SINGULARLY PERTURBED PROBLEM WITH DOUBLE BOUNDARY LAYER

2021 ◽  
Vol 1 (1) ◽  
pp. 102-109
Author(s):  
Dilmurat Abdillazhanovich Tursunov ◽  
Gulbayra Abdimalikovna Omaralieva ◽  
Makhfuzakhon Ibrakhimzhanovna Mamatbuvaeva ◽  
Shahzadakhan Adylzhanovna Ramankulova
Keyword(s):  
Boundary Layer ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem

scholarly journals On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 189 ◽  
Author(s):  
Stephane Malek
Keyword(s):  
Boundary Layer ◽  
Laplace Transform ◽  
Asymptotic Expansions ◽  
Perturbation Parameter ◽  
Fourier Integral ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Irregular Singularity ◽  
Gevrey Order ◽  
Logarithmic Dependence

We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

scholarly journals Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 464
Author(s):  
Jichao Wang ◽  
Ting Yu
Keyword(s):  
Poisson Equation ◽  
Existence Result ◽  
Critical Growth ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Number Of Solutions ◽  
Concentration Behavior ◽  
The Relationship ◽  
Concentration Phenomenon

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.

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