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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

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 & 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,, , ().   Google Scholar

[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.  Google Scholar

[3]

T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. angew. Math. Phys., 51 (2000), 366-384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Reed and B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S), 7 (1982), 447-526.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

T. Bartsch and M. Parnet, Nonlinear Schrödinger equations near an infinite potential well,, , ().   Google Scholar

[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.  Google Scholar

[3]

T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. angew. Math. Phys., 51 (2000), 366-384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Reed and B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S), 7 (1982), 447-526.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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