Multiple solutions of critical singular degenerate elliptic system with concave-convex nonlinearities

2018 ◽  
Vol 11 (3) ◽  
pp. 157-174
Author(s):  
Chang-Mu Chu ◽  
Lin Li
Keyword(s):  
Elliptic System ◽  
Multiple 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.

scholarly journals Picone’s identity for -Laplace operator and its applications

2021 ◽  
Vol 73 (4) ◽  
pp. 515-522
Author(s):  
D. T. Luyen
Keyword(s):  
Elliptic System ◽  
Laplace Operator ◽  
Type Inequality ◽  
Strict Monotonicity ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Degenerate Elliptic System ◽  
Picone’S Identity ◽  
Degenerate Elliptic ◽  
Nonlinear Analogue

UDC 517.9We prove a nonlinear analogue of Picone's identity for -Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.  

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