Singular behavior of an electrostatic–elastic membrane system with an external pressure

We analyze nonnegative solutions of the nonlinear elliptic problem [Formula presented], 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=B₁(0)⊂R² 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.