Singular behavior of an electrostatic–elastic membrane system with an external pressure. (January 2020)

• Record Type: Journal Article
• Authors: Guo, Yujin; Zhang, Yanyan; Zhou, Feng
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: 35J75 -- 35A01 -- 35C20 -- 74K15 -- 74F15; Electrostatic MEMS -- Classification -- Singular solution -- Anisotropic singularity -- Łojasiewicz–Simon method -- Convergence rate; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111611
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We analyze nonnegative solutions of the nonlinear elliptic problem Δ u = λ f ( x ) u 2 + P, where λ > 0 and P ≥ 0 are constants, on a bounded domain Ω of R N ( N ≥ 1 ) with a Dirichlet boundary condition. This equation models an electrostatic–elastic membrane system with an external pressure P ≥ 0, where λ > 0 denotes the applied voltage. First, we completely address the existence and nonexistence of positive solutions. The classification of all possible singularities at x = 0 for nonnegative solutions u ( x ) satisfying u ( 0 ) = 0 is then analyzed for the special case where Ω = B 1 ( 0 ) ⊂ R 2 and f ( x ) = | x | α with α > − 2 . In particular, we show that for some α, u ( x ) admits only the "isotropic" singularity at x = 0, and otherwise u ( x ) may admit the "anisotropic" singularity at x = 0 . When u ( x ) admits the "isotropic" singularity at x = 0, the refined singularity of u ( x ) at x = 0 is further investigated, depending on whether P > 0, by applying Fourier analysis.