Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
-
Xin Qiu
Abstract
In this article, we study the following Kirchhoff equation with combined nonlinearities:
where
1 Introduction and main results
In this article, we consider the existence, nonexistence, and asymptotic behavior of solutions to the following Kirchhoff problem:
with the
where
Equation (1.1) is related to the stationary analog of the equation
which is proposed first by Kirchhoff [17] in 1883 to stretch the original wave equation by describing the transversal oscillations of a stretched string. Ma and Munõz Rivera [25] studied a transmission problem concerning a system of two nonlinear elliptic equations of Kirchhoff-type. We can also see [1,2,6,8,11,27,34] and references therein. Mathematically, equation (1.1) is not a pointwise identity as a result of the emergence of the term
In the past few years, many authors have widely studied Kirchhoff-type equations. We point out that the parameter
When
It is well known that
When
where
For
we have
for any given
Most results in the literature for the Kirchhoff problem consider
To obtain the solutions of equation (1.1), one usually look for critical points of functional
constrained on the
with the inner product
and the norm
We say that
In order to obtain the ground-state solution, as in [30], we introduce the Pohozaev manifold
where
with
Due to the
then
We see that the monotonicity and convexity properties of
Thus, we can see that
for
Then, we introduce the main results of this article as Theorems 1.1–1.7.
Theorem 1.1
Let
where
(i) If
for a suitable
(ii) There exists a number
Remark 1.1
There are a few points to say about Theorem 1.1.
In the proof of Theorem 1.1, we use the idea of [29] to constrain the functional
Compared with the case
Theorem 1.1 improves and extends the results of [16]. When
Theorem 1.2
Assume that
Then (1.1) has a positive normalized ground-state solution of mountain pass type on
Remark 1.2
For
The case of Theorem 1.2 is even more special due to the simultaneous occurrence of three critical exponents:
The corresponding fiber map
In the previous articles, we usually construct a Palais-Smale sequence of
Theorem 1.3
Let
then (1.1) has a positive normalized ground-state solution of mountain pass type on
Remark 1.3
For the case
Theorem 1.3 means that
We still look for the critical point on the Pohozaev set
Next, we consider the defocusing perturbation case, i.e.,
Theorem 1.4
Let
If u is a critical point for
The problem
with
Now, we study the asymptotic behavior of the normalized ground-states as
Theorem 1.5
Assume that
If
Theorem 1.6
Assume that
if
for some
Finally, we discuss the asymptotic behavior of the normalized ground-states as
Theorem 1.7
Assume that
If
If
We introduce some notations that will clarify what follows:
The arrows ‘
2 Preliminaries
In this section, we show several results that will be often used throughout the rest of this article. We start by introducing the famous Sobolev inequality and Gagliardo-Nirenberg inequality. For
It is well known [32] that the optimal constant is achieved by
which are the only positive classical solutions to the critical Lane-Emden equation
By (2.1), we have
It is easy to find that
Let
Define
The following estimates can be deduced by standard arguments (see [9,13]): as
and
where
Lemma 2.1
Let
Then, as
where
Lemma 2.2
Suppose that
then
Proof
If
we have
for
If
for
If
for
Since
for every
which implies that
Proposition 2.1
Suppose that
then, up to a subsequence, one of the following alternatives holds:
(i) either
for some
(ii) or
Proof
The proof is divided into two main steps.
Step 1:
Since
It follows that
due to
Substituting (2.14) into (2.13), we have
Step 2: One of the alternatives
for every
Then, we can obtain the corresponding Pohozaev identity
Letting
By the Brézis-Lieb lemma in [3], we have
Rewrite
From
The Sobolev inequality (2.1) implies that
We deduce that
Let us suppose at first that
where
which implies that alternative
3 Proof of Theorem 1.1
We first consider the existence of the local minimizer for
for every
Since
Since
Lemma 3.1
Under assumption (1.12), then the function
Proof
Define
Direct computation shows that
due to assumption (1.12). Then,
As a direct consequence,
Clearly,
Using again (1.12), we can also prove as in [29, Lemma 5.2] that
Lemma 3.2
Let
there holds
and
the maps
Again, the proof is completely analogous to the one in [29, Lemma 5.3].
Now, for
From Lemma 3.2, it follows that
for
Recall that in the proof of Lemma 3.1, we define
Next, we show the monotonicity of
Lemma 3.3
Let
Proof
It is easy to show that
We assume by contradiction that
By the assumption
Lemma 3.4
The map
If
Moreover, if
Proof
for
for
and hence,
for
When
Consequently, on the basis of
and if, in addition,
Since
Since
with a strict inequality if
Remark 3.1
As a consequence of Lemma 3.4 (ii) combined with
Proof of Theorem 1.1(i) for the local minimizer
Let
in this way, we obtain a new minimizing sequence
By Lemma 2.2,
and
We shall prove that
Step 1:
By Lemma 3.4(i),
We also have
which is absurd. If
which is also a contradiction. Therefore, we have
Step 2:
for
which combined with (3.10) and
Thus, we have
Next, we focus now on the existence of the second critical point for
Lemma 3.5
For
The first part of Theorem 1.1 states that there exists
is not empty. Now, we consider the minimax value
Lemma 3.6
There holds
The proof can argue in the same way as that of [10, Lemmas 4.9 and 4.10] and hence omit the details.
