morse theory
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2022 ◽  
Vol 40 ◽  
pp. 1-10
Author(s):  
Duong Trong Luyen ◽  
Le Thi Hong Hanh

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.


2021 ◽  
pp. 1-24
Author(s):  
D. Fernández-Ternero ◽  
E. Macías-Virgós ◽  
D. Mosquera-Lois ◽  
J. A. Vilches

We develop Morse–Bott theory on posets, generalizing both discrete Morse–Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik–Schnirelmann theorem for general matchings on posets, in particular, for Morse–Bott functions.


2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Karran Pandey ◽  
Talha Bin Masood ◽  
Saurabh Singh ◽  
Ingrid Hotz ◽  
Vijay Natarajan ◽  
...  

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Anthony Tromba

Abstract We prove for the first time that classical Morse theory applies to functionals of the form 𝒥 ⁢ ( u ) = 1 2 ⁢ ∫ Ω A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ∂ ⁡ u i ∂ ⁡ x α ⁢ ∂ ⁡ u j ∂ ⁡ x β ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x \displaystyle\mathcal{J}(u)=\frac{1}{2}\int_{\Omega}A^{ij}_{\alpha\beta}(x)% \frac{\partial u^{i}}{\partial x^{\alpha}}\frac{\partial u^{j}}{\partial x^{% \beta}}\,dx+\int_{\Omega}G(x,u)\,dx where u : Ω → ℝ N {u:\Omega\to\mathbb{R}^{N}} , Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} compact with C ∞ {C^{\infty}} boundary ∂ ⁡ Ω {\partial\Omega} , u | ∂ ⁡ Ω = φ {u|_{\partial\Omega}=\varphi} , and we argue that this is the largest class to which Morse theory applies.


2021 ◽  
Vol 40 (6) ◽  
pp. 1473-1487
Author(s):  
Rafael Galeano Andrades ◽  
Joel Torres del Valle

In this paper we study the unidimensional Stationary Boltzmann Equation by an approach via Morse theory. We define a functional J whose critical points coincide with the solutions of the Stationary Boltzmann Equation. By the calculation of Morse index of J’’0(0)h and the critical groups C2(J, 0) and C2(J, ∞) we prove that J has two different critical points u1 and u2 different from 0, that is, solutions of Boltzmann Equation.


2021 ◽  
Vol 212 ◽  
pp. 112466
Author(s):  
Desheng Li ◽  
Mo Jia

Author(s):  
Jun Wang ◽  
li wang ◽  
Qiao Zhong

This paper is devoted to the following fractional Schrödinger-Poisson systems: \begin{equation*} \left\{\aligned &(-\Delta)^{s} u+V(x)u+\phi(x)u= f(x,u) \,\,\,&\text{in } \mathbb{R}^3, \\ & (-\Delta)^{t} \phi(x)=u^2 \,\,\,&\text{in } \mathbb{R}^3, \endaligned \right. \end{equation*} where $(-\Delta)^{s}$ is the fractional Lapalcian, $s, t \in (0, 1),$ $V : \R^3 \to \R$ is continuous. In contrast to most studies, we consider that the potentials $V$ is indefinite. With the help of Morse theory, the existence of nontrivial solutions for the above problem is obtained.


2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Roberto Livrea

AbstractWe consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.


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