in many geometric problems by virtue of the complete knowledge of sharp constants and extremal functions. For example, the existence of extremal functions of HLS inequality (1.2) was considered on the Euclidean space [4,18,19,30], on Riemannian manifolds [24], and on the Heisenberg group [17].
In a ground-breaking work [20], Frank and Lieb obtained the sharp constant and the explicit form of extremal functions for Folland-Stein inequality on the Heisenberg group [17]. We recall some basic results concerning the Heisenberg group. The Heisenberg group
where
and the distance between
The natural dilation of the Heisenberg group is given by
Throughout this article, the homogeneous dimension of
with the equality if and only if
for some
More recently, Hang and Wang [25] gave a simpler proof of inequality (1.4), which bypasses the sophisticated proof for the existence of minimizers and Hersch-type arguments.
The
where
Moreover, they proved the following
for some constant
Recently, Han and Zhang [23] established the reverse version of inequality (1.4), i.e., for
The sharp constant satisfies
where
Different from the proof of [20], Han and Zhang proved the existence of extremal functions for inequality (1.7) by using the subcritical approach and renormalization method.
In the last two decades, various extensions of HLS inequality have been investigated. We focus on the typical examples of reverse HLS inequalities with weights. Chen et al. [6] established the following reverse Stein-Weiss inequality:
where
They obtained the existence of extremal functions for inequality (1.8). Later, Chen et al. [5] established reverse Stein-Weiss inequalities on the upper half space and proved the existence of their extremal functions. Dai et al. [8] obtained the sharp reversed HLS inequality with an extended kernel on the upper half space, and computed the sharp constants. More recently, Dou et al. [13] proved the reversed HLS inequality with vertical weights on the upper half space and discussed the extremal functions by the renormalization method. For more results about (weighted) HLS-type inequalities, we refer the reader to [9,10,14,15,21,31,32] and the references therein.
In contrast, not much is known about reverse HLS-type inequalities on the Heisenberg group or the CR sphere. The main motivation of this article is to introduce the reverse HLS inequality with weights on the Heisenberg group. We first obtain the following reverse HLS-type inequality with
Assume that
where
Theorem 1.1
Let
holds. Moreover, the upper and lower bounds of constant
where
and
Define operators
and
By duality and the reversed Hölder’s inequality, we readily check that the inequality (1.10) is equivalent to the following two inequalities.
Corollary 1.2
Let
or
holds for any nonnegative function
We also obtain an improved form of inequality (1.10) without assumptions
Theorem 1.3
Let
if one of the following conditions is satisfied:
and for case
where
To prove Theorems 1.1 and 1.3, we adopt the standard approach, based on the reverse integral version of Hardy inequalities on the Heisenberg group (see [29]). The reverse integral version of Hardy inequalities on the upper half space was established by Chen et al. [5] and Tao [35]. Once we establish inequality (1.10), it is natural to consider the attainability of sharp constant
For the case
We obtain the following result for the extremal problem of inequality (1.10), which, to the best of our knowledge, is new.
Theorem 1.4
Let
Remark 1.5
Note that for
where
For
In this article, we focus on the less investigated inequality (1.10) and the existence of extremal functions. We verify the existence of minimizers for inequality (1.10) by combining the subcritical approach (see [10,16]) and the renormalization method (see [13,23,36]). We cannot simply follow the same line of [13,16,23,36] to prove the existence of extremal functions for inequality (1.10) due to the loss of translation invariance of reverse weighted HLS inequality and the complex structure of the Heisenberg group. To overcome these difficulties, we use the subcritical approach to construct the extremal sequences of inequality (1.15), then combine the renormalization method and rotation invariance to obtain the expected attainability of the best constant. We believe this method can also be applied to obtain the existence of extremal function for several other weighted integral inequalities on the Heisenberg group.
We also investigate the reverse Stein-Weiss inequality in the Heisenberg group.
Theorem 1.6
Let
if one of the following conditions is satisfied:
and for case
where
where
Note that for
Similar to Theorem 1.1, we can also prove Theorem 1.6. Hence, we omit the details of the proof of Theorem 1.6.
Define operators
It is easy to verify that the inequality (1.16) is equivalent to the following two inequalities.
Corollary 1.7
Under the hypothesis of Theorem
1.6, there exists a constant
or
holds for any nonnegative function
This article is organized as follows: Section 2 is devoted to establishing the reverse weighted HLS inequality (1.10); in Section 3, combining the subcritical approach with the renormalization method, we prove the existence of extremal function of the reverse weighted HLS inequality (1.10).
Throughout the article, positive constants are denoted by
2 The reverse weighted HLS inequality on the Heisenberg group
In this section, we prove the reverse weighted HLS inequalities on the Heisenberg group. The following lemma is the reverse integral version of Hardy inequalities on general homogeneous groups in the study by Kassymov et al. [29], which will be crucial in our proof of Theorem 1.2.
