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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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N2$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N2$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

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