  Using $4.5~\mathrm{fb}^{-1}$ of $e^+e^-$ annihilation data samples collected
at the center-of-mass energies ranging from 4.600~GeV to 4.699~GeV with the
BESIII detector at the BEPCII collider, a first study of the semileptonic
decays $\Lambda_c^+\rightarrow pK^-e^+\nu_e$, $\Lambda_c^+\rightarrow
\Lambda(1520) e^+\nu_e$ and $\Lambda_c^+\rightarrow \Lambda(1405) e^+\nu_e$ is
performed. The $\Lambda_c^+\rightarrow pK^-e^+\nu_e$ decay is observed with a
significance of $8.2\sigma$ and the branching fraction is measured to be
$\mathcal{B}(\Lambda_c^+\rightarrow pK^- e^+\nu_e)=(0.88\pm0.17_{\rm
stat.}\pm0.07_{\rm syst.})\times 10^{-3}$. We also report evidence of
$\Lambda_c^+\rightarrow \Lambda(1520)e^+\nu_e$ and $\Lambda_c^+\rightarrow
\Lambda(1405)e^+\nu_e$ with significances of $3.3\sigma$ and $3.2\sigma$,
respectively, and measure $\mathcal B(\Lambda^+_c\rightarrow
\Lambda(1520)e^+\nu_e)=(1.02\pm0.52_{\rm stat.}\pm0.11_{\rm
syst.})\times10^{-3}$ and $\mathcal B(\Lambda^+_c\rightarrow
\Lambda(1405)[\rightarrow pK^-]e^+\nu_e)=(0.42\pm0.19_{\rm stat.}\pm0.04_{\rm
syst.})\times10^{-3}$. Combining these with the inclusive semileptonic
$\Lambda_c^+$ branching fraction measured by BESIII, the relative fraction is
determined to be $[\mathcal{B}(\Lambda_c^+\rightarrow
pK^-e^+\nu_e)/\mathcal{B}(\Lambda_c^+\rightarrow X e^+\nu_e)]=(2.1\pm0.4_{\rm
stat.}\pm0.2_{\rm syst.})\%$, which provides a clear confirmation that
semileptonic $\Lambda_c^+$ decays are not saturated by the $\Lambda
\ell^+\nu_{\ell}$ final state.
