Twin positive solutions for singular nonlinear elliptic equations - Chen, Jianqing; Papageorgiou, Nikolaos S.; Rocha, Eugénio M. (Topological Methods in Nonlinear Analysis, 2010)
For a bounded domain $Z\subseteq{\mathbb{R}}^N$ with a $C^2$-boundary, we prove the existenceof an ordered pair of smooth positive strong solutions for the nonlinear Dirichlet problem$$-Δ_p x(z) = β(z)x(z)^{-η}+f(z,x(z))\quad \text{a.e on } Z\text{ with } x\in W^{1,p}_0(Z),$$which exhibits the combined effects of a singular term ($η≥ 0$) and a $(p-1)$-linear term $f(z,x)$ near $+∞$, by using a combination of variational methods, with upper-lower solutions and with suitable truncation techniques.