scholarly journals The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Ge Dong ◽  
Xiaochun Fang
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Nonlinear Term ◽  
Extremal Solutions ◽  
Quasilinear Elliptic Equations ◽  
Quasilinear Elliptic Problem ◽  
Analysis Theory ◽  
Carathéodory Function ◽  
Quasilinear Elliptic ◽  
Differential Part

We prove the existence of extremal solutions of the following quasilinear elliptic problem -∑i=1N∂/∂xiai(x,u(x),Du(x))+g(x,u(x),Du(x))=0 under Dirichlet boundary condition in Orlicz-Sobolev spaces W01LM(Ω) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term g:Ω×R×RN→R is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.

Quasilinear elliptic equations with slowly growing principal part and critical Orlicz–Sobolev nonlinear term

2009 ◽  
Vol 139 (1) ◽  
pp. 73-106 ◽  
Author(s):  
Nobuyoshi Fukagai ◽  
Masayuki Ito ◽  
Kimiaki Narukawa
Keyword(s):  
Growth Rate ◽  
Elliptic Equations ◽  
Variational Approach ◽  
Principal Part ◽  
Nonlinear Term ◽  
Quasilinear Elliptic Equations ◽  
Lower Term ◽  
Negative Solution ◽  
Fréchet Differentiable ◽  
Quasilinear Elliptic

A variational problem for a functional with slowly growing principal part and involving critical Orlicz–Sobolev lower term with respect to the principal part is discussed. The principal part of the functional is not Fréchet differentiable. The lack of differentiability and the critical growth rate of the lower term demand a precise compactness argument in the variational approach. A non-negative solution for the Euler equation is given.

scholarly journals Existence of Renormalized Solutions for Some Anisotropic Quasilinear Elliptic Equations

2020 ◽  
Vol 44 (4) ◽  
pp. 617-637
Author(s):  
T. AHMEDATT ◽  
A. AHMED ◽  
H. HJIAJ ◽  
A. TOUZANI
Keyword(s):  
Dirichlet Problem ◽  
Growth Condition ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Renormalized Solutions ◽  
Carathéodory Function ◽  
Regularity Results ◽  
Quasilinear Elliptic

In this paper, we consider a class of anisotropic quasilinear elliptic equations of the type ( | ∑N { − ∂ia (x, u, ∇u ) + |u|s(x )− 1u = f (x,u ), in Ω, i |( i=1 u = 0 on ∂ Ω, where f(x,s) is a Carathéodory function which satisfies some growth condition. We prove the existence of renormalized solutions for our Dirichlet problem, and some regularity results are concluded.

Existence of Solutions in Weighted Sobolev Spaces for Some Degenerate Quasilinear Elliptic Equations

2008 ◽  
Vol 15 (4) ◽  
pp. 627-634
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Quasilinear Elliptic Equations ◽  
Open Set ◽  
Quasilinear Elliptic

Abstract We prove an existence result for the Dirichlet problem associated to some degenerate quasilinear elliptic equations in a bounded open set Ω in in the setting of weighted Sobolev spaces .

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