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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term
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by Francisco Odair de Paiva, Wojciech Kryszewski and Andrzej Szulkin PDF
Proc. Amer. Math. Soc. 145 (2017), 4783-4794 Request permission

Abstract:

We study the Schrödinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb {R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega \subset \mathbb {R}^N$. We assume that $f$ is superlinear but of subcritical growth and $u\mapsto f(x,u)/|u|$ is nondecreasing. In $\mathbb {R}^N$ we also assume that $V$ and $f$ are periodic in $x_1,\ldots ,x_N$. We show that these equations have a ground state and that there exist infinitely many solutions if $f$ is odd in $u$. Our results generalize those by Szulkin and Weth [J. Funct. Anal. 257 (2009), 3802–3822], where $u\mapsto f(x,u)/|u|$ was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis.
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Additional Information
  • Francisco Odair de Paiva
  • Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos, Brazil
  • MR Author ID: 719980
  • Email: odair@dm.ufscar.br
  • Wojciech Kryszewski
  • Affiliation: Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
  • MR Author ID: 107160
  • Email: wkrysz@mat.umk.pl
  • Andrzej Szulkin
  • Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
  • MR Author ID: 210814
  • Email: andrzejs@math.su.se
  • Received by editor(s): September 15, 2016
  • Received by editor(s) in revised form: December 5, 2016
  • Published electronically: May 30, 2017
  • Additional Notes: The first author was supported by FAPESP under the grant 2015/10545-0. This work was done while he was visiting the mathematics department of Stockholm University. He would like to thank the members of the department for their hospitality and a stimulating scientific atmosphere
    The second author was partially supported by the Polish National Science Center under grant 2013/09/B/ST1/01963
  • Communicated by: Joachim Krieger
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4783-4794
  • MSC (2010): Primary 35J20, 35J60, 58E30
  • DOI: https://doi.org/10.1090/proc/13609
  • MathSciNet review: 3691995