Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
-
Sami Baraket
,
Anis Ben Ghorbal
Abstract
This work investigates the existence of singular limit solutions for nonlinear elliptic systems. Our main approach focuses on using the nonlinear domain decomposition method to establish a new Liouville-type result.
1 Introduction and statement of the results
We consider the following problem:
where
This article aims to establish the existence of solutions
In all the following, we can use by deduction the fact that
Using the transformation
then
where
Yamabe system has attracted considerable interest in recent years due to its presence in various physical phenomena, such as nonlinear optics. Here, the coupled solution
In [7], Chanillo and Kiessling established a strict isoperimetric inequality and a Pohozaev-Rellich identity for the system
under the finite mass conditions
where
They proved that all solutions
Incidentally, equation (8) is the simplest special case of the L-system, which is invariant under translation, rotation and dilation in the Euclidean plane. Moreover, it shares invariance under the Kelvin transform. In [9], Chen and Li proved the following important classification result.
Theorem 1
[9] Let
Then u is radially symmetric and decreasing about some point in
This result plays a crucial role in obtaining the complete solution of (8) under (9) as it simplifies the problem to a straightforward ordinary differential equation (ODE) problem. Thus, Chen and Li concluded
that all solutions of (8), which verify (9), are given by
where
Furthermore, similar to Liouville’s equation, the system (5) bears significant geometric implications. A solution
The system (5)–(7) raises some interesting problems. First, there is always the solution family
In higher dimensions
It is well known that as the parameter
and let
Theorem 2
[17] Let
then there exists
Several generalizations are available in [18–21].
The blow-up analysis for the equation (11) is well understood, thanks to the results of Suzuki [22] and Nagasaki and Suzuki [23]. Their work enabled the localisation of the blow-up set of singular limit solutions (up to a subsequence) as critical points of a function given by Green’s functions. In several cases, if the solution is blow-up, then it must have a concentration [24]. In contrast, the blow-up analysis for the Liouville system (1) is almost open.
In [25], Abid et al. considered the following problem:
where
A natural question that arises is as follows: Can a singular limit
where
where
In this article, we will prove the existence of singular limit solutions of (2). More precisely, we will construct a singular limit
In the following, we will assume that
In the following lemma, we give a necessary condition on the position of the points
Lemma 1
Let
and
then
Before stating our main result, we define an auxiliary function
Theorem 3
Let
and
By introducing a new assumption on
Theorem 4
Let
Then there exists
and
Remark 1
The conditions of Theorem 4 are certainly not valid on all domains of
The outline of this article is as follows: In Section 2, we establish a necessary condition on the position of the points
2 Proof of Lemma 1
Let
Similarly, by multiplying the equation
By making use of the identity, we obtain the following:
Then, by multiplying (18) by
We insert the profile of the limits of the solutions in the identity (20) as
So we obtain
By applying the same argument as in the proof of Theorem 3 in [27], we conclude that
3 Proof of Theorem 3
3.1 Construction of the approximate solution
We denote by
Let
which is a solution of
in
is also a solution of (22). Finally, we denote by
3.1.1 Linearized operators
Initially, we establish certain definitions and notation.
Definition 1
Given
is finite. Similarly, for a given
is finite. Finally, set
is finite.
We define the linear second-order elliptic operator
which is the linearized operator of
For all
Proposition 1
[17] All bounded solutions of
Moreover, for
We denote by
Similarly, by using the fact that any bounded harmonic function in
Proposition 2
Let
We denote by
Finally, we consider punctured domains. Given that
Definition 2
Let
Furthermore, for
We recall the following result.
Proposition 3
[18] Let
We denote by
Before proceeding to the various stages of constructing the solution to the problem (2), we will provide a brief overview of the technique employed and its intended purpose. First, we construct an approximate solution of the system (2) excluding the gradient term. Then, prior to constructing an approximate solution of problem (2), we perform a suitable transformation to work in a larger ball. To give a certain degree of freedom to the edge data, we establish an exact solution of our problem inside small balls centered on singularities with arbitrary data at the edge of each small ball. Next, we build an exact solution outside the balls with arbitrary data at the edge of the balls. Finally, with a suitable choice of these data at the edges, we combine interior and exterior solutions to derive a global solution across the whole domain
3.1.2 Ansatz and first estimates
For all
Here,
Now, we define an ansatz solution for the system (2) without the gradient term. More precisely, we define an ansatz solution for the following problem:
Let
and
Therefore, in
Then, for
where
If
If
On the other hand, in
Finally, in
Then, for
Now, we define an approximate solution for the following system near
By using the following transformations, we obtain
and
System (33) can be written as follows:
and
with
In
This amounts to solving the equation
We denote by
To find a solution of (38), it is enough to find a fixed point
Then, we have the following result.
