Existence of axially symmetric solutions for a kind of planar
Schr\”{o}dinger-Poisson system
Abstract
In this paper, we study the following kind of
Schr\”{o}dinger-Poisson system in
${\R}^{2}$ \begin{equation*}
\left\{\begin{array}{ll}
-\Delta u+V(x)u+\phi
u=K(x)f(u),\ \ \
x\in{\R}^{2},\\
-\Delta \phi=u^{2},\
\ \ \ \
\ \ \ \
\ \ \ \
\ \ \ \
\ \ \ \
\ \ \ \
\ \ \ \
\ \ \
x\in{\R}^{2},
\end{array}\right.
\end{equation*} where $f\in
C({\R}, {\R} )$, $V(x)$ and
$K(x)$ are both axially symmetric functions. By constructing a new
variational framework and using some new analytic techniques, we obtain
an axially symmetric solution for the above planar system. our result
improves and extends the existing works.