Morse theory and local linking for a class of boundary value problems with impulsive effects

2015 ◽  
Vol 51 (1-2) ◽  
pp. 353-365 ◽  
Author(s):  
Hongxia Shi ◽  
Haibo Chen ◽  
Hongliang Liu
Keyword(s):  
Boundary Value Problems ◽  
Morse Theory ◽  
Boundary Value ◽  
Impulsive Effects ◽  
Local Linking

scholarly journals Existence and Multiplicity of Solutions to Discrete Conjugate Boundary Value Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-26
Author(s):  
Bo Zheng
Keyword(s):  
Nonlinear Analysis ◽  
Boundary Value Problems ◽  
Morse Theory ◽  
Boundary Value ◽  
Asymptotically Linear ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Linear Condition ◽  
Concrete Computation ◽  
Special Case

We consider the existence and multiplicity of solutions to discrete conjugate boundary value problems. A generalized asymptotically linear condition on the nonlinearity is proposed, which includes the asymptotically linear as a special case. By classifying the linear systems, we define index functions and obtain some properties and the concrete computation formulae of index functions. Then, some new conditions on the existence and multiplicity of solutions are obtained by combining some nonlinear analysis methods, such as Leray-Schauder principle and Morse theory. Our results are new even for the case of asymptotically linear.

scholarly journals Parameter Dependence of Positive Solutions for Second-Order Singular Neumann Boundary Value Problems with Impulsive Effects

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Xuemei Zhang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Second Order ◽  
Impulsive Effects ◽  
Parameter Dependence ◽  
Neumann Boundary Value Problem ◽  
Neumann Boundary Value Problems

The author considers the Neumann boundary value problem-y′′t+Myt=λωtft,yt,  t∈J,    t≠tk,  -Δy′|t=tk=λIktk,ytk,   k=1,2,…,m,  y′(0)=y′(1)=0and establishes the dependence results of the solution on the parameterλ, which cover equations without impulsive effects and are compared with some recent results by Nieto and O’Regan.

scholarly journals Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

2022 ◽  
Vol 40 ◽  
pp. 1-10
Author(s):  
Duong Trong Luyen ◽  
Le Thi Hong Hanh
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Bounded Domain ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Nontrivial Solutions ◽  
Subcritical Growth ◽  
Subelliptic Operator

In this paper, we study the existence of multiple solutions for the boundary value problem\begin{equation}\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad u=0 \quad \mbox{ on } \partial \Omega, \notag\end{equation}where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$ and $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N), $$the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.

scholarly journals Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guangcun Lu
Keyword(s):  
Boundary Value Problems ◽  
Morse Theory ◽  
Bifurcation Theory ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Abstract Theory ◽  
Elliptic Boundary Value ◽  
Potential Operators ◽  
Linear Elliptic Boundary

<p style='text-indent:20px;'>This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.</p>

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