Lemma 2.1
[29] Let
holds for some
Moreover, the biggest constant
In addition, inequality
holds for some
Moreover, the biggest constant
Remark 2.2
Note that for
Proof of Theorem 1.1
By reversed Hölder’s inequality, we have
Let
Since
we have
Therefore, we obtain
Similarly, one has
Combining equations (2.5) and (2.6), we have
Hence, it suffices to show that
We divide into two cases to discuss.
(1) If
Then, for any
This implies that
so that
Therefore, we have
Set
where
That is,
where
Moreover, since
Then, we obtain
Combining equations (2.9) and (2.10) yields
where in the second equality we have used the fact
(2) If
where
Let us verify the condition (2.4). Using equation (1.9), we have
and
Combining equations (2.12) and (2.13), we know that
and hence, equation (2.11) is proved.
Now we give the estimate of constant
where
This completes the proof of Theorem 1.1.□
3 Existence of extremal functions for reverse HLS inequality with horizontal weights
This section is devoted to proving the existence of extremal functions for inequality (1.10) by combining the subcritical approach and the renormalization method.
To better illustrate our results, we first introduce some well-known notations. The sphere
The standard Euclidean volume element of
The Heisenberg group
and with the inverse
The Jacobian determinant of this transformation is given by
which implies that
Under Cayley transform, a simple computation gives
where
For any
such that
respectively.
For
where
We define the extremal function of inequality (3.1) as
Then, we have
Theorem 3.1
3.1 Subcritical reverse weighted HLS inequality on CR sphere
The aim of this subsection is to establish the subcritical reverse weighted HLS inequality on
Theorem 3.2
Assume
For
We can easily see that the inequality (3.2) is equivalent to
for any nonnegative function
Now we consider the extremal problem of inequality (3.2):
By equations (3.2) and (3.4), we know that
The following proposition will be used in the proof of Theorem 1.4.
Proposition 3.3
Furthermore,
Proof
(i). We first show that
We choose a pair of nonnegative minimizing sequence
and
We divide the proof of part (i) into three steps.
Step 1. Our first step is to claim that
Indeed, we know from equation (1.11) that there exist constants
Then, using reversed Hölder’s inequality, we obtain
Since
where
We now choose
which helps to conclude that
Hence, there exists
From this, we know that
Step 2. There exist two subsequences of
Without loss of generality, we may assume that
Since
Step 3: We show that
By equation (3.7) and the interpolation inequality, we obtain
It follows that
Then, for any fixed
and for any fixed
Thus, as
Similar to the arguments in the study by Dou et al. [11], we know that the above convergences are uniformly convergent for all
Therefore, for any
where in the last inequality, we have used Hölder’s inequality. Noting that
we arrive at
From the above convergence and equation (3.11), we obtain
Again by Hölder’s inequality, we have
We can then readily check that
Combining the preceding estimate with equation (3.4) gives
Therefore, we deduce that
(ii). We show that
To see this, we first prove
where in the last equality, we have used the mean value theorem. Then, applying equation (3.12) and Fatou’s lemma, we have
Now we claim
Then, using equation (3.12), we derive
This gives us a contradiction when
(iii). We prove
Since
Let
Using the boundedness of
where
3.2 Minimizer of reverse weighted HLS inequality on CR sphere
In this subsection, we prove Theorem 3.1. The method we shall use is similar to that used by Dou et al. [13], but we have to set up a framework to fit the complex structure of Heisenberg group. We first give the following lemma, which is crucial in our proof.
Lemma 3.4
Let
and the minimizer pair
Proof
Set
This implies
Let
Define
Then, by equation (3.17) and dominated convergence theorem, we obtain
Letting
Therefore, by equations (3.16) and (3.18), we have
Then, using Hölder’s inequality yields
Hence, we obtain equation (3.15).□
Now we are ready to give the proof of Theorem 3.1.
Proof of Theorem 3.1
We start our proof by letting
There are two possible cases as follows.
Case 1. For some subsequences
By repeating the same argument as equations (3.6) and (3.14), we deduce that
This implies
Then, combining equation (3.5) with Lemma 3.4 gives
as
Case 2. For any subsequences
Case (i).
From this and equation (3.5), we have
Then, applying Cayley transform and dilation on
Now we take
By a direct computation, one finds that
Note that
We claim that there exists a constant
Once the inequality (3.24) holds and by equations (1.3) and (3.22) and polar decomposition, we find
where
In the last part of Case (i), we prove that the inequality (3.24) holds. Using equations (3.19) and (3.21) gives
This implies
Then, there exist
with
On the one hand, for any
It follows that there exists a constant
Similarly, there exists a constant
On the other hand, we know that
To simplify, we write
It follows that for any
Combining equations (3.26) and (3.29), we obtain equation (3.24).
