scholarly journals On the Neumann problem for an elliptic system of equations involving the critical Sobolev exponent

2001 ◽  
Vol 90 (1) ◽  
pp. 19-35 ◽  
Author(s):  
J. Chabrowski ◽  
Jianfu Yang
Keyword(s):  
Neumann Problem ◽  
Elliptic System ◽  
Critical Sobolev Exponent ◽  
System Of Equations ◽  
The Neumann Problem ◽  
Sobolev Exponent

scholarly journals The Neumann Problem for a Degenerate Elliptic System Near Resonance

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Yu-Cheng An ◽  
Hong-Min Suo
Keyword(s):  
Neumann Problem ◽  
Elliptic System ◽  
Linear Part ◽  
Degenerate System ◽  
Normal Vector ◽  
Coefficient Function ◽  
Degenerate Elliptic System ◽  
Near Resonance ◽  
Degenerate Elliptic ◽  
The Neumann Problem

This paper studies the following system of degenerate equations-divpx∇u+qxu=αu+βv+g1x,v+h1x,x∈Ω,-div(p(x)∇v)+q(x)v=βu+αv+g2(x,u)+h2(x),x∈Ω,∂u/∂ν=∂v/∂ν=0,x∈∂Ω.HereΩ⊂Rnis a boundedC2domain, andνis the exterior normal vector on∂Ω. The coefficient functionpmay vanish inΩ¯,q∈Lr(Ω)withr>ns/(2s-n),  s>n/2. We show that the eigenvalues of the operator-div(p(x)∇u)+q(x)uare discrete. Secondly, when the linear part is near resonance, we prove the existence of at least two different solutions for the above degenerate system, under suitable conditions onh1,h2,g1, andg2.

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