Normalized solutions for the double-phase problem with nonlocal reaction
-
Li Cai
Abstract
In this article, we consider the double-phase problem with nonlocal reaction. For the autonomous case, we introduce the methods of the Pohozaev manifold, Hardy-Littlewood Sobolev subcritical approximation, adding the mass term to prove the existence and nonexistence of normalized solutions to this problem. For the nonautonomous case, we show the existence of normalized solutions to the double-phase problem by using the Pohozaev restrict method and describing the relationship between the energy of this problem and its limit problem. Moreover, we study the existence of normalized solutions to the double-phase problem involving double Hardy-Littlewood-Sobolev critical exponents.
1 Introduction and preliminary results
In this article, we are concerned with the solutions of the following double-phase problem
with the normalized constraint
where
for each
The study of equation (1.1) is motivated by recent fundamental progress in the mathematical analysis of many nonlinear patterns with unbalanced growth and nonlocal reaction. The main novelty of (1.1) is the combination of a double-phase operator and two nonlocal Choquard reaction terms. Since equation (1.1) is closely concerned with unbalanced double-phase problems and nonlocal Choquard problems. We briefly introduce in what follows the related background and applications and recall some pioneering contributions in these fields. Problem (1.1) combines an interesting phenomenon that the operator involved in (1.1) is the so-called double-phase operator whose behavior switches between two different elliptic situations, which generates a double-phase associated energy. Originally, the idea to treat such operators comes from Zhikov [61] who introduced such classes to provide models of strongly anisotropic materials, see also the monograph of Zhikov et al. [62]. We refer to the remarkable works initiated by Marcellini [37,38], where the author investigated the regularity and existence of solutions of elliptic equations with unbalanced growth conditions. We also mention the recent paper by Mingione and Rădulescu [39], which is a comprehensive overview of the recent developments concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators.
The double-phase problem (1.1) is motivated by numerous models arising in mathematical physics. For instance, we can refer to the following Born-Infeld equation [11] that appears in electromagnetism:
Indeed, by the Taylor formula, we have
By taking
We also refer to the following fourth-order relativistic operator
which describes large classes of phenomena arising in relativistic quantum mechanics. Again, by Taylor’s formula, we have
This shows that the fourth-order relativistic operator can be approximated by the following autonomous double-phase operator
For more details on the physical background and other applications, we refer the readers to Bahrouni et al. [6] (for phenomena associated with transonic flows) and to Benci et al. [9] (for models arising in quantum physics).
In the past few decades, the double-phase problem has been the subject of extensive mathematical studies. Using various variational and topological arguments, many authors studied the existence and multiplicity results of nontrivial solutions, ground state solutions, nodal solutions, and some qualitative properties of solutions, respectively, such as [21,43,46] in case of bounded domains. In this classical setting, we recall the seminal papers by Ni and Wei [44], Li and Nirenberg [29], del Pino and Felmer [18], del Pino et al. [19], and Ambrosetti and Malchiodi [5]. The regularity, existence of solutions, and multiplicity of the double-phase problem on the whole space can be found in [3,24,58]. The qualitative and asymptotic analysis of solutions for some related elliptic problems can refer to [13,14,28,47,60].
The second interesting phenomenon in equation (1.1) is the appearance of Choquard reaction terms, which generate the nonlocal characteristic. The nonlocal problem goes back to the description of the quantum theory of a polaron at rest by Pekar in [48] and the modeling of an electron trapped in its own hole in the work of Choquard, as a certain approximation to Hartree-Fock theory of one-component plasma [30]. In some particular cases, the Choquard equation is also known as the Schrödinger-Newton equation, which was introduced by Penrose in his discussion on the selfgravitational collapse of a quantum mechanical wave function [49]. The existence and qualitative properties of ground state solutions of Choquard equation have been widely studied in the last decades. See, for example, [32,35,40–42].
Recently, physicists are often interested in the existence of normalized solutions. Indeed, prescribed mass appears in nonlinear optics and the theory of Bose-Einstein condensates, see [20,36] and the references, therein. In particular, when
Cazenave and Lions in [16] proved the existence and stability of normalized solutions for (1.2) with
where
where
under the normalized constraint
They proved several nonexistence and existence results by introducing some new arguments. In particular, they first considered the existence of normalized solutions to (1.5) involving double critical exponents and described some qualitative properties of the solutions with prescribed mass and of the associated Lagrange multipliers
On the other hand, when
In particular, when
holds uniformly for
where
In the
Inspired by the aforementioned literature, we want to study the normalized solutions of the double-phase problem with nonlocal reaction (1.1).
The features of equation (1.1) are the following:
The presence of several differential operators with different growth, which generates a double-phase associated energy.
The equation combines the multiple effects generated by two nonlocal terms and a variable potential.
Due to the unboundedness of the domain, the Palais-Smale sequence does not have the compactness property.
