The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
-
Guotao Wang
Abstract
In this article, we first introduce a new fractional
where
1 Introduction
In recent years, fractional Laplacian operator has attracted more and more attentions. It has many applications, such as anomalous diffusion, spectral graph theory, quasi-geostrophic flows, minimal surfaces, water waves, error analysis of images, and relativistic quantum mechanics of stars [2,12,18,19,24]. The fractional Laplacian operator has also become one of the powerful tools to describe large-scale irregular diffusion phenomena such as dislocation phenomena in dynamical systems, combustion phenomena and disorder phenomena in crystal structures, etc. [7,25]. In 2019, Bonder and Salort introduced the fractional
where
The properties of
Many results have been obtained regarding the fractional
The operator
Problems with fractional Monge-Ampère operator [8], introduced by Caffarelli and Charro, appear in areas such as analysis, geometry, and applied mathematics. The research about nonlinear problems with the fractional Monge-Ampère operators is a hot problem in domestic and international research. Many results involving fractional Monge-Ampère have been derived. For instance, Dai and Li obtained the equivalence condition of the existence of subsolutions to the equation involving the Monge-Ampère operator [16]. Radially symmetric boundary value problems for solutions to the equation involving the Monge-Ampère operator for real and complex ellipsoids were studied by Delanoë [17]. However, the nonlocality of fractional Monge-Ampère operator adds difficulties to the study, and various feasible methods have been introduced by domestic and international experts and scholars to solve this challenge. Chen and Li introduced the direct method of moving planes [11], which acts directly on the fractional Monge-Ampère operator and is more efficient and convenient for studying nonlinear problems with fractional Monge-Ampère operator. Therefore, using the direct method of moving planes to solve nonlinear problems with fractional Laplacian operator and fractional Monge-Ampère operator has attracted increasing attention of scholars at home and abroad, and a number of fruitful results were obtained. For example, the radial symmetry of the standing wave solutions of the nonlinear fractional Hardy-Schrödinger equation is derived through the use of direct method of moving planes [10]. For more literature about the direct methods of moving planes, please see [13,20,21,32].
The first primary purpose of this study is to introduce a new operator called fractional
with
where
Central to the innovations of our consequence are that we applied different methods to overcome the nonlinearity of the operator. Therefore, the obtained results encompass a wide range and can be applied to non-local operators. And it is worth mentioning that it allows for more general behavior than powers, a case in point is that
We further emphasize that the specificity of the properties of
In this article, we can use the direct method of moving planes to derive various properties of solutions to equations involving the fractional
Theorem 1.1
(The maximum principle) Assume that
Then
Furthermore, if there exists a point in
Notations. Let
Denote
we have
The specific location relationship is as follows:
For convenience, we will replace
Theorem 1.2
(The maximum principle for anti-symmetric function) Let
where
Furthermore, if there exists a point x in T such that
For
Theorem 1.3
(The narrow region principle) Suppose that the bounded narrow region
where
These conclusions are also valid in unbounded domain if we further assume that
Theorem 1.4
(Decay at infinity) Assume that
with
where
then
We can derive various properties of solutions involving the operator
In the article, the following inequality
plays an important role in the proof of the theorems. Next, we consider the following ellipsoid region:
where
Theorem 1.5
Suppose that
where f is a Lipschitz continuous function. Then v must be radially symmetric.
In the following, we extend the range of
Theorem 1.6
Assume that
Suppose for some
and
Then v must be radially symmetric.
We reduce some of the restrictions of
Theorem 1.7
Assume that
Suppose that
Then v must be radially symmetric.
Next, we consider the space
We are able to prove the following Theorems 1.8 and 1.9 under different conditions.
Theorem 1.8
Assume that
Suppose for some
and
where f is a Lipschitz continuous function, and
We reduce some of the restrictions of
Theorem 1.9
Assume that
where f is a Lipschitz continuous function, and
Then
Lemma 1.1
(Strong maximum principle) Assume that
Theorem 1.10
Suppose that
Let the Lipschitz continuous function f satisfy
Then
Theorem 1.11
(Maximum principle in unbounded open sets) Let
where
for
Then
Lemma 1.2
Assume that
and
Suppose f satisfies
Next, we present the following equation:
To make sure that the proof goes on well, we provide the following assumptions about
Lemma 1.3
Let
Theorem 1.12
Assume that
Suppose that f is a continuous function satisfying assumption (a), (b), and (c). Then,
In Section 2, we give detailed proofs of Theorems 1.1 to 1.4. In Section 3, we use the direct method of moving planes to prove Theorems 1.5–1.9, and we derive that the solution is symmetric about a point through the coordinate plane when the solution is on a given ellipsoid region or
2 The maximum principle
In this section, we give specific proofs of the above maximum principles, the narrow region principle, and the decay estimates at infinity concerning the operator
2.1 Proof of Theorem 1.1
Assume that condition (1.4) is not correct. Because
According to (1.3), we have
It follows from (2.2) that
Since
By relevant calculations, we have
Thus,
It is obvious that
By using the properties of the
This creates the contradiction. Therefore, we can derive that
Further, we assume that there exists a point
Since
let
In the unbounded domain, we give the condition
to make sure that the minimum of
In the following, we will give specific proof of Theorem 1.2.
