American Institute of Mathematical Sciences

Advanced
August  2015, 35(8): 3393-3415. doi: 10.3934/dcds.2015.35.3393

Multi-bump solutions for Schrödinger equation involving critical growth and potential wells

Yuxia Guo 1, and  Zhongwei Tang 2,

1. 

Department of Mathematics, Tsinghua University, Beijing, 100084
2. 

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

Received  October 2014 Revised  December 2014 Published  February 2015

In this paper, we consider the following Schrödinger equation with critical growth $$-\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u \quad \hbox{ in } \mathbb{R}^N, $$ where $N\geq 5$, $2^*$ is the critical Sobolev exponent, $\delta>0$ is a constant, $a(x)\geq 0$ and its zero set is not empty. We will show that if the zero set of $a(x)$ has several isolated connected components $\Omega_1,\cdots,\Omega_k$ such that the interior of $\Omega_i (i=1, 2, ..., k)$ is not empty and $\partial\Omega_i (i=1, 2, ..., k)$ is smooth, then for any non-empty subset $J\subset \{1,2,\cdots,k\}$ and $\lambda$ sufficiently large, the equation admits a solution which is trapped in a neighborhood of $\bigcup_{j\in J}\Omega_j$. Our strategy to obtain the main results is as follows: By using local mountain pass method combining with penalization of the nonlinearities, we first prove the existence of single-bump solutions which are trapped in the neighborhood of only one isolated component of zero set. Then we construct the multi-bump solution by summing these one-bump solutions as the first approximation solution. The real solution will be obtained by delicate estimates of the error term, this last step is done by using Contraction Image Principle.
Keywords: critical exponent, multi-bump solutions, Schrödinger equation, potential wells..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393
References:
[1]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch.Ration.Mech.Anal., 140 (1997), 285.  doi: 10.1007/s002050050067.  Google Scholar

[2]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, Arch.Ration.Mech.Anal., 159 (2001), 253.  doi: 10.1007/s002050100152.  Google Scholar

[3]

T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential,, Discrete Contin. Dyn. Syst, 33 (2013), 7.   Google Scholar

[4]

T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation,, Z. Angew. Math.Phys., 51 (2000), 366.  doi: 10.1007/PL00001511.  Google Scholar

[5]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}$ in $\mathbbR^N,$, J. Funct. Anal., 88 (1990), 90.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar

[6]

J. Byeon and Z. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, Arch. Ration. Mech. Anal., 165 (2002), 295.  doi: 10.1007/s00205-002-0225-6.  Google Scholar

[7]

J. Byeon and Z. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II,, Calc.Var.P. D. E., 18 (2003), 207.  doi: 10.1007/s00526-002-0191-8.  Google Scholar

[8]

A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent,, Ann.Inst.Henri Poincaré, 2 (1985), 463.   Google Scholar

[9]

J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica., 57 (2000), 273.   Google Scholar

[10]

J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent,, Z. Angew. Math. Phys., 49 (1998), 276.  doi: 10.1007/PL00001485.  Google Scholar

[11]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, J. Diff.Equat., 160 (2000), 118.  doi: 10.1006/jdeq.1999.3662.  Google Scholar

[12]

S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations,, Proc. Royal Soc. Edinburgh., 128 (1998), 1249.  doi: 10.1017/S030821050002730X.  Google Scholar

[13]

M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, Ann.Inst.Henri Poincaré, 15 (1998), 127.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[14]

M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245.  doi: 10.1006/jfan.1996.3085.  Google Scholar

[15]

Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials,, J. Funct. Anal., 251 (2007), 546.  doi: 10.1016/j.jfa.2007.07.005.  Google Scholar

[16]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[17]

M. Grossi, A nondegeneracy result for a nonlinear elliptic equation,, Nonlinear Differ. Equ. Appl., 12 (2005), 227.  doi: 10.1007/s00030-005-0010-y.  Google Scholar

[18]

W. M. Ni, X. Pan and I. Takagi, Singular behavior of least energy solutions of a smilinear Neumannn problem involving critcial Sobolev exponents,, Duke Math.J., 67 (1992), 1.  doi: 10.1215/S0012-7094-92-06701-9.  Google Scholar

[19]

Y.-G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223.  doi: 10.1007/BF02161413.  Google Scholar

[20]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_a$,, Comm. Part. Diff. Equat., 13 (1988), 1499.  doi: 10.1080/03605308808820585.  Google Scholar

[21]

O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent,, J. Funct. Anal., 89 (1990), 1.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[22]

Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials,, Commun. Pure Appl. Anal., 13 (2014), 237.  doi: 10.3934/cpaa.2014.13.237.  Google Scholar

[23]

J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth,, Preprint., ().   Google Scholar

show all references

References:
[1]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch.Ration.Mech.Anal., 140 (1997), 285.  doi: 10.1007/s002050050067.  Google Scholar

[2]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, Arch.Ration.Mech.Anal., 159 (2001), 253.  doi: 10.1007/s002050100152.  Google Scholar

[3]

T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential,, Discrete Contin. Dyn. Syst, 33 (2013), 7.   Google Scholar

[4]

T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation,, Z. Angew. Math.Phys., 51 (2000), 366.  doi: 10.1007/PL00001511.  Google Scholar

[5]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}$ in $\mathbbR^N,$, J. Funct. Anal., 88 (1990), 90.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar

[6]

J. Byeon and Z. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, Arch. Ration. Mech. Anal., 165 (2002), 295.  doi: 10.1007/s00205-002-0225-6.  Google Scholar

[7]

J. Byeon and Z. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II,, Calc.Var.P. D. E., 18 (2003), 207.  doi: 10.1007/s00526-002-0191-8.  Google Scholar

[8]

A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent,, Ann.Inst.Henri Poincaré, 2 (1985), 463.   Google Scholar

[9]

J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica., 57 (2000), 273.   Google Scholar

[10]

J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent,, Z. Angew. Math. Phys., 49 (1998), 276.  doi: 10.1007/PL00001485.  Google Scholar

[11]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, J. Diff.Equat., 160 (2000), 118.  doi: 10.1006/jdeq.1999.3662.  Google Scholar

[12]

S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations,, Proc. Royal Soc. Edinburgh., 128 (1998), 1249.  doi: 10.1017/S030821050002730X.  Google Scholar

[13]

M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, Ann.Inst.Henri Poincaré, 15 (1998), 127.  doi: 10.1016/S0294-1449(97)89296-7.  Google Scholar

[14]

M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245.  doi: 10.1006/jfan.1996.3085.  Google Scholar

[15]

Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials,, J. Funct. Anal., 251 (2007), 546.  doi: 10.1016/j.jfa.2007.07.005.  Google Scholar

[16]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[17]

M. Grossi, A nondegeneracy result for a nonlinear elliptic equation,, Nonlinear Differ. Equ. Appl., 12 (2005), 227.  doi: 10.1007/s00030-005-0010-y.  Google Scholar

[18]

W. M. Ni, X. Pan and I. Takagi, Singular behavior of least energy solutions of a smilinear Neumannn problem involving critcial Sobolev exponents,, Duke Math.J., 67 (1992), 1.  doi: 10.1215/S0012-7094-92-06701-9.  Google Scholar

[19]

Y.-G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223.  doi: 10.1007/BF02161413.  Google Scholar

[20]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_a$,, Comm. Part. Diff. Equat., 13 (1988), 1499.  doi: 10.1080/03605308808820585.  Google Scholar

[21]

O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent,, J. Funct. Anal., 89 (1990), 1.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[22]

Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials,, Commun. Pure Appl. Anal., 13 (2014), 237.  doi: 10.3934/cpaa.2014.13.237.  Google Scholar

[23]

J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth,, Preprint., ().   Google Scholar

[1]

Weiming Liu, Lu Gan. Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2016, 15 (2) : 413-428. doi: 10.3934/cpaa.2016.15.413
[2]

Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058
[3]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[4]

Claudianor O. Alves, Olímpio H. Miyagaki, Sérgio H. M. Soares. Multi-bump solutions for a class of quasilinear equations on $R$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 829-844. doi: 10.3934/cpaa.2012.11.829
[5]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587
[6]

Miaomiao Niu, Zhongwei Tang. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1215-1231. doi: 10.3934/cpaa.2016.15.1215
[7]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[8]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[9]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[10]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[11]

Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137
[12]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[13]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[14]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[15]

Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273
[16]

Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883
[17]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[18]

Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1
[19]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[20]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences