INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHR&#214;DINGER EQUATIONS

Wei Zhang; Gui-Dong Li; Chun-Lei Tang

doi:10.11948/2018.1475

2018 Volume 8 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(5): 1475-1493

Next Article Previous Article

Wei Zhang, Gui-Dong Li, Chun-Lei Tang. INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1475-1493. doi: 10.11948/2018.1475

Citation:

Wei Zhang, Gui-Dong Li, Chun-Lei Tang. INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1475-1493. doi: 10.11948/2018.1475

INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER EQUATIONS

School of mathematics and statistics, Southwest University, Chongqing 400715, China

Fund Project:

Abstract

Abstract

In this paper, we investigate the Schrödinger equation, which satisfies that the potential is asymptotical 0 at infinity in some measure-theoretic and the nonlinearity is sublinear growth. By using variant symmetric mountain lemma, we obtain infinitely many solutions for the problem. Moreover, if the nonlinearity is locally sublinear defined for |u| small, we can also get the same result. In which, we show that these solutions tend to zero in L^∞(R^N) by the Brézis-Kato estimate.
- Sublinear Schrödinger equation /
- local sublinear nonlinearities /
- symmetric mountain lemma /
- variational methods
MSC: 35A15;35D30;35J61

FullText(HTML)

References

[1]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 1973, 14(4), 349-381. Google Scholar
[2]	C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differ. Equ., 2013, 254(4), 1977-1991. Google Scholar
[3]	A. Bahrouni, Infinite solutions for a class of Brézis-Nirenberg equations with an indefinite linear and nonlinear terms in sign, Appl. Math. Comput., 2013, 219(24), 11198-11205. Google Scholar
[4]	M. Benrhouma, Study of multiplicity and uniqueness of solutions for a class of nonhomogeneous sublinear elliptic equations, Nonlinear Anal., 2011, 74(7), 2682-2694. Google Scholar
[5]	V. Benci and G. Cerami, Existence of Positive Solutions of the Equation -△u + a(x)u=u^(N+2)/(N-2) in R^N, J. Funct. Anal., 1990, 88(1), 90-117. Google Scholar
[6]	M. Balabane, J. Dolbeault and H. Ounaies, Nodal solutions for a sublinear elliptic equation, Nonlinear Anal., 2003, 52(1), 219-237. Google Scholar
[7]	G. Bao and Z. Han, Infinitely many solutions for a resonant sublinear Schrödinger equation, Math. Methods Appl. Sci., 2015, 37(17), 2811-2816. Google Scholar
[8]	H. Brézis and S. Kamin, Sublinear elliptic equations in Rn, Manuscr. Math., 1992, 74(1), 87-106. Google Scholar
[9]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 1983, 82(4), 313-345. Google Scholar
[10]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 1983, 82(4), 347-375. Google Scholar
[11]	M. Benrhouma and H. Ounaies, Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equations, J. Math. Anal. Appl., 2009, 358(2), 307-319. Google Scholar
[12]	M. Benrhouma and H. Ounaies, Existence of solutions for a perturbation sublinear elliptic equation in RN, Nonlinear Differ. Equ. Appl., 2010, 17(5), 647-662. Google Scholar
[13]	A. Bahrouni, H. Ounaies and V.D. Rǎdulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh., 2015, 145(3), 445-465. Google Scholar
[14]	T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 2001, 3(4), 549-569. Google Scholar
[15]	J. Chen and X.-H. Tang, Infinitely many solutions for a class of sublinear Schrödinger equations, Taiwanese J. Math., 2015, 19(2), 381-396. Google Scholar
[16]	Y. Deng, Existence of multiple positive solution for a semilinear equation with critical exponent, Proc. Roy. Soc. Edinburgh., 1992, 122(1-2), 161-175. Google Scholar
[17]	Y. Ding, Infinitely many entire solutions of an elliptic system with symmetry, Topol. Methods Nonlinear Anal., 1997, 9(2), 313-323. Google Scholar
[18]	Y. Ding and L. Cheng, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differ. Equ., 2006, 222(1), 137-163. Google Scholar
[19]	C. Dellacherie and P. A. Meyer, Probabilités et Potential, Hermann, Paris, 1983. Google Scholar
[20]	R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 2005, 225(2), 352-370. Google Scholar
[21]	A. Kristály, Multiple solutions of a sublinear Schrödinger equation, Nonlinear differ. equ. appl., 2007, 14(3-4), 291-301. Google Scholar
[22]	J. Liu, J.-F. Liao and C.-L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 2016, 71(4), 965-976. Google Scholar
[23]	Z. Liu, J. Sun and T. Weth, Compactness results for Schrödinger equations with asymptotically linear terms, J. Differ. Equ., 2006, 231(2), 501-512. Google Scholar
[24]	Z.-L. Liu and Z.-Q. Wang, On the Ambrosetti-Rabinowitz Superlinear Condition, Adv. Nonlinear Stud., 2004, 4(4), 563-574. Google Scholar
[25]	C.-Y. Liu, Z. Wang and H.-S. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differ. Equ., 2008, 245(1), 201-222. Google Scholar
[26]	L. Li and X. Zhong, Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl., 2016, 435(1), 955-967. Google Scholar
[27]	J.-J. Sun and C.-L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 2011, 74(4), 1212-1222. Google Scholar
[28]	H. Tehrani, Existence results for an indefinite unbounded perturbation of a resonant Schrödinger equation, J. Differ. Equ., 2007, 236(1), 1-28. Google Scholar
[29]	J. J. C. Teixeira-Dias, The Schrödinger Equation, Molecular Physical Chemistry, Springer International Publishing, Berlin, 2017. Google Scholar
[30]	Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear differ. equ. appl., 2001, 8(1), 15-33. Google Scholar
[31]	Q.-Y. Zhang and Q. Wang, Mulitiple solutions for a class of sublinear Schrödinger equations, J. Math. Anal. Appl., 2012, 389(1), 511-518. Google Scholar