Guo, Zongming; Wei, Juncheng On a fourth order nonlinear elliptic equation with negative exponent. (English) Zbl 1175.35144 SIAM J. Math. Anal. 40, No. 5, 2034-2054 (2009). Summary: We consider the following nonlinear fourth order equation: \(T\Delta u-D\Delta^2u=\frac{\lambda}{(L+u)^2}\), \(-L<u<0\), in \(\Omega\), \(u=0\), \(\Delta u=0\) on \(\partial\Omega\), where \(\lambda>0\) is a parameter. This nonlinear equation models the deflection of charged plates in electrostatic actuators under the pinned boundary condition [F. Lin and Y. Yang, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 463, No. 2081, 1323–1337 (2007; Zbl 1143.78001)]. Lin and Yang proved that there exists a \(\lambda_c>0\) such that for \(\lambda>\lambda_c\) there is no solution, while for \(\lambda<\lambda_c\) there is a branch of maximal solutions. In this paper, we show that in the physical domains (two or three dimensions) the maximal solution is unique and regular at \(\lambda=\lambda_c\). In a two-dimensional (2D) convex smooth domain, we also establish the existence of a second mountain-pass solution for \(\lambda\in(0,\lambda_c)\). The asymptotic behavior of the second solution is also studied. The main difficulty is the analysis of the touch-down behavior. Cited in 1 ReviewCited in 33 Documents MSC: 35Q60 PDEs in connection with optics and electromagnetic theory 35B45 A priori estimates in context of PDEs 35J40 Boundary value problems for higher-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35J35 Variational methods for higher-order elliptic equations 78A30 Electro- and magnetostatics Keywords:electrostatic actuation; touch-down; pull-in threshold Citations:Zbl 1143.78001 PDFBibTeX XMLCite \textit{Z. Guo} and \textit{J. Wei}, SIAM J. Math. Anal. 40, No. 5, 2034--2054 (2009; Zbl 1175.35144) Full Text: DOI