An Overview on Singular Nonlinear Elliptic Boundary Value Problems

Abstract

We give a survey of old and recent results concerning existence and multiplicity of positive solutions (classical or weak) to nonlinear elliptic equations with singular nonlinear terms of the form

$$\displaystyle \left \{ \begin {array}{ll} -\varDelta _p u= f(x,u)+ u^{-\gamma }, & \mbox{ in }\ \varOmega \\ u>0, & \mbox{ in }\ \varOmega \\ u=0, & \mbox{ on }\ \partial \ \varOmega , \end {array} \right . $$

where Ω is a bounded domain in $\mathbb {R}^N$ (N ≥ 2) with sufficiently smooth boundary ∂Ω, Δ _p u = div(|∇u|^p−2∇u) (1 < p < ∞), f : Ω × [0, +∞) → [0, +∞) is a Carathéodory function and γ > 0. In some cases and in order to control more carefully the nonlinearity, we need to multiply the singular term u ^−γ or f(⋅, u) by positive parameters. The main difficulty which arises in the study of such problems is the lack of differentiability of the corresponding energy functional which represents an obstacle to the application of classical critical point theory.