Some results for a class of quasilinear elliptic equations with singular nonlinearity

Abstract

In this paper, motivated by recent works on the study of the equations which model the electrostatic MEMS devices, we study the quasilinear elliptic equation involving a singular nonlinearity {−(rα|u′(r)|βu′(r))′=λrγf(r)(1−u(r))2,r∈(0,1),0≤u(r)<1,r∈(0,1),u′(0)=u(1)=0. According to the choice of the parameters α,β and γ, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the p-Laplacian and the k-Hessian. In this work we present conditions over which we can assert regularity for solutions, including the case λ=λ∗, where λ∗ is a critical value for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists another solution of mountain pass type for λ close to the critical one. In addition, we use the Shooting Method to prove uniqueness of solutions for λ in a neighborhood of 0.