Touchdown solutions in general MEMS models

Abstract We study general problems modeling electrostatic microelectromechanical systems devices (Pλ ) φ ( r , − u ′ ( r ) ) = λ ∫ 0 r f ( s ) g ( u ( s ) ) d s , r ∈ ( 0 , 1 ) , 0 < u ( r ) < 1 , r ∈ ( 0 , 1 ) , u ( 1 ) = 0 , \left\{\begin{array}{ll}\varphi (r,-u^{\prime} \left(r))=\lambda \underset{0}{\overset{r}{\displaystyle \int }}\frac{f\left(s)}{g\left(u\left(s))}{\rm{d}}s,\hspace{1.0em}& r\in \left(0,1),\\ 0\lt u\left(r)\lt 1,\hspace{1.0em}& r\in \left(0,1),\\ u\left(1)=0,\hspace{1.0em}\end{array}\right. where φ \varphi , g g , and f f are some functions on [ 0 , 1 ] \left[0,1] and λ > 0 \lambda \gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to ( P λ {P}_{\lambda } ) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ ( r , v ) = r α ∣ v ∣ β v \varphi \left(r,v)={r}^{\alpha }{| v| }^{\beta }v , i.e., when the associated differential equation involves the operator r − γ ( r α ∣ u ′ ∣ β u ′ ) ′ {r}^{-\gamma }\left({r}^{\alpha }{| u^{\prime} | }^{\beta }u^{\prime} )^{\prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.