Variational methods and almost solvability of semilinear equations

1981 ◽  
pp. 377-384
Author(s):  
Michel Willem
Keyword(s):  
Variational Methods ◽  
Semilinear Equations

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
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