Multibumpsolutions of nonlinear Schrödinger equations with steep potential welland indefinite potential

Thomas Bartsch; Zhongwei Tang

doi:10.3934/dcds.2013.33.7

Article Contents

2013, Volume 33, Issue 1: 7-26. Doi: 10.3934/dcds.2013.33.7

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Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential

Thomas Bartsch^1, and
Zhongwei Tang^2,

1.
Mathematisches Institut, University of Giessen, Arndtstr. 2 35392 Giessen
2.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received: August 2011

Revised: December 2011

Published: January 2013

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Abstract

We are concerned with the existence of single- and multi-bump solutions of the equation $-\Delta u+(\lambda a(x)+a_0(x))u=|u|^{p-2}u$, $x\in{\mathbb R}^N$; here $p>2$, and $p<\frac{2N}{N-2}$ if $N\geq 3$. We require that $a\geq 0$ is in $L^\infty_{loc}({\mathbb R}^N)$ and has a bounded potential well $\Omega$, i.e. $a(x)=0$ for $x\in\Omega$ and $a(x)>0$ for $x\in{\mathbb R}^N$\$\bar{\Omega}$. Unlike most other papers on this problem we allow that $a_0\in L^\infty({\mathbb R}^N)$ changes sign. Using variational methods we prove the existence of multibump solutions $u_\lambda$ which localize, as $\lambda\to\infty$, near prescribed isolated open subsets $\Omega_1,\dots,\Omega_k\subset\Omega$. The operator $L_0:=-\Delta+a_0$ may have negative eigenvalues in $\Omega_j$, each bump of $u_\lambda$ may be sign-changing.

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Mathematics Subject Classification: 35J61, 35J91, 35J20, 35B40, 47J30.

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References

[1]	T. Bartsch and M. Parnet, Nonlinear Schrödinger equations near an infinite potential well, arXiv:1205.1345.
[2]	T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear ellipticequation $\mathbbR^N$, Comm. Part. Diff. Eq., 20 (1995), 1725-1741.doi: 10.1080/03605309508821149.
[3]	T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. angew. Math. Phys., 51 (2000), 366-384.
[4]	T. Bartsch, A. Pankov and Z. Q.Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.doi: 10.1142/S0219199701000494.
[5]	Y. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödingerequation, Manuscripta Math., 112 (2003), 109-135.doi: 10.1007/s00229-003-0397-x.
[6]	Y. Ding and A. Szulkin, Existence and number of solutions for a class of semilinearSchrödinger equation, Progr. Nonlin. Diff. Equ. Appl., 66 (2006), 221-231.doi: 10.1007/3-7643-7401-2_15.
[7]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1983.doi: 10.1007/978-3-642-61798-0.
[8]	A. Pankov, Periodic nonlinear Schrödinger equation with application to photoniccrystals, Milan J. Math., 73 (2005), 563-574.doi: 10.1007/s00032-005-0047-8.
[9]	M. Reed and B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S), 7 (1982), 447-526.
[10]	Y. Sato and K. Tanaka, Sign-changingmulti-bump solutions for nonlinear Schrödinger equations withsteep potential wells, Trans. Amer. Math. Soc., 361 (2009), 6205-6253.doi: 10.1090/S0002-9947-09-04565-6.
[11]	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.
[12]	Z. P. Wang and H. S. Zhou, Positive solutions for nonlinear Schrödinger equations withdeepening potential well, J. Europ. Math. Soc., 11 (2009), 545-573.doi: 10.4171/JEMS/160.