A sharp eigenvalue theorem for fractional elliptic equations View Full Text


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Article Info

DATE

2017-04-28

AUTHORS

Giovanni Molica Bisci, Vicenţiu D. Rădulescu

ABSTRACT

By using variational methods, in this paper we study a nonlinear elliptic problem defined in a bounded domain Ω ⊂ ℝN, with smooth boundary ∂Ω, involving fractional powers of the Laplacian operator together with a suitable nonlinear term f. More precisely, we prove a characterization theorem on the existence of one weak solution for the elliptic problem {(−Δ)α/2μ=λf(μ)inΩ,u>0inΩ,u=0in∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^{\alpha /2}}\mu = \lambda f(\mu )in\Omega ,} \\ {u > 0in\Omega ,} \\ {u = 0in\partial \Omega ,} \end{array}} \right.$$\end{document}, where α ∈ (0, 2), N > α, λ > 0 and (−Δ)α/2 denotes the nonlocal fractional Laplacian operator. Our result extends to the nonlocal setting recent theorems for ordinary and classical elliptic equations, as well as a characterization for elliptic problems on certain non-smooth domains. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. More... »

PAGES

331-351

References to SciGraph publications

  • 2015-04-14. Nonlocal Equations with Measure Data in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2010-10-12. The Brezis–Nirenberg type problem involving the square root of the Laplacian in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2014-01-31. A note on spherical maxima sharing the same Lagrange multiplier in FIXED POINT THEORY AND ALGORITHMS FOR SCIENCES AND ENGINEERING
  • 2015. Multiple Solutions for an Eigenvalue Problem Involving Non-local Elliptic p-Laplacian Operators in GEOMETRIC METHODS IN PDE’S
  • 2010-07-10. Regularity Results for Nonlocal Equations by Approximation in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2015-06-30. Ground state solutions of scalar field fractional Schrödinger equations in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
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    http://dx.doi.org/10.1007/s11856-017-1482-2

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