Abstract
We study the regularity of the extremal solution of the semilinear biharmonic equation \({{\Delta^2} u=\frac{\lambda}{(1-u)^2}}\), which models a simple micro-electromechanical system (MEMS) device on a ball \({B\subset{\mathbb{R}}^N}\), under Dirichlet boundary conditions \({u=\partial_\nu u=0}\) on \({\partial B}\). We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u λ with 0 < u λ < 1 exists for \({\lambda\in (0,\lambda^*)}\), while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution \({u_{\lambda^*}}\) is regular \({({\rm sup}_B u_{\lambda^*} <1 )}\) provided \({N \leqq 8}\) while \({u_{\lambda^*}}\) is singular \({({\rm sup}_B u_{\lambda^*} =1)}\) for \({N \geqq 9}\), in which case \({1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}}\) on the unit ball, where \({C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}}\) and \({\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}\).
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Communicated by D. Kinderlehrer
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Cowan, C., Esposito, P., Ghoussoub, N. et al. The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity. Arch Rational Mech Anal 198, 763–787 (2010). https://doi.org/10.1007/s00205-010-0367-x
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DOI: https://doi.org/10.1007/s00205-010-0367-x