American Institute of Mathematical Sciences

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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

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