High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models
-
Alberto Boscaggin
,
Guglielmo Feltrin
und
Elisa Sovrano
Abstract
We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation
where
1 Introduction and Statement of the Results
In this paper, we investigate existence and multiplicity of non-constant positive solutions for the parameter-dependent second-order ordinary differential equation
where λ and μ are positive real parameters,
and the superlinear growth condition at zero
Following a terminology popularized in [32], we refer to (($\mathscr{E}_{\lambda,\mu}$)) as an indefinite equation, meaning that the weight function
The mathematical questions we address here are motivated by the study of the spatial effects on the variation in the genetic material along environmental gradients. In population genetics, when individuals of a continuously distributed population mate at random in their habitat, and no genetic drift nor new mutations appear, the evolution of the frequencies of two alleles,
where
In this context, available theory also assumes that migration is homogeneous and isotropic, namely,
where Δ denotes the Laplace operator and
The existence of a unique non-constant and globally asymptotically stable solution of (1.2) is proved in [12, 31, 37] for sufficiently small d provided that
In this paper, we deal with migration-selection models in a unidimensional habitat. We also assume that
Hence, the selection is
We are interested in periodically changes in genotype within a population as a function of spatial location. Thus we assume that
These models are appropriate in the case of populations living in circular habitats (e.g., around a lake or along the shore of an island), as well as for ring species, for instance, around the arctic.
To state our main results, we introduce the following condition on the weight function
There exist
and
In the above condition, the symbol
and notice that
With this notation, our first result reads as follows.
Theorem 1.1.
Let
More precisely, fixed an arbitrary constant
Let us notice that, when
for
The second, and main, part of our investigation is focused on the appearance of high multiplicity phenomena for solutions of (($\mathscr{E}_{\lambda,\mu}$)). In this regard, the fact that the weight function
To state our result precisely, we introduce the condition
and notice that it is satisfied whenever
Theorem 1.2.
Let
More precisely, fixed an arbitrary constant
Let us notice that the number of solutions provided by Theorem 1.2 is strongly related with the nodal behavior of the weight function
The possibility of providing, in the context of indefinite boundary value problems, high multiplicity results by playing with the nodal behavior of the weight function was first suggested in [27]; therein, an interesting analogy was proposed with the papers [14, 15], giving, in the PDE setting, multiplicity of solutions depending on the shape of the domain. Later on, along this line of research, several contributions followed [5, 6, 7, 11, 19, 20, 21, 22, 23, 25, 26, 47, 45]. In particular, dealing with equation (($\mathscr{E}_{\lambda,\mu}$)), with
Our last result concerns the dynamics of equation (($\mathscr{E}_{\lambda,\mu}$)) on the whole real line. Precisely, having defined the intervals
we provide globally defined positive solutions of (($\mathscr{E}_{\lambda,\mu}$)), whose behavior in each of the above intervals can be coded, as in Theorem 1.2, by a bi-infinite (possibly non-periodic) sequence
Theorem 1.3.
Let
In particular, if the sequence
For the proofs of Theorem 1.1 and Theorem 1.2, we adopt a functional analytic approach based on topological degree theory in Banach spaces (cf. [22] and the subsequent papers [10, 11, 23]). In particular, we follow the general strategies developed in [10, 11], dealing with a nonlinear term
The paper is structured as follows. In Section 2, we describe the abstract degree setting and we prove some technical estimates on the solutions of (($\mathscr{E}_{\lambda,\mu}$)) (and of some related equations). Based on this, in Section 3 and Section 4, we give the proofs of Theorem 1.1 and Theorem 1.2, respectively. The proof of Theorem 1.3 is then presented, together with some comments about the existence of subharmonic solutions, in Section 5. The paper ends with Section 6, discussing some related results: subharmonic solutions via the Poincaré–Birkhoff theorem, Dirichlet/Neumann boundary value problems, stability issues, and an asymptotic analysis of the solutions for
2 Abstract Degree Setting and Technical Lemmas
The aim of this section is to present the main tools used in the proofs of our theorems as well as some preliminary technical lemmas.
Before doing this, we introduce the following notation employed throughout the paper:
where
Notice that, due to the P-periodicity, we have assumed without loss of generality that
2.1 Coincidence Degree Framework
In this section we recall Mawhin’s coincidence degree theory (cf. [24, 40, 41]) and we present two lemmas for the computation of the degree (cf. [11]).