Lemma 2
Given
Provided
Proof
We have
By making use of Proposition 1 together with (26), for
For the second estimate, we have
where
If
To derive the third estimate, for
Using the following estimates
together with condition
By making use of Proposition 1 together with (26) and using the condition
On the other hand, we have
Using the following estimates
in conjunction with condition
Applying Proposition 1 together with (26) and using the condition
which proves Lemma 2.□
Reducing
is a contraction from the ball
into itself. Hence, within this set, a unique fixed point
Proposition 4
Given
Hence,
In
This amounts to solve the equation
We denote
To find a solution of (45), it is enough to find a fixed point
Then, we have the following result.
Lemma 3
Given
Provided
Proof
We have
For
For the second estimate, we have
To establish the third estimate, for
By using Proposition 1 along with (26) and considering condition
Similarly, we obtain the estimate for
which proves the Lemma 3.□
We note that there exists
is a contraction from the ball
(46) as the unique fixed point
Proposition 5
Given
Hence,
Our objective is to obtain a solution of (36) in
This enables to solve the equation
We denote
To obtain a solution of (52), it suffices to identify a fixed point
By using the same reasoning as in the proof of Proposition 4, we establish
Proposition 6
Given
Therefore,
3.1.3 Harmonic extensions
In this subsection, we will study the properties of interior and exterior harmonic extensions. Given
We will use also the following definition.
Definition 3
Given
is finite.
We denote
Lemma 4
[18] There exists
we have
then
If
Lemma 5
[18] The mapping
is an isomorphism.
3.2 The nonlinear interior problem
Here, we are interested to study the system
and
With
In
This involves solving the equation
We denote
To find a solution of (62), it is sufficient to locate a fixed point
Given
In consequence, we obtain the following result.
Lemma 6
Given
Given
Proof
The initial estimate stems from Lemma 4, along with the assumption regarding the norm of the boundary data
On the other hand, for
Using Proposition 1 and (26), according to condition
For the second estimate, we have
By using the same argument as previously mentioned, we obtain
To derive the third estimate, for
By making use of Proposition 1 together with (26) and using the condition
Similarly, using the condition
The proof of Lemma 6 is achieved.□
By reducing
is a contraction from the ball
into itself and hence a unique fixed point
Proposition 7
Given
that is
Thus,
In
This is equivalent to solving the equation
We designate
Finding a fixed point
Given
Then, we have the following result.
Lemma 7
Given
Provided
Proof
We have
By using Proposition 1, (26) and condition
For the second estimate, we have
To establish the third estimate, for
On the basis of Proposition 1, inequality (26), and condition
Similarly, we obtain the estimate for
This concludes the proof of Lemma 7.□
We note that there exists
is a contraction from the ball
into itself. Consequently, a unique fixed point
Proposition 8
Given
Therefore,
In
This facilitates solving the equation
We denote
To find a solution of (78), it is enough to find a fixed point
Given
On the basis of the same arguments of the proof of Proposition 7, we prove the following proposition.
Proposition 9
Given
Thus,
Also, we remark that the functions
3.3 The nonlinear exterior problem
Given
Here,
in the domain
This amounts to solve in
For all
Here,
We fix
where
We denote
Given
Then, the following result holds.
Lemma 8
Under the aforementioned assumptions, there exists a constant
and
Provided that
Proof
Similarly for the interior problem, the initial two estimates stem from the asymptotic behavior of
On the other hand,
We will estimate
In
Hence, for
In
where
In
Thus, for
In
where
In
Hence, for
In
where
In
For
Then,
Now, let’s focus on the second equation of the system (86).
In
Hence, for
In
where
In
Therefore, for
In
where
In
Accordingly, for
In
where
In
So for
Then,
By using Proposition 3 and the inequality (84), we conclude that
For proving of the third estimate, let
and
Then, for
and
The proof is completed.□
Reducing
is a contraction from the ball
into itself. Hence, there exists a unique fixed point
We summarize the previous results to derive the following proposition.
