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April 2018, 38(4): 1889-1933. doi: 10.3934/dcds.2018077
Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$
Juntao Sun 1,2,, , Tsung-Fang Wu 3, and Zhaosheng Feng 4,
School of Mathematics and Statistics, Shandong University of Technology Zibo 255049, China
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, Texas 78539, USA
Received November 2016 Revised October 2017 Published January 2018
We study the existence of positive solutions for the non-autonomous Schrödinger-Poisson system:
$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$
$\lambda >0$
$2 < p \le 4$
and both
$K\left( x\right) $
$a\left( x\right) $
are nonnegative functions in
$\mathbb{R}^{3}$
, which satisfy the given conditions, but not require any symmetry property. Assuming that
$% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$
$\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$
, we explore the existence of positive solutions, depending on the parameters
$\lambda$
. More importantly, we establish the existence of ground state solutions in the case of
$3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$
Keywords: Positive solution, Schrödinger-Poisson system, variational method, ground state, fibering maps, Sobolev embedding theorem, concentration-compactness principle.
Mathematics Subject Classification: Primary: 35J20, 35J61, 35A01; Secondary: 35B40.
Citation: Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077
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Juntao Sun Tsung-Fang Wu Zhaosheng Feng
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