First of all, we remark that solving the P-periodic problem associated with (($\mathscr{E}_{\lambda,\mu}$)) is equivalent to looking for solutions
on
and we denote by
The coincidence degree theory ensures that the P-periodic problem associated with
is equivalent to the coincidence equation
or to the fixed point problem
where
In this framework, if
the coincidence degree
and it satisfies the standard properties of the topological degree, such as additivity, excision, homotopic invariance.
Our goal is to construct open and bounded sets
The first inequality follows from a simple convexity argument (the so-called maximum principle). Indeed, if
To construct the sets Λ as above, we need to introduce some auxiliary sets where we will compute the degree. Given three constants
With this notation, the following lemmas hold.
Lemma 2.1.
Let
If
(2.3)with
There exists
Then it holds that
Lemma 2.2.
Let
If
(2.4)with
Then it holds that
The proofs of Lemmas 2.1 and 2.2 follow the argument of the ones of [11, Lemma 3.1] and [11, Lemma 3.2], respectively (even with some simplifications, due to the fact that the sets considered in the present paper are bounded, differently from the case treated in [11]). We point out that in [11] only the case
Property.
If
The above property is a direct consequence of the Hopf maximum principle (see, for instance, [33, Theorem 1.2]); alternatively it can be obtained by arguing as in [23, Remark 3.4].
We notice that, for
An application of property (2.5) together with the excision property of the degree allows us to compute the degree on the open ball
Corollary 2.1.
Let
If
There exists
Then it holds that
Corollary 2.2.
Let
If
Then it holds that
2.2 Finding the Constant λ *
In the following lemma we provide the constant
Lemma 2.3.
Let
with
The proof is essentially the same as in [10, Section 3.1] and, since we need to slightly refine the estimates, we just provide a sketch.
Proof.
We fix
is well defined and positive.
Let
Arguing as in [10, Step 1 and Step 2 of Section 3.1] (with the care of replacing the constant T therein with
and that
We define
Then, by integrating equation (2.6) on
Therefore, non-negative P-periodic solutions
2.3 Some Estimates for Small Solutions
The following lemma gives a lower bound for positive P-periodic solutions of (2.4) that will be exploited in the proof of the existence result in Theorem 1.1.
Lemma 2.4.
Let
Proof.
Let
An easy computation shows that
Moreover,
We claim that
We suppose by contradiction that this is not true. Then we can find a maximal interval
an integration on
from which
Passing to the limit as
Now, we integrate (2.10) on
and so a contradiction is reached using the fact that
The next lemma gives us some estimates for positive solutions of (2.4) which will be used to prove the multiplicity result in Theorem 1.2. To state it, let us introduce the following notation. For any constant
Furthermore, recalling (a${}_{*}$) and the positions in (2.1), for all
Lemma 2.5.
Let
If
If
Proof.
From condition ($g_{0}$) we can fix a constant
We give the proof when
Step 1. Let
We notice that if
Suppose first that
integrating between
Then
and
This inequality, together with the maximality of
implying
Furthermore, by integrating (2.13) on
On the other hand, if we suppose that
and
Thus, in any case, (2.14) and (2.15) hold, and so we can proceed with the second part of the proof.
Step 2. We consider the interval
Therefore, integrating on
where the last inequality follows from (2.12). On the other hand, integrating
on
In particular,
Finally, a further integration and condition (2.15) provide
where the last inequality follows from (2.12). Thus the proof is completed. ∎
2.4 Some Estimates for Large Solutions
We start by introducing the following auxiliary result.
Lemma 2.6.
Let
that satisfies
Proof.
Given
First of all we notice that either
ensured by condition ($g_{1}$).
In the first case the thesis follows straightforwardly. In the second case, we compute
From the previous equality and since by ($g_{1}$) we can fix
Hence, by an integration of the above inequality from
As a consequence, it follows that
for all
The following lemma gives an upper bound for positive P-periodic solutions of (2.4) which will be used to prove the existence result in Theorem 1.1.
Lemma 2.7.
Let
Proof.
By contradiction we assume that there exists a sequence
By applying Lemma 2.6 with the choice of
Through the change of variable introduced in (2.9) and an integration of (2.10) on
When
The next lemma gives us some estimates for positive solutions of (2.4) which will be used to prove the multiplicity result in Theorem 1.2. To state it, we recall the definition of
where
Lemma 2.8.
Let
Proof.
Given
We now apply Lemma 2.6 with the choice of
Notice that
Let
Let
We claim that
The inequality in
Then an integration gives
where the last inequality follows from the choice of ε. A similar argument applies in the interval
Recalling that
implying
Therefore
As a consequence, in the interval
An integration of the above inequality, together with the estimate for
where the last inequality follows from (2.12). Thus the proof is completed. ∎
Remark 2.1.
