scholarly journals An elliptic system with logarithmic nonlinearity

2017 ◽  
Vol 8 (1) ◽  
pp. 928-945 ◽  
Author(s):  
Claudianor Alves ◽  
Abdelkrim Moussaoui ◽  
Leandro Tavares
Keyword(s):  
Elliptic System ◽  
Bifurcation Theory ◽  
Existence Of Solutions ◽  
Singular Systems ◽  
Quasilinear Equations ◽  
Singular Terms ◽  
Laplacian Operators

Abstract In the present paper, we study the existence of solutions for some classes of singular systems involving the {\Delta_{p(x)}} and {\Delta_{q(x)}} Laplacian operators. The approach is based on bifurcation theory and the sub-supersolution method for systems of quasilinear equations involving singular terms.

scholarly journals ON AN ELLIPTIC SYSTEM OF P(X)-KIRCHHOFF-TYPE UNDER NEUMANN BOUNDARY CONDITION

2012 ◽  
Vol 17 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Zehra Yucedag ◽  
Mustafa Avci ◽  
Rabil Mashiyev
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Variational Principle ◽  
Variational Method ◽  
Elliptic System ◽  
Neumann Boundary Condition ◽  
Existence Of Solutions ◽  
Neumann Boundary ◽  
Kirchhoff Type ◽  
Direct Variational Method

In the present paper, by using the direct variational method and the Ekeland variational principle, we study the existence of solutions for an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition and show the existence of a weak solution.

Bifurcation of steady-state solutions in predator-prey and competition systems

1984 ◽  
Vol 97 ◽  
pp. 21-34 ◽  
Author(s):  
J. Blat ◽  
K. J. Brown
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Existence Of Solutions ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Predator Prey ◽  
Interacting Species ◽  
Global Bifurcation Theory ◽  
Steady State Solutions

SynopsisWe discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

scholarly journals Existence of Solutions to a Class of Semilinear Elliptic Problem with Nonlinear Singular Terms and Variable Exponent

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Ying Chu ◽  
Yanchao Gao ◽  
Wenjie Gao
Keyword(s):  
Variable Exponent ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Value Problem ◽  
Lebesgue Spaces ◽  
Singular Terms ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Existence And Regularity Results ◽  
Regularity Results

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation- div(M(x)∇un)=f(x)/uα(x)withf∈Lm(Ω)  (m⩾…1)andα(x)>0. The results show the dependence of the summability offin some Lebesgue spaces and on the values ofα(x).

Homogenization of a nonlinear elliptic system with a transport term in L 2

2021 ◽  
pp. 1-16
Author(s):  
Juan Casado-Díaz
Keyword(s):  
Elliptic System ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic System ◽  
Convection Term ◽  
Renormalized Solutions ◽  
Linear Diffusion ◽  
Nonlinear Elliptic ◽  
Non Linear ◽  
Non Linear System ◽  
Transport Term

We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in L 2 and a right-hand side which is only in L 1 . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45(7) (2020) 690–713). Here, we prove its stability by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system.

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