# American Institute of Mathematical Sciences

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

 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.
Citation: Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete and Continuous Dynamical Systems, 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), 285-300. doi: 10.1007/s002050050067. [2] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch.Ration.Mech.Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152. [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), 7-26. [4] T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math.Phys., 51 (2000), 366-384. doi: 10.1007/PL00001511. [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), 90-117. doi: 10.1016/0022-1236(90)90120-A. [6] J. Byeon and Z. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6. [7] J. Byeon and Z. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II, Calc.Var.P. D. E., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8. [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), 463-470. [9] J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugaliae Mathematica., 57 (2000), 273-284. [10] J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Z. Angew. Math. Phys., 49 (1998), 276-293. doi: 10.1007/PL00001485. [11] S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff.Equat., 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662. [12] S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Royal Soc. Edinburgh., 128 (1998), 1249-1260. doi: 10.1017/S030821050002730X. [13] M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, Ann.Inst.Henri Poincaré, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7. [14] M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085. [15] Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572. doi: 10.1016/j.jfa.2007.07.005. [16] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0. [17] M. Grossi, A nondegeneracy result for a nonlinear elliptic equation, Nonlinear Differ. Equ. Appl., 12 (2005), 227-241. doi: 10.1007/s00030-005-0010-y. [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), 1-20. doi: 10.1215/S0012-7094-92-06701-9. [19] Y.-G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. doi: 10.1007/BF02161413. [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), 1499-1519. doi: 10.1080/03605308808820585. [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), 1-52. doi: 10.1016/0022-1236(90)90002-3. [22] Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Commun. Pure Appl. Anal., 13 (2014), 237-248. doi: 10.3934/cpaa.2014.13.237. [23] J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, Preprint.

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##### References:
 [1] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch.Ration.Mech.Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067. [2] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch.Ration.Mech.Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152. [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), 7-26. [4] T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math.Phys., 51 (2000), 366-384. doi: 10.1007/PL00001511. [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), 90-117. doi: 10.1016/0022-1236(90)90120-A. [6] J. Byeon and Z. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6. [7] J. Byeon and Z. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II, Calc.Var.P. D. E., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8. [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), 463-470. [9] J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugaliae Mathematica., 57 (2000), 273-284. [10] J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Z. Angew. Math. Phys., 49 (1998), 276-293. doi: 10.1007/PL00001485. [11] S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff.Equat., 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662. [12] S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Royal Soc. Edinburgh., 128 (1998), 1249-1260. doi: 10.1017/S030821050002730X. [13] M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, Ann.Inst.Henri Poincaré, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7. [14] M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085. [15] Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572. doi: 10.1016/j.jfa.2007.07.005. [16] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0. [17] M. Grossi, A nondegeneracy result for a nonlinear elliptic equation, Nonlinear Differ. Equ. Appl., 12 (2005), 227-241. doi: 10.1007/s00030-005-0010-y. [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), 1-20. doi: 10.1215/S0012-7094-92-06701-9. [19] Y.-G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. doi: 10.1007/BF02161413. [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), 1499-1519. doi: 10.1080/03605308808820585. [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), 1-52. doi: 10.1016/0022-1236(90)90002-3. [22] Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Commun. Pure Appl. Anal., 13 (2014), 237-248. doi: 10.3934/cpaa.2014.13.237. [23] J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, Preprint.
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