Singular extremal solutions for supercritical elliptic equations in a ball

2018 ◽  
Vol 265 (7) ◽  
pp. 2842-2885 ◽  
Author(s):  
Yasuhito Miyamoto ◽  
Yūki Naito
Keyword(s):  
Elliptic Equations ◽  
Extremal Solutions ◽  
Singular Extremal

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.

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