American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions
  • DCDS Home
  • This Issue
  • Next Article
    Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures
July  2015, 35(7): 2921-2948. doi: 10.3934/dcds.2015.35.2921

Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity

Mingwen Fei 1, and  Huicheng Yin 1,

1. 

Department of Mathematics & IMS, Nanjing University, Nanjing, 210093, China, China

Received  May 2013 Revised  October 2014 Published  January 2015

We will focus on the existence and concentration of nodal solutions to the following critical nonlinear Schrödinger equations in $\Bbb R^2$ $$ -\epsilon^2\triangle u_{\epsilon}+V(x)u_{\epsilon}=K(x) |u_{\epsilon}|^{p-2}u_{\epsilon}e^{\alpha_{0}|u_{\epsilon}| ^{2}},\quad u_{\epsilon}\in H^1(\Bbb R^2), $$ where $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, and $\epsilon>0$ is a small constant. For the positive potential $V(x)$ which decays at infinity like $(1+|x|)^{-\alpha}$ with $0 < \alpha \le 2$, we will show that a nodal solution with one positive and one negative peaks exists, and concentrates around local minimum points of the related ground energy function $G(\xi)$ of the Schrödinger equation $ -\triangle u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{2}}$.
Keywords: multi-peak, critical exponent, concentration-compactness., Nonlinear Schrödinger equation, nodal solution.
Mathematics Subject Classification: Primary: 35B33, 35J60; Secondary: 35Q5.
Citation: Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921
References:
[1]

C. O. Alves and S. H. M. Soares, On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations, J. Math. Anal. Appl., 296 (2004), 563-577. doi: 10.1016/j.jmaa.2004.04.022.  Google Scholar

[2]

C. O. Alves and S. H. M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differential Equations, 234 (2007), 464-484. doi: 10.1016/j.jde.2006.12.006.  Google Scholar

[3]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144. doi: 10.4171/JEMS/24.  Google Scholar

[4]

A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332.  Google Scholar

[5]

T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. doi: 10.1007/BF02787822.  Google Scholar

[6]

T. Bartsch, C. Mónica and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation, Math. Ann., 338 (2007), 147-185. doi: 10.1007/s00208-006-0071-1.  Google Scholar

[7]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[8]

J. Byeon and Z.-Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials, J. Eur. Math. Soc., 8 (2006), 217-228. doi: 10.4171/JEMS/48.  Google Scholar

[9]

D. Cao, Nontrivial solutions of semilinear elliptic equation with critical exponent in $\mathbbR^{2}$, Comm. Partial Differential Equations, 17 (1992), 407-435. doi: 10.1080/03605309208820848.  Google Scholar

[10]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. doi: 10.1007/BF01189950.  Google Scholar

[11]

J. M. do Ó and M. A. S. Souto, On a class of nonlinear Schrödinger equations in $\mathbbR^{2}$ involving critical growth, J. Differential Equations, 174 (2001), 289-311. doi: 10.1006/jdeq.2000.3946.  Google Scholar

[12]

M. Fei and H. Yin, Existence and concentration of bound states of nonlinear Schrödinger equations with compactly supported and competing potentials, Pacific. J. Math., 244 (2010), 261-296. doi: 10.2140/pjm.2010.244.261.  Google Scholar

[13]

D. G. de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure. Appl. Math., 55 (2002), 135-152. doi: 10.1002/cpa.10015.  Google Scholar

[14]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbbR^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153. doi: 10.1007/BF01205003.  Google Scholar

[15]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin-New York, 1998. Google Scholar

[17]

Z. Liu and Z.-Q. Wang, Sign-changing solutions of nonlinear elliptic equations, Front. Math. China, 3 (2008), 221-238. doi: 10.1007/s11464-008-0014-0.  Google Scholar

[18]

E. S. Noussair and J. Wei, On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems, Indiana Univ. Math. J., 46 (1997), 1255-1271. doi: 10.1512/iumj.1997.46.1401.  Google Scholar

[19]

E. S. Noussair and J. Wei, On the location of spikes and profile of nodal solutions for a singularly perturbed Neumann problem, Comm. Partial Differential Equations, 23 (1998), 793-816. doi: 10.1080/03605309808821366.  Google Scholar

[20]

Y. Sato, Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency, Commun. Pure. Appl. Anal., 7 (2008), 883-903. doi: 10.3934/cpaa.2008.7.883.  Google Scholar

[21]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.  Google Scholar

[22]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655. doi: 10.1137/S0036141095290240.  Google Scholar

[23]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[24]

H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity, J. Differential Equations, 247 (2009), 618-647. doi: 10.1016/j.jde.2009.03.002.  Google Scholar

show all references

References:
[1]

C. O. Alves and S. H. M. Soares, On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations, J. Math. Anal. Appl., 296 (2004), 563-577. doi: 10.1016/j.jmaa.2004.04.022.  Google Scholar

[2]

C. O. Alves and S. H. M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differential Equations, 234 (2007), 464-484. doi: 10.1016/j.jde.2006.12.006.  Google Scholar

[3]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144. doi: 10.4171/JEMS/24.  Google Scholar

[4]

A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332.  Google Scholar

[5]

T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. doi: 10.1007/BF02787822.  Google Scholar

[6]

T. Bartsch, C. Mónica and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation, Math. Ann., 338 (2007), 147-185. doi: 10.1007/s00208-006-0071-1.  Google Scholar

[7]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[8]

J. Byeon and Z.-Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials, J. Eur. Math. Soc., 8 (2006), 217-228. doi: 10.4171/JEMS/48.  Google Scholar

[9]

D. Cao, Nontrivial solutions of semilinear elliptic equation with critical exponent in $\mathbbR^{2}$, Comm. Partial Differential Equations, 17 (1992), 407-435. doi: 10.1080/03605309208820848.  Google Scholar

[10]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. doi: 10.1007/BF01189950.  Google Scholar

[11]

J. M. do Ó and M. A. S. Souto, On a class of nonlinear Schrödinger equations in $\mathbbR^{2}$ involving critical growth, J. Differential Equations, 174 (2001), 289-311. doi: 10.1006/jdeq.2000.3946.  Google Scholar

[12]

M. Fei and H. Yin, Existence and concentration of bound states of nonlinear Schrödinger equations with compactly supported and competing potentials, Pacific. J. Math., 244 (2010), 261-296. doi: 10.2140/pjm.2010.244.261.  Google Scholar

[13]

D. G. de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure. Appl. Math., 55 (2002), 135-152. doi: 10.1002/cpa.10015.  Google Scholar

[14]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbbR^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153. doi: 10.1007/BF01205003.  Google Scholar

[15]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.  Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin-New York, 1998. Google Scholar

[17]

Z. Liu and Z.-Q. Wang, Sign-changing solutions of nonlinear elliptic equations, Front. Math. China, 3 (2008), 221-238. doi: 10.1007/s11464-008-0014-0.  Google Scholar

[18]

E. S. Noussair and J. Wei, On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems, Indiana Univ. Math. J., 46 (1997), 1255-1271. doi: 10.1512/iumj.1997.46.1401.  Google Scholar

[19]

E. S. Noussair and J. Wei, On the location of spikes and profile of nodal solutions for a singularly perturbed Neumann problem, Comm. Partial Differential Equations, 23 (1998), 793-816. doi: 10.1080/03605309808821366.  Google Scholar

[20]

Y. Sato, Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency, Commun. Pure. Appl. Anal., 7 (2008), 883-903. doi: 10.3934/cpaa.2008.7.883.  Google Scholar

[21]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.  Google Scholar

[22]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655. doi: 10.1137/S0036141095290240.  Google Scholar

[23]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[24]

H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity, J. Differential Equations, 247 (2009), 618-647. doi: 10.1016/j.jde.2009.03.002.  Google Scholar

[1]

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
[2]

Jaeyoung Byeon, Louis Jeanjean. Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 255-269. doi: 10.3934/dcds.2007.19.255
[3]

Xudong Shang, Jihui Zhang. Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3183-3201. doi: 10.3934/dcds.2015.35.3183
[4]

Minbo Yang, Jianjun Zhang, Yimin Zhang. Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (2) : 493-512. doi: 10.3934/cpaa.2017025
[5]

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
[6]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[7]

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
[8]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[9]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[10]

Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
[11]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[12]

Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3331-3355. doi: 10.3934/cpaa.2021108
[13]

Myeongju Chae, Sunggeum Hong, Sanghyuk Lee. Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 909-928. doi: 10.3934/dcds.2011.29.909
[14]

Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359
[15]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[16]

Yuta Ishii, Kazuhiro Kurata. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1633-1679. doi: 10.3934/cpaa.2021035
[17]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[18]

Vincenzo Ambrosio. The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5659-5705. doi: 10.3934/dcds.2021092
[19]

Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences