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

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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

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

Received May 2013 Revised October 2014 Published January 2015

Abstract
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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.
[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.
[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.
[4]	A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, Differential Integral Equations, 18 (2005), 1321.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (1998).
[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.
[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.
[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.
[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.
[21]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229. 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. doi: 10.1137/S0036141095290240.
[23]	M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and their Applications, (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. 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. 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. 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. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (1998).
[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.
[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.
[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.
[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.
[21]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229. 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. doi: 10.1137/S0036141095290240.
[23]	M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and their Applications, (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. doi: 10.1016/j.jde.2009.03.002.

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