## Abstract and Applied Analysis

### Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in ${\Bbb R}^{3}$

#### Abstract

We study the following Schrödinger-Poisson system: $-\mathrm{\Delta }u+V(x)u+\varphi u=|u{|}^{p-\mathrm{1}}u$, $-\mathrm{\Delta }\varphi ={u}^{\mathrm{2}}$, ${\mathrm{lim}}_{|x|\to +\infty }\varphi (x)=\mathrm{0}$, where $u,\varphi :{\Bbb R}^{\mathrm{3}}\to \Bbb R$ are positive radial functions, $p\in (\mathrm{1},+\mathrm{\infty })$, $x=({x}_{\mathrm{1}},{x}_{\mathrm{2}},{x}_{\mathrm{3}})\in {\Bbb R}^{\mathrm{3}}$, and $V(x)$ is allowed to take two different forms including $V(x)=\mathrm{1}/({x}_{\mathrm{1}}^{\mathrm{2}}+{x}_{\mathrm{2}}^{\mathrm{2}}+{x}_{\mathrm{3}}^{\mathrm{2}}{)}^{\alpha /\mathrm{2}}$ and $V(x)=\mathrm{1}/({x}_{\mathrm{1}}^{\mathrm{2}}+{x}_{\mathrm{2}}^{\mathrm{2}}{)}^{\alpha /\mathrm{2}}$ with $\alpha >\mathrm{0}$. Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the $\alpha -p$ plane where the system has no nontrivial solutions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 890126, 6 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393443520

Digital Object Identifier
doi:10.1155/2013/890126

Mathematical Reviews number (MathSciNet)
MR3096834

Zentralblatt MATH identifier
07095458

#### Citation

Jiang, Yongsheng; Zhou, Yanli; Wiwatanapataphee, B.; Ge, Xiangyu. Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in ${\Bbb R}^{3}$. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 890126, 6 pages. doi:10.1155/2013/890126. https://projecteuclid.org/euclid.aaa/1393443520

#### References

• N. J. Mauser, “The Schrödinger-Poisson-\emphX $_{\alpha }$ equation,” Applied Mathematics Letters, vol. 14, no. 6, pp. 759–763, 2001.
• D. Ruiz, “The Schrödinger-Poisson equation under the effect of a nonlinear local term,” Journal of Functional Analysis, vol. 237, no. 2, pp. 655–674, 2006.
• Ó. Sánchez and J. Soler, “Long-time dynamics of the Schrödinger-Poisson-Slater system,” Journal of Statistical Physics, vol. 114, no. 1-2, pp. 179–204, 2004.
• J. C. Slater, “A simplification of the Hartree-Fock method,” Physical Review, vol. 81, no. 3, pp. 385–390, 1951.
• T. D'Aprile and D. Mugnai, “Non-existence results for the coupled Klein-Gordon-Maxwell equations,” Advanced Nonlinear Studies, vol. 4, no. 3, pp. 307–322, 2004.
• Y. Jiang and H. -S. Zhou, “Nonlinear Schrödinger-Poisson equations with singular potentials or cylindrical potentials in ${\mathbb{R}}^{3}$,” Submitted.
• M. Badiale, M. Guida, and S. Rolando, “Elliptic equations with decaying cylindrical potentials and power-type nonlinearities,” Advances in Differential Equations, vol. 12, no. 12, pp. 1321–1362, 2007.