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Volume 25, Issue 1
Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials

Tingjian Luo & Zhengping Wang

DOI: 10.4208/jpde.v25.n1.6

J. Part. Diff. Eq., 25 (2012), pp. 79-89.

Published online: 2012-03

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  • Abstract

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1].

  • Keywords

Nonlinear Schrödinger equation nodal type bound state decaying potential

  • AMS Subject Headings

35J20, 35J60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

luotj2008@126.com (Tingjian Luo)

wangzp@wipm.ac.cn (Zhengping Wang)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-25-79, author = {Luo , Tingjian and Wang , Zhengping}, title = {Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials}, journal = {Journal of Partial Differential Equations}, year = {2012}, volume = {25}, number = {1}, pages = {79--89}, abstract = {

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1].

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v25.n1.6}, url = {http://global-sci.org/intro/article_detail/jpde/5176.html} }
TY - JOUR T1 - Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials AU - Luo , Tingjian AU - Wang , Zhengping JO - Journal of Partial Differential Equations VL - 1 SP - 79 EP - 89 PY - 2012 DA - 2012/03 SN - 25 DO - http://doi.org/10.4208/jpde.v25.n1.6 UR - https://global-sci.org/intro/article_detail/jpde/5176.html KW - Nonlinear Schrödinger equation KW - nodal type bound state KW - decaying potential AB -

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1].

Luo , Tingjian and Wang , Zhengping. (2012). Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials. Journal of Partial Differential Equations. 25 (1). 79-89. doi:10.4208/jpde.v25.n1.6
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