Touchdown Solutions in General MEMS Models

We study general problems modeling electrostatic microelectromechanical systems devices {phi(r,-u'(r) = lambda integral(r)(0) f(s)/g(u(s)) ds, r is an element of (0, 1), 0 < u(r) < 1, r is an element of (0, 1), (P-lambda) u(1) = 0, where phi, g, and f are some functions on [0, 1] and lambda > 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to (P-lambda) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when phi(r, v) = r(alpha)vertical bar v vertical bar(beta)v, i.e., when the associated differential equation involves the operator r(-gamma) (r(alpha)vertical bar u'vertical bar(beta)u')', we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.