The present work is devoted to analyze the Dirichlet problem forquasilinear elliptic equation related to someCaffarelli-Kohn-Nirenberg inequalities. Precisely the problemunder study is, -div ( |x|-pγ||ᵖ⁻²∇ u)=f(x, u)∈ L¹(Ω), x∈ Ωu(x)=0$ on $∂ Ω,$ where $-∞<γ<N-p/p$, $Ω$ is abounded domain in $ R^N$ such that $0∈Ω$ and $f(x,u)$ is aCaratheodory function under suitable conditions that will bestated in each section.
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Abdellaoui et al. (2003) studied this question.