Multiple positive solutions for singular elliptic problems involving concave-convex nonlinearities and sign-changing potential

In this paper, we are interested in considering the following singular elliptic problem with concaveconvex nonlinearities {-Δu-µ|x|2u=f(x)|u|p−2u+g(x)|u|q-2u,inΩ\{0},u=0,on2Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\begin{array}{*{20}{l}} { - \Delta u - \frac{\mu }{{|x{|^2}}}u = f(x)|u{|^{p - 2}}u + g(x)|u{|^{q - 2}}u,}&{in\;\;\Omega \backslash \{ 0\} ,} \\ {u = 0,}&{on\;\;2\Omega ,} \end{array}} \right.$$\end{document} where Ω ⊂ ℝN(N ≥ 3) is a smooth bounded domain with 0 ∈ Ω, 0<µ<µ¯=(N-2)24\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \mu < \bar\mu = \frac{{{{(N - 2)}^2}}}{4}$$\end{document}, 1 < q < 2 < p < 2* and 2*=2NN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2* = \frac{{2N}}{{N - 2}}$$\end{document} is the Sobolev critical exponent, the coefficient functions f, g may change sign on Ω. By the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.