Ground state solutions for magnetic Schrödinger equations with polynomial growth
-
Yan Wu
Abstract
In this article, we investigate the following nonlinear magnetic Schrödinger equations:
where
1 Introduction
1.1 Background and motivations
In this article, we are concerned with the following magnetic Schrödinger equations:
where
where
Over the past few decades, significant progress has been made in understanding the existence, multiplicity, and dynamic behavior of the Schrödinger system without magnetic fields (i.e.,
which has been extensively investigated in the literature based on various assumptions on the potentials
with exponential growth, and they proved the existence of ground state solutions both in the indefinite case with the nonlinear function is subcritical exponential growth and in the definite case with the nonlinear function is critical exponential growth. To overcome the difficulty brought about by the presence of a magnetic field, they used subtle estimates and established a new energy estimate inequality in complex field. For the situation of the Schrödinger equation with a single magnetic field potential, we refer the readers to previous studies [1,17,18] and the references therein.
For the Schrödinger equations, Ding and Liu [11] explored the Schrödinger equations with a magnetic field:
They studied the existence and concentration phenomena of semiclassical solutions with critical growth of nonlinear terms. Zhang et al. [30] studied the following nonlinear Schrödinger system:
They proved the existence of semiclassical solutions for nonlinearity, which is both super-quadratic and subcritical by using generalized linking theorems for strongly indefinite functionals. For more related results about the magnetic Schrödinger system, one can refer to previous studies [14,19,20,31] and the references therein.
Inspired by the aforementioned works [9,10,27,28], we will use non-Nehari approach developed recently by Tang et al. [27] to demonstrate the ground state solutions for the magnetic Schrödinger system (1.1) under three types of polynomial growth conditions: super-quadratic, asymptotically quadratic, and local super-quadratic at infinity. Throughout this article, we assume the following basic hypotheses:
(V)
where
(A)
(F1)
where
(F2)
(F3)
1.2 Main difficulties and ideas
We point out that the strongly indefinite features of the problem (1.1) together with the magnetic potential
the existence and compactness of (PS) sequence become more difficult to demonstrate due to the destruction of the mountain-pass geometry of the energy functional corresponding to the strongly indefinite potential;
our main obstacle in the case of asymptotically quadratic or locally super-quadratic functions is to achieve the linking geometry and boundedness of Cerami sequences since their arguments become invalid when the nonlinearity is not globally super quadratic;
due to the lack of the strict monotonicity condition, seeking a ground state solution of (1.1) in the Nehari-Pankov manifold is difficult, which needs some new techniques and method.
It is worth noting that in the case of local super-quadratic that we are studying, the local super-quadratic condition allows for nonlinearity to be super-quadratic in some domain and asymptotically quadratic in others, and in this case, we have no global information on the nonlinearity, so the non-Nehari manifold method seems not to work for our problem under the local super-quadratic condition. To prove the existence of ground state solutions in this case about (1.1), motivated by Tang et al. [27], we present a perturbation approach by adding a perturbation term of power function. The idea is to obtain existence of critical points of perturbation functional and to establish suitable estimates for the critical points so that we may pass to the limit to get solutions of the original problem.
1.3 Organization of the paper
Section 2 is dedicated to studying strongly indefinite structure and give some preliminaries. We will recall the abstract linking theorem in the study by Li and Szulkin [15], which will be used to prove the existence of solutions in the three cases we consider. Section 3 focuses on the super quadratic case. To achieve this, we will decompose the space
Notation
2 The variational setting and preliminaries
Let
Then for any
Under the assumption (V),
and
respectively. For any
To establish the variational structure of problem (1.1), we set
where
where
In such situation,
We define the following inner product on
Then
Under assumptions (V1), (F1), and (F2), the corresponding energy functional of problem (1.1) can be defined as follows:
Set
Using (F1) and (F2), for any
In view of (2.3),
Moreover, for any
To find a ground state solution of (1.1), we define the following Nehari-Pankov manifold introduced by Pankov [21]:
Lemma 2.1
[15] Let X be a real Hilbert space,
Suppose that the following assumptions are satisfied:
There exist
where
Then for some
3 Super-quadratic elliptic systems
3.1 Main assumptions and results
(F3’)
where
(F4)
Theorem 3.1
Assume that (V), and (A), (F1)–(F4) are satisfied, then system (1.1) has at least a ground state solution, i.e., it has at least a solution
Theorem 3.2
Assume that (V), (A), (F1), (F2), (F3’), and (F4) are satisfied, then system (1.1) has at least a ground state solution, i.e., it has at least a solution
Example 3.3
Remark 3.4
As far as we know, there have been no scholars who have conducted a systematic study on the existence of ground state solutions for the (1.1) Schrödinger equation system with a magnetic field.
3.2 Proof of the main results
As a direct result of [28, Lemma 3.1], we have the following lemma.
Lemma 3.5
Assume that (F1)–(F3) are satisfied, for any
Lemma 3.6
Assume that (V) and (F1)–(F3) are satisfied, Then
Proof
Suppose that
Next, we show that
Firstly, we show that
From (F1) and the Hölder inequality, one has
Similarly, we have
Moreover, the absolute continuity of integrals implies that there exists
for any
where
and
Passing to a subsequence, we have
Moreover,
Therefore, it is easy to see from the Lebesgue dominated convergence theorem that
Noting that
Due to the arbitrariness of
Similarly,
which implies
Hence,
Lemma 3.7
Assume that (V) and (F1)–(F3) are satisfied, Then for all
Proof
By using Lemma 3.7, some important corollaries are given as follows, and the proof process will be omitted.
Corollary 3.8
Assume that (V) and (F1)–(F3) are satisfied. Then for all
Corollary 3.9
Assume that (V) and (F1)–(F3) are satisfied. Then for all
Lemma 3.10
Assume that (V), (F1)–(F3) are satisfied. Then
There exists
Proof
(i) By (F1) and (F2), there exist
By (3.8), we have
and
From (3.9), (3.10), and Corollary 3.8, one has
This shows that there exists a
(ii) By Lemma 3.6,
which implies that
Lemma 3.11
Assume that (V), (F1), (F2), and (F4) are satisfied. Then for every
Proof
Arguing indirectly, assume that for some sequence
Case (1)
This implies
Case (2)
which is a contradiction.□
Corollary 3.12
Assume that (V), (F1), (F2), and (F4) are satisfied. Let
Lemma 3.13
Assume that (V), (F1), (F2), (F3), and (F4) are satisfied. Then for any
Lemma 3.14
Assume that (V), (F1), (F2), and (F4) are satisfied. Then there exists a constant
where Q is defined in (3.12).
Proof
Combining with Lemmas 2.1, 3.6, 3.10, and Corollary 3.12, it is easy to verify Lemma 3.14. The proof will be omitted.□
Lemma 3.15
Assume that (V), (F1), (F2), (F3), and (F4) are satisfied. Then there exists a constant
Lemma 3.16
Assume that (V), (A), (F1), (F2), (F3), and (F4) are satisfied. Then for any
Proof
To prove the boundedness of
Then the Lions’ concentration compactness lemma implies
which, together with (3.14) and (2.2), yields that
for
Let
This leads to a contradiction, so
In view of (
which means
and
where we note that
Put
which is a contradiction. Hence, the statement of Lemma 3.16 is proved.□
Proof of Theorem 3.1
In light of Lemma 3.16, there exists a bounded sequence
If
then
On the basis of the aforementioned discussion, we have
Thus,
which is a contradiction. Then
Passing to the subsequence, through a similar argument of the proof of Lemma 3.16, we may assume that there exists
Set
Due to the periodic assumption of
Thus, passing to the subsequence, suppose that
which implies
Proof of Theorem 3.2
By applying Lemmas 3.14 and 3.16, we deduce that there exists a bounded sequence
Note that
and
where
From (3.13), (3.18), (3.19), and (3.20), one can obtain that
Analogous to the proof of Theorem 3.1, through a similar argument of the proof of Lemma 3.16, we can prove that there exists
Set
Passing to a subsequence, we have
In the same way as the last part of the proof of Theorem 3.1, we can prove that
It is easy to show that
This contradiction implies that
4 Asymptotically quadratic elliptic systems
4.1 Main assumptions and results
(F5)
(F5’)
Theorem 4.1
Assume that (V), (A), (F1)–(F3), and (F5) are satisfied, and then system (1.1) has at least a ground state solution, i.e., it has at least a solution
Theorem 4.2
Assume that (V), (A), (F1)–(F3), and (F5’) are satisfied, and then system (1.1) has at least a ground state solution, i.e., it has at least a solution
Example 4.3
Example 4.4
Remark 4.5
We point out that, as a consequence of Theorem 3.1, the minimax characterization of the least energy value
where
4.2 Proof of the main results
Employing a standard argument, one can easily check the following lemma.
Lemma 4.6
Assume that (V) and (F1)–(F3) are satisfied, Then
Lemma 4.7
Assume that (V) and (F1)–(F3) are satisfied. Then for all
Corollary 4.8
Assume that (V) and (F1)–(F3) are satisfied. Then for all
Corollary 4.9
Assume that (V) and (F1)–(F3) are satisfied. Then for all
Lemma 4.10
Assume that (V), (F1), (F2), and (F5) hold. Then, for any
Proof
In view of (2.1) and (2.4), we have
which shows that (4.1) holds. From Lemma 4.10, we have the following corollary immediately.
Lemma 4.11
Assume that (V), (F1), (F2), and (F5) hold. Then, for any
Similar to the proof of Lemma 3.10, we have the following lemma.
Lemma 4.12
Assume that (V), (F1)–(F3), and (F5) are satisfied. Then
there exists
Define a set
Lemma 4.13
Suppose that (V), (F1), (F2), and (F5) hold, then for any
Proof
It is sufficient to show that
Clearly, from (4.3), together with
Let
Clearly,
Hence,
Lemma 4.14
Suppose that (V), (F1), (F2), and (F5) hold, let
Similar to Lemma 3.13, we have the following lemma.
Lemma 4.15
Assume that (V), (F1)–(F3), and (F5) are satisfied. Then for any
Lemma 4.16
Assume that (V), (F1), (F2), and (F4) are satisfied. Then there exists a constant
where Q is defined in (4.5).
Proof
Choose
By Lemma 4.12, we have
Let
which yields that
Then, by Lemma 3.9, there exist a constant
In virtue of Corollary 4.8, we obtain
Since
We can choose
Set
Lemma 4.17
Assume that (V), (A), (F1)–(F3), and (F5) are satisfied. Then there exists a constant
is bounded in E.
Proof
To prove the boundedness of
then by Lions’ concentration compactness principle,
Thus,
for
Set
which implies that
Going if necessary to a subsequence, through a similar argument of the proof of Lemma 3.16, we may assume that there exists
Set
Passing to a subsequence if necessary, we have
Now we define
which, together with (4.8), yields
Note that
Hence,
Thus,
Proof of Theorem 4.1
By applying Lemmas 4.16 and 4.17, we deduce that there exists a bounded sequence
Set
We have
Passing to a subsequence if necessary, we have
This shows that
Similar to the proof of Theorem 4.1, we can prove Theorem 4.2, so we omit its proof process.
5 Local super-quadratic elliptic systems
5.1 Main assumptions and results
With the purpose of studying the local super-linear quadratic case, the following assumptions on the nonlinear
(F6) There exists a domain
(F7)
and for some
(F7’)
and there exists some
Theorem 5.1
Assume that (V), (A), (F1)–(F3), (F6), and (F7) are satisfied, and then system (1.1) has at least a ground state solution, i.e., it has at least a solution
Theorem 5.2
Assume that (V), (A), (F1), (F2), (F6), and (F7’) are satisfied, and then system (1.1) has a solution
Remark 5.3
Assume that (F1), (F2), (F3), and (F7) hold. Then (F7’) holds also [27].
Example 5.4
Let
It is easy to see that
5.2 Proof of the main results
Employing a standard argument, one can easily check the following lemma.
