Nonlinear Scalar Field Equations with L2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches
-
Jun Hirata
Abstract
We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in
where
0 Introduction
In this paper, we study the existence of radially symmetric solutions of the following nonlinear scalar field equations in
where
Solutions of
where
When
See also Ruppen [15] and Stuart [18] for earlier works. The paper [7] dealt with
For
There exists some
Shibata [16] showed:
There exists
It is important to check
(0.2)(0.3)
We remark that the authors of [7, 16] also studied orbital stability of the minimizer. We also refer to Jeanjean [11] and Bartsch and de Valeriola [2] for the study of the
We note that conditions (g1)–(g4) are related to those in [4, 5] (see also [3, 9]) as almost necessary and sufficient conditions for the existence of solutions of nonlinear scalar field equations
More precisely, replacing (g2) by
We remark that if
In [7, 16], to show the achievement of
which ensures compactness of minimizing sequences for
In this paper, we take another approach to
That is, we obtain solutions
Theorem 0.1.
Assume (g1)–(g4). Then the following statements hold:
There exists
In the setting of (i)–(ii), a solution is obtained through a mountain pass minimax method:
See ( 0.7 ) below for the definition of I and see also Section 5 for a precise definition of the minimax class
where
We will give a presentation of
We also deal with the existence of infinitely many solutions assuming oddness of
Theorem 0.2.
Assume (g1)–(g4) and (g5). Then the following statements hold:
For any
To show Theorem 0.2, we develop a version of symmetric mountain pass methods, in which genus plays an important role.
In the following sections, we give proofs to our Theorems 0.1 and 0.2. Since the existence part of Theorem 0.1 is already known by [7, 16], we mainly deal with Theorem 0.2 in Sections 1–4.
In Section 1, first we give a variational formulation of problem
We also setup function spaces. Second for a fixed
Behavior of
See (1.5) for the definition of
In Sections 2–3, we find that
In Section 4, we develop a new deformation argument to justify the minimax methods in Section 2. Usually deformation theories are developed under the so-called Palais–Smale condition. However, under conditions (g1)–(g4), it is difficult to check the standard Palais–Smale condition for
We note that
In Section 5, we deal with Theorem 0.1 and we study the minimizing problem (0.1). Applying the mountain pass approach to
Our new deformation argument is applicable for problems with suitable scaling properties. In Section 6, we give a typical example and we deal with nonlinear scalar field equations (0.4). We show that the corresponding functional
satisfies our (PSP) condition and our deformation argument works also for
plays an important role and our (PSP) condition for
Condition (PSP).
If a sequence
then
We note that
We believe that our argument is based on an essential property of our problem –
1 Preliminaries
1.1 Functional Settings
In Sections 1–4, we deal with Theorem 0.2 and we assume (g1)–(g5). We denote by
We also write
In what follows, we denote by p the
In particular, we have
which we will use repeatedly in this paper.
For technical reasons, we set
It is easy to see that
The following functionals will play important roles in our argument:
We note that:
For a fixed
(1.3)It is easy to see that
(1.4)
1.2 Some Estimates for I ^ ( λ , u )
First we observe that for
and set
Lemma 1.1.
The following statements hold:
For any
When
In particular,
Proof.
By (g1)–(g5) and definition (1.5) of
For
We note that
We also have
For the behavior of
Lemma 1.2.
If
(1.9)then for any
Proof.
(i) Choose
from which we have
Setting
We note that
has the symmetric mountain pass geometry and thus there exists an odd continuous map
Thus for
Since
(ii) By (1.9) and (g3), there exists
Thus we have
As in (i),
from which we deduce that
Thus (ii) holds. ∎
We define for
By (1.8), we have
In what follows, we fix
As a corollary to Lemma 1.2, we have the following result, which is analogous to (0.2)–(0.3).
1.3 An Estimate from Below
By (g2) and (g3), for any
Then we also have
Setting
we have
The functional
In what follows, we denote by
Lemma 1.4.
For
Proof.
Let
is a least energy solution of
By (1.1), it is easy to see that for
Thus we get (1.14) from (1.13). Noting
and that for
2 Minimax Methods for I ( λ , u )
2.1 Symmetric Mountain Pass Methods
We fix
where
We choose
and take
we set
We note that
In particular,
Using Lemma 1.4, we have:
Lemma 2.1.
The following statements hold:
Proof.
Note that
By Lemma 1.4,
By our choice (2.2) of
We introduce a family of minimax methods. For
where
There exists
We note that:
For
From these facts, we have
We set
Proposition 2.2.
The following statements hold:
To show Proposition 2.2, we need:
Lemma 2.3.
We have
Proof.
First we note that by (ii) of Lemma 2.1 that
Second we remark that we may assume for
In fact, for
from which we deduce that if
Thus, for a given
we find for
Thus we may assume (2.5) for
Next we show (2.4). For
Then for R large, we have
Since
Now we give a proof to Proposition 2.2.
Proof of Proposition 2.2.
Thus
Since
(ii) By Lemma 2.3, for any
Since
In Section 3, we will see that
2.2 Symmetric Mountain Pass Methods Using Genus
In this subsection, we use an idea from Rabinowitz [14] to define another family of minimax methods. Here the notion of genus plays a role.
