Taiwanese Journal of Mathematics

Nehari Type Ground State Solutions for Asymptotically Periodic Schrödinger-Poisson Systems

Sitong Chen and Xianhua Tang

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Abstract

This paper is dedicated to studying the following Schrödinger-Poisson system\[  \begin{cases}  -\Delta u + V(x)u + K(x) \phi(x)u = f(x,u), &x \in \mathbb{R}^{3}, \\  -\Delta \phi = K(x) u^2, &x \in \mathbb{R}^{3},  \end{cases}\]where $V(x)$, $K(x)$ and $f(x,u)$ are periodic or asymptotically periodic in $x$. We use the non-Nehari manifold approach to establish the existence of the Nehari type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions $\lim_{|t| \to \infty} \left( \int_0^t f(x,s) \, \mathrm{d}s \right)/|t|^3 = \infty$ uniformly in $x \in \mathbb{R}^3$ and\[  \left[ \frac{f(x,\tau)}{\tau^3} - \frac{f(x,t\tau)}{(t\tau)^3} \right]    \operatorname{sign}(1-t)    + \theta_0 V(x) \frac{|1-t^2|}{(t\tau)^2}  \geq 0, \quad \forall\, x \in \mathbb{R}^3, \; t \gt 0, \; \tau \neq 0\]with constant $\theta_0 \in (0,1)$, instead of $\lim_{|t| \to \infty} \left( \int_0^t f(x,s) \, \mathrm{d}s \right)/|t|^4 = \infty$ uniformly in $x \in \mathbb{R}^3$ and the usual Nehari-type monotonic condition on $f(x,t)/|t|^3$.

Article information

Source
Taiwanese J. Math., Volume 21, Number 2 (2017), 363-383.

Dates
First available in Project Euclid: 29 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498750957

Digital Object Identifier
doi:10.11650/tjm/7784

Mathematical Reviews number (MathSciNet)
MR3632520

Zentralblatt MATH identifier
06871322

Subjects
Primary: 35J10: Schrödinger operator [See also 35Pxx] 35J20: Variational methods for second-order elliptic equations

Keywords
Schrödinger-Poisson system Nehari type ground state solution asymptotically periodic

Citation

Chen, Sitong; Tang, Xianhua. Nehari Type Ground State Solutions for Asymptotically Periodic Schrödinger-Poisson Systems. Taiwanese J. Math. 21 (2017), no. 2, 363--383. doi:10.11650/tjm/7784. https://projecteuclid.org/euclid.twjm/1498750957


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