Singular behavior of an electrostatic–elastic membrane system with an external pressure. (January 2020) - Guo, Yujin; Zhang, Yanyan; Zhou, Feng
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.