Quasilinear Elliptic Equations with Singular Nonlinearity

Abstract

Abstract In this paper, motivated by recent works on the study of the equations which model electrostatic MEMS devices, we study the quasilinear elliptic equation (Pλ) { - ( r α ⁢ | u ′ | β ⁢ u ′ ) ′ = λ ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ⁢ ( r ) < 1 , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 . ${}\begin{cases}-(r^{\alpha}|u^{\prime}|^{\beta}u^{\prime})^{\prime}=\dfrac{% \lambda r^{\gamma}f(r)}{(1-u)^{2}},&r\in(0,1),\\ 0\leq u(r)<1,&r\in(0,1),\\ u^{\prime}(0)=u(1)=0.&\end{cases}$ 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. We prove the existence of an extremal parameter λ ∗ > 0 ${\lambda^{\ast}>0}$ such that, for λ ∈ ( 0 , λ ∗ ) ${\lambda\in(0,\lambda^{\ast})}$ , there exists a minimal solution u ¯ λ ${\underline{u}_{\lambda}}$ and, for λ > λ ∗ ${\lambda>\lambda^{\ast}}$ , there exists no solution of any kind. We also study the behavior of the minimal branch of solutions and we prove uniqueness of solutions of ( P λ ∗ $P_{\lambda^{\ast}}$ ) for β > - 1 ${\beta>-1}$ .