Existence of positive solutions of Schrödinger equations with vanishing potentials

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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 and 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.
[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.
[3]	A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332.
[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.
[5]	H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, New York: Springer, 2011.
[6]	R. de Marchi, Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 745-757. doi: 10.1017/S0308210515000104.
[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.
[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.
[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.
[10]	W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-471.
[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.
[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.
[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.
[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.
[15]	C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 15 (2011), 569-588.
[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.

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.
[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.
[3]	A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332.
[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.
[5]	H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, New York: Springer, 2011.
[6]	R. de Marchi, Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 745-757. doi: 10.1017/S0308210515000104.
[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.
[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.
[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.
[10]	W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-471.
[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.
[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.
[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.
[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.
[15]	C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 15 (2011), 569-588.
[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.

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