Lemma 5.5
Assume that (V) and (F1)–(F3) are satisfied, Then
Lemma 5.6
Assume that (V) and (F1)–(F3) are satisfied, Then there exists
We omit the proof here since it is standard. Suppose that satisfying
then
Lemma 5.7
Assume that (V), (F1), (F2), and (F6) are satisfied, Then
Proof
Arguing indirectly, assume that for some sequence
Case (1)
This implies
Case (2)
If (5.3) is not true, one has
Since
This contradiction shows (5.3) holds. Now it follows from (F1), (F6), and Fatou’s lemma that
which is a contradiction.□
Lemma 5.8
Assume that (V), (F1), (F2), and (F7) hold. Then
Proof
By Lemma 5.7, there exists
To prove Theorems 5.1 and 5.2, we define
Let
Similar to Lemma 5.8, for
Lemma 5.9
For any
Proof
Define a function
If
Assume that
Set
Case (1)
Case (2)
which implies
This implies
Lemma 5.10
Assume that (V), (F1), (F2), and (F3) hold. Then
Proof
From (2.1), (2.5), (3.1), (5.4), and (5.7), we have
The proof is completed.□
By using Lemma 5.10, some important corollaries are given as follows, and the proof process will be omitted.
Corollary 5.11
Assume that (V) and (F1)–(F3) are satisfied. Then for all
Corollary 5.12
Assume that (V), and (F1)–(F3) are satisfied. Then for all
Lemma 5.13
Assume that (V), (F1), (F2), and (F3) are satisfied. Then, for
There exists
(5.11)
Proof
By (F1) and (F2), there exists a constant
Then
In virtue of (5.10) and (5.12), for any
Choose
(ii) Similar to Lemma 3.10 (ii), we can prove Lemma 5.13 (ii) hold.□
Lemma 5.14
Assume that (V), (A), (F1), (F2), and (F3) hold. Then for any
Proof
By virtue of [23], there exists a bounded sequence
where
Then the Lions’ concentration compactness lemma implies
Thus,
That is a contradiction, so we have
Going if necessary to a subsequence, through a similar argument of the proof of Lemma 3.16, we may assume that there exists
Set
Hence,
Going if necessary to a subsequence, we have
This shows that
Lemma 5.15
Assume that (V), (F1), (F2), and (F3) are satisfied. Then for any
We can easily prove this lemma in a similar way as Lemma 3.13, so we omit it.
Proof of Theorem 5.1
Consider the case
By Lemma 5.8,
Choose a sequence
There are there steps to prove Theorem 5.1.
Step 1: We prove that
Arguing by contradiction, suppose that
In view of Sobolev embedding theorem, there exists a constant
then by Lions’ concentration compactness principle,
Let
which is a contradiction, and then
Going if necessary to a subsequence, through a similar argument of the proof of Lemma 3.16, we may assume that there exists
set
Hence,
Define
which implies
By virtue of (F1), (F2), (F7), (5.19), and the Hölder inequality, one can obtain that
It follows from (5.22) and (5.23) that
In view of
This implies that
Step 2: We prove that there exists
By using Lions’ concentration compactness principle, we can prove that there exist a constant
Let
Thus there exists
This implies that
Step 3: we prove that
In view of (5.4), (5.25), and Fatou’s lemma, we have
Let
Thus,
Since
And the case
Lemma 5.16
Assume that (V), (F1), (F2), (F6), and (F7’) hold, Then
Proof
We only consider the case where
(i) Similar to [23], we have
From (F1), (F2), and (5.27), one has
Taking
Then
This implies that (i) holds.
(ii) Let
Let
By using
Moreover, by virtue of (F7’), (5.28), (5.29), and the Hölder inequality, one can obtain
From (5.31) and (5.32), we have
Then, we can obtain that
Proof of Theorem 5.2
Let
-
Funding information: This work is supported by Natural Science Foundation of China (11301297), Natural Science Foundation of Hubei Province (No. 2024AFB730) and Yichang City (No. A-24-3-008), and Open Research Fund of Key Laboratory of Nonlinear Analysis & Applications (Central China Normal University), Ministry of Education, P. R. China.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. The first author prepared the manuscript, and the second author prepared and checked it.
-
Conflict of interest: The authors declare that they have no conflict of interest.
-
Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
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- 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”
Artikel in diesem Heft
- 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”
