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

Abstract

In this paper, we are interested in considering the following singular elliptic problem with concaveconvex nonlinearities $$\left\{ {\begin{array}{*{20}{l}} { - \Delta u - \frac{\mu }{{|x{|^2}}}u = f(x)|u{|^{p - 2}}u + g(x)|u{|^{q - 2}}u,}&{in\;\;\Omega \backslash \{ 0\} ,} \\ {u = 0,}&{on\;\;2\Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN(N ≥ 3) is a smooth bounded domain with 0 ∈ Ω, $$0 < \mu < \bar\mu = \frac{{{{(N - 2)}^2}}}{4}$$, 1 < q < 2 < p < 2* and $$2* = \frac{{2N}}{{N - 2}}$$ 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.