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For a given continuous function $g:~\Omega\rightarrow\mathbb{C}$, weestablish some Schwarz type Lemmas for mappings $f$ in $\Omega$ satisfying the{\rm PDE}: $\Delta f=g$, where $\Omega$ is a subset of the complex plane$\mathbb{C}$. Then we apply these results to obtain a Landau type theorem,which is a partial answer to the open problem in \cite{CP}. %The obtainedresults are

The Schwarz type Lemmas and the Landau type theorem of mappings satisfying Poisson's equations