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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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  • [1] Amina Chabi and Alain Haraux, Un théorème de valeurs intermédiaires dans les espaces de Sobolev et applications, Ann. Fac. Sci. Toulouse Math. (5) 7 (1985), no. 2, 87-100 (French, with English summary). MR 842764
  • [2] Jan Chabrowski, Variational methods for potential operator equations, With applications to nonlinear elliptic equations, De Gruyter Studies in Mathematics, vol. 24, Walter de Gruyter and Co., Berlin, 1997. MR 1467724
  • [3] Kung Ching Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102-129. MR 614246
  • [4] F. H. Clarke, Optimization and nonsmooth analysis, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1989. Reprint of the 1983 original. MR 1019086
  • [5] Aleksander Ćwiszewski and Wojciech Kryszewski, Equilibria of set-valued maps: a variational approach, Nonlinear Anal. 48 (2002), no. 5, Ser. A: Theory Methods, 707-746. MR 1868111
  • [6] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. MR 0346619
  • [7] Shibo Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations 45 (2012), no. 1-2, 1-9. MR 2957647
  • [8] Jarosław Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations 41 (2016), no. 9, 1426-1440. MR 3551463
  • [9] A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259-287. MR 2175045
  • [10] Michael Struwe, Variational methods, Applications to nonlinear partial differential equations and Hamiltonian systems, Springer-Verlag, Berlin, 1990. MR 1078018
  • [11] Andrzej Szulkin and Tobias Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), no. 12, 3802-3822. MR 2557725
  • [12] Andrzej Szulkin and Tobias Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, Int. Press, Somerville, MA, 2010, pp. 597-632. MR 2768820
  • [13] X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud. 14 (2014), no. 2, 361-373. MR 3194360
  • [14] X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math. 18 (2014), no. 6, 1957-1979. MR 3284041
  • [15] Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007
  • [16] X. Zhong and W. Zou, Ground state and multiple solutions via generalized Nehari manifold, Nonlinear Anal. 102 (2014), 251-263. MR 3182813

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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

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