  In this paper, we establish the following Leray--Adams type inequality on a
bounded domain $\Omega$ in $\mathbb R^{4}$ containing the origin, \[ \sup_{u\in
C_0^\infty(\Omega), \tilde I_4[u,\Omega,R] \leq 1} \int_\Omega
\exp\left(c\left(
\frac{|u|}{E_2^{\beta}\left(\frac{|x|}R\right)}\right)^2\right) dx \leq C
|\Omega| \] for some constants $c >0$ and $C >0$, where $\beta\geq 1$, $R \geq
\sup_{x\in \Omega} |x|$, $ \tilde I_4[u,\Omega,R]:= \int_\Omega |\Delta u|^2 dx
- \int_\Omega \frac{|u|^2}{|x|^{4} E_1^2\left(\frac{|x|}R\right)} dx, $ and
$E_1(t) = 1-\ln t$, $E_2(t) = \ln (eE_1(t))$ for $t \in (0,1]$. This extends
the Leray--Trudinger inequality recently established by Psaradakis and Spector
\cite{PS2015} and Mallick and Tintarev \cite{MT2018} to the case of Laplacian
operator. In the higher dimensions or higher order derivatives, we prove the
Leray--Adams type inequality for radial function on the ball $B_r$ (with center
at origin and radius $r >0$) in $\mathbb R^n$.
