scholarly journals Classification for positive solutions of degenerate elliptic system

2019 ◽  
Vol 39 (3) ◽  
pp. 1457-1475
Author(s):  
Yuxia Guo ◽  
◽  
Jianjun Nie ◽  
Keyword(s):  
Elliptic System ◽  
Positive Solutions ◽  
Degenerate Elliptic System ◽  
Degenerate Elliptic

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.

Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

2017 ◽  
Vol 8 (1) ◽  
pp. 661-678 ◽  
Author(s):  
Cung The Anh ◽  
Bui Kim My
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Smooth Boundary ◽  
Infinitely Many Solutions ◽  
Degenerate Elliptic System ◽  
Degenerate Elliptic ◽  
Degenerate Operator ◽  
Strongly Degenerate

Abstract We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system \left\{\begin{aligned} &\displaystyle{-}\Delta_{\lambda}u=\lvert v\rvert^{p-1}% v&&\displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle{-}\Delta_{\lambda}v=\lvert u\rvert^{q-1}u&&\displaystyle% \phantom{}\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\phantom{}\text{on }\partial\Omega,\end{% aligned}\right. in a bounded domain {\Omega\subset\mathbb{R}^{N}} with smooth boundary {\partial\Omega} . Here {p,q>1} , and {\Delta_{\lambda}} is the strongly degenerate operator of the form \Delta_{\lambda}u=\sum^{N}_{j=1}\frac{\partial}{\partial x_{j}}\Bigl{(}\lambda% _{j}^{2}(x)\frac{\partial u}{\partial x_{j}}\Bigr{)}, where {\lambda(x)=(\lambda_{1}(x),\dots,\lambda_{N}(x))} satisfies certain conditions.

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