scholarly journals A generic multiplicity result for a Yamabe-type equation via Morse theory

2020 ◽  
Author(s):  
Gustavo de Paula Ramos
Keyword(s):  
Morse Theory ◽  
Type Equation ◽  
Multiplicity Result

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

2020 ◽  
pp. 2050031
Author(s):  
Mingzheng Sun ◽  
Jiabao Su ◽  
Binlin Zhang
Keyword(s):  
Critical Exponents ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Nonlinear Term ◽  
Type Equation ◽  
Critical Groups ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
The First Eigenvalue

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

An indefinite and critical concave—convex-type equation

2013 ◽  
Vol 143 (1) ◽  
pp. 169-184 ◽  
Author(s):  
Humberto Ramos Quoirin ◽  
Pedro Ubilla
Keyword(s):  
Bounded Domain ◽  
Type Equation ◽  
Nehari Manifold ◽  
Real Parameter ◽  
Linear Case ◽  
Energy Estimates ◽  
Multiplicity Result ◽  
Image Position

We analyse the multiplicity of non-negative solutions for the critical concave—convex-type equationwhere Ω is a bounded domain of ℝN, 1 < r < p and λ > 0, and μ is a real parameter. Combining minimization on the underlying Nehari manifold with energy estimates, we show that, under suitable conditions on a and b, three non-negative solutions may exist when λ is positive and sufficiently small and μ is in a right neighbourhood of μ1, the first weighted eigenvalue of the p-Laplacian. To the best of our knowledge, our multiplicity result is new, even in the semi-linear case p = 2.

A multiplicity result for asymptotically linear Kirchhoff equations

2017 ◽  
Vol 8 (1) ◽  
pp. 267-277 ◽  
Author(s):  
Chao Ji ◽  
Fei Fang ◽  
Binlin Zhang
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Type Equation ◽  
Ground State Solution ◽  
Mountain Pass ◽  
Multiplicity Result ◽  
Asymptotically Linear ◽  
Type Solution ◽  
Kirchhoff Equations ◽  
Kirchhoff Type

Abstract In this paper, we study the following Kirchhoff type equation: -\bigg{(}1+b\int_{\mathbb{R}^{N}}\lvert\nabla u|^{2}\,dx\biggr{)}\Delta u+u=a(% x)f(u)\quad\text{in }\mathbb{R}^{N},\qquad u\in H^{1}(\mathbb{R}^{N}), where {N\geq 3} , {b>0} and {f(s)} is asymptotically linear at infinity, that is, {f(s)\sim O(s)} as {s\rightarrow+\infty} . By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on {a(x)} .

scholarly journals A multiplicity result for a (p, q)-Schrödinger–Kirchhoff type equation

2021 ◽  
Author(s):  
Vincenzo Ambrosio ◽  
Teresa Isernia
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Category Theory ◽  
Type Equation ◽  
Multiplicity Result ◽  
Positive Potential ◽  
Subcritical Growth ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Penalization Techniques

AbstractIn this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.

The build of nonlinear quantum mechancs and variations of feature of microscopic particles as well as their experimental affirmation

2014 ◽  
Vol 5 (3) ◽  
pp. 871-981 ◽  
Author(s):  
Pang Xiao Feng
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Lagrange Equation ◽  
Minimum Principle ◽  
Superposition Principle ◽  
Type Equation ◽  
Experimental Results ◽  
Traditional Concept ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics

We establish the nonlinear quantum mechanics due to difficulties and problems of original quantum mechanics, in which microscopic particles have only a wave feature, not corpuscle feature, which are completely not consistent with experimental results and traditional concept of particle. In this theory the microscopic particles are no longer a wave, but localized and have a wave-corpuscle duality, which are represented by the following facts, the solutions of dynamic equation describing the particles have a wave-corpuscle duality, namely it consists of a mass center with constant size and carrier wave, is localized and stable and has a determinant mass, momentum and energy, which obey also generally conservation laws of motion, their motions meet both the Hamilton equation, Euler-Lagrange equation and Newton-type equation, their collision satisfies also the classical rule of collision of macroscopic particles, the uncertainty of their position and momentum is denoted by the minimum principle of uncertainty. Meanwhile the microscopic particles in this theory can both propagate in solitary wave with certain frequency and amplitude and generate reflection and transmission at the interfaces, thus they have also a wave feature, which but are different from linear and KdV solitary wave’s. Therefore the nonlinear quantum mechanics changes thoroughly the natures of microscopic particles due to the nonlinear interactions. In this investigation we gave systematically and completely the distinctions and variations between linear and nonlinear quantum mechanics, including the significances and representations of wave function and mechanical quantities, superposition principle of wave function, property of microscopic particle, eigenvalue problem, uncertainty relation and the methods solving the dynamic equations, from which we found nonlinear quantum mechanics is fully new and different from linear quantum mechanics. Finally, we verify further the correctness of properties of microscopic particles described by nonlinear quantum mechanics using the experimental results of light soliton in fiber and water soliton, which are described by same nonlinear Schrödinger equation. Thus we affirm that nonlinear quantum mechanics is correct and useful, it can be used to study the real properties of microscopic particles in physical systems.

β Grain Growth Kinetics of a New Metastable β Titanium Alloy

2013 ◽  
Vol 747-748 ◽  
pp. 844-849 ◽  
Author(s):  
Yue Fei ◽  
Xin Nan Wang ◽  
Zhi Shou Zhu ◽  
Jun Li ◽  
Guo Qiang Shang ◽  
...  
Keyword(s):  
Titanium Alloy ◽  
Grain Growth ◽  
Kinetic Equations ◽  
Type Equation ◽  
Growth Exponent ◽  
Solution Treated ◽  
Growth Activation ◽  
Grain Growth Kinetics ◽  
Kinetics Of ◽  
Strength And Ductility

Ti-Mo-Nb-Cr-Al-Fe-Si alloy is a new metastable β titanium alloy with excellent combination of strength and ductility. The β grain-growth exponent and the activation energies for β grain growth for the investigated alloy at specified temperature were computed by the kinetic equations and the Arrhenius-type equation. The rate of β grain growth decreases with elongating solution treated time and increases with the increasing solution-treated temperature. The β grain-growth exponents, n, are 0.461, 0.464 and 0.469 at 1113, 1133 and 1153K, respectively. The β grain growth activation energy is determined to be 274 KJ/mol.

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