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We study a class of gauged nonlinear Schr��dinger equations in the plane. We obtain existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, we prove existence of infinitely many solutions for odd nonlinearities., 22 pages, minor revision, accepted in DCDS-A

The present study is concerned with the following fractional Schr��dinger-Poisson system with steep potential well: $$ \left\{% \begin{array}{ll} (-��)^s u+ \la V(x)u+K(x)��u= f(u), & x\in\R^3, (-��)^t ��=K(x)u^2, & x\in\R^3, \end{array}% \right. $$ where $s,t\in(0,1)$ with $4s+2t>3$, and $\la>0$ is a parameter. Under certain assumptions on $V(x)$, $K(x)$ and $f(u)$ behaving like $|u|^{q-2}u$ with $2, 21 pages