ON STRONGLY INDEFINITE SCHRÖDINGER EQUATIONS WITH NON-PERIODIC POTENTIAL

Yue Wu; Wei Chen

doi:10.11948/20210036

2023 Volume 13 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(1): 1-10

Next Article Previous Article

Yue Wu, Wei Chen. ON STRONGLY INDEFINITE SCHRÖDINGER EQUATIONS WITH NON-PERIODIC POTENTIAL[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 1-10. doi: 10.11948/20210036

Citation:

Yue Wu, Wei Chen. ON STRONGLY INDEFINITE SCHRÖDINGER EQUATIONS WITH NON-PERIODIC POTENTIAL[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 1-10. doi: 10.11948/20210036

ON STRONGLY INDEFINITE SCHRÖDINGER EQUATIONS WITH NON-PERIODIC POTENTIAL

Yue Wu¹,
Wei Chen^1, ,

1.
School of Mathematics and Statistics, Linyi University, Linyi 276100, China

Corresponding author: Email: cwei1990@126.com(W. Chen)
Fund Project: The authors were supported by National Natural Science Foundation of China (Nos. 12201277, 12271232, 11701251), the Natural Science Foundation of Shandong Province (Nos. ZR2022QA008, ZR2017BA015)

Abstract

Abstract

This paper is concerned with the non-periodic superlinear Schrödinger equation $ -\Delta u+V(x)u=f(x, u) $, $ u\in H^{1}(\mathbb{R}^{N}) $. Here, the Shrödinger operator $ -\Delta + V $ is strongly indefinite, that is, possesses a infinite dimensional negative space, which leads to more difficulty in verifying the compactness conditions. We prove the existence, as well as multiplicity provided $ f(x, t) $ is odd in $ t $, of solutions via variational methods.
- Schrödinger equations /
- superlinear /
- variational methods /
- strongly indefinite functionals
MSC: 35J20, 35J60, 58E05

FullText(HTML)

References

[1]	S. Alama and Y. Y. Li, On "multibump" bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 1992, 41(4), 983-1026. doi: 10.1512/iumj.1992.41.41052 CrossRef Google Scholar
[2]	T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal., 1993, 20(10), 1205-1216. doi: 10.1016/0362-546X(93)90151-H CrossRef Google Scholar
[3]	T. Bartsch and Y. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 1999, 313(1), 15-37. doi: 10.1007/s002080050248 CrossRef Google Scholar
[4]	T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations, 2004, 29(1-2), 25-42. Google Scholar
[5]	C. Joël Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 2013, 405(2), 438-452. doi: 10.1016/j.jmaa.2013.04.018 CrossRef Google Scholar
[6]	W. Chen, Z. Fu and Y. Wu, Positive solutions for nonlinear Schrödinger –Kirchhoff equations in $\mathbb{R}^3$, Appl. Math. Lett., 2020, 104, 106274. doi: 10.1016/j.aml.2020.106274 CrossRef 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, 2006, 222(1), 137-163. doi: 10.1016/j.jde.2005.03.011 CrossRef Google Scholar
[8]	L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $\Bbb R^ N$ autonomous at infinity, ESAIM Control Optim. Calc. Var., 2002, 7, 597-614 (electronic). doi: 10.1051/cocv:2002068 CrossRef Google Scholar
[9]	W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 1998, 3(3), 441-472. Google Scholar
[10]	Q. Li, X. Du and Z. Zhao, Existence of sign-changing solutions for nonlocal Kirchhoff-Schrödinger-type equations in ${R}^3$, J. Math. Anal. Appl., 2019, 477(1), 174-186. doi: 10.1016/j.jmaa.2019.04.025 CrossRef Google Scholar
[11]	S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 2012, 45(1-2), 1-9. doi: 10.1007/s00526-011-0447-2 CrossRef Google Scholar
[12]	S. Liu and S. Mosconi, On the schrödinger-poisson system with indefinite potential and 3-sublinear nonlinearity, J. Differential Equations, 2020, 269(1), 689-712. doi: 10.1016/j.jde.2019.12.023 CrossRef Google Scholar
[13]	S. Liu and Z. Zhao, Solutions for fourth order elliptic equations on $r^n$ involving $u\delta(u^2)$ and sign-changing potentials, J. Differential Equations, 2019, 267(3), 1581-1599. doi: 10.1016/j.jde.2019.02.017 CrossRef Google Scholar
[14]	S. Liu and J. Zhou, Standing waves for quasilinear Schrödinger equations with indefinite potentials, J. Differential Equations, 2018, 256(9), 3970-3987. Google Scholar
[15]	Z. Liu, J. Su and T. Weth, Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations, 2006, 231(2), 501-512. doi: 10.1016/j.jde.2006.05.007 CrossRef Google Scholar
[16]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 1992, 43(2), 270-291. doi: 10.1007/BF00946631 CrossRef Google Scholar
[17]	Z. Shen and S. Liu, On asymptotically linear elliptic equations in $\Bbb R^N$, J. Math. Anal. Appl., 2012, 392(1), 83-88. doi: 10.1016/j.jmaa.2012.02.059 CrossRef Google Scholar
[18]	B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S. ), 1982, 7(3), 447-526. doi: 10.1090/S0273-0979-1982-15041-8 CrossRef Google Scholar
[19]	A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 2009, 257(12), 3802-3822. doi: 10.1016/j.jfa.2009.09.013 CrossRef Google Scholar
[20]	C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations, 1996, 21(9-10), 1431-1449. doi: 10.1080/03605309608821233 CrossRef Google Scholar
[21]	M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston Inc., Boston, MA, 1996. Google Scholar
[22]	M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 2003, 52(1), 109-132. doi: 10.1512/iumj.2003.52.2273 CrossRef Google Scholar
[23]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${\bf R}^n$, Comm. Pure Appl. Math., 1992, 45(10), 1217-1269. doi: 10.1002/cpa.3160451002 CrossRef Google Scholar
[24]	F. Zhao, L. Zhao and Y. Ding, Existence and multiplicity of solutions for a non-periodic Schrödinger equation, Nonlinear Anal., 2008, 69(11), 3671-3678. doi: 10.1016/j.na.2007.10.024 CrossRef Google Scholar