This paper is devoted to a class of elliptic problems

with 0 < u(x) < 1 in a bounded domain
 RN(N  1): Hence  and p are positive
parameters and f(x) 2 C(
) is a nonnegative function. By the sub-super solutions
method we show that there exists a critical parameter  = (; p; f) such that (S)
has at least one solution including a unique minimal solution for  2 (0; ) and no
solution for  > : Moreover we get the rigorous bounds on : We also discuss
properties of minimal solutions of (S) including regularity and monotonicity.

1 Introduction.

2 The pull-in voltage.
2-1 Existence of the pull-in voltage.
2-2 Monotonicity results for pull-in voltage.

3 Estimates for the pull-in voltage.
3.1 Lower bounds for λ*
3.2 Upper bounds for λ*

4 The branch of minimal solutions.
4-1 Spectral properties of minimal solutions.

References
[1] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximumprinciple, Commun. Math. Phys. 68 (1979) 209-243.
[2] C. Bandle, Isoperimetric inequalities and applications, in: Monographs and Studiesin Mathematics, Pitman, Boston, MA, London, 1980.
[3] D. Bernstein, P. Guidotti, J.A. Pelesko, Analytic and numerical analysis of elec-trostatically actuated MEMS devices, Proc. Model. Simul. Microsyst. 2000 (2000)489-492.
[4] D.D. Joseph, T.S. Lundgren, Quasilinear Dirichlet problems driven by positive
sources, Arch. Ration. Mech. Anal. 49 (1973) 241-268.
[5] E. Koizumi, K. Schmitt, Ambrosetti-Prodi-type problems for quasilinear elliptic equa-tions, Differential Integral Equat. 18 (3) (2005) 241-262.
[6] G. Flores, G. A. Mercado, And J. A. Pelesko, Dynamics and touchdown in elec-
trostatic MEMS, in Proceedings of the Internationsl Conference on MEMS, Nano,
and Smart Systems (ICMENS 2003) Ban, AB, 2003, IEEE Computer Soicety Press,
Piscataway, NJ, pp. 182-187.
[7] G.I. Taylor, The coalescence of closely spaced drops when they are at different electric potentials, Proc. R. Soc. A 306 (1968) 423-434.
[8] H.C. Nathanson, W.E. Newell, R.A. Wickstrom, J.R. Davis, The resonant gate tran-sistor, IEEE Trans. Electron. Dev. 14 (1967) 117-133.
[9] I. Stackgold, Green's Functions and Boundary Value Problems, Wiley, New York,
1998.
[10] J.A. Pelesko, A.A Triolo, Nonlocal problems in MEMS device control, J. Eng. Math.41 (4) (2001) 345-366.
[11] J.A. Pelesko, D.H. Bernstein, Modeling MEMS and MEMS, Chapman Hall and CRC
Press, 2002.
[12] J.A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math. 62 (3) (2002) 888-908.
[13] M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal. 52 (1973) 161-180.
[14] M.G. Crandall, P.H. Rabinowitz, Some continuation and variational methods for
positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal.58 (1975) 207-218.
[15] M.T.A. Saif, B.E. Analyical modeling electrostatic membrane actuator micropumps,IEEE J. MEMS (8) (1999) 335-344.
[16] N. Ghoussoub, Y. Guo, On the partial dierential equations of electrostatic MEMS devices: stationary case, preprint (2005).
[17] N. Ghoussoub, Y. Guo, On the partial dierential equations of electrostatic MEMS devices II: dynamic case, NoDEA Nonlinear Dierential Equations. Appl, to appear.
[18] N. Ghoussoub, Y. Guo, on the partial of electrostatic MEMS devices III: rened
touchdown behavior, in preparation.
[19] P. Esposito, N. Ghoussoub, Y. Guo, Compactness along the rst branch of unstable solutions for an elliptic problem with a singular nonlinearity, preprint (2005).
[20] R.E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Equations, Uer's Guide 8.0, Software, Environments, and Tools, SIAM, Philadelphia, PA, 1998, xi+110 pages.
[21] S. Filippas, R.V. Kohn, Rened asymptotics for the blow up of ut