## 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.
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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 $u\mapsto f(x,u)/|u|$ 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.## References

- 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** - Jan Chabrowski,
*Variational methods for potential operator equations*, De Gruyter Studies in Mathematics, vol. 24, Walter de Gruyter & Co., Berlin, 1997. With applications to nonlinear elliptic equations. MR**1467724**, DOI 10.1515/9783110809374 - 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**, DOI 10.1016/0022-247X(81)90095-0 - F. H. Clarke,
*Optimization and nonsmooth analysis*, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1989. Reprint of the 1983 original. MR**1019086** - 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**, DOI 10.1016/S0362-546X(00)00210-8 - I. Ekeland,
*On the variational principle*, J. Math. Anal. Appl.**47**(1974), 324–353. MR**346619**, DOI 10.1016/0022-247X(74)90025-0 - Shibo Liu,
*On superlinear Schrödinger equations with periodic potential*, Calc. Var. Partial Differential Equations**45**(2012), no. 1-2, 1–9. MR**2957647**, DOI 10.1007/s00526-011-0447-2 - 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**, DOI 10.1080/03605302.2016.1209520 - A. Pankov,
*Periodic nonlinear Schrödinger equation with application to photonic crystals*, Milan J. Math.**73**(2005), 259–287. MR**2175045**, DOI 10.1007/s00032-005-0047-8 - Michael Struwe,
*Variational methods*, Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems. MR**1078018**, DOI 10.1007/978-3-662-02624-3 - Andrzej Szulkin and Tobias Weth,
*Ground state solutions for some indefinite variational problems*, J. Funct. Anal.**257**(2009), no. 12, 3802–3822. MR**2557725**, DOI 10.1016/j.jfa.2009.09.013 - 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** - 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**, DOI 10.1515/ans-2014-0208 - X. H. Tang,
*Non-Nehari manifold method for superlinear Schrödinger equation*, Taiwanese J. Math.**18**(2014), no. 6, 1957–1979. MR**3284041**, DOI 10.11650/tjm.18.2014.3541 - Michel Willem,
*Minimax theorems*, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR**1400007**, DOI 10.1007/978-1-4612-4146-1 - X. Zhong and W. Zou,
*Ground state and multiple solutions via generalized Nehari manifold*, Nonlinear Anal.**102**(2014), 251–263. MR**3182813**, DOI 10.1016/j.na.2014.02.018

## 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