Nodal solutions for nonlinear Schrödinger equations with decaying potential

Zuji  Guo

doi:10.3934/cpaa.2016.15.1125

Article Contents

2016, Volume 15, Issue 4: 1125-1138. Doi: 10.3934/cpaa.2016.15.1125

This issue Previous Article Nonlinear noncoercive Neumann problems Next Article Existence and uniqueness for $\mathbb{D}$-solutions of reflected BSDEs with two barriers without Mokobodzki's condition

Nodal solutions for nonlinear Schrödinger equations with decaying potential

Zuji Guo^1,

1.
School of Mathematics, Taiyuan University of Technology, Taiyuan 030024

Received: November 2014

Revised: June 2015

Published: July 2016

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Abstract

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

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J65.

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References

[1]	R. Adams, Sobolev Space, Academic Press, New York, 1975.
[2]	A. Amborosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potential vanishing at infinity, J. Eur. Math. Soc. (JEMS), 7 (2005), 117-144.doi: 10.4171/JEMS/24.
[3]	A. Ambrositti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$, Birkhäuser Verlag, 2006.
[4]	A. Amborosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potential vanishing at infinity, J. Anal. Math., 8 (2006), 317-348.
[5]	A. Amborosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332.
[6]	C. O. Alves and S. 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.
[7]	S. Bae and J. Byeon, Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity, Commun. Pure Appl. Anal., 12 (2013), 831-850.doi: 10.3934/cpaa.2013.12.831.
[8]	T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation, Proc. London Math. Soc.(3), 91 (2005), 129-152.doi: 10.1112/S0024611504015187.
[9]	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.
[10]	T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\mathbbR^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.doi: 10.1007/BF00953069.
[11]	H. Berestycki and P. L. Lions, Nonlinear scalar fields equations I, Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.doi: 10.1007/BF00250555.
[12]	M.-F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49.doi: 10.1007/BF02788105.
[13]	J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316.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, J. Eur. Math. Soc.(JEMS), 8 (2006), 217-228.doi: 10.4171/JEMS/48.
[15]	M. Del Pino and P. Felmer, Local mountainpasses for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.doi: 10.1007/BF01189950.
[16]	Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math., 112 (2003), 109-135.doi: 10.1007/s00229-003-0397-x.
[17]	A. Farina, On the classification of solutions of the Lane-Emden equation on undounded domains of $\R^N$, J. Math. Pures Appl., (9) 87 (2007), 537-561.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, Comm. Math. Phys., 68 (1979), 209-243.
[19]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.doi: 10.1002/cpa.3160340406.
[20]	D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Grundlehren, 224, Berlin-Heidelgerg-New York-Tokyo: Springer, 1983.doi: 10.1007/978-3-642-61798-0.
[21]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u +u^p=0$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.doi: 10.1007/BF00251502.
[22]	P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case: Parts 1,2, Ann. Inst. H.Poincaré Anal. Non linéaire, 1 (1984), 109-145; Ann. Inst. H. Poincaré Anal. Non linéaire, 2 (1984), 223-283.
[23]	V. Moroz and J.V. Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials, Calc. Var. Partial Differential Equations, 37 (2010), 1-27.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, Comm. Math. Phys., 131 (1990), 223-253.
[25]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.doi: 10.1007/BF00946631.
[26]	H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potential tending to zero at infinity, J. Differential Equations, 247 (2009), 618-647.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, SIAM J. Math. Anal., 28 (1997), 633-655.doi: 10.1137/S0036141095290240.
[28]	M. Willem, Minimax Theorems, Birkhäiuser, Basel, 1996.doi: 10.1007/978-1-4612-4146-1.