Upper and lower bounds for the pull-in voltage and the pull-in distance for a generalized MEMS problem

We study upper and lower bounds for the pull-in voltage and the pull-in distance for the one-dimensional prescribed mean curvature problem arising in MEMS $ \begin{equation*} \left \{\begin{array}{l} - \left( \frac{u^{ \prime } (x)}{\sqrt{1 +\left (u^{ \prime } (x)\right )^{2}}} \right)^{ \prime } = \frac{\lambda }{(1 -u)^{p}} , \ \ u <1 , \ \ -L <x <L, \\ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $ where $ \lambda > 0 $ is a bifurcation parameter, and $ p, L > 0 $ are two evolution parameters. We further study monotonicity properties and asymptotic behaviors for the pull-in voltage and pull-in distance with respect to positive parameters $ p $ and $ L $.