On a Class of Weighted p-Laplace Equation with Singular Nonlinearity

This article deals with the existence of the following quasilinear degenerate singular elliptic equation: (Pλ){-div(w(x)|∇u|p-2∇u)=gλ(u),u>0inΩ,u=0on∂Ω,where Ω ⊂ Rⁿ is a smooth bounded domain, n≥ 3 , λ> 0 , p> 1 , and w is a Muckenhoupt weight. Using variational techniques, for g_λ(u) = λf(u) u^-q and certain assumptions on f, we show existence of a solution to (P_λ) for each λ> 0. Moreover, when g_λ(u) = λu^-q+ u^r, we establish existence of at least two solutions to (P_λ) in a suitable range of the parameter λ. Here, we assume q∈ (0 , 1) and r∈(p-1,ps∗-1).