2.2 Proof of Theorem 1.2
Assume that condition (1.7) is not correct. Because
According to the condition (1.6) we can derive
We have
where
Therefore, we derive
Similar to (2.3), we have
Then, from (2.4), we can obtain
According to the fact that
when
This leads to the contradiction.
Assume that
According to the known condition that
in (1.6), we have
when
In the unbounded domain, we present the condition
to make sure that the minimum of
2.3 Proof of Theorem 1.3
Assume that condition (1.9) is not correct. Because
According to the condition (1.8) that
Let
Then, we have
Notice that
This leads to the contradiction.
In the unbounded domain, we give the condition
to make sure that the minimum of
2.4 Proof of Theorem 1.4
If we assume that there is some point
For each fixed
Then, we have
Since
When
By summarizing the aforementioned, we can derive that there exists a constant
then
3 Symmetry and nonexistence of positive solutions
In this section, we will use the direct method of moving planes to derive specific proofs of the symmetry for solutions about equation (1.1) in the ellipsoid region
3.1 Proof of Theorem 1.5
In the following proof, we choose the
Step 1. Let
First, we prove that when
Obviously, we have
where
Because
By (1.5) and Theorem 1.3, we proved in Section 2, we can see that (3.1) is established. This is the starting point of moving plane.
Step 2. Define
Under the condition that (3.1) holds, the plane
We assume that (3.4) does not hold, then we have
First, we consider the region
Thus for any
Since
It is obvious that
This together with (3.5) indicates that for all
This yields a confliction. Thus,
Now, we move the plane
Combining this with (3.6), we obtain that
Indeed, if we move the plane vertically along the other
3.2 Proof of Theorem 1.6
Step 1. We start with the case
Through the mean value theorem, we have
where
Let
Since
and
Therefore, the conclusion is obtained easily that
then
Combining Theorem 1.4, if
That is when
Step 2. We can define
Under condition (3.7), we move the plane
We assume that (3.11) does not hold, then we can continue to move the plane
where
It remains to show (3.12). Since
When
where A is given in (3.9). There exists
For sufficiently small
According to (3.13), (3.14), and the conclusion we derived in step 1, we have (3.12) holds, which is a contradiction to (3.10). Thus, (3.11) is correct.
If we move the plane vertically along the other
3.3 Proof of Theorem 1.7
Step 1. We are able to verify that when
Assume that the condition (3.15) is not correct, we can derive that there is a point
By using the mean value theorem, we have
where
On the other hand, by (2.6), we have
which yields a confliction with (3.16). Therefore, we prove the inequality (3.15) presented in the Step 1.
Step 2. Define
Under condition (3.15), the plane
We assume that (3.17) does not hold, then we can continue to move the plane
where
If there exists one point
Then for all
For all sufficiently small
For sufficiently small
According to (3.19), (3.20) and the conclusion we derived in step 1, we have (3.18) holds, which is a contradiction to the definition of
Therefore, if we move the plane vertically along the other
3.4 Proof of Theorem 1.8
First, we prove either
Case 1. If there exists a point
Taking
Combining the properties of
Case 2. Suppose
Step 1. When
This provides the starting point of moving planes.
Step 2. Define
Keep moving the plane
Suppose
Let
Since
this conflicts with hypothesis
Since inequality (3.21) holds for all
3.5 Proof of Theorem 1.9
Combining the condition
Step 1. When
From this, we obtain the starting point of the moving planes.
Step 2. Define
We are able to prove that
Suppose
We can prove that
4 Monotonicity
In this section, by the direct method of moving planes, we prove monotonicity result.
4.1 Proof of Lemma 1.1
Since
Therefore,
4.2 Proof of Theorem 1.10
Because
Next, we will prove
Step 1. We will prove that there exists
Assume that the condition (4.2) is not correct,that is, there exists a sequence
Thus, we can derive that there exists a point
Therefore, by combining with (1.25), we have
where
When
which contradicts with (4.3). Thus, there exists
By combining with Lemma 1.1, we can obtain
Step 2. Define
In this step, we show that
Assume that the aforementioned condition above is not correct, then
In fact, according to
Because
in which
where
When
Assume that the condition (4.13) is false, that is, there exists a point
Thus, there exists one point
Therefore, by combining (1.25), we obtain
where
Obviously,
which contradicts with (4.15). Thus, when
Therefore,
where
Therefore,
This is the proof of Theorem 1.10.
5 Asymptotic property
In this part, we will apply the maximum principles which are introduced in Section 1 to investigate the asymptotic property of solutions.
5.1 Proof of Theorem 1.11
Suppose the conclusion is not correct, that is, there exists a point
There exists a sequence
Let
Easily we can derive that
Define
Take
Therefore, there exists
Further, we can obtain that
Then, by (5.5), we can derive
where
Subsequently, the lower bound of
In fact, we can derive
By combining (5.7) and (5.8), we derive
Then
Since
5.2 Proof of Lemma 1.2
We first show that
Combining the equation (1.28) given in Lemma 1.2, when
From (1.27), we can obtain that
Then, we can conclude that
5.3 Proof of Lemma 1.3
Proof
Denote the principle eigenvalue of
with
By assumption (b), for any
Assume that there exists a point
Then, we can derive
For
Then, we will prove
Assume that the condition (5.17) is false, that is, there exist
Since
which is contradictory to (5.18). Therefore, we can obtain that (5.17) is valid. When
Since
Now, we prove Theorem 1.12. From Lemma 1.2, we can obtain that
5.4 Proof of Theorem 1.12
According to the assumption (a) and
For
Let
Because
Let
be the maximum point. Since
According to the definition of
By combining (5.21), we obtain
Also, we can derive that
By combining (5.20), (5.22), and (5.23), we have
Thus, when
The aforementioned is the proof of Theorem 1.12.
Acknowledgements
The authors would like to express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions that led to the improvement of the original manuscript.
-
Funding information: The work was supported by National Natural Science Foundation of China (No. 12001344), NSF of Shanxi Province, China (No. 20210302123339) and Postgraduate Education Innovation Program of Shanxi Normal University, China (No. 2024YJSKCSZSFK-06).
-
Author contributions: Guotao Wang: writing – review and editing, supervision, and funding acquisition. Rui Yang: writing – review and editing and writing – original draft. Lihong Zhang: conceptualization, and supervision.
-
Conflict of interest: The authors have no relevant financial or non-financial interests to disclose.
-
Data availability statement: No data were used for the research described in the article.
References
[1] A. Alberico, A. Cianchi, L. Pick, and L. Slavíková, On the limit as s→0+ of fractional Orlicz-Sobolev spaces, J. Fourier Anal. Appl. 26 (2020), 80. 10.1007/s00041-020-09785-zSearch in Google Scholar
[2] S. Bhattarai, On fractional Schrodinger systems of Choquard type, J. Differential Equations 263 (2017), 3197–3229. 10.1016/j.jde.2017.04.034Search in Google Scholar
[3] S. Bahrouni and H. Ounaies, Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems, Discrete Contin. Dyn. Syst. 40 (2020), 06584. 10.3934/dcds.2020155Search in Google Scholar
[4] S. Bahrouni, H. Ounaies, and A. Salort, Variational eigenvalues of the fractional g-Laplacian, Complex Var. Elliptic Equ. 68 (2023), 1021–1044. 10.1080/17476933.2022.2034152Search in Google Scholar
[5] J. F. Bonder and A. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal. 277 (2019), 333–367. 10.1016/j.jfa.2019.04.003Search in Google Scholar
[6] J. F. Bonder, A. Salort, and H. Vivas, Interior and up to the boundary regularity for the fractional g-Laplacian: the convex case, Nonlinear Anal. 223 (2022), 113060. 10.1016/j.na.2022.113060Search in Google Scholar
[7] G. Chaussonnet, S. Braun, L. Wieth, and H. Bauer, Influence of particle disorder and smoothing length on SPH operator accuracy, 10th International SPHERIC Workshop, Italy, 2015, pp. 16–18. Search in Google Scholar
[8] L. Caffarelli and F. Charro, On a fractional Monge-Ampère operator, Ann PDE 42 (2015), 1–4. 10.1007/s40818-015-0005-xSearch in Google Scholar
[9] T. Cheng, G. Huang, and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19 (2017), 1750018. 10.1142/S0219199717500183Search in Google Scholar
[10] W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math 335 (2018), 735–758. 10.1016/j.aim.2018.07.016Search in Google Scholar
