October  2020, 40(10): 5831-5843. doi: 10.3934/dcds.2020248

Existence of positive solutions of Schrödinger equations with vanishing potentials

1. 

Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330, Minas Gerais, Brazil

2. 

Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

* Corresponding author: Pedro Ubilla

Received  October 2019 Revised  March 2020 Published  June 2020

Fund Project: The first author gratefully acknowledges financial support from Universidad de Santiago de Chile, Usach. Agradecimientos Proyecto POSTDOC_DICYT, Código 041733UL_POSTDOC, Vicerrectoría de Investigación, Desarrollo e Innovación. Partially supported by FAPEMIG CEX APQ 01745/18. The second author was supported by FONDECYT grants 1181125, 1161635 and 1171691

We prove the existence of at least one positive solution for a Schrödinger equation in
$ \mathbb{R}^N $
of type
$ - \Delta u + V(x) u = f(x, u) \ \ \text{in} \ \mathbb{R}^N $
with a vanishing potential at infinity and subcritical nonlinearity
$ f $
. Our hypotheses allow us to consider examples of nonlinearities which do not verify the Ambrosetti-Rabinowitz condition, neither monotonicity conditions for the function
$ \frac{f(x, s)}{s} $
. Our argument requires new estimates in order to prove the boundedness of a Cerami sequence.
Citation: Eduard Toon, Pedro Ubilla. Existence of positive solutions of Schrödinger equations with vanishing potentials. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5831-5843. doi: 10.3934/dcds.2020248
References:
[1]

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, Journal of Differential Equations, 254 (2013), 1977-1991.  doi: 10.1016/j.jde.2012.11.013.  Google Scholar

[2]

A. AmbrosettiV. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS), 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[3]

A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332.   Google Scholar

[4]

D. Bonheure and J. Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl., 189 (2010), 273-301.  doi: 10.1007/s10231-009-0109-6.  Google Scholar

[5]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, New York: Springer, 2011.  Google Scholar

[6]

R. de Marchi, Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 745-757.  doi: 10.1017/S0308210515000104.  Google Scholar

[7]

Y. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), 137-163.  doi: 10.1016/j.jde.2005.03.011.  Google Scholar

[8]

D. Gilbard and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

Q. Han, Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations, Bull. Sci. Math., 141 (2017), 46-71.  doi: 10.1016/j.bulsci.2015.11.005.  Google Scholar

[10]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-471.   Google Scholar

[11]

G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar

[12]

Y. LiZ.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

[13]

S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  Google Scholar

[14]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[15]

C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 15 (2011), 569-588.   Google Scholar

[16]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

show all references

References:
[1]

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, Journal of Differential Equations, 254 (2013), 1977-1991.  doi: 10.1016/j.jde.2012.11.013.  Google Scholar

[2]

A. AmbrosettiV. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS), 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[3]

A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332.   Google Scholar

[4]

D. Bonheure and J. Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl., 189 (2010), 273-301.  doi: 10.1007/s10231-009-0109-6.  Google Scholar

[5]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, New York: Springer, 2011.  Google Scholar

[6]

R. de Marchi, Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 745-757.  doi: 10.1017/S0308210515000104.  Google Scholar

[7]

Y. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), 137-163.  doi: 10.1016/j.jde.2005.03.011.  Google Scholar

[8]

D. Gilbard and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

Q. Han, Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations, Bull. Sci. Math., 141 (2017), 46-71.  doi: 10.1016/j.bulsci.2015.11.005.  Google Scholar

[10]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-471.   Google Scholar

[11]

G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar

[12]

Y. LiZ.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

[13]

S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  Google Scholar

[14]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[15]

C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 15 (2011), 569-588.   Google Scholar

[16]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

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