# American Institute of Mathematical Sciences

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:
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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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