On a class of singular elliptic equation arising in MEMS modeling

Abstract

Motivated by recent works on the study of the equations that model the electrostatic MEMS devices, we study the radial solutions of some quasilinear elliptic equations with nonlinearity of inverse square type. According to the choice of the parameters on the problem, the differential operator which we are dealing with corresponds to the radial form of p-Laplacian (p > 1) and k-Hessian. We prove the existence of an extremal parameter λ* > 0 such that for λ ∈ (0, λ*) there exists a minimal solution uλ and for λ > λ* there is no solution of any considered kind. Via Shooting Method, we prove uniqueness of solutions for λ close to 0. We also study the behavior of the minimal branch of solutions. Concerning the case λ = λ*, we prove uniqueness of solutions and present a regularity result. In addition, we present conditions over which we can assert regularity for the critical solution with respect to the parameter λ for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists other solutions of mountain pass type for λ close to λ*.