American Institute of Mathematical Sciences

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

Existence of positive solutions of Schrödinger equations with vanishing potentials

Eduard Toon 1, and  Pedro Ubilla 2,,

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.
Keywords: Cerami sequences, variational methods, Schrödinger equation, nonautonomous nonlinearity, vanishing potentials.
Mathematics Subject Classification: 35J20, 35J10, 35J91, 35J15, 35B09.
Citation: Eduard Toon, Pedro Ubilla. Existence of positive solutions of Schrödinger equations with vanishing potentials. Discrete & Continuous Dynamical Systems, 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. Ambrosetti, V. 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. Li, Z.-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. Ambrosetti, V. 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. Li, Z.-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

[1]

Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[2]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[3]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723
[4]

Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
[5]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392
[6]

Bartosz Bieganowski, Jaros law Mederski. Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (1) : 143-161. doi: 10.3934/cpaa.2018009
[7]

Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921
[8]

Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[9]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[10]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031
[11]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102
[12]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83
[13]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047
[14]

Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems & Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038
[15]

Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic & Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505
[16]

Russell Johnson, Luca Zampogni. Some examples of generalized reflectionless Schrödinger potentials. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1149-1170. doi: 10.3934/dcdss.2016046
[17]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[18]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[19]

J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445
[20]

Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (172)
  • HTML views (77)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences