Perturbation and Variational Methods in Nonextensive Tsallis Statistics

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].

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On the variational methods of solution of problems of steady-state creep in plates and shells in the case of finite displacements

Journal of Applied Mathematics and Mechanics ◽

10.1016/0021-8928(62)90040-0 ◽

1962 ◽

Vol 26 (3) ◽

pp. 733-739

Author(s):

I.G. Teregulov

Keyword(s):

Steady State ◽

Variational Methods ◽

Steady State Creep ◽

Finite Displacements ◽

Plates And Shells

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Variational Methods for Machine Learning with Applications to Deep Networks

10.1007/978-3-030-70679-1 ◽

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

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

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0151 ◽

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.

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