We consider the anisotropic quasilinear elliptic Dirichlet problem $$\left\{\begin{array}{cc}{\displaystyle -\sum _{i=1}^N{D}^i{a}_i(x,u,\nabla u)+{\left|u\right|}^{s\left(x\right)-1}u=f+\lambda \frac{{\left|u\right|}^{p_0\left(x\right)-2}u}{{\left|x\right|}^{p_0\left(x\right)}}}\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is an open bounded subset of ${ℝ}^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^1\left(\Omega \right)$ and $\lambda$ is a positive constant.

We study the following singular elliptic equation with critical exponent ⎧$-\Delta u=Q\left(x\right)u^{2*-1}+\lambda u^{-\gamma }$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.