### Twin positive solutions for singular nonlinear elliptic equations

#### Abstract

For a bounded domain $Z\subseteq{\mathbb{R}}^N$ with a $C^2$-boundary, we prove the existence

of an ordered pair of smooth positive strong solutions for the nonlinear Dirichlet problem

$$

-\Delta_p x(z) = \beta(z)x(z)^{-\eta}+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 ($\eta\geq 0$) and a $(p-1)$-linear term $f(z,x)$ near

$+\infty$, by using a combination of variational methods, with upper-lower solutions and with suitable truncation techniques.

of an ordered pair of smooth positive strong solutions for the nonlinear Dirichlet problem

$$

-\Delta_p x(z) = \beta(z)x(z)^{-\eta}+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 ($\eta\geq 0$) and a $(p-1)$-linear term $f(z,x)$ near

$+\infty$, by using a combination of variational methods, with upper-lower solutions and with suitable truncation techniques.

#### Keywords

Singular nonlinearity; positive solutions; variational methods; truncation techniques; upper-lower solutions

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