Proposition 10
Given
solve (82) in
3.4 The nonlinear Cauchy-data matching
We will gather the results of previous sections. By using the previous notations, we assume that
are given (the values of
in
and
where, for i = 1, 2, 3 and
Similarly, given boundary data
with
on
Suppose equation (95) is satisfied, indicating that for sufficiently small
Before advancing, let’s address the following remarks. First, it is worth noting that the function
On
Next, although all functions are defined on
Since the boundary data are chosen to satisfy either (57) or (56). We make the following decomposition:
where
Using equality (97), for
Then, the projection of the equations (98) over
where
The system (100) can be simply written as follows:
We are now in a position to define
Hence, it is enough to choose
Consider now the projection of (98) over
With these notations in mind, we obtain the system
Finally, we consider the projection onto
Thanks to Lemma 5, the above system can be rewritten as follows:
If we define the parameters
then the system we need to solve reads
As usual, the terms
By using the fact that
with
yield to
On
We denote
Then, we have
Next, despite all functions being defined on
Since the boundary data are chosen to satisfy (57) or (56). We decompose as follows:
where
We insist that, for
Then, the projection of the equations (106) over
where
The system (108) can be simplified to:
We are now in a position to define
Hence, it is enough to choose
Consider now the projection of (106) over
Taking into account these notations, we derive the system
Finally, we consider the projection onto
By Lemma 5, we can express this final system as follows:
If we define the parameters
Thus, the system to be solved can be restated as follows:
where the terms
We use the fact that
with
implies that
On
Alternatively, although all functions are defined on the boundary of
Given that the boundary data are chosen to fulfill (57) or (56). We decompose as follows:
where
By using (113), we have for
Then, the projection of the equations (114) over
where
The system (116) can be simply expressed as follows:
We are now able to define
Therefore, it suffices to select
Consider now the projection of (114) over
Taking these notations into consideration, we derive the system.
Finally, we consider the projection onto
Thanks to Lemma 5, we can rewrite this final system as follows:
If we define the parameters
then the system we need to solve can be stated as follows:
As usual, the terms
Recalling that
By combining (102), (103), (111), (112), and (118), we obtain
Furthermore, the nonlinear mapping appearing on the right-hand side of (119) is continuous and compact. Moreover, by reducing
4 Proof of Theorem 4
4.1 Construction of the approximate solution
Here, we will construct another approximate solution for
Using the following transformations
and
The system (120) can be expressed as follows:
and
with
In
Proposition 11
Given
solves (121) in
Proposition 12
Given
solves (123) in
In
This involves solving the equation
We denote
To find a solution of (125), it suffices to find a fixed point
Then, we obtain the following result.
Lemma 9
Given
Provided
Proof
We have
Using Proposition 1 in conjunction with (26) and condition (17), for
For the second estimate, we have
To obtain the third estimate, for
By making use of Proposition 1 together with (26) and using the condition
Similarly, we obtain the estimate for
This completes the proof of Lemma 9.□
There exists
is a contraction from the ball
into itself. Consequently, a unique fixed point
Proposition 13
Given
Therefore,
4.2 The nonlinear interior problem
Here, we are interested in studying the system
and
with
In
Proposition 14
Given
solves (131) in
Proposition 15
Given
solves (133) in
In
This entails solving the equation
We denote by
To find a solution of (135), it suffices to locate a fixed point
Given
Then, the following result holds.
Lemma 10
Given
Provided
Proof
The first estimate arises from Lemma 4, alongside the assumption regarding the norm of the boundary data
On the other hand for
By Proposition 1, inequality (26) and condition
For the second estimate, we have
To obtain the third estimate, for
By making use of Proposition 1 together with (26) and using the condition
Similarly, we obtain the estimate
This concludes the proof of Lemma 10.□
There exist
is a contraction from the ball
into itself, and hence, a unique fixed point
Proposition 16
Given
Hence,
Also, we remark that the functions
4.3 The nonlinear exterior problem
By precisely reproducing the same nonlinear exterior problem as in the proof of Theorem 3, we establish the following proposition using the same notations.
Proposition 17
Given
which solves (82) in
4.4 The nonlinear Cauchy-data matching
We will gather the results from previous sections. Employing the prior notations, suppose
are given (the values of
in
and
where, for i = 1, 2, 3 and
Similarly, given boundary data
Here,
on
Suppose that (143) is verified, this implies that for each
Before we proceed, it is worth noting the following remarks. First, it will be convenient to observe that the function
On
where
Also, using the fact that
where
on
where
Finally, by using the following equalities
with
on
The terms
On
and
Subsequently, despite all functions being defined on
Since the boundary data are chosen to satisfy (57) or (56), we perform the following decomposition:
where
We insist that, for
Then, projecting equations (149) over
where
The system (151) can be simply written as follows:
We are now in a position to define
Therefore, it suffices to choose
Consider now the projection of (149) over
Keeping these notations, we derive the system
Finally, we consider the projection onto
Thanks to Lemma 5, we can rewrite this final system as follows:
If we define the parameters
Then, the system we need to solve reads:
The terms
On
with
On other hand, according to the same proof and since when
where
Also, by using the fact that
where
on
where
The terms
Recall that
By combining (145), (146), (153), (154), and (155), we obtain
The nonlinear mapping appearing on the right-hand side of (156) is continuous and compact. In addition, reducing
-
Funding information: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (Grant Number IMSIU-DDRSP2503).
-
Author contributions: All authors contributed equally to this work.
-
Conflict of interest: The authors state that there is no conflict of interest.
-
Ethical approval: Not applicable.
-
Data availability statement: Not applicable.
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- Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
- On the sum form functional equation related to diversity index
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
- Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
- Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
- Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
- Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
- New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
- Special Issue on Approximation Theory and Special Functions 2024 conference
- Ulam-type stability for Caputo-type fractional delay differential equations
- Faster approximation to multivariate functions by combined Bernstein-Taylor operators
- (λ, ψ)-Bernstein-Kantorovich operators
- Some special functions and cylindrical diffusion equation on α-time scale
- (q, p)-Mixing Bloch maps
- Orthogonalizing q-Bernoulli polynomials
- On better approximation order for the max-product Meyer-König and Zeller operator
- Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
- Special Issue on Variational Methods and Nonlinear PDEs
- A note on mean field type equations
- Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
- Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
Articles in the same Issue
- Research Articles
- On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
- Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
- Grand Triebel-Lizorkin-Morrey spaces
- Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
- Proofs of two conjectures involving sums of normalized Narayana numbers
- On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
- New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
- Some inequalities for rational function with prescribed poles and restricted zeros
- Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
- Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
- Fixed point results for contractions of polynomial type
- Estimation for spatial semi-functional partial linear regression model with missing response at random
- Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
- New forms of bilateral inequalities for K-g-frames
- Rate of pole detection using Padé approximants to polynomial expansions
- Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
- Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
- Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
- New topologies derived from the old one via operators
- Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
- Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
- Nonparametric expectile shortfall regression for functional data
- Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
- A generalized p-Laplacian problem with parameters
- A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
- Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
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- Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
- Solutions of several general quadratic partial differential-difference equations in ℂ2
- Inequalities for the generalized trigonometric functions with respect to weighted power mean
- Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
- Hankel determinants for q-starlike functions connected with q-sine function
- System of partial differential hemivariational inequalities involving nonlocal boundary conditions
- A new family of multivalent functions defined by certain forms of the quantum integral operator
- A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
- Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
- Behavior of spatial curves under different transformations in Euclidean 4-space
- Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
- A new kind of Durrmeyer-Stancu-type operators
- A study of generalized Mittag-Leffler-type function of arbitrary order
- On the approximation of Kantorovich-type Szàsz-Charlier operators
- Split quaternion Fourier transforms for two-dimensional real invariant field
- Review Article
- Characterization generalized derivations of tensor products of nonassociative algebras
- Special Issue on Differential Equations and Numerical Analysis - Part II
- Existence and optimal control of Hilfer fractional evolution equations
- Persistence of a unique periodic wave train in convecting shallow water fluid
- Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
- Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
- Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
- Stability and bifurcation analysis of a modified chemostat model
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
- Analytic solutions of a generalized complex multi-dimensional system with fractional order
- Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
- Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
- On the sum form functional equation related to diversity index
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
- Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
- Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
- Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
- Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
- New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
- Special Issue on Approximation Theory and Special Functions 2024 conference
- Ulam-type stability for Caputo-type fractional delay differential equations
- Faster approximation to multivariate functions by combined Bernstein-Taylor operators
- (λ, ψ)-Bernstein-Kantorovich operators
- Some special functions and cylindrical diffusion equation on α-time scale
- (q, p)-Mixing Bloch maps
- Orthogonalizing q-Bernoulli polynomials
- On better approximation order for the max-product Meyer-König and Zeller operator
- Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
- Special Issue on Variational Methods and Nonlinear PDEs
- A note on mean field type equations
- Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
- Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
