Existence of positive solutions for elliptic systems with nonstandard -growth conditions via sub-supersolution method

2007 ◽  
Vol 67 (4) ◽  
pp. 1055-1067 ◽  
Author(s):  
Qihu Zhang
Keyword(s):  
Positive Solutions ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
Nonstandard Growth ◽  
Existence Of Positive Solutions ◽  
Nonstandard Growth Conditions

scholarly journals On Solutions of a Parabolic Equation with Nonstandard Growth Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Growth Condition ◽  
Boundary Value ◽  
Growth Conditions ◽  
Boundary Value Condition ◽  
Nonstandard Growth ◽  
Partial Boundary Value Condition ◽  
Nonstandard Growth Condition ◽  
Nonstandard Growth Conditions ◽  
The Stability

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient bx,t on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω×0,T. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.

Positive solutions of certain elliptic systems with density-dependent diffusions

1995 ◽  
Vol 125 (5) ◽  
pp. 1031-1050 ◽  
Author(s):  
Inkyung Ahn ◽  
Lige Li
Keyword(s):  
Positive Solutions ◽  
Growth Rates ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Biological Interactions ◽  
Bounded Region ◽  
Density Dependent ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Increasing Functions

Results are obtained on the existence of positive solutions to the following elliptic system:in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

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