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Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term

Authors: Francisco Odair de Paiva, Wojciech Kryszewski and Andrzej Szulkin
Journal: Proc. Amer. Math. Soc. 145 (2017), 4783-4794
MSC (2010): Primary 35J20, 35J60, 58E30
Published electronically: May 30, 2017
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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)/\vert u\vert$ 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)/\vert u\vert$ 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

Wojciech Kryszewski
Affiliation: Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Andrzej Szulkin
Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

Keywords: Generalized Nehari manifold, Schr\"odinger equation, strongly indefinite functional, Clarke's subdifferential
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
Article copyright: © Copyright 2017 American Mathematical Society