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Advanced Nonlinear Studies

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Volume 19, Issue 1

Issues

Sign-Changing Solutions for a Class of Zero Mass Nonlocal Schrödinger Equations

Vincenzo AmbrosioORCID iD: https://orcid.org/0000-0003-3439-1428 / Giovany M. Figueiredo / Teresa IserniaORCID iD: https://orcid.org/0000-0002-6215-3219 / Giovanni Molica Bisci
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  • Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio Calabria, Italy
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Published Online: 2018-07-07 | DOI: https://doi.org/10.1515/ans-2018-2023

Abstract

We consider the following class of fractional Schrödinger equations:

(-Δ)αu+V(x)u=K(x)f(u)in N,

where α(0,1), N>2α, (-Δ)α is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

Keywords: Fractional Laplacian; Potential Vanishing at Infinity; Nehari Manifold; Sign-Changing Solutions; Deformation Lemma

MSC 2010: 35A15; 35J60; 35R11; 45G05

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About the article


Received: 2017-12-16

Revised: 2018-02-26

Accepted: 2018-06-03

Published Online: 2018-07-07

Published in Print: 2019-02-01


The manuscript was realized within the auspices of the INdAM–GNAMPA projects 2017 titled “Teoria e modelli per problemi non locali”.


Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 1, Pages 113–132, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2018-2023.

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