Nodal solutions for nonlinear Schrödinger equations with decaying potential

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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

Abstract
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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., $\lim\limits_{|x|\rightarrow\infty}V(x)=0$.

Keywords: nonlinear Schrödinger equations, nodal solution, Harnack inequality, vanishing potential., Hardy inequality.

Mathematics Subject Classification: Primary: 35J20; Secondary: 35J6.

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:

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[5]	A. Amborosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations}, 18 (2005), 1321. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[18]	B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, (1992), 270. doi: 10.1007/BF00946631. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

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References:

[1]	R. Adams, Sobolev Space,, Academic Press, (1975). Google Scholar
[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. Google Scholar
[3]	A. Ambrositti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$,, Birkh$\ddota$user Verlag, (2006). Google Scholar
[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. Google Scholar
[5]	A. Amborosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations}, 18 (2005), 1321. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, (1992), 270. doi: 10.1007/BF00946631. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

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