Throughout this article, for any
and
For equation (1.1), we introduce the working space
In addition, for given
which remains in
To study equation (1.1) variationally, we give the following Hardy-Littlewood-Sobolev inequality and the Gagliardo-Nirenberg inequality.
Lemma 1.1
[31] Let
If
In this case, there is equality in (1.9) if and only if
for some given constants
Remark 1.2
In particular, if
is well defined if
Then for
Thus, power
Lemma 1.3
[1] Let
Moreover,
Remark 1.4
By applying Lemmas 1.1 and 1.3, we see that
Lemma 1.5
[3] The space E is embedded continuously into
Lemma 1.6
[2,45] The following results hold:
For every
(1.13)where
Let
(1.14)where
Based on the aforementioned facts, we shall consider the existence and nonexistence of normalized solutions to equation (1.1). It is well known that the normalized solution of equation (1.1) can be obtained by looking for a critical point of the following functional
constrained on
and
for some positive constants
Next, we define the ground state in the following sense.
Definition 1.1
We say that
In addition, when the functional
In order to search for critical points of
and
First, we study the existence and nonexistence of normalized solutions to equation (1.1) in case of
Obviously, for any
Now we state our main results about the autonomous problem as follows.
Theorem 1.7
Assume that
is attained by
Remark 1.8
To prove that
Next, we make most of Pohozaev equality, Hardy-Littlewood-Sobolev inequality, and Gagliardo-Nirenberg inequality (1.16) about the nonlocal Choquard term, we give the following nonexistence result.
Theorem 1.9
Assume that
then equation (1.1) has no solution for any
Theorem 1.10
Assume that
where
Remark 1.11
To show the existence of normalized solutions to equation (1.1) with
Theorem 1.12
Assume that
and
then there exists
Remark 1.13
In Theorem 1.12, in order to overcome the technical difficulties caused by the emergence of double Hardy-Littlewood-Sobolev critical terms, we introduce the new methods of adding mass term and the Hardy-Littlewood-Sobolev subcritical approximation to prove that
From the analysis of Theorems 1.7, 1.10 and 1.12, we in addition obtain the following properties of
Theorem 1.14
The following results hold:
Suppose that the assumptions of Theorem 1.7 hold. Then the mapping
Secondly, we shall study the existence of normalized solutions to equation (1.1) in case of
Let
and
such that for all
Let
such that for all
Let
and
such that for all
Based on Theorems 1.10, 1.12, and 1.14, we treat equation (1.1) in case of
Theorem 1.15
Assume that
and for
and
Remark 1.16
To our best knowledge, it seems to be the first work on the existence of normalized solutions for the double-phase problem with nonlocal critical reaction, potential, and mass supercritical perturbation. The appearance of different operators with different growth and potential term affect the geometry of this problem. Moreover, since
This article is organized as follows. In Section 2, we study the existence of normalized solutions to (1.1) in case of
2 The low critical leading term with focusing
L
p
-subcritical perturbation
In this section, we consider the existence of normalized solutions to (1.1) in case of
Lemma 2.1
Assume that
Let
Proof
which yields that
Then combining the condition that
This means that
In fact, for any
for
This is impossible and
Hence,
Define
for
Lemma 2.2
Assume that
Proof
By Lemma 2.1(i), we know that
By assuming by contradiction and by Lemma
which contradicts Lemma 2.1(iii) by the condition that
we derive that
where
In view of Lemma 2.1(ii), letting
which is a contradiction. Thus,
This means that
Proof of Theorem 1.7
Based on Lemma 2.1(i), let
one has
which implies that
3 The low critical leading term with focusing
L
p
-critical perturbation
In this section, we mainly show the nonexistence of normalized solutions to (1.1) in case of
Proof of Theorem 1.9
Assume that
This means that
Remark 3.1
From Theorem 1.9, if we let
This means that
then equation (1.1) has no solution for any
4 The low critical leading term with focusing
L
p
-supercritical perturbation
In this section, we consider the existence of normalized solutions to (1.1) in case of
Lemma 4.1
Assume that
Proof
First, for any
Then
Then
So the proof of Lemma 4.1 is completed.□
Lemma 4.2
Assume that
Proof
For
which implies that
By combining (1.11), (4.1), and (4.2), we deduce that
provided that
and
Hence, we complete the proof of Lemma 4.2.□
Lemma 4.3
Assume that
where
Proof
On the one hand, since
On the other hand, we need to show that
It follows from Lemma 4.1 that
Fix
Obviously,
Combining (4.4), we obtain that
To sum up, in view of Lemma 4.2,
Hence, the proof of Lemma 4.3 is completed.□
Next, by applying the implicit function theorem on the
Lemma 4.4
Assume that
Suppose that
then there exists a Palais-Smale sequence
Based on Lemma 4.3, we obtain that
Then applying Lemma 4.4, we easily obtain the following lemma.