Definition.
Let E be a Banach space. For a closed set
We refer to [14] for fundamental properties of the genus.
Our setting is different from [14]; our functional is invariant under the following
that is,
We define our second family of minimax sets as follows:
Here we summarize fundamental properties of
Lemma 2.4.
The following statements hold:
Let
(2.7)(2.8)Then for
For
Proof.
Parts (i) and (ii) follow from the definition of
(iii) Suppose
(iv) Following and modifying the argument in [14, Sections 7–8], we can show (iv). For the sake of completeness, we give a proof in the Appendix.
(v) Suppose
from which we have
In particular,
As fundamental properties of
Lemma 2.5.
The following statements hold:
Proof.
(i) By Lemma 2.4 (v), we have for any
which implies
(ii) It is easy to see
In the following section, we use a special deformation lemma to show
3 Deformation Argument and Existence of Critical Points
In this section we introduce a deformation result for
3.1 Deformation Result for I ( λ , u )
For
Here
We have the following deformation result.
Proposition 3.1.
Assume (g1)–(g5) and
For any open neighborhood
If
Writing
for all
Such a deformation result is usually obtained under the Palais–Smale compactness condition. However, it seems difficult to verify the standard Palais–Smale condition under (g1)–(g4). In Section 4, we introduce a new version (PSP) of Palais–Smale condition and we develop a new deformation argument to prove Proposition 3.1. We postpone a proof of Proposition 3.1 until Section 4 and in this section we show
Remark 3.2.
Our deformation flow
For
We note that the above properties (i)–(ii) hold for the standard deformation flow. See [14] and see also Remark 4.10 in Section 4. We also note that properties (1)–(6) in Proposition 3.1 enable us to apply minimax arguments and the genus arguments.
3.2 Existence of Critical Points
As an application of our Proposition 3.1 we show the following proposition.
Proposition 3.3.
The following statements hold:
For
If
In particular,
Proof.
(ii) Suppose that
there exists
Here we denote a δ-neighborhood of a set
where
and
By Proposition 3.1, there exist
We note that
We take
If
Equations (3.2) and (3.3) imply
Now we can show:
Proof of Theorem 0.2 (ii).
Under condition (0.6), we have
Arguing indirectly, we assume
and choose
As in the proof of Proposition 3.3, there exist
We choose
Equations (3.4) and (3.5) imply
4 (PSP) Condition and Construction of a Flow
In this section we give a new type of deformation argument for our functional
4.1 (PSP) Condition
Since it is difficult to verify the standard Palais–Smale condition for
Definition.
For
Condition (PSP)${}_{b}$.
If a sequence
then
First we observe that
Proof.
Let
Step 1:
we have from (4.1) and (4.4) that
Thus
Step 2:
it follows from (4.2) and Step 1 that
Step 3:
By (g2) and (g3), for any
Thus
Since
Thus it follows from (4.5) that
where
Choosing
Step 4: Conclusion. By Steps 1–3,
Thus, we deduce from
which implies
Remark 4.2.
For
But
As a corollary to Proposition 4.1, we have:
Corollary 4.3.
For
Proof.
The set
4.2 Functional J ( θ , λ , u )
To construct a deformation flow, we need an augmented functional
We introduce
We will construct our deformation flow for
The functional
Lemma 4.4.
For all
Proof.
We compute that
Thus we have (4.7)–(4.9). Equation (4.10) follows from (4.6). ∎
To analyze
and we introduce a metric
for
It is easily seen that
We also have for all
We denote a natural distance induced by the metric
By property (4.13), we have for all
Using notation
we have:
Lemma 4.5.
For
Proof.
By Lemma 4.4, we have
Noting (4.12), the conclusion of Lemma 4.5 follows from the definition (4.11). ∎
For
By (4.6)–(4.9), we observe that
We also use notation for
From condition
Proposition 4.6.
For
Condition (${\widetilde{\textup{PSP}}}$)${}_{b}$.
For any sequence
we have
Proof.
Suppose that
Thus by Proposition 4.1, there exists a subsequence — still denoted by
Note that
as
As a corollary to Proposition 4.6, we have the following uniform estimate of
Corollary 4.7.
Assume
When
When
We note that
4.3 Deformation Flow for J ( θ , λ , u )
In this subsection we give a deformation result for
We have the following deformation result.
Proposition 4.8.
Assume
If
We write
Proof.
Let
We can also have the following:
For a given
We choose a locally Lipschitz continuous function
We note that
For
We consider the following ODE in M:
For
We can check properties (1)–(3) easily and we use them in what follows. We also note that
For
If
Case 1:
Case 2:
First we consider Case 1. By (4.18) we have
By our choice of φ and ψ, we have
Thus we have
If Case 2 takes a place, we can find an interval
By (4.20),
Thus,
Choosing
In the following section, we can construct a deformation flow for
4.4 Deformation Flow for I ( λ , u )
In this subsection, we construct a deformation flow for
We use the following maps:
and we construct a deformation flow
as a composition
As fundamental properties of π and ι, we have
Clearly
Lemma 4.9.