As a direct conclusion of Lemma 3.6, there holds the function
Remark 3.2
Using the same arguments as the first part (i) of Theorem 1.1, we can also conclude that (1.1) with
where
Lemma 3.7
There exists a number
Proof
Let
where
and
Choosing
that
In view of (3.12) and (3.13), we rewrite
Note that
Then, there exists
We also see from (3.15) that
Hence, by (2.5), there exists
Finally, we consider
and
Then, by (3.15), (2.5), and (2.6), we have
for
The proof is complete.□
Lemma 3.8
Suppose that
where
Proof
We introduce the augmented functional
Set
It is easy to see that if
By the definition of
Replacing
Using Ekeland’s variational principle as in [15, Lemma 2.3] yields the existence of a sequence
Note that
for
Taking
For
Proof of Theorem 1.1(ii) for the mountain pass type
By Lemma 3.8, there exists a sequence
If alternative (i) in Proposition 2.1 occurs, then
In the case
yielding a contradiction with Lemma 3.7; however, in the case
which also contradicts the result of Lemma 3.7. Therefore, alternative (ii) in Proposition 2.1 holds true, and hence,
4 Proof of Theorem 1.2
Next we prove Theorem 1.2 for the case
We have
due to assumption (1.13) and
Lemma 4.1
Proof
For
Since
Moreover, for
which means
Lemma 4.2
For every
The map
Proof
For every
Lemma 4.3
Proof
Lemma 4.2 indicates that
Thus, we have the desired result.□
Lemma 4.4
There exists
where
Proof
By the Sobolev inequality (2.1), the Gagliardo-Nirenberg inequality (2.3), and assumption (1.13), we have
if
We introduce the minimax class
then
Lemma 4.5
There holds
Proof
For any
Then, there exists
Lemma 4.6
Proof
Let
which combined with Lemma 2.1 leads to
Letting
Proof of Theorem 1.2
Similar to 3.8 and by Lemma 4.5, there exists a sequence
According to Proposition 2.1, one of the two alternatives in Proposition 2.1 occurs. Suppose that alternative (i) occurs, then
Then, we can obtain the corresponding Pohozaev identity
Moreover,
By assumption (1.13), we have
5 Proof of Theorem 1.3
Next, we prove Theorem 1.3 for the case
Lemma 5.1
Proof
For
which implies that
Moreover, for
which means
Lemma 5.2
For every
The map
Proof
For every
We can see that the function
Lemma 5.3
Proof
Lemma 5.2 indicates that
which combined with (5.1) concludes the proof.□
Lemma 5.4
There exists
where
Proof
By the Sobolev inequality (2.1), the Gagliardo-Nirenberg inequality (2.3), and
if
Lemma 5.5
There holds
Proof
The proof is same as Lemma 4.5, and we omit it.□
Lemma 5.6
Proof
Let
which combined with Lemma 2.1 leads to
Letting
Proof of Theorem 1.3
Similar to 3.8 and by Lemma 5.5, there exists a sequence
According to Proposition 2.1, one of the two alternatives in Proposition 2.1 occurs. Suppose that alternative (i) occurs, then
Then, we can obtain the corresponding Pohozaev identity:
Moreover,
Since
6 Nonexistence result
In this section, we prove Theorem 1.4. The main tool is the Hadamard three spheres theorem [28].
Proof of Theorem 1.4(i)
Let
Combined with
we have
Since
where we used that
Thus, we have
Moreover,
where we used that
Proof of Theorem 1.4(ii)
Suppose that
Thus, there exists
Therefore,
for all
for all
Since
for all
for all
where we used that
7 Asymptotic behavior of normalized solutions
In this section, we investigate the asymptotic behavior of normalized ground states for (1.1)–(1.2) and give the proofs of Theorems 1.5–1.7. Let us start by considering
Proof of Theorem 1.5(i)
Let
as
Next, we consider the case where
Lemma 7.1
There holds
Proof
We see from Lemmas 4.2 and 5.2 that
and if
The proof is complete.□
Lemma 7.2
We have
Proof
Similar to Lemma 7.1, we have
Lemma 2.1 indicates that
It follows from the definition of
Hence, we have
Lemma 7.3
Let
Proof
Let
The proof is complete.□
Proof of Theorem 1.5(ii) and (iii)
Let
Assume up to a subsequence that
from which it follows that either
Then, there must be
and then
which has only trivial solution in
Next, we consider the case when
Lemma 7.4
There holds
Lemma 7.5
If
Proof
We see from Lemma 3.1 that
The proof is complete.□
Lemma 7.6
Proof
By (1.15) and [30, Theorem 1.1], there exists
and
Hence,
Thus, the proof is complete.□
Proof of Theorem 1.6 for 2 < q < 3
Let
By Lemma 7.5, it is easy to verify that
If
This contradicts the result of Lemma 7.4. Thus,
which means that
It is standard to show that
Setting
from which it follows that
Combining this with
where we used (7.4) and Lemma (2.3). Defining
we see from (1.15) that
Then
Next, we consider the case where
Lemma 7.7
We have
and
Lemma 7.8
The function
Proof
Let
which completes the proof.□
Lemma 7.9
Proof
It is similar to that of Lemma 7.6 and now is based on Lemmas 4.2 and 5.2.□
Proof of Theorem 1.6 for 3 ≤ q < 4
In view of Lemmas 7.7–7.9, we can repeat the arguments in the proof of Theorem 1.6 for
Proof of Theorem 1.7
The proof is similar to that of Theorem 1.5, so we omit it.□
Acknowledgement
The authors are very grateful for the anonymous reviewers for their careful reading of the manuscript and valuable comments.
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Funding information: This research was supported by Innovation Research 2035 Pilot Plan of Southwest University (SWU-XDPY22015), National Natural Science Foundation of China (No. 12371120) and Chongqing Postgraduate Research and Innovation Programme (CYS240130).
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Author contributions: Xin Qiu: Writing-original draft, Writing-review & editing; Zeng-Qi Ou: Supervision, Formal Analysis; Chun-Lei Tang: Methodology, Supervision; Ying Lv: Writing-review & editing.
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Conflict of interest: The authors state no conflict of interest.
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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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- 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
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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”