Lemma 4.5
Assume that
Next, we will establish the existence of a Palais-Smale sequence
Lemma 4.6
Assume that
Proof
Since
Take
which implies that
and
which along with
Next, we show that
which together with
Then
which contradicts with Lemma 4.3. So
In the last, we claim that
and
Moreover,
Hence, in view of
In addition, we observe that
Then
we denote that
Repeating the same argument as proving
Proof of Theorem 1.10
In view of Lemmas 4.2 and 4.5, there exists a bounded Palais-Smale sequence
Then we know that
This means that
Thus the proof of Theorem 1.10 is completed.□
Next we study the existence of normalized solutions to (1.1) in case of
Lemma 4.7
Assume that
for
Proof
By the definition of
Then, the Lebesgue dominated convergence theorem implies that
is continuous on
for
for any
To clearly determine the weak convergence of nonlocal terms, we first give some necessary lemmas as follows.
Lemma 4.8
[54] Let
Lemma 4.9
[10] If
where
Lemma 4.10
Assume that
Then there exists
Proof
Let
which implies that
Take
Combining Lemma 1.1 and the Sobolev embedding theorem, we see that
This means that
Moreover, we have
Since
as
By Lemma 1.1 and the Lebesgue-dominated convergence theorem,
By combining this with (4.7), we deduce that
as
and
one has
Based on the fact that
By using Lemma 4.7, (4.9), and the fact that
In addition, by using (1.10), we let
Obviously,
which yields that
By direct calculation, it follows from (4.11) that there exists
This means that
Then combining (4.10), we derive that
However, since
one has
This is a contradiction. So
Next we show that
and
By combining the fact that
Then we deduce that
On the other hand, similar to (4.13), we also have
Then
Therefore, in view of (4.14), one infers that
provided that
To sum up, if
Here, using the condition
we find that for
That is,
Similarly,
By combining
whcih implies that
Hence, the proof of Lemma 4.10 is completed.□
Proof of Theorem 1.12
In view of Lemma 4.10, we know that
5 Qualitative properties of the mappings
a
c
and
m
c
0
,
l
In this section, we discuss the continuity and monotonicity of
Proof of Theorem 1.14
(i) Let
Suppose that
In summary, we obtain that
This means that
and
Thus,
This means that
6 The nonautonomous problem
In this section, we consider the existence of normalized solutions to equation (1.1) in case of
The following lemma helps us to show that
Lemma 6.1
Assume that
Proof
For any
which implies that
Since
Lemma 6.2
Assume that
where
then
Proof
For any
which along with (1.11), (6.1), and the condition
Then we see that
Next, similar to the aforementioned sections, we consider the decomposition of
where
and for any
Now we give some properties of
Lemma 6.3
Assume that
Proof
For any
In addition, by direct calculation, one has
Then, by combining (6.2) and (6.3), in view of
This means that
Lemma 6.4
Assume that
Proof
Note that
Obviously,
This means that
It follows that
In particular,
Now based on (6.4) and (6.7), we let
and
It is clear that
by
Thus, we know that
In the last, we shall show that the map
Lemma 6.5
Assume that
and
where
Proof
From
By combining the condition
This implies that
So we complete the proof of Lemma 6.5.□
Lemma 6.6
Assume that
Proof
In view of Theorems 1.10 and 1.12, we know that there exists
Thus
Lemma 6.7
Assume that
Proof
If
which means that
Similarly, the corresponding Pohozaev identity is given by
On the other hand, since
By combining (6.8) and (6.9), we obtain that
It follows from (6.3) that
Then
Proof of Theorem 1.15
To show that there exists
Step
Let
Then it is clear that
with the associated minimax level
Fix
is a path in
By Lemma 6.5, since
which gives that
which yields that
By applying Section 5 in [22], we easily find that
In addition, it follows from
where
Obviously,
Setting
where
Next we show that
Indeed,
Hence, in virtue of
namely,
In addition, in view of
Then on the basis of the proof of Lemma 6.2, we know that
for every
which along with the boundedness of
In the following, we claim that
one has
Then in view of
This means that
which along with the fact that
which contradicts with Lemma 6.6. This give that
As a consequence,
Step
Hence, applying Lemma 4.1, we know that for any
Then in view of Lemma 6.6, one has
Step
From
This gives that
In the last, we observe that
which implies that
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Funding information: The research of Li Cai is partially supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX21_0076). Li Cai would like to thank the China Scholarship Council for its support (No. 202206090124) and the Embassy of the People’s Republic of China in Romania. The research of Fubao Zhang is partially supported by National Natural Science Foundation of China (No. 11671077) and National Scientific Research Program Cultivation Fund of Chengxian College of Southeast University (No. 2022NCF008).
-
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 Cai prepared the manuscript, and the corresponding author Zhang prepared and checked it.
-
Conflict of interest: The authors state no conflict of interest.
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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”
