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: |
[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.![]() ![]() ![]() |