Case (ii)
Case (iii).
Therefore,
Repeating the arguments as equation (3.24), we also deduce that there exist constants
Then, for any given
Thus,
A similar argument as equation (3.14) also shows that
Since
Since
and
uniformly on any compact domain, we know from equation (3.30) that
Similarly, we have
Now we set
and
Since
Therefore,
Acknowledgments
The authors would like to thank the referee for his/her careful reading of the manuscript and valuable suggestions.
-
Funding information: This project is supported by the National Natural Science Foundation of China (Grant No. 12101380), China Postdoctoral Science Foundation (Grant No. 2021M700086), Youth Innovation Team of Shaanxi Universities, and the Fundamental Research Funds for the Central Universities (Grant Nos. GK202307001 and GK202202007).
-
Conflict of interest: The authors state that there is no conflict of interest.
References
[1] W. Beckner, Sharp inequalities and geometric manifolds, in: J. Fourier Anal. Appl. 3 (1997), 825–836, Special Issue: Proceedings of the Conference Dedicated to Professor Miguel de Guzmán, El Escorial, 1996. 10.1007/BF02656488Search in Google Scholar
[2] W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math. 20 (2008), 587–606. 10.1515/FORUM.2008.030Search in Google Scholar
[3] W. Beckner, Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008), 1871–1885. 10.1090/S0002-9939-07-09216-7Search in Google Scholar
[4] E. Carlen and M. Loss, Extremals of functionals with competing symmetries, J. Funct. Anal. 88 (1990), 437–456. 10.1016/0022-1236(90)90114-ZSearch in Google Scholar
[5] L. Chen, G. Lu, and C. Tao, Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals, Adv. Nonlinear Stud. 19 (2019), 475–494. 10.1515/ans-2018-2038Search in Google Scholar
[6] L. Chen, Z. Liu, G. Lu, and C. Tao, Reverse Stein-Weiss inequalities and existence of their extremal functions, Trans. Amer. Math. Soc. 370 (2018), 8429–8450. 10.1090/tran/7273Search in Google Scholar
[7] L. Chen, G. Lu, and C. Tao, Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, J. Funct. Anal. 277 (2019), 1112–1138. 10.1016/j.jfa.2019.01.002Search in Google Scholar
[8] W. Dai, Y. Hu, and Z. Liu, Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel, Studia Math. 271 (2023), 1–38, https://doi.org/10.4064/sm220323-26-1. Search in Google Scholar
[9] J. Dou, Weighted Hardy-Littlewood-Sobolev inequalities on the upper half space, Comm. Cont. Math. 18 (2016), 1550067. 10.1142/S0219199715500674Search in Google Scholar
[10] J. Dou, Q. Guo, and M. Zhu, Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math. 312 (2017), 1–45. 10.1016/j.aim.2017.03.007Search in Google Scholar
[11] J. Dou, Q. Guo, and M. Zhu, Negative power nonlinear integral equations on bounded domains, J. Differential Equations 269 (2020), 10527–10557. 10.1016/j.jde.2020.07.021Search in Google Scholar
[12] X. Han, Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group, Indiana Univ. Math. J. 62 (2013), 737–751. 10.1512/iumj.2013.62.4976Search in Google Scholar
[13] J. Dou, Y. Hu, and J. Ma, Reversed Hardy-Littlewood-Sobolev inequalities with vertical weights on the upper half space, preprint, arXiv:2311.03933.Search in Google Scholar
[14] J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. 3 (2015), 651–687. 10.1093/imrn/rnt213Search in Google Scholar
[15] J. Dou and M. Zhu, Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. 19 (2015), 9696–9726. 10.1093/imrn/rnu241Search in Google Scholar
[16] J. Dou and M. Zhu, Integral equations on bounded domains, J. Funct. Anal. 277 (2019), 111–134. 10.1016/j.jfa.2018.05.020Search in Google Scholar
[17] G. B. Folland and E. M. Stein, Estimates for the ∂¯b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. 10.1002/cpa.3160270403Search in Google Scholar
[18] R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations 39 (2010), 85–99. 10.1007/s00526-009-0302-xSearch in Google Scholar
[19] R. L. Frank and E. H. Lieb, A new rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, spectral theory, function spaces and inequalities, in: B. M. E. A Brown, ed., Oper. Theory Adv. Appl. vol. 219, Birkh auser, Basel, 2012, pp. 55–67. 10.1007/978-3-0348-0263-5_4Search in Google Scholar
[20] R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. Math. 176 (2012), 349–381. 10.4007/annals.2012.176.1.6Search in Google Scholar
[21] M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal. 278 (2020), no. 1, 108082, 46 pp. 10.1016/j.jfa.2018.08.012Search in Google Scholar
[22] X. Han and G. Lu, Stein-Weiss inequalities and integral systems on the Heisenberg group, Nonlinear Anal. 75 (2012), 4296–4314. 10.1016/j.na.2012.03.017Search in Google Scholar
[23] Y. Han and S. Zhang, Reversed Hardy-Littlewood-Sobolev inequality on Heisenberg group Hn and CR sphere S2n+1, arXiv:2202.09241. Search in Google Scholar
[24] Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications, J. Differential Equations 260 (2016), 1–25. 10.1016/j.jde.2015.06.032Search in Google Scholar
[25] F. Hang and X. Wang, A simpler proof of Frank and Lieb’s sharp inequality on the Heisenberg group, arXiv: 2211.10301v1. Search in Google Scholar
[26] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, Math. Z. 27 (1928), 565–606. 10.1007/BF01171116Search in Google Scholar
[27] I. W. Herbst, Spectral theory of the operator (p2+m2)1⁄2−Ze2⁄r2, Comm. Math. Phys. 53 (1977), no. 3, 285–294. 10.1007/BF01609852Search in Google Scholar
[28] A. Kassymov, M. Ruzhansky, and D. Suragan, Reverse Stein-Weiss, Hardy-Littlewood-Sobolev, Hardy, Sobolev and Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups, Forum Math. 34 (2022), 1147–1158. 10.1515/forum-2021-0110Search in Google Scholar
[29] A. Kassymov, M. Ruzhansky, and D. Suragan, Reverse integral Hardy inequality on metric measure spaces, Ann. Fenn. Math. 47 (2022), 39–55. 10.54330/afm.112455Search in Google Scholar
[30] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), 349–374. 10.2307/2007032Search in Google Scholar
[31] Q. A. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on Rn, Israel J. Math. 220 (2017), 189–223. 10.1007/s11856-017-1515-xSearch in Google Scholar
[32] Q. A. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space R+n, Int. Math. Res. Not. 2017 (2017), no. 20, 6187–6230. 10.1093/imrn/rnw108Search in Google Scholar
[33] S. L. Sobolev, On a theorem in functional analysis (in Russian), Mat. Sb 4 (1938), 471–497. Search in Google Scholar
[34] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. 10.1512/iumj.1958.7.57030Search in Google Scholar
[35] C. Tao, Reversed Stein-Weiss inequalities with Poisson-type kernel and qualitative analysis of extremal functions, Adv. Nonlinear Stud. 21 (2021), 167–187. 10.1515/ans-2020-2112Search in Google Scholar
[36] S. Zhang and Y. Han, Rearrangement free method for Hardy-Littlewood-Sobolev inequalities on Sn, Anal. Theory Appl. 3x (2022), 1–26. Search in Google Scholar
© 2023 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Regular Articles
- On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
- On the critical Choquard-Kirchhoff problem on the Heisenberg group
- On the local behavior of local weak solutions to some singular anisotropic elliptic equations
- Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
- Double-phase parabolic equations with variable growth and nonlinear sources
- Logistic damping effect in chemotaxis models with density-suppressed motility
- Bifurcation diagrams of one-dimensional Kirchhoff-type equations
- Standing wave solution for the generalized Jackiw-Pi model
- Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
- Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
- Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
- Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
- Bautin bifurcation with additive noise
- Small solitons and multisolitons in the generalized Davey-Stewartson system
- Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
- A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
- On a strongly damped semilinear wave equation with time-varying source and singular dissipation
- Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
- Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
- Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
- Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
- Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
- Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
- Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
- Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
- On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
- Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
- Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
- Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
- Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
- Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
- Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
- Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
- Approximations of center manifolds for delay stochastic differential equations with additive noise
- 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
Articles in the same Issue
- Regular Articles
- On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
- On the critical Choquard-Kirchhoff problem on the Heisenberg group
- On the local behavior of local weak solutions to some singular anisotropic elliptic equations
- Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
- Double-phase parabolic equations with variable growth and nonlinear sources
- Logistic damping effect in chemotaxis models with density-suppressed motility
- Bifurcation diagrams of one-dimensional Kirchhoff-type equations
- Standing wave solution for the generalized Jackiw-Pi model
- Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
- Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
- Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
- Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
- Bautin bifurcation with additive noise
- Small solitons and multisolitons in the generalized Davey-Stewartson system
- Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
- A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
- On a strongly damped semilinear wave equation with time-varying source and singular dissipation
- Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
- Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
- Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
- Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
- Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
- Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
- Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
- Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
- On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
- Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
- Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
- Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
- Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
- Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
- Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
- Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
- Approximations of center manifolds for delay stochastic differential equations with additive noise
- 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
