  We investigate radial solutions for the problem \[ \begin{cases}
\displaystyle -\Delta U=\frac{\lambda+\delta|\nabla U|^2}{1-U},\; U>0 &
\textrm{in}\ B,\\ U=0 & \textrm{on}\ \partial B, \end{cases} \] which is
related to the study of Micro-Electromechanical Systems (MEMS). Here, $B\subset
\mathbb{R}^N$ $(N\geq 2)$ denotes the open unit ball and $\lambda, \delta>0$
are real numbers. Two classes of solutions are considered in this work: (i)
{\it regular solutions}, which satisfy $0<U<1$ in $B$ and (ii) {\it rupture
solutions} which satisfy $U(0)=1$, and thus make the equation singular at the
origin. Bifurcation with respect to parameter $\lambda>0$ is also discussed.
