scholarly journals The Existence of Entire Positive Solutions to Monge-Amp`ere Type Equations

2021 ◽  
pp. 95-103
Author(s):  
Shuangshuang Bai
Keyword(s):  
Electric Power ◽  
Laplace Operator ◽  
Positive Solutions ◽  
Mathematical Analysis ◽  
North China ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Break Line ◽  
Sufficient Condition ◽  
Mathematics And Physics

Aims/ Objectives: In this paper, we study the Monge-Amp`ere type equation det D2u + α∆u = p(|x|)f(u)(x ∈ Rn).  In the previous articles, the equation with Monge-Amp`ere operator or Laplace operator has been studied extensively. However, the research about the combination of two kinds of operator is scarce. We would like do some research on this topic. We obtain a sufficient condition of the existence of entire positive solutions for the equation.  Study Design: Study on the existence of solutions.  Place and Duration of Study: Department of Mathematics and Physics, North China Electric Power University, September 2019 to present.  Methodology: We prove the existence of the solution by constructing Euler’s break line, combining the idea of transformation and the method of mathematical analysis. Results: We obtain a sufficient condition of the existence of entire positive solutions for the equation. Conclusion: We prove the existence of entire positive solutions to Monge-Amp`ere type equation det D2u + α∆u = p(|x|)f(u)(x ∈ Rn) and obtain the sufficient condition for the existence of solutions.

Positive Solutions to Second Order Singular Differential Equations Involving the One-Dimensional M-Laplace Operator

1999 ◽  
Vol 6 (4) ◽  
pp. 347-362
Author(s):  
Tomoyuki Tanigawa
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Laplace Operator ◽  
Positive Solutions ◽  
Existence Of Solutions ◽  
Second Order ◽  
Singular Differential Equations ◽  
One Dimensional ◽  
The One ◽  
Positive Half

Abstract We consider a class of second order quasilinear differential equations with singular ninlinearities. Our main purpose is to investigate in detail the asymptotic behavior of their solutions defined on a positive half-line. The set of all possible positive solutions is classified into five types according to their asymptotic behavior near infinity, and sharp conditions are established for the existence of solutions belonging to each of the classified types.

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

scholarly journals Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

2020 ◽  
Vol 20 (2) ◽  
pp. 373-384
Author(s):  
Quoc-Hung Nguyen ◽  
Nguyen Cong Phuc
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Linear Range ◽  
Measure Data ◽  
Regularity Conditions ◽  
Compactly Supported ◽  
Singular Data ◽  
Quasilinear Elliptic

AbstractWe characterize the existence of solutions to the quasilinear Riccati-type equation\left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.with a distributional or measure datum σ. Here {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ({p>1}), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that {p>1} and {q>p}. For measure data, we assume that they are compactly supported in Ω, {p>\frac{3n-2}{2n-1}}, and q is in the sub-linear range {p-1<q<1}. We also assume more regularity conditions on {\mathcal{A}} and on {\partial\Omega\Omega} in this case.

Electric Turret-Traversing Mechanism for Tanks

1945 ◽  
Vol 12 (4) ◽  
pp. A228-A240
Author(s):  
Stanley J. Mikina
Keyword(s):  
Electric Power ◽  
Mathematical Analysis ◽  
Mass Production ◽  
Pearl Harbor ◽  
System Stability ◽  
Design Criteria ◽  
North African ◽  
Power Drive ◽  
The North ◽  
Acceptance Tests

Abstract This paper describes the development of a new electric-power drive for 360-deg traverse of tank turrets that was undertaken early in 1941. The development was brought to a successful conclusion with the satisfactory completion of Army Ordnance acceptance tests on December 6, 1941, and mass production of these traverse units was achieved shortly after Pearl Harbor Day, in time to equip the General Sherman tanks that contributed to the victories in the North African campaigns. A description of the drive and its positionally regulated system of control is given together with the mathematical analysis of system stability upon which the design criteria of the control were based.

scholarly journals The Cauchy problem for the Finsler heat equation

2020 ◽  
Vol 13 (3) ◽  
pp. 257-278 ◽  
Author(s):  
Goro Akagi ◽  
Kazuhiro Ishige ◽  
Ryuichi Sato
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Heat Equation ◽  
Laplace Operator ◽  
Sufficient Condition ◽  
Smooth Functions ◽  
Dual Norm ◽  
The Cauchy Problem

AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H. Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}. In this paper we prove that the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}}-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.

scholarly journals Solutions of Nonlocal -Laplacian Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Mustafa Avci ◽  
Rabil Ayazoglu (Mashiyev)
Keyword(s):  
Laplace Operator ◽  
Variational Approach ◽  
Nonlocal Problem ◽  
Type Equation ◽  
Multiplicity Of Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Equation ◽  
Laplacian Equations

In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving -Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.

scholarly journals Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

2021 ◽  
Vol 11 (1) ◽  
pp. 598-619
Author(s):  
Guofeng Che ◽  
Tsung-fang Wu
Keyword(s):  
Variational Principle ◽  
Positive Solutions ◽  
Type Equation ◽  
Nehari Manifold ◽  
Sobolev Embedding ◽  
Multiple Positive Solutions ◽  
Kirchhoff Type ◽  
Indefinite Nonlinearities ◽  
The Nehari Manifold ◽  
Constraint Method

Abstract We study the following Kirchhoff type equation: − a + b ∫ R N | ∇ u | 2 d x Δ u + u = k ( x ) | u | p − 2 u + m ( x ) | u | q − 2 u     in     R N , $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N=3, a , b > 0 $ a,b \gt 0 $ , 1 < q < 2 < p < min { 4 , 2 ∗ } $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2N/(N − 2), k ∈ C (ℝ N ) is bounded and m ∈ L p/(p−q)(ℝ N ). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H 1 ( R N ) ↪ L r ( R N ) ( 2 ≤ r < 2 ∗ ) $ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

scholarly journals Micro-lecture Teaching for Improving the Learning Effect of Non-English Majors at North China Electric Power University

2019 ◽  
Vol 12 (6) ◽  
pp. 209 ◽  
Author(s):  
Jia-ling HAN
Keyword(s):  
Statistical Analysis ◽  
Electric Power ◽  
Questionnaire Survey ◽  
Learning Effect ◽  
North China ◽  
Learning Motivation ◽  
College English ◽  
Academic Achievements ◽  
English Majors

The present study aimed to investigate the effect of micro-lecture teaching on non-English majors&rsquo; academic achievements and learning motivation. One hundred and twenty-two non-English majors studying the college English course in North China Electric Power University participated in the study. The micro-lecture teaching and traditional PPT teaching were implemented in the two classes respectively. Statistical analysis showed a significant improvement in the academic achievements of the participants (t = 3.128, p &lt; 0.05) between the pre and post measurements in favor of the post measurement. The results of the questionnaire survey undertaken in the experimental class showed that participants&rsquo; learning motivations were improved. Accordingly, the researcher concluded that micro-lecture teaching was significantly effective in improving non-English majors&rsquo; academic achievements and enhancing their learning motivation.

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