Infinitely many solutions for semilinear Δλ-Laplace equations with sign-changing potential and nonlinearity

2017 ◽  
Vol 54 (4) ◽  
pp. 536-549 ◽  
Author(s):  
Jianhua Chen ◽  
Xianhua Tang ◽  
Zu Gao
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Boundary Value ◽  
Quadratic Growth ◽  
Infinitely Many Solutions ◽  
Type Condition ◽  
Degenerate Elliptic Operator ◽  
Laplace Equations ◽  
Degenerate Elliptic ◽  
Strongly Degenerate

In this paper, we prove the existence of infinitely many solutions for the following class of boundary value elliptic problems where Ω is a bounded domain in RN (N ≥ 2), Δλ is a strongly degenerate elliptic operator, V (x) is allowing to be sign-changing and f is a function with a more general super-quadratic growth, which is weaker than the Ambrosetti-Rabinowitz type condition.

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 On Boundary Value Problems for Elliptic Equations in a Singular Domain

1969 ◽  
Vol 36 ◽  
pp. 99-115
Author(s):  
Kazunari Hayashida
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Tangent Line ◽  
Boundary Operators ◽  
One To One ◽  
Singular Domain

1. Let Ω be a bounded domain in the plane and denotes its closure and boundary by Ω̅ and ∂Ω, respectively. We shall say that the domain Ω is regular, if every point P ∈ ∂û has an 2-dimensional neighborhood U such that dΩ ∩ U can be mapped in a one-to-one way onto a portion of the tangent line through P by a mapping T which together with its inverse is infinitely differentiable. Let L be an elliptic operator of order 2m defined in Ω̅ and let be a normal set of boundary operators of orders mf <2m. If f is a given function in Ω, the boundary value problem II(L,f,Bj) will be to find a solution u ofsatisfyingBju = 0 on ∂Ω, j = 1, …, m.

Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains

1987 ◽  
Vol 105 (1) ◽  
pp. 205-213 ◽  
Author(s):  
Donato Fortunato ◽  
Enrico Jannelli
Keyword(s):  
Boundary Value Problem ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Boundary Value ◽  
Sobolev Embedding ◽  
Infinitely Many Solutions ◽  
Positive Parameter ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Image Position

SynopsisWe consider the boundary value problemwhere Ω ⊂ ℝn is a bounded domain, n≧3, 2* = 2n/(n − 2) is the critical exponent for the Sobolev embedding and λ is a real positive parameter. We prove the existence of infinitely many solutions of (*) when Ω exhibits suitable symmetries.

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