Variational Methods

1996 ◽  
Author(s):  
Michael Struwe
Keyword(s):  
Variational Methods

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

Variational Methods for Machine Learning with Applications to Deep Networks

2021 ◽  
Author(s):  
Lucas Pinheiro Cinelli ◽  
Matheus Araújo Marins ◽  
Eduardo Antônio Barros da Silva ◽  
Sérgio Lima Netto
Keyword(s):  
Machine Learning ◽  
Variational Methods ◽  
Deep Networks

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Quantum mechanical algebraic variational methods for inelastic and reactive molecular collisions

1988 ◽  
Vol 92 (11) ◽  
pp. 3202-3216 ◽  
Author(s):  
David W. Schwenke ◽  
Kenneth Haug ◽  
Meishan Zhao ◽  
Donald G. Truhlar ◽  
Yan Sun ◽  
...  
Keyword(s):  
Variational Methods ◽  
Quantum Mechanical ◽  
Molecular Collisions

scholarly journals Fermat Metrics

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1422
Author(s):  
Antonio Masiello
Keyword(s):  
Variational Methods ◽  
Morse Theory ◽  
Sagnac Effect ◽  
Variational Theory ◽  
Global Hyperbolicity ◽  
Fermat Principle ◽  
Multiple Image ◽  
Light Rays ◽  
Stationary Spacetimes

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

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}} .

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