scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

scholarly journals A Lotka-Volterra Competition Model with Cross-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wenyan Chen ◽  
Ya Chen
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Cross Diffusion ◽  
Existence Of Positive Solutions

A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

The Bahri–Coron Theorem for Fractional Yamabe-Type Problems

2018 ◽  
Vol 18 (2) ◽  
pp. 393-407 ◽  
Author(s):  
Wael Abdelhedi ◽  
Hichem Chtioui ◽  
Hichem Hajaiej
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Laplacian Operator ◽  
Fractional Laplacian Operator

AbstractWe study the following fractional Yamabe-type equation:\left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.Here Ω is a regular bounded domain of{\mathbb{R}^{n}},{n\geq 2}, and{A_{s}},{s\in(0,1)}, represents the fractional Laplacian operator{(-\Delta)^{s}}in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

scholarly journals Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

2008 ◽  
Vol 2 (2) ◽  
pp. 158-174 ◽  
Author(s):  
Qianqiao Guo ◽  
Pengcheng Niu ◽  
Jingbo Dou
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

scholarly journals Blow-up and energy decay for a class of wave equations with nonlocal Kirchhoff-type diffusion and weak damping

2021 ◽  
Author(s):  
Menglan Liao ◽  
Zhong Tan
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Initial Data ◽  
Dirichlet Boundary Condition ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Kirchhoff Type ◽  
Weak Damping

The purpose of this paper is to study the following equation driven by a nonlocal integro-differential operator $\mathcal{L}_K$: \[u_{tt}+[u]_s^{2(\theta-1)}\mathcal{L}_Ku+a|u_t|^{m-1}u_t=b|u|^{p-1}u\] with homogeneous Dirichlet boundary condition and initial data, where $[u]^2_s$ is the Gagliardo seminorm, $a\geq 0,~b>0,~0

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