scholarly journals Multiple solutions for an impulsive boundary value problems on the halfline via Morse theory

2015 ◽  
pp. 1 ◽  
Author(s):  
Karima Ait-Mahiout ◽  
Smail Djebali ◽  
Toufik Moussaoui
Keyword(s):  
Boundary Value Problems ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Impulsive Boundary Value Problems

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 Multiple solutions for discrete boundary value problems

2009 ◽  
Vol 356 (2) ◽  
pp. 418-428 ◽  
Author(s):  
Alberto Cabada ◽  
Antonio Iannizzotto ◽  
Stepan Tersian
Keyword(s):  
Boundary Value Problems ◽  
Multiple Solutions ◽  
Boundary Value
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