July  2016, 15(4): 1125-1138. doi: 10.3934/cpaa.2016.15.1125

Nodal solutions for nonlinear Schrödinger equations with decaying potential

1. 

School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

Received  November 2014 Revised  June 2015 Published  April 2016

This paper concerns the following nonlinear Schrödinger equations: \begin{eqnarray} \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\Delta u +V(x)u= |u|^{p_+-2}u^++|u|^{p_--2}u^-,\ x\in\mathbb{R}^N,\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0, \\ \end{array} \right. \end{eqnarray} where $N\geq 3$ and $2 < p_{\pm} < \frac{2N}{N-2}$. We obtain nodal solutions for the above nonlinear Schrödinger equations with decaying and vanishing potential at infinity, i.e., $\lim\limits_{|x|\rightarrow\infty}V(x)=0$.
Citation: 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
References:
[1]

R. Adams, Sobolev Space,, Academic Press, (1975).

[2]

A. Amborosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Eur. Math. Soc. (JEMS)}, 7 (2005), 117. doi: 10.4171/JEMS/24.

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A. Ambrositti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$,, Birkh$\ddota$user Verlag, (2006).

[4]

A. Amborosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Anal. Math.}, 8 (2006), 317.

[5]

A. Amborosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations}, 18 (2005), 1321.

[6]

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

[7]

S. Bae and J. Byeon, Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 831. doi: 10.3934/cpaa.2013.12.831.

[8]

T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation,, \emph{Proc. London Math. Soc.(3)}, 91 (2005), 129. doi: 10.1112/S0024611504015187.

[9]

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

[10]

T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\mathbbR^N$,, \emph{Arch. Ration. Mech. Anal.}, 124 (1993), 261. doi: 10.1007/BF00953069.

[11]

H. Berestycki and P. L. Lions, Nonlinear scalar fields equations I, Existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555.

[12]

M.-F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems,, \emph{J. Anal. Math.}, 84 (2001), 1. doi: 10.1007/BF02788105.

[13]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Ration. Mech. Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6.

[14]

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

[15]

M. Del Pino and P. Felmer, Local mountainpasses for semilinear elliptic problems in unbounded domains,, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121. doi: 10.1007/BF01189950.

[16]

Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation,, \emph{Manuscripta Math.}, 112 (2003), 109. doi: 10.1007/s00229-003-0397-x.

[17]

A. Farina, On the classification of solutions of the Lane-Emden equation on undounded domains of $\R^N$,, \emph{J. Math. Pures Appl.}, 87 (2007), 537. doi: 10.1016/j.matpur.2007.03.001.

[18]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.

[19]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406.

[20]

D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order,, 2$^{nd}$ edition, 224 (1983). doi: 10.1007/978-3-642-61798-0.

[21]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u +u^p=0$,, \emph{Arch. Ration. Mech. Anal.}, 105 (1989), 243. doi: 10.1007/BF00251502.

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case: Parts 1,2,, \emph{Ann. Inst. H.Poincar\'e Anal. Non lin\'eaire}, 1 (1984), 109.

[23]

V. Moroz and J.V. Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 1. doi: 10.1007/s00526-009-0249-y.

[24]

Y. G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential,, \emph{Comm. Math. Phys.}, 131 (1990), 223.

[25]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, (1992), 270. doi: 10.1007/BF00946631.

[26]

H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potential tending to zero at infinity,, \emph{J. Differential Equations}, 247 (2009), 618. doi: 10.1016/j.jde.2009.03.002.

[27]

X. Wang and B. Zeng, On the concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions,, \emph{SIAM J. Math. Anal.}, 28 (1997), 633. doi: 10.1137/S0036141095290240.

[28]

M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

R. Adams, Sobolev Space,, Academic Press, (1975).

[2]

A. Amborosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Eur. Math. Soc. (JEMS)}, 7 (2005), 117. doi: 10.4171/JEMS/24.

[3]

A. Ambrositti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$,, Birkh$\ddota$user Verlag, (2006).

[4]

A. Amborosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Anal. Math.}, 8 (2006), 317.

[5]

A. Amborosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations}, 18 (2005), 1321.

[6]

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

[7]

S. Bae and J. Byeon, Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 831. doi: 10.3934/cpaa.2013.12.831.

[8]

T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation,, \emph{Proc. London Math. Soc.(3)}, 91 (2005), 129. doi: 10.1112/S0024611504015187.

[9]

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

[10]

T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\mathbbR^N$,, \emph{Arch. Ration. Mech. Anal.}, 124 (1993), 261. doi: 10.1007/BF00953069.

[11]

H. Berestycki and P. L. Lions, Nonlinear scalar fields equations I, Existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555.

[12]

M.-F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems,, \emph{J. Anal. Math.}, 84 (2001), 1. doi: 10.1007/BF02788105.

[13]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Ration. Mech. Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6.

[14]

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

[15]

M. Del Pino and P. Felmer, Local mountainpasses for semilinear elliptic problems in unbounded domains,, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121. doi: 10.1007/BF01189950.

[16]

Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation,, \emph{Manuscripta Math.}, 112 (2003), 109. doi: 10.1007/s00229-003-0397-x.

[17]

A. Farina, On the classification of solutions of the Lane-Emden equation on undounded domains of $\R^N$,, \emph{J. Math. Pures Appl.}, 87 (2007), 537. doi: 10.1016/j.matpur.2007.03.001.

[18]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.

[19]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406.

[20]

D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order,, 2$^{nd}$ edition, 224 (1983). doi: 10.1007/978-3-642-61798-0.

[21]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u +u^p=0$,, \emph{Arch. Ration. Mech. Anal.}, 105 (1989), 243. doi: 10.1007/BF00251502.

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case: Parts 1,2,, \emph{Ann. Inst. H.Poincar\'e Anal. Non lin\'eaire}, 1 (1984), 109.

[23]

V. Moroz and J.V. Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 1. doi: 10.1007/s00526-009-0249-y.

[24]

Y. G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential,, \emph{Comm. Math. Phys.}, 131 (1990), 223.

[25]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, (1992), 270. doi: 10.1007/BF00946631.

[26]

H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potential tending to zero at infinity,, \emph{J. Differential Equations}, 247 (2009), 618. doi: 10.1016/j.jde.2009.03.002.

[27]

X. Wang and B. Zeng, On the concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions,, \emph{SIAM J. Math. Anal.}, 28 (1997), 633. doi: 10.1137/S0036141095290240.

[28]

M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1.

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