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.
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