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

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January 2013, 33(1): 7-26. doi: 10.3934/dcds.2013.33.7

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, Germany
2.	School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received August 2011 Revised December 2011 Published September 2012

Abstract
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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.

Keywords: single-bump standing waves, variational methods., multi-bump solutions, Nonlinear Schrödinger equation, potential well.

Mathematics Subject Classification: 35J61, 35J91, 35J20, 35B40, 47J3.

Citation: Thomas Bartsch, Zhongwei Tang. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 7-26. doi: 10.3934/dcds.2013.33.7

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 $\mathbb{R}^N2$, 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.

show all references

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 $\mathbb{R}^N2$, 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.

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