For any
Moreover,
Proof.
For
In fact, for any
In particular, since
Thus
On the other hand, since
Since
We set
Then
Since
which implies (4.24). Noting
Now we can give a proof of Proposition 3.1 (ii).
Proof of Proposition 3.1 (ii).
Let
Since
By (4) of Proposition 4.8,
By the definition of π and (4.6),
Remark 4.10.
By our construction,
is of class
is continuous but not of class
is continuous but not of class
5 Minimizing Problem
In this section we assume (g1)–(g4) (without g5) and we deal with Theorem 0.1. Under the condition
5.1 Mountain Pass Approach
Under conditions (g1)–(g4), as in Sections 1–2, we define
For
When
We set for
We note that if (g5) holds,
We set
As in Sections 1–4, we can show the following theorem.
Theorem 5.1.
Assume (g1)–(g4). Suppose
where
Here
As a corollary, we have:
Proof.
The critical point
Thus
We also have:
Theorem 5.3.
Under the assumption of Theorem 5.1, there exists
Proof.
Let
As in the proof of Lemma 2.3, we may assume
and
we find the desired path
5.2 Mountain Pass Characterization of ℐ m
Next we consider problem
Shibata [16] showed the following:
Theorem 5.4 ([16]).
There exists
If
In what follows, we will show that
For
First we show the minimizer of
Lemma 5.5.
Suppose
Proof.
First we deal with (ii). We can verify that the Pohozaev identity (ii) holds for
We set
that is,
Since
Thus (ii) follows from (5.3) and (5.4).
Next we show (i). By (ii), we have
Thus we have
By Lemma 5.5, setting
Proposition 5.6.
Suppose
Proof.
(i) It suffices to show
Then we have for some
In fact, when
We also have from
Thus (5.5) holds with
(i) holds.
(ii) By the result of [12], the mountain pass minimax value
(iii) Note that (ii) implies
Thus
Proposition 5.7.
Suppose
Proof.
As in Lemma 2.3, we can show
Thus, by (ii) of Proposition 5.6, for a critical point
On the other hand, it follows from (iii) of Proposition 5.6 that
and so
Therefore we have
Corollary 5.8.
We have
6 Deformation Lemma for Scalar Field Equations
In this section we study the following nonlinear scalar field equations:
where
Here we use notation different from previous sections. We also write
In this section we give a new deformation result for (6.1) using ideas in Sections 3–4.
A key of our argument is the following proposition.
Proposition 6.1.
For any
Condition ($\textup{PSP}^{\prime}$)${}_{b}$.
If a sequence
then
Proof.
First we note by (g2)–(g3) with
The proof consists of several steps. Here we follow essentially the argument in [9, Propositions 5.1 and 5.3].
Step 1:
From now on we prove that
We set
Step 2:
Setting
Thus
from which we have
Thus
Step 3:
Thus, by (6.5),
Step 4: Conclusion. By Step 1 and Step 3,
Thus
Arguing as in Sections 3–4, we obtain the following proposition.
Proposition 6.2.
Under the assumption of Proposition 6.1, for any
For any open neighborhood
If
Here we use notation
for
Using Proposition 6.2, we can show that
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 25287025
Award Identifier / Grant number: 17H02855
Funding statement: The second author is partially supported by JSPS Grants-in-Aid for Scientific Research (B) (Grant No. 25287025) and (B) (Grant No. 17H02855).
A Proof of Lemma 2.4 (iv)
Suppose that a closed set Z is invariant under
For
In fact,
Conversely, since
Thus (A.1) follows from (A.2) and (A.3). Since
Thus,
Therefore, by (A.1) we have
Acknowledgements
The authors would like to thank Professor Tohru Ozawa for helpful discussions and comments. They also thank Professor Louis Jeanjean for valuable advices and Professor Norihisa Ikoma for a careful reading of the manuscript and valuable advices and comments.
References
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Articles in the same Issue
- Frontmatter
- Sharp Singular Trudinger–Moser Inequalities Under Different Norms
- Nonlinear Scalar Field Equations with L2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches
- Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent
- Existence and Multiplicity of Periodic Solutions to Indefinite Singular Equations Having a Non-monotone Term with Two Singularities
- Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem
- Ground State Solutions for Kirchhoff Type Quasilinear Equations
- Blow-Up Results for Higher-Order Evolution Differential Inequalities in Exterior Domains
- Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems
- A Variational Approach for the Neumann Problem in Some FLRW Spacetimes
- Shadowing for Nonautonomous Dynamics
Articles in the same Issue
- Frontmatter
- Sharp Singular Trudinger–Moser Inequalities Under Different Norms
- Nonlinear Scalar Field Equations with L2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches
- Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent
- Existence and Multiplicity of Periodic Solutions to Indefinite Singular Equations Having a Non-monotone Term with Two Singularities
- Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem
- Ground State Solutions for Kirchhoff Type Quasilinear Equations
- Blow-Up Results for Higher-Order Evolution Differential Inequalities in Exterior Domains
- Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems
- A Variational Approach for the Neumann Problem in Some FLRW Spacetimes
- Shadowing for Nonautonomous Dynamics
