Title: On a Fourth Order Nonlinear Elliptic Equation with Negative Exponent
Abstract: 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 (Lin and Yang [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), pp. 1323–1337]). 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.
Publication Year: 2009
Publication Date: 2009-01-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 61
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