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Variational Methods for Nonlocal Fractional Problems
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TLDR
A thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators can be found in this paper, where the authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of equations, plus their application to various processes arising in the applied sciences.Abstract:
This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.read more
Citations
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Journal ArticleDOI
Ground state solutions of scalar field fractional Schrödinger equations
TL;DR: In this article, the existence of multiple ground state solutions for a class of parametric fractional Schrodinger equations whose simplest prototype is (−�) s u + V (x)u = λ f (x, u) in R n, where n > 2, s stands for the fractional Laplace operator of order s ∈ (0, 1),a ndλ is a positive real parameter.
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Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations
TL;DR: In this paper, the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in ℝN was investigated by using variational methods and topological degree theory.
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Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
TL;DR: In this paper, the existence and multiplicity of solutions for the Schrődinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent were studied.
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On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent
TL;DR: In this article, a variational analysis of a class of nonlocal fractional problems with variable exponents was performed in the context of non-homogeneous Laplace operators, and the abstract results established in this paper are applied in the variational analyses of a related non-local operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator.
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Existence of solutions for perturbed fractional p-Laplacian equations
TL;DR: In this article, the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p-Laplacian operator was investigated and the existence and multiplicity results for the above-mentioned equations depending on λ and according to the integrability properties of the ratio a q − p / b r − p.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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Elliptic Problems in Nonsmooth Domains
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.
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Encyclopedia of Mathematics and its Applications.
William B. Jones,W. J. Thron +1 more
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Dual variational methods in critical point theory and applications
TL;DR: In this paper, general existence theorems for critical points of a continuously differentiable functional I on a real Banach space are given for the case in which I is even.