[11] W. Chen, C. Li, and Y. Li, A direct method of moving planes for fractional Laplacian, Adv. Math. 308 (2017), 404–437. 10.1016/j.aim.2016.11.038Search in Google Scholar
[12] A. Córdoba and D. Martlllnez, A pointwise inequality for fractional laplacians, Adv. Math. 280 (2015), 79–85. 10.1016/j.aim.2015.02.018Search in Google Scholar
[13] M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Discrete Contin Dynam Syst A 38 (2018), 4603–4615. 10.3934/dcds.2018201Search in Google Scholar
[14] W. Chen and L. Wu, A maximum principle on unbounded domains and a Liouville theorem for fractional p-harmonic functions. arxiv, 2019, 09986. Search in Google Scholar
[15] W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equ. 260 (2016), 4758–4785. 10.1016/j.jde.2015.11.029Search in Google Scholar
[16] L. Dai and H. Li, Entire subsolutions of Monge-Ampère type equations, Commun. Pure Appl. Anal. 19 (2020), 19–30. 10.3934/cpaa.2020002Search in Google Scholar
[17] P. Delanoë, Radially symmetric boundary value problems for real and complex elliptic Monge-Ampère equations, J. Differ. Equ. 58 (1985), 318–344. 10.1016/0022-0396(85)90003-8Search in Google Scholar
[18] A. Fiscella, P. Pucci, and S. Saldi, Existence of entire solutions for Schrödinger-Hardy systems involving two fractional operators, Nonlinear Anal. 158 (2017), 109–131. 10.1016/j.na.2017.04.005Search in Google Scholar
[19] Z. Guo, S. Luo, and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl 446 (2017), 681–706. 10.1016/j.jmaa.2016.08.069Search in Google Scholar
[20] B. Gidas and L. Nirenberg, Symmetry of positive solutions of nonlinear Monge-Ampère equations, Adv. Math. 7 (1981), 369–402. Search in Google Scholar
[21] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian system, Nonlinear Anal. 146 (2016), 120–135. 10.1016/j.na.2016.08.022Search in Google Scholar
[22] M. Lai and W. Wei, Gelfand problem and hemisphere rigidity, Commun. Pure Appl. Anal. 22 (2023), 1226–1238. 10.3934/cpaa.2023026Search in Google Scholar
[23] S. Molina, H. Vivas, and A. Salort, Maximum principles, Liouville theorem and symmetry results for the fractional g-Laplacian, Nonlinear Anal. 212 (2021), 112465. 10.1016/j.na.2021.112465Search in Google Scholar
[24] P. Ma and J. Zhang, Existence and multiplicity of solutions for fractional Choquard equations, Nonlinear Anal. 164 (2017), 100–117. 10.1016/j.na.2017.07.011Search in Google Scholar
[25] J. L. Vázquez, Nonlinear Diffusion with Fractional Laplacian Operators, Springer Verlag, Germany, 2010, pp. 271–298. 10.1007/978-3-642-25361-4_15Search in Google Scholar
[26] A. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, J. Differential Equations 26 (2020), 5413–5439. 10.1016/j.jde.2019.11.027Search in Google Scholar
[27] A. Salort, B. Schvager, and A. Silva, Nonstandard growth optimization problems with volume constraint, Differ. Integral Equ. 36 (2023), 573–592. 10.57262/die036-0708-573Search in Google Scholar
[28] A. Salort and H. Vivas, Fractional eigenvalues in Orlicz spaces with no Δ2 condition, J. Differ. Equ. 327 (2022), 166–188. 10.1016/j.jde.2022.04.029Search in Google Scholar
[29] G. Wang, X. Ren, Z. Bai, and W. Hou, Radial symmetry of standing waves for nonlinearfractional Hardy-Schrödinger equation, Appl. Math. Lett. 96 (2019), 131–137. 10.1016/j.aml.2019.04.024Search in Google Scholar
[30] L. Zhang, B. Ahmad, G. Wang, and X. Ren, Radial symmetry of solution for fractional p-Laplacian system, Nonlinear Analysis 196 (2020), 111801. 10.1016/j.na.2020.111801Search in Google Scholar
[31] L. Zhang and W. Hou, Standing waves of nonlinear fractional p-Laplacian Schöodinger equation involving logarithmic nonlinearity, Appl. Math. Lett. 102 (2020), 106149. 10.1016/j.aml.2019.106149Search in Google Scholar
[32] L. Zhang, W. Hou, J. J. Nieto, and G. Wang, An anisotropic tempered fractional p-Laplacian model involving logarithmic nonlinearity, Evol. Equ. Control Theory 13 (2024), no. 1, 1–11. https://doi.org/10.3934/eect.2023033. Search in Google Scholar
© 2024 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
- 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”
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”
