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On a fourth order nonlinear elliptic equation with negative exponent. (English) Zbl 1175.35144

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.

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

Citations:

Zbl 1143.78001
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