UID:
almafu_9960117470302883
Umfang:
1 online resource (xvi, 383 pages) :
,
digital, PDF file(s).
ISBN:
1-316-57301-X
,
1-316-57070-3
,
1-316-57268-4
,
1-316-57367-2
,
1-316-57334-6
,
1-316-57499-7
,
1-316-28239-2
Serie:
Encyclopedia of mathematics and its applications ; volume 162
Inhalt:
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.
Anmerkung:
Title from publisher's bibliographic system (viewed on 08 Mar 2016).
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Cover; Half-title; Series page; Title page; Copyright information; Dedication; Table of contents; Foreword; Preface; Part I Fractional Sobolev spaces; Part II Nonlocal subcritical problems; Part III Nonlocal critical problems; Bibliography; Index; 1 Fractional framework; 2 A density result for fractional Sobolev spaces; 3 An eigenvalue problem; 4 Weak and viscosity solutions; 5 Spectral fractional Laplacian problems; 6 Mountain pass and linking results; 7 Existence and localization of solutions; 8 Resonant fractional equations; 9 A pseudoindex approach to nonlocal problems
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10 Multiple solutions for parametric equations11 Infinitely many solutions; 12 Fractional Kirchhoff-type problems; 13 On fractional Schrödinger equations; 14 The Brezis-Nirenberg result for the fractional Laplacian; 15 Generalization of the Brezis-Nirenberg result; 16 The Brezis-Nirenberg result in low dimension; 17 The critical equation in the resonant case; 18 The Brezis-Nirenberg result for a general nonlocal equation; 19 Existence of multiple solutions; 20 Nonlocal critical equations with concave-convex nonlinearities; 1.1 Fourier transform of tempered distributions
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1.2 Fractional Sobolev spaces1.3 The fractional Laplacian operator; 1.4 The fractional Sobolev space H0s(Ω); 1.5 Other fractional Sobolev-type spaces; 2.1 The main theorems; 2.2 Some preliminary lemmas; 2.3 Proof of Theorem 2.2 (and of Remark 2.3); 2.4 Proof of the main result; 2.5 Note on the partition of unity; 3.1 Eigenvalues and eigenfunctions of -LK; 3.2 A direct approach; 3.3 Another variational characterization of the eigenvalues; 3.4 A regularity result for the eigenfunctions; 3.5 Nodal set of the eigenfunctions of (-Δ)s: 1D case
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4.1 Viscosity solutions4.2 Perron method and existence theory for viscosity solutions; 4.3 Regularity theory for weak solutions; 4.4 Proof of Theorem 4.1; 4.5 On the boundedness of weak solutions; 5.1 Two different fractional operators; 5.2 A comparison between the eigenfunctions of As and (-Δ)s; 5.3 The spectrum of As and (-Δ)s; 5.4 One-dimensional analysis; 5.5 The first eigenvalue of As and that of (-Δ)s; 6.1 Hypotheses and statements; 6.2 Estimates on the nonlinearity and its primitive; 6.3 Proofs of the main theorems
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6.4 Comments on the sign of the solutions6.5 A remark on the case λ=0; 7.1 Existence of one weak solution; 7.2 A doubly parametric problem; 8.1 A saddle point result; 8.2 Eigenvalues for linear problems with weights; 8.3 Some technical lemmas; 8.4 The main result; 9.1 A multiplicity result; 9.2 A pseudoindex theorem; 9.3 The Palais-Smale condition; 9.4 Some preparatory lemmas; 9.5 k -h+1 distinct pairs of solutions; 10.1 Two abstract critical points results; 10.2 Three weak solutions; 10.3 Two weak solutions; 11.1 The main results; 11.2 Abstract approach; 11.3 Some compactness conditions
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11.4 Existence of infinitely many solutions
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English
Weitere Ausg.:
ISBN 1-107-11194-3
Sprache:
Englisch
URL:
https://doi.org/10.1017/CBO9781316282397