Lemma 2.8 will be exploited in Section 4.1, while verifying the assumptions of Lemma 2.1 and Lemma 2.2. We stress that only the assertion on
3 Existence of Two Solutions
In this section we give the proof of Theorem 1.1.
Proof of Theorem 1.1.
Given
We claim that Corollary 2.1 applies with the choice of
First, we verify assumption ($\widetilde{H}_{1}$). From property (2.5), since
Therefore, ($\widetilde{H}_{2}$) follows for α sufficiently large. Summing up, from Corollary 2.1, we thus obtain
Now, we use Lemma 2.4 and Lemma 2.7 to fix
The additivity property of the coincidence degree implies
As a consequence, there exist a P-periodic solution
Remark 3.1.
By a careful checking of the proof, one can realize that Theorem 1.1 is still valid if
(cf. [10, Section 4.3]). At last, we mention that, by arguing as in [10], one could also weaken assumption (a${}_{*}$), so as to cover some situations when the weight function
We end this section by stating the following straightforward corollary, dealing with the one-parameter equation (1.5).
Corollary 3.1.
Let
4 High Multiplicity of Solutions
In this section we give the proof of Theorem 1.2.
Proof of Theorem 1.2.
Given
We apply Lemma 2.5 to find
Moreover, we apply Lemma 2.8, with the choice of
We claim that there exists
We define the open and bounded sets
and so from (4.1) and the combinatorial argument in [11, Appendix A], we obtain that
As a consequence of the existence property for the coincidence degree, we thus obtain the existence of a P-periodic solution of (2.2) in each of these
Summing up, for each choice of
To achieve the conclusion of Theorem 1.2, we observe that, given any finite string
so that
4.1 Finding the Constant μ * ( λ , r , R )
The constant
where
Checking the Assumptions of Lemma 2.1
Let
Verification of (${H}_{1}$).
Let
There is an index
There is an index
There is an index
Suppose that ($h_{1}^{1}$) holds. Since
Thus, taking
we obtain
Now, we suppose that ($h_{2}^{1}$) holds. In this case a contradiction is immediately obtained by Lemma 2.3 (no assumption on
At last, we assume that ($h_{3}^{1}$) holds. As for the case ($h_{1}^{1}$) we have
Taking
we obtain
We conclude that (${H}_{1}$) holds for
Verification of (${H}_{2}$).
Let
Let
This yields a contradiction if
Checking the Assumptions of Lemma 2.2
Let
Verification of (${H}_{3}$).
By contradiction, suppose that there exists a P-periodic solution
There is an index
There is an index
Suppose that ($h_{1}^{3}$) holds. We consider at first the case
for all
Proving (4.5) in I i + 1 +
Taking
and so, from Lemma 2.5, we obtain
Integrating (2.4) on
Notice that the first of the above inequalities requires
we finally obtain that
inequalities in (4.5) hold.
Proving (4.5) in I i + 1 -
Using the monotonicity of the map
Thus the conclusion follows, since
Proving (4.5) in I i + 2 +
Integrating equation (2.4) on
in particular,
On the other hand, integrating equation (2.4) on
for all
Then the solution
Proving (4.5) in [ 0 , P ]
This is easily achieved by repeating the argument just described in order to cover a P-periodicity interval. This eventually requires
Having dealt with the case
with
and
Thus the contradiction
Taking into account all the possible situations we conclude that case ($h_{1}^{3}$) never occurs if
To conclude the proof, we suppose now that ($h_{2}^{3}$) holds. By applying Lemma 2.8, the contradiction
We conclude that case ($h_{2}^{3}$) never occurs if
Summing up, we can apply Lemma 2.2 for
and therefore formula (2.4) is verified. ∎
5 Globally Defined Solutions and Symbolic Dynamics
In this section we prove Theorem 1.3. Actually, we are going to give just a sketch of the argument, which follows the same schemes of the one for the proof of [23, Theorem 4.5]. We also remark that one could adapt to the present setting also the discussion developed in [11, Section 6], in order to show that the existence of non-periodic bounded solutions coded by sequences of three symbols implies semiconjugation of a suitable map induced by (($\mathscr{E}_{\lambda,\mu}$)) with the Bernoulli shift.
Proof of Theorem 1.3.
Given
Let
An application of Theorem 1.2 on the interval
for every
A compactness argument (cf. [23, Section 4.3]) ensures the existence of a solution
for every
To conclude the proof we have to show that the above inequalities are strict. This can be done using on one hand Lemma 2.3 (ensuring that
Remark 5.1.
Given an integer
Given an integer
where
6 Related Results and Remarks
We conclude the paper with some complementary results and remarks.
6.1 Subharmonic Solutions
In the context of Theorem 1.1, if we further suppose that
has, for every
This follows from [8, Theorem 3.3], after having observed that the constant
6.2 Dirichlet and Neumann Boundary Conditions
A suitable variant of Theorem 1.2 is valid when equation (($\mathscr{E}_{\lambda,\mu}$)) is coupled with Dirichlet boundary conditions
or Neumann boundary conditions
Let us recall that, in both cases, with a standard change of variable we can assume
In this context, it is possible to consider a slightly more general sign condition, with respect to (a${}_{*}$), for the
The proofs require just minor modifications with respect to the ones given for the periodic problem. Precisely, the appropriate abstract setting for Dirichlet and Neumann boundary conditions is described in [11, Remark 2.1]; with this in mind, the general strategy in Section 4 remains the same. In order to verify the assumptions of the degree lemmas in Section 2.1, the estimates given in Section 2.2 can still be exploited, since they are of local nature, and the boundary condition at
As standard corollaries, one can give multiplicity results for radially symmetric positive solutions of elliptic BVPs on annular domains (cf. [11, Section 7.3] and [21, Section 3]).
Figure 1High multiplicity of positive solutions for the indefinite Neumann boundary value problem associated with (($\mathscr{E}_{\lambda,\mu}$)).
![(a)
Graph of the weight term defined as a(x)=2sin(2x)-max{0,sin(x)}{a(x)=2\sin(2x)-\max\{0,\sin(x)\}} on [0,2π]{{[0,2\pi]}} and a(x)=0.2{a(x)=0.2} on [2π,8]{{[2\pi,8]}}.](/document/doi/10.1515/ans-2020-2094/asset/graphic/j_ans-2020-2094_fig_159.png)
Graph of the weight term defined as
Graph of the nonlinear term
Graphs of
6.3 Stability Issues
Dealing with equation (6.1) and assuming further that
satisfies
Taking into account the above discussion, one can apply [8, Lemma 4.2] ensuring that
In the same way we can also deduce that the small solution
6.4 Asymptotic Analysis
By using the arguments described in [11, Section 5] and in [23, Section 3.5], it is possible to investigate the asymptotic behavior for
Similarly, one can discuss the case of Dirichlet and Neumann boundary conditions (in the Neumann case, whenever
It is worth mentioning that, for the one-parameter equation
Asymptotic analysis for the indefinite Neumann boundary value problem associated with (($\mathscr{E}_{\lambda,\mu}$)) with respect to the parameters λ and μ.
Graph of the “large” solution
Graph of the “large” solution
Funding statement: The paper has been written under the auspices of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first two authors have been supported by the project ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT”. The first author has been supported by INdAM-GNAMPA project “Il modello di Born–Infeld per l’elettromagnetismo nonlineare: esistenza, regolarità e molteplicità di soluzioni”. The third author has been supported by INdAM project “Problems in Population Dynamics: from Linear to Nonlinear Diffusion”.
Acknowledgements
The authors are grateful to the referee for her/his useful remarks on improving the presentation of the paper.
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Artikel in diesem Heft
- Frontmatter
- Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent
- The ε - εβ Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets
- Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential
- Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth
- Extremals for Fractional Moser–Trudinger Inequalities in Dimension 1 via Harmonic Extensions and Commutator Estimates
- Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions
- New Results About the Lambda Constant and Ground States of the 𝑊-Functional
- A Critical Point Theorem for Perturbed Functionals and Low Perturbations of Differential and Nonlocal Systems
- High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models
- Eigenvalue Problems for Fredholm Operators with Set-Valued Perturbations
- Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian
Artikel in diesem Heft
- Frontmatter
- Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent
- The ε - εβ Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets
- Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential
- Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth
- Extremals for Fractional Moser–Trudinger Inequalities in Dimension 1 via Harmonic Extensions and Commutator Estimates
- Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions
- New Results About the Lambda Constant and Ground States of the 𝑊-Functional
- A Critical Point Theorem for Perturbed Functionals and Low Perturbations of Differential and Nonlocal Systems
- High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models
- Eigenvalue Problems for Fredholm Operators with Set-Valued Perturbations
- Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian
