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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 - A, 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. 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. doi: 10.1016/j.jde.2006.12.006. Google Scholar

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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. doi: 10.4171/JEMS/24. Google Scholar

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

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

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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. doi: 10.1007/s00208-006-0071-1. Google Scholar

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

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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. 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. 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. 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. 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. 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. doi: 10.1002/cpa.10015. Google Scholar

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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. 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, (1981), 369. Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (1998). Google Scholar

[17]

Z. Liu and Z.-Q. Wang, Sign-changing solutions of nonlinear elliptic equations,, Front. Math. China, 3 (2008), 221. 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. 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. 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. 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. 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. doi: 10.1137/S0036141095290240. Google Scholar

[23]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and their Applications, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (1981), 369. Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (1998). Google Scholar

[17]

Z. Liu and Z.-Q. Wang, Sign-changing solutions of nonlinear elliptic equations,, Front. Math. China, 3 (2008), 221. 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. 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. 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. 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. 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. doi: 10.1137/S0036141095290240. Google Scholar

[23]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and their Applications, (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. doi: 10.1016/j.jde.2009.03.002